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TRANSCRIPT
DEVELOPMENT OF A VISCOELASTIC
TUNED MASS DAMPER TO REDUCE
WALKING INDUCED VIBRATIONS IN
BUILDING FLOORS
by
Ibrahim Saidi
A thesis submitted in total fulfilment of the requirements
of the degree of Doctor of Philosophy
February 2012
Faculty of Engineering and Industrial Sciences
Swinburne University of Technology
i
ABSTRACT
Excessive floor vibration due to human excitations has become a significant
serviceability concern for office floors in the last two decades. The use of
lightweight and high strength materials, nominal furniture and open layouts have
resulted in lighter floors with less damping and hence an increase in vibration
related problems. In office floors, this may lead to unacceptable floor response
due to walking excitation and hence annoyance problems.
When excessive vibrations are encountered in existing floors, there are few
options available as remedial actions. If a change of floor layout or stiffening is
not practical, the use of a Tuned Mass Damper (TMD) to increase floor damping
is one of the attractive possible solutions. Currently, conventional viscous Tuned
Mass Dampers (TMDs) to treat floors experiencing small displacements due to
walking is not practically available.
The main aim of this research was to develop a cost effective and simple TMD
that can deal with floor systems with small vibration displacements. A new
innovative TMD in the form of a cantilever beam with an end mass was
developed to suppress floor vibrations. The cantilever beam consists of a
viscoelastic layer constrained by two constraining layers. The new viscoelastic
TMD can be fitted within the false floor or false ceiling spaces.
Based on extensive analytical modelling, Finite Element (FE) analysis, laboratory
testing and investigation of a real office floor, the following conclusions could be
made:
i) the new viscoelastic TMD is simple and cost effective as it can be
constructed from commercially available rubber.
ii
ii) an analytical model was developed which can be used to design the new
TMD for floor applications. As expected, the performance of the
viscoelastic TMD is highly dependent on the geometry and properties of
the materials used in its construction particularly the dissipation loss
factor of the rubber as it acts as the damping element of the TMD.
iii) prototype viscoelastic TMDs were tested on simply supported steel and
concrete beams and it was found that the viscoelastic TMDs were very
efficient in reducing the levels of vibration generated from a mechanical
shaker and human excitations.
iv) for full scale floors where a single damper may be too large to fit in the
available space, multiple damper solutions were developed and tested.
The multiple dampers can be located at one spot or distributed.
v) the viscoelastic TMD solution in the form of multiple distributed dampers
was successfully applied to a real office floor experiencing excessive
vibrations. The new TMD was able to reduce the vibrations by at least a
factor of 1/3.
The new viscoelastic damper offers design engineers a non intrusive and cost
efficient solution to reduce excessive vibrations in floor systems. In a distributed
form, the TMDs can be reasonably small in size and easily tuned.
iii
DECLARATION
This is to certify that:
the thesis comprises only my original work, except where
acknowledgement is made in the text.
to the best of my knowledge, contains no material previously published or
written by another person, except where due reference is made, and
this material has not been submitted, either in whole or in part, for a
degree at this or any other academic institution.
Signature: _______________________
Ibrahim Saidi
iv
ACKNOWLEDGEMENTS
I would like to give all my thanks to Allah who loves me and gave me the ability
to complete this research.
I would like to thank my supervisors Prof. Emad Gad, Prof. John Wilson and
A/Prof. Nicholas Haritos for their patience and guidance throughout my time as a
Ph.D. student. Their kindness and interest in the work of all their students is
encouraging.
I would like to thank Dr. Adnan Mohammad from the University of Technology,
Iraq, for his contributions to my work. He offered valuable assistance for which I
am truly grateful. I would like to thank Tuan Nguyen and Ari Wibowo for helping
me with my experimental tests and analysis. I would like also to thank Dr. Igor
Sbarski and Header Haddad for their assistance in material testings.
I would especially like to thank my wife, Asma, for her encouragement, patience,
and support. A great big thanks to my lovely daughters who were a powerful
source of inspiration and energy to me. I would also like to express my deepest
gratitude to my parents for their support throughout my education.
v
LIST OF PUBLICATIONS
The following is a list of papers arising from the research presented in this
thesis.
Published Papers
Saidi, I, Haritos, N, Gad, EF & Wilson, JL 2006, 'Floor vibrations due to
human excitation : damping perspective', Annual Technical Conference of
the Australian Earthquake Engineering Society : Earthquake Engineering
in Australia, Canberra, pp. 257-264.
Saidi, I, Mohammed, AD, Gad, EF, Wilson, JW & Haritos, N 2007,
'Optimum design for passive tuned mass dampers using viscoelastic
materials', Proceedings of the Australian Earthquake Engineering Society
Conference (AEES 2007), Wollongong, New South Wales, Australia,
Paper no. 47.
Saidi, I, Mohammed, A, Gad, E, Wilson, J & Haritos, N 2008,
'Development of a Viscoelastic Tuned Mass Damper for Floor Vibration
Applications', Australian Structural Engineering Conference, Melbourne,
Vic., pp. 884-892.
Saidi, I, Gad, EF, Wilson, JL & Haritos, N 2008, 'Innovative passive
viscoelastic damper to suppress excessive floor vibrations', Proceedings of
the Earthquake Engineering in Australia Conference (AEES 2008),
Ballarat, Victoria, Australia, Paper no. 42.
vi
Saidi, I, Gad, EF, Wilson, JL & Haritos, N 2010, 'Rectification of floor
vibrations using viscoelastic tuned mass damper', Concrete in Australia,
vol. 36, no. 2, pp. 27-31.
Saidi, I, Gad, EF, Wilson, JL & Haritos 2011, 'Development of passive
viscoelastic damper to attenuate excessive floor vibrations', Engineering
Structures, doi:10.1016/j.engstruct.2011.05.017.
Accepted Papers
Nguyen, T, Saidi, I, Gad, EF, Wilson, JL & Haritos, N, 'Performance of
distributed multiple viscoelastic tuned mass dampers for floor vibration
applications', Advances in Structural Engineering, reviewer comments
received on 16 May 2011.
vii
Table of Contents
List of Figures ........................................................................................................ xi
List of Tables ..................................................................................................... xviii
1. Introduction..........................................................................................................1
1.1. Background ...................................................................................................1
1.2. Research Aim and Objectives.......................................................................3
1.3. Thesis Outline ...............................................................................................5
2. Literature Review ................................................................................................7
2.1. Introduction...................................................................................................7
2.2. Walking Excitation .......................................................................................7
2.2.1. Single Person Walking Excitation .............................................................9
2.2.2. Group Walking Excitation .......................................................................16
2.3. Assessment of Floor Response to Walking Excitation ...............................16
2.3.1. AISC DG11 Method ................................................................................16
2.3.1.1. Acceptance Criteria for Human Comfort..............................................17
2.3.1.2. Evaluation of Floor’s Peak Response to Walking ................................19
2.3.1.3. Idealisation of Single Degree of Freedom Systems..............................20
2.3.1.4. Floor Damping Value for AISC DG11 Method ...................................24
2.3.1.5. Summary of AISC DG11 Procedure ....................................................27
2.3.2. SCI P354 Method.....................................................................................28
2.3.3. Hivoss Method.........................................................................................32
2.3.3.1. Damping Estimation Using Hivoss Method .........................................32
2.3.3.2. Determination of Natural Frequency and Mass of Floor Using Hivoss
Method ...............................................................................................................33
2.3.3.3. Hivoss Acceptance Criteria...................................................................36
2.3.4. CCIP-016 Method....................................................................................39
2.4. Damping Estimation from Measured Data .................................................45
2.4.1. Logarithmic Decrement Analysis (LDM)................................................45
2.4.2. Half-Power Bandwidth (HPB).................................................................46
viii
2.4.3. Circle-Fit Method ....................................................................................48
2.4.4. Random Decrement Technique (Randec) ................................................50
2.5. Remedial Measures to Suppress Floor Vibrations......................................52
2.6. Dampers ......................................................................................................54
2.6.1. Passive Tuned Mass Dampers (TMD).....................................................54
2.6.1.1. Damping Elements................................................................................58
2.6.1.2. Application of Passive TMDs on Floor Systems..................................60
2.6.2. Semiactive Control Dampers ...................................................................65
2.6.3. Active Control Dampers ..........................................................................68
2.7. Concluding Remarks...................................................................................69
3. Viscoelastic Damper ..........................................................................................72
3.1. Introduction.................................................................................................72
3.2. Damping Using Viscoelastic Materials ......................................................72
3.3. The Concept of New Viscoelastic Damper.................................................74
3.4. Development of Analytical Model .............................................................76
3.5. Approximate Analytical Method ................................................................78
3.6. Design of Proposed Viscoelastic Damper ..................................................80
3.7. Determination of Viscoelastic Material Properties.....................................83
3.7.1. Dynamic Mechanical Analyser (DMA)...................................................84
3.7.2. Back Calculation from Prototype Testing ...............................................87
3.8. Validation of Analytical Model Using Prototype Dampers........................88
3.8.1. Validation of Analytical Model Using Physical Testing .........................89
3.8.2. Validation of Analytical Model Using FE Analysis ................................92
3.9. Concluding Remarks...................................................................................98
4. Performance of Viscoelastic Dampers.............................................................101
4.1. Introduction...............................................................................................101
4.2. Case Study 1 – Steel Beam .......................................................................101
4.2.1. Measurement of the Steel Beam Dynamic Properties ...........................102
4.2.2. Prediction of Steel Beam Response without and with Damper .............104
4.2.3. Development of Viscoelastic Damper for Steel Beam ..........................107
4.2.4. Measurement of Steel Beam Response..................................................108
4.2.5. FE Modeling for the Steel Beam ...........................................................110
ix
4.2.6. Sensitivity of Steel Beam Response to TMD Natural Frequency .........114
4.2.7. Sensitivity of TMD Performance to Damping Ratio of TMD and Beam
.........................................................................................................................115
4.2.8. Sensitivity of Steel Beam Response to TMD Point of Attachment .......117
4.3. Case Study 2 – Concrete T beam..............................................................119
4.3.1. Experimental Modal Analysis ...............................................................122
4.3.2. Viscoelastic Damper Design for T Beam ..............................................129
4.3.3. Application of TMD to T Beam.............................................................131
4.3.4. Performance of the Damper Due to Heel Drop Excitation ....................131
4.3.5. Performance of the Damper Due to Walking Excitation .......................133
4.3.6. FE Model for Walking Excitation .........................................................135
4.3.7. Sensitivity of the T Beam Response to TMD Natural Frequency .........141
4.3.8. Sensitivity of the T Beam Response to TMD Damping Ratio ..............143
4.4. Concluding Remarks.................................................................................144
5. Application of Multiple Dampers ....................................................................148
5.1. Introduction...............................................................................................148
5.2. Multiple Tuned Mass Dampers (MTMD) ................................................149
5.3. Multiple Dampers at the Mid-span of the T beam....................................151
5.3.1. Distributed Damper Systems .................................................................154
5.3.2. Multiple Dampers Distributed on T beam .............................................156
5.4. Multiple Viscoelastic Dampers on an Office Floor ..................................161
5.4.1. Determination of Floor Natural Frequency ...........................................165
5.4.2. Determination of the Peak Acceleration Due to Walking .....................165
5.4.3. Determination of Floor Damping Ratio.................................................168
5.4.3.1. Dynamic Testing of Building Floors ..................................................170
5.4.3.1.1. Modal Testing of Floors without Measuring the Excitation Force..170
5.4.3.1.2. Modal Testing of Floors with Measurements Made of the Excitation
Force ................................................................................................................171
5.4.4. FE Model to Determine Floor Dynamic Properties ...............................171
5.4.4.1. FE Model for Problematic Bay...........................................................172
5.4.4.2. FE Model for Problematic Bay with Adjoining Bays.........................173
5.4.4.3. FE Model for Entire Floor ..................................................................175
x
5.4.5. Peak Acceleration of the Floor Using Analytical and FE Models.........177
5.4.5.1. Peak Acceleration Obtained From Analytical Solution......................177
5.4.5.2. Peak Acceleration Obtained from FE Model......................................178
5.4.6. Development of Multiple Viscoelastic Dampers...................................180
5.4.7. Preliminary Estimation of the Retrofitted Floor Response....................182
5.4.8. Installation of Viscoelastic Dampers and Testing .................................183
5.4.9. In-situ Vibration Measurements ............................................................184
5.4.9.1. Retrofitted Floor Response to Walking Excitation.............................184
5.4.9.2. Retrofitted Floor Response to Shaker Excitation ...............................185
5.4.10. FE Model for Floor with Dampers ......................................................186
5.5. Concluding Remarks.................................................................................187
6. Parametric Analyses ........................................................................................190
6.1. Introduction...............................................................................................190
6.2. Performance of MTMD to Variations in Floor Damping Ratio ...............190
6.3. Performance of MTMD to Variations in Damper Damping Ratio ...........192
6.4. Performance of MTMD to the Variation in Mass Ratio ...........................193
6.5. Performance of MTMD to Variations in Damper’s Frequency................195
6.6. Performance of MTMD to Departure from Location of Maximum
Response ..........................................................................................................198
6.7. Concluding Remarks.................................................................................200
7. Conclusions......................................................................................................203
7.1. Floor Assessment and Rectification Methods ..........................................203
7.2. Concept of the New Damper ....................................................................204
7.3. Development of Analytical Model ...........................................................204
7.4. Validation of Analytical Model ................................................................204
7.5. Application of Viscoelastic TMD on Simple Beams................................205
7.6. Application of New TMD in Multiple Form ............................................209
7.7. Application of Multiple TMD on a Real Floor.........................................210
7.8. Sensitivity of Floor Response to variations in Floor and TMD Properties
.........................................................................................................................212
7.9. Recommendations for Future Work .........................................................213
References............................................................................................................215
xi
List of Figures
Figure 2-1 Human walking force model................................................................10
Figure 2-2 Footfall overlap function during walking at 2 pace/sec .......................10
Figure 2-3 Walking force for several frequencies .................................................11
Figure 2-4 DLFs for the first four harmonics for (a) walking, (b) running and (c)
jumping force.........................................................................................................14
Figure 2-5 Third order polynomial fit to the first harmonic ..................................14
Figure 2-6 A comparison between walking force of two suggested values of phase
angles .....................................................................................................................15
Figure 2-7 Modified Reiher-Meister Scale ............................................................17
Figure 2-8 Acceptability criteria for vertical floor accelerations ..........................18
Figure 2-9 Schematic of single degree of freedom system....................................21
Figure 2-10 Types of office floors .........................................................................26
Figure 2-11 Weighting factor for human perception of vibrations........................29
Figure 2-12 A procedure to calculate the natural frequency and modal mass of
isotropic plates .......................................................................................................34
Figure 2-13 Typical composite floor consists of a slab and beams .......................35
Figure 2-14 The OS-RMS90-values as a function of step frequency and body mass
...............................................................................................................................36
Figure 2-15 OS-RMS90 application for floors with 3% damping ratio..................38
Figure 2-16 Classification of floor response and recommendation for the
application of classes .............................................................................................39
Figure 2-17 Baseline RMS acceleration ................................................................43
Figure 2-18 Response for all walking frequencies ................................................45
Figure 2-19 Half-Power Bandwidth ......................................................................47
Figure 2-20 Nyquist plot for a system with viscous damping ...............................50
Figure 2-21 Basic concept of the Random Decrement Technique
...............................................................................................................................52
Figure 2-22 Stiffening technique for steel joists and beams..................................53
xii
Figure 2-23 Typical Representation of Two Degree of Freedom Tuned Mass
Dampers .................................................................................................................55
Figure 2-24 Example showing the effects of attaching a TMD to a SDOF system
...............................................................................................................................56
Figure 2-25 Pendulum Tuned Mass Damper .........................................................57
Figure 2-26 Example of viscous damper ...............................................................58
Figure 2-27 Viscoelastic damper ...........................................................................59
Figure 2-28 Liquid damper ....................................................................................60
Figure 2-29 Ballroom floor long section with a TMD...........................................61
Figure 2-30 Liquid TMD .......................................................................................62
Figure 2-31 Acceleration responses of the floor due to walking without and with
liquid TMDs...........................................................................................................63
Figure 2-32 TMD with viscoelastic damping element ..........................................64
Figure 2-33 Walking induced response of the office floor without and with TMDs
...............................................................................................................................65
Figure 2-34 schematic of semiactive tuned mass damper .....................................66
Figure 2-35 Typical magneto- rheological damper ...............................................66
Figure 2-36 Semiactive tuned mass damper ..........................................................67
Figure 2-37 Test rig and primary components.......................................................68
Figure 3-1 Resotec product installation .................................................................73
Figure 3-2 Performance of resotec product ...........................................................74
Figure 3-3 Viscoelastic damper compared with viscous damper ..........................75
Figure 3-4 Proposed viscoelastic damper installed within false ceilings and false
floors ......................................................................................................................75
Figure 3-5 Typical sandwich beam........................................................................76
Figure 3-6 The DMA machine with dual cantilever clamp mode .........................85
Figure 3-7 Example of the DMA test result for a rubber sample using the
frequency sweep ....................................................................................................86
Figure 3-8 Example of the DMA test results for rubber samples using the strain
sweep .....................................................................................................................87
Figure 3-9 Prototype 1 vibrational test using non-contact accelerometer .............89
Figure 3-10 Time history for the Prototype 1 without and with an end mass .......90
xiii
Figure 3-11 Time history for the Prototype 2 without and with an end mass .......92
Figure 3-12 FE model of the prototype damper ....................................................94
Figure 3-13 Response of the Prototype 1 without and with an end mass using FE
harmonic analysis ..................................................................................................95
Figure 3-14 Time history for Prototype 1 without and with an end mass using FE
transient analysis....................................................................................................96
Figure 3-15 Response of Prototype 2 without and with an end mass using FE
harmonic analysis ..................................................................................................96
Figure 3-16 Time history for Prototype 2 without and with an end mass using FE
transient analysis....................................................................................................97
Figure 4-1 Normalised steel beam response due to the pluck test .......................103
Figure 4-2 The measured natural frequency of the steel beam............................103
Figure 4-3 Prediction of steel beam response without and with damper due to 1 N
harmonic force using Equations (2-5) & (4-6) ....................................................106
Figure 4-4 Viscoelastic damper attached to a vibrating steel beam ....................108
Figure 4-5 Viscoelastic damper attached to the steel beam.................................109
Figure 4-6 Steel beam response without and with damper attached....................109
Figure 4-7 Response of the steel beam with and without damper attached in the
frequency domain using FE analysis ...................................................................112
Figure 4-8 Response of the steel beam with and without damper in time domain
using transient analysis in FE analysis ................................................................114
Figure 4-9 Steel beam response due to the variation in damper natural frequency
using analytical, FE and experimental results .....................................................115
Figure 4-10 Steel beam response due to the variation in the damping ratio of the
damper using Equations (2-5) & (4-6) and FE analysis ......................................116
Figure 4-11 Reduction in the steel beam response for different damping ratio of
the beam and damper using Equations (2-5) & (4-6) ..........................................117
Figure 4-12 Variation in damper point of attachment along the length of the steel
beam.....................................................................................................................119
Figure 4-13 Cross-section of T beam floor used in Case Study 2 .......................119
Figure 4-14 T beam supports ...............................................................................120
Figure 4-15 T beam response due to heel drop excitation...................................121
xiv
Figure 4-16 Normalised T beam response to heel drop using Randec technique121
Figure 4-17 Fundamental mode shape for a simply supported T beam...............122
Figure 4-18 Typical mode shapes for simply supported floor system.................123
Figure 4-19 T beam grid points ...........................................................................124
Figure 4-20 Distribution of accelerometers for three rounds ..............................125
Figure 4-21 First mode shape of the T beam obtained from experimental modal
analysis.................................................................................................................125
Figure 4-22 Estimation of the T beam natural frequency and damping ratio using
ARTeMIS.............................................................................................................126
Figure 4-23 T beam mode shapes obtained from experimental modal analysis..126
Figure 4-24 Acceleration response of the T beam without and with optimum
damper for different beam damping due to 1 kN harmonic excitation force using
Equations (2-5) and (4-6).....................................................................................128
Figure 4-25 Acceleration response of the T beam with and without optimum
damper in frequency domain due to 1 kN harmonic force using Equations (2-5)
and (4-6)...............................................................................................................129
Figure 4-26 Response of damper developed for the T beam due to a pluck test .130
Figure 4-27 Tuned mass viscoelastic damper attached to the experimental T beam
.............................................................................................................................131
Figure 4-28 T beam response due to heel drop excitation...................................132
Figure 4-29 Response of the T beam without and with damper due to heel drop
excitation with measured damping ratios using log decay method .....................132
Figure 4-30 T beam response in frequency domain for cases without and with as
built damper due to 1 kN harmonic force based on FE analyses .........................133
Figure 4-31 T beam measured acceleration response due to walking excitation
based on averaging of 24 records for cases with and without damper ................134
Figure 4-32 Peak accelerations of T beam with and without the damper from 24
walking excitation records ...................................................................................135
Figure 4-33 Equivalent walking force function according to the mode shape ....137
Figure 4-34 Equivalent time dependent walking force for the bare T beam using
Equation (4-14) ....................................................................................................138
xv
Figure 4-35 T beam loading and response without damper due to on-the-spot
walking and walking along the length of the beam using FE models .................139
Figure 4-36 T beam loading and response with damper due to on-the-spot walking
and walking along the length of the beam using FE models ...............................140
Figure 4-37 T beam response without and with damper due to walking along the
length of the beam using FE models....................................................................140
Figure 4-38 Reduction factor of the T-beam response with attached TMD due to
variations in the natural frequency of the damper using Equations (2-5) and (4-6)
.............................................................................................................................141
Figure 4-39 Sensitivity of the damper due to variation in the natural frequency 142
Figure 4-40 Sensitivity of damper performance to the variation in its natural
frequency using FE model ...................................................................................143
Figure 4-41 Reduction factor in the T beam response with damper attached for
different damping ratios of the damper using Equations (2-5) & (4-6) and FE
model ...................................................................................................................143
Figure 5-1 Schematic five degrees of freedom system........................................149
Figure 5-2 Prototype viscoelastic damper developed for multiple damper system
to replace the single large damper .......................................................................152
Figure 5-3 Four viscoelastic tuned mass dampers at the centre of the T beam ...153
Figure 5-4 Response due to heel drop with four dampers attached at the centre of
T beam .................................................................................................................153
Figure 5-5 Response of the T beam with four dampers at the centre due to heel
drop excitation .....................................................................................................154
Figure 5-6 T beam with four spatially distributed dampers.................................155
Figure 5-7 T beam response with four spatially distributed dampers..................157
Figure 5-8 T beam response with four distributed dampers attached..................157
Figure 5-9 Deactivated damper............................................................................158
Figure 5-10 Identical dampers located at a distance of 1 m away from the point of
maximum response ..............................................................................................160
Figure 5-11 Plan of the floor and problematic bay..............................................162
Figure 5-12 Pathway ‘A’ of the problematic bay along the secondary beams ....163
Figure 5-13 Pathway ‘B’ of the problematic bay crossing the secondary beams 163
xvi
Figure 5-14 Natural frequency of the bare floor obtain from heel drop excitation
.............................................................................................................................165
Figure 5-15 Response acceleration of the bare floor due to walking along Pathway
‘A’........................................................................................................................166
Figure 5-16 Measured peak accelerations of the floor for the eight walking records
.............................................................................................................................166
Figure 5-17 Measured original floor response acceleration due to on-the-spot
walking.................................................................................................................167
Figure 5-18 Response of the floor due to the heel drop excitations using LDM.168
Figure 5-19 Damping ratio of the floor obtained from heel drop excitation using
Randec .................................................................................................................169
Figure 5-20 Fundamental mode shape for the model of the problematic bay only
.............................................................................................................................173
Figure 5-21 First eight mode shapes for the model of the problematic bay with
adjoined bays .......................................................................................................174
Figure 5-22 First eight mode shapes and corresponding frequencies for the model
of entire floor .......................................................................................................176
Figure 5-23 Floor acceleration response due to equivalent walk along the critical
path and response at mid-span using FE analysis................................................179
Figure 5-24 Floor response from FE analysis due to on-the-spot walking at the
centre of problematic bay ....................................................................................179
Figure 5-25 Plan view of TMD configurations ...................................................181
Figure 5-26 Damper response to pluck test .........................................................182
Figure 5-27 Viscoelastic dampers installed within false floor ............................183
Figure 5-28 Peak acceleration responses for the floor without and with the MTMD
system due to walking excitation.........................................................................184
Figure 5-29 Typical traces for acceleration responses of the floor without and with
the MTMD system due to walking excitation .....................................................185
Figure 5-30 Typical acceleration responses for the floor without and with the
MTMD system due to shaker excitation..............................................................186
Figure 5-31 Peak acceleration response for the floor without and with the MTMD
system due to shaker excitation ...........................................................................186
xvii
Figure 5-32 Acceleration responses of the floor without and with dampers due to
walking excitation based on FE analyses ............................................................187
Figure 6-1 Acceleration response of the office floor without and with as built
MTMD system to variations in the floor damping ratio using FE analysis.........192
Figure 6-2 Maximum acceleration response of the office floor to variations in the
damping ratio of the MTMD system using FE analysis for floor with 3% damping
ratio ......................................................................................................................193
Figure 6-3 Acceleration response of the 3% damping ratio office floor to
variations in the MTMD system mass ratio using FE analysis............................194
Figure 6-4 Maximum acceleration response of the retrofitted office floor to
variations in the natural frequency of MTMD system using FE analysis ...........196
Figure 6-5 MTMD system distributed apart from the point of maximum response
.............................................................................................................................199
Figure 6-6 Acceleration response of the 3% damping ratio office floor to different
MTMD system locations using FE analysis ........................................................199
Figure 6-7 First mode shape of the problematic bay without damper .................200
xviii
List of Tables
Table 2-1 DLFs for single person force model for different authors.....................13
Table 2-2 Dynamic load factors for walking .........................................................15
Table 2-3 Multiplying factors for low probability of adverse comment ...............19
Table 2-4 Examples of damping values for the fundamental mode of floors in
buildings.................................................................................................................24
Table 2-5 Damping ratios for different type floors................................................27
Table 2-6 Damping ratios for various floor types..................................................29
Table 2-7 Recommended multiplying factors based on single person excitation .31
Table 2-8 Vibration dose value limits....................................................................31
Table 2-9 Estimation of floor’s damping design values ........................................33
Table 2-10 Suggested damping values for different structures .............................41
Table 2-11 Response factor calculation for walking at 2.18 Hz............................44
Table 3-1 Analytical model, FE analysis and experimental results for prototype
dampers 1 & 2 without and with an end mass .......................................................98
Table 4-1 Viscoelastic damper properties for steel beam based on available rubber
.............................................................................................................................107
Table 4-2 Predicted viscoelastic damper properties for the T beam....................130
Table 5-1 Properties of each viscoelastic damper in the MTMD configuration .151
Table 5-2 Sensitivity analysis of distributed dampers .........................................159
Table 5-3 Dynamic properties of modified distributed dampers .........................160
Table 5-4 Acceleration response of T beam to walking excitations for different
damper setup ........................................................................................................161
Table 6-1 Comparison floor acceleration response to variations in the MTMD
mass ratio .............................................................................................................195
Table 6-2 Floor response due to the tuning of damper sets to different frequencies
.............................................................................................................................197
1
1. Introduction
1.1. BackgroundHigh levels of vibration can occur in floor systems due to excitation from human
activities such as walking and aerobics. In building floors, excessive vibrations
are generally not a safety concern but a cause of annoyance and discomfort.
Excessive vibrations typically occur in:
a) lightweight floors;
b) floor systems with low stiffness where the floor dominant natural frequency is
close to (or coincides with) a harmonic of the excitation frequency; and
c) floors with low damping.
Floors are subjected to dynamic forces induced by people when they walk, run,
jump or dance. The latter three typically take place when a building contains
facilities such as exercise rooms, dance floors or gymnasia. The excitation from
these live loads can be classified into the two broad categories of in-situ and
moving. Periodic jumping to music, sudden standing of a crowd, and random in-
place movements are examples of in-situ activities, whilst walking, marching, and
running are examples of moving activities (Ebrahimpour & Sack 2005).
Annoying levels of floor vibrations due to human movements such as walking and
running have become more common in the last two decades. One of the main
factors contributing to this problem is the decrease in damping due to fewer
partitions and items of furniture. The problem can also occur due to decrease in
the floor mass resulting from the use of high strength building materials and
composite systems. The other important factors are the decrease in the floor
natural frequency due to longer floor spans and increase in the number of
rhythmic human activities such as aerobics (Setareh et al. 2006).
2
While the floor mass and stiffness are generally constant during the life of the
structure and can be estimated with a high degree of accuracy, damping is more
difficult to predict because it is mostly associated with non-structural components
such as partitions, false floors, suspended ceilings and ducts as well as furniture
such as filing cabinets and book shelves.
For traditional offices with book shelves and filing cabinets, the damping could be
as high as 5% (Murray et al. 1997). On the other hand, for modern electronic
offices with hardly any book shelves and a limited number of filing cabinets, the
damping level could be about 2-2.5% (Hewitt & Murray 2004). Consequently,
because of the decrease in damping, the maximum response due to walking
excitation increases and may exceed the acceptable limit of response. In extreme
cases, excessive vibrations can render a floor totally unusable by the occupants
based solely on levels of human comfort (Alvis 2001). Furthermore, for long span
floors, walking can produce vibration limits up to steady state, which may not be
realised in traditional shorter span floors. Indeed, long span floors are becoming
more common because of the functional demands of owners and tenants as well as
the availability of new construction materials and technology. For example, new
concrete mixes have higher strengths and the standard yield stress of reinforcing
steel in Australia has increased to 500 MPa compared to 400 MPa about 10 years
ago (Salzmann 2002). Furthermore, there is a massive increase in the use of
lightweight composite floors, which employ thin concrete slabs, steel decking,
and steel beams. In such systems, lightweight concrete may also be used with a
density reduced to 1800 3mkg .
Few options are available to remedy a floor with excessive vibrations. Additional
damping can be achieved by installing full-height partitions but in most cases, this
option is not possible due to architectural and functional requirements. For floors
with low fundamental frequency, increasing the floor stiffness can reduce human
induced vibration because it increases the natural frequency of the floor and hence
shifting the resonance to higher and less significant harmonics. However, this
option normally requires significant structural modification to the floors to
3
achieve a significant shift in the natural frequency. Such modifications are often
not feasible as they require expensive decommissioning of office space to install
structural members. In addition, space may not be available to install such
members. Adding mass can reduce the vibration level but in most cases it is not
practical as it may create overstress in structural members. Furthermore, if the
additional mass is not carefully considered, it could reduce the natural frequency
of the floor to a level which makes it more vulnerable.
One other available option to rectify existing floors experiencing excessive
vibration is to use tuned mass dampers (TMDs). A conventional TMD consists of
mass, spring and dashpot. This remedial measure was successfully used in a
concert floor experiencing excessive vibration due to dancing (Webster &
Vaicaitis 1992). This concert floor was excited with a maximum vibration
displacement of 3.3 mm. The damper used was extended from the roof beam to
the point of maximum response of the problematic floor. However, such access is
often not available in typical office floors. Furthermore, in typical office floors
experiencing vibration, the displacement is too small (in the order of 0.1 mm) to
be treated with a conventional viscous damper. Viscous dampers can be effective
where there is a large motion associated with the vibrations such as in footbridges
and stadia floors where the vibration energy can be dissipated through their
typical dashpot systems.
Given that typical viscous dampers would not be effective in reducing vibration in
typical office floors, a number of attempts were made to develop other forms of
complex passive, semiactive and active dampers. However, there is a
demonstrated need to develop a cost effective and simple TMD specifically for
floor applications.
1.2. Research Aim and ObjectivesThe overall aim of this study is to develop a new damper to reduce floor
vibrations due to walking in existing buildings. This damper is to be used as a
passive tuned mass damper (TMD) on typical floors where the vibration
4
displacements are too small for conventional viscous dampers. In order to achieve
this aim, the following objectives are to be accomplished:
1. Perform a detailed literature review to present state of the art methods of
design for floor vibrations and provide a critique of available rectification
methods using dampers.
2. Develop a concept of a new passive TMD specifically for floor applications
where the damper is to be fitted within the false floor or false ceiling space.
3. Develop an analytical model to design the new TMD to suit any given floor
application. The model is to predict the TMD dynamic properties based on
basic material and geometric properties.
4. Design and build prototype dampers using commercially available materials to
validate the developed analytical model and also demonstrate construction
viability.
5. Assess the effectiveness of the new TMD in reducing the level of vibration of
simple beams using physical tests and finite element (FE) analysis.
6. Extend the application of the developed damper into a distributed multiple
damper system to rectify floors with limited space for a single large damper.
7. Assess the performance of the multiple damper system on a real office floor.
8. Conduct a sensitivity analysis using validated FE models to establish the
performance limits of the newly developed damper.
The above objectives will be achieved using analytical modelling, physical testing
in laboratory, FE analysis and investigation of a real office floor.
5
1.3. Thesis OutlineThe chapters of this thesis are arranged according to the workflow and are
summarised as follows;
Chapter two presents a detailed literature review covering human
excitation, human comfort criteria, determination of floor dynamic
properties and assessment methods. It covers damping estimation methods
and rectification measures for floors with excessive levels of vibration.
Chapter three discusses the concept of the new TMD and the development
of the analytical model to design the proposed TMD. This chapter also
presents the methods used to obtain the relevant material properties for the
TMD. This chapter covers the validation of the analytical model of the
TMD in terms of natural frequency, damping and modal mass. The
analytical model is validated using experimental tests and FE analysis.
Chapter four discusses two case studies for the TMD. A small size
prototype TMD was developed and installed on a simply supported steel
beam. The dynamic properties of the steel beam were determined
analytically and experimentally in addition to using FE analysis. This
chapter presents an evaluation of the performance of the proposed TMD.
The response of the steel beam was also investigated for variations in the
TMD location, TMD natural frequency, TMD damping ratio and the
damping of the steel beam.
The second case study was of a larger size prototype TMD which was
installed on a 9.5 m concrete beam. The dynamic properties and maximum
response of the concrete beam were evaluated using experimental tests in
addition to using FE analysis. The performance of the proposed damper
was evaluated by comparing the response of the concrete beam for the
configurations without and with the TMD. The performance of the TMD
was assessed for variations in the damper natural frequency, damper
6
damping ratio and the damping ratio of the retrofitted beam. The
performance of the proposed TMD was investigated using heel drop and
walking excitations.
Chapter five discusses the concept of a multiple tuned mass damper
(MTMD) system and the development of an analytical model to obtain the
response of a floor with a MTMD system. Four TMDs were developed
and installed on the concrete beam that was investigated in Chapter four.
This chapter covers two types of installation; (a) four TMDs at the point of
maximum response and (b) TMDs in a spatially distributed form at four
locations. The performance of the MTMD system in reducing the level of
vibration for the two configurations was assessed using analytical and FE
models as well as physical testing with both heel drop and walking
excitations.
This chapter also discusses the performance of the distributed MTMD
system on a real office floor. It presents the evaluation of the maximum
response of the original floor and compares it with human comfort criteria.
The office floor was retrofitted by the distributed MTMD system and its
response due to human and shaker excitations was evaluated using
experimental tests in addition to using analytical and FE analyses.
Chapter six presents a sensitivity analysis using validated FE models for
the response of an office floor to variations in the mass and damping of the
floor in addition to variations in the damping ratio, natural frequency and
points of attachment of the MTMD system.
Chapter seven presents the major findings and conclusions of this research
in addition to recommendations for future work.
7
2. Literature Review
2.1. IntroductionThis chapter reviews the human dynamic loading associated with design methods
for floor vibrations. The evaluation of dynamic properties of floor systems such as
the modal mass, natural frequency, damping and peak response are presented.
Methods used to estimate the damping of a floor system are demonstrated. This
chapter also discusses the available tuned mass dampers (TMDs) to retrofit floor
systems with excessive vibrations such as passive, semiactive and active TMDs in
addition to the advantages and disadvantages of each type of dampers.
2.2. Walking ExcitationOccupants excite floors from their activities such as walking, dancing and
jumping. Such forces are particularly problematic because they can not be easily
isolated from the floor and they occur frequently (Hanagan & Murray 1997).
Walking pedestrians can induce considerable vertical and horizontal rhythmic
impulsive dynamic loads that are dominated by the pacing rate. Typical pacing
rates for walking are between 1.6 and 2.4 steps per second, i.e. 1.6-2.4 Hz (slow to
fast walk) whilst for jogging the pace rate is about 2.5 Hz and running occurs at
pace rates up to about 3 Hz (Collette 2004).
Although the force produced from pedestrians is dominated by the pacing rate, it
also includes higher harmonic components caused by the impulsive nature of the
load with frequencies corresponding to an integer multiple of the pacing rate. One
pedestrian walking at a pacing rate of 2 Hz will therefore excite a floor with a
force composed of harmonic components at 2 Hz (1st harmonic), 4 Hz (2nd
harmonic), 6 Hz (3rd harmonic), etc. A floor may be prone to resonance induced
by pedestrian walking, if one or more of its natural frequencies are within the
ranges 1.6-2.4 Hz (1st harmonic), 3.2-4.8 Hz (2nd harmonic) and 4.8-7.2 Hz (3rd
8
harmonic). The fourth harmonic in some cases can also cause significant
excitation, however higher harmonics components for walking seldom induce
unacceptable vibrations (Collette 2004).
Generally, the first three or four harmonics comprise the main dynamic
components of walking forces (Rainer & Pernica 1986). It is worth to mention
that the floor natural frequency usually coincides with the second or third
harmonics of the walking force. Floors could also be excited through resonance of
the fourth harmonic, however the forcing function is typically greatly reduced.
For example, a floor response due to the fourth harmonic force is half of a floor
response due to the third harmonic force because Fourier coefficient of the third
harmonic is about 0.1 while it is about 0.05 for the fourth harmonic (Da Silva et
al. 2007 ; Murray et al. 1997).
The annoying vibration amplitudes are caused by a coincidence of the natural
frequency ( nf ) of the floor with one of the harmonics of the excitation force such
as walking excitation. This problem may be avoided by keeping these frequencies
away from each other. This strategy is called High Tuning Method (HTM)
(Bachmann 1995 ; Naeim 1991), which for a highly damped floor system
( 5%), the lowest ( nf ) of the floor should be above the frequency range of the
second harmonic of walking (i.e. above 4.8 Hz) and for a floors with low damping
( 2%), it is recommended that the lowest resonance frequency should be above
the third harmonic of walking (i.e. above 7.2 Hz). Since the natural frequencies of
most floors are greater than 3 Hz (they often fall between 4 Hz and 8 Hz),
problems are most likely to occur as a result of the second and third harmonics.
However, the lower the harmonic the larger the vibration produced by resonance
(Allen & Pernica 1998). To allow for some scatter in the accuracy of estimating
the parameters, fn 7.5 Hz should be targeted. This HTM is a simple rule of thumb
for general evaluation of systems but it is not accurate enough. The high-
frequency tuning method was popular in the past as analytical calculation of
actual floor responses was difficult (Pavic & Reynolds 2002). However, many
floors can not be simply designed for such a high frequency and the method does
9
not take account of damping explicitly or the effect of a large participating mass.
As a consequence, some floors with a fundamental frequency less than the 7.5 Hz
criterion can perform quite satisfactory to walking (Bachmann 1995). On the other
hand, composite floors with very low damping ( 2%), can experience high
levels of vibration even if their first natural frequency is above 7.5Hz (Haritos et
al. 2005).
2.2.1. Single Person Walking ExcitationFor normal walking, the forcing function has a saddle shape as shown in Figure
2-1. The first maximum corresponds to the impact of the heel onto the floor and
the second corresponds to the thrust of the sole of the foot.
The forcing function of walking depicted in Figure 2-1 is represented by the
contact peaks of heel and the tip of the toe, and the decrease in between is due to
the reduced impact forces from stiffness effects of the legs in cases with slow
walking. As walking pacing rate increases, the time interval between the heel peak
and the peak due to the tip of the toe decreases and the difference between the
maximum and minimum increases. These walking force functions can be
represented by using quarter sine wave and one cosine wave. The quarter sine
waves are for the heel and tip of the toe peaks and the cosine wave assigned to the
impact decrease part (Obata & Miyamori 2006). A typical time force function is a
series of footfall overlap as illustrated in Figure 2-2.
10
Figure 2-1 Human walking force model (Obata & Miyamori 2006)
Figure 2-2 Footfall overlap function during walking at 2 pace/sec (Newland 2004)
11
The pace rate (step frequency) has an impact on the magnitude of the walking
force. Figure 2-3 shows the effect of the walking frequency on the amplitude of
the walking force. The figure illustrates that the magnitude of the walking force is
proportional to the walking frequency (i.e. force increased when the pace rate
increased).
Figure 2-3 Walking force for several frequencies (Waarts & Van Duin 2006)
The periodic forcing function tF of walking can be defined using Equation
(2-1) (Brownjohn et al. 2004 ; Ellis et al. 2000 ; Ellis & Littler 2004 ; Murray et
al. 1997 ; Setra 2006);
N
iistepi tifSinrFtF
10 21 2-1
where
0F = the static force (person’s weight);
i = the order of harmonic of walking rate ( i = 1, 2, … N);
ir = the i th dynamic load factor (DLF), which is the Fourier coefficient of the
dynamic forcing function normalised by the static weight of person;
stepf = the walking frequency in Hz;
12
t = the time variable in seconds;
i = the phase angle of the i th harmonic in relation to the first one in radians/sec;
and
N = the number of harmonics taken into account.
The Fourier coefficient of the dynamic forcing function design values of a vertical
footfall force as a function of the excitation frequency for walking excitation in
Equation (2-1) can be estimated using values demonstrated in Table 2-1. This
table summarises the dynamic load factors (DLFs) for vertical force from single
person as suggested by different authors for different human activities.
Generally as demonstrated in Table 2-1, the DLF decreases when harmonics
increase and the fourth harmonic DLF is usually small enough and for simplicity,
it may be omitted from Equation (2-1).
13
Table 2-1 DLFs for single person force model for different authors (Zivanovic et al. 2005)
Author DLF Comment Activity
Blanchard et al.
(1977)257.01 r
DLF is lessen from 4 Hz to 5
Hzwalking
Rainer et al.
(1988)Figure 2-4
DLFs are frequency
dependent
Walking, jumping
and running
Allen (1990a,
1990b)
5.11 r 2-2.75 Hz
Aerobics6.02 r 4-5.5 Hz
1.03 r 6-8.25 Hz
Bachmann
(1995)
5.04.01 r Between 2 Hz and 2.4 HzWalking
1.032 rr At approximately 2 Hz
2.0,7.0,6.1 321 rrr From 2 Hz to 3 Hz Running
7.0,3.1,8.1 321 rrr Normal jump 2 Hz
Jumping5.0,1.1,7.1 321 rrr Normal jump 3 Hz
1.1,6.1,9.1 321 rrr High jump 2 Hz
8.0,3.1,8.1 321 rrr High jump 3 Hz
Kerr & Bishop
(2001)06.0,07.0& 321 rrr
r is frequency dependent
(Figure 2-5)Walking
Willford et al.
(2005)Table 2-2
Mean values for DLFs is
frequency dependentWalking
ISO 10137
(2007)
)1(37.01 stepfr 1.2 to 2.4 Hz
Walking
1.02 r 2.4 to 4.8 Hz
06.03 r 3.6 to 7.2 Hz
06.04 r 4.8 to 9.6 Hz
06.05 r 6.0 to 12.0 Hz
14
Figure 2-4 DLFs for the first four harmonics for (a) walking, (b) running and (c) jumping
force (Rainer et al. 1988 ; Zivanovic et al. 2005)
Figure 2-5 Third order polynomial fit to the first harmonic (Kerr & Bishop 2001)
15
Table 2-2 Dynamic load factors for walking (Willford et al. 2005)
Harmonic
( i ) stepf (Hz) Design value (DFL)
1 1-2.8 5.095.041.0 stepf
2 2-5.6 stepf0056.0069.0
3 3-8.4 stepf0064.0033.0
4 4-11.2 stepf0065.0013.0
4i >11.2 0
The relative phase angles ( i ), in Equation 2-1), were determined to be 0 , 2 ,
and 0 for i = 1, 2 and 3, respectively, for walking rate between 2 and 2.4 Hz
(Rainer & Pernica 1986). Bachmann (1995) and Setra (2006) suggested
alternative values for the phase angle for a step frequency of 2 Hz. These values
are 01 and 232 .
The magnitude of maximum walking force using Bachmann (1995) and Setra
(2006) phase angle values is about 10% greater than the magnitude of walking
force obtained from Rainer & Pernica (1986) values as illustrated in Figure 2-6.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
200
400
600
800
1000
1200
Time (s)
Wal
king
Exc
itatio
n (N
)
Rainer and Pernica (1988)Setra (2006)
Figure 2-6 A comparison between walking force of two suggested values of phase angles
16
2.2.2. Group Walking ExcitationStudies were performed for group walking tests on a building floor and it was
found that the peak acceleration resulting from groups of size up to 32 was
approximately twice the corresponding result produced by a 75 kg individual.
Subsequently, a factor of 2 could be added to approximate the resultant
acceleration for groups up to 32 (Brand et al. 2007 ; Ellis 2003). The floor
response increased with increasing of group size while groups of people walking
at a pace to generate resonance do not always produce the largest response for a
given group size. Although there is a large difference in acceleration levels for
resonant and off resonant loading for an individual this is not always the case for
the larger groups (Ellis 2003). This situation of large number of group walking is
not normally relevant to floor systems especially with office environments. It is
more relevant for pedestrian bridges, stadia and shopping centres.
2.3. Assessment of Floor Response to Walking ExcitationSeveral methods are available to evaluate a floor response to walking excitation.
Each method uses different criteria to assess the peak response of a floor system
and different acceptance criteria. The four methods commonly used worldwide
are discussed. These are AISC design guide 11 (AISC DG11), SCI P354,
European Hivoss and British Cement and Concrete Industry Publication no. 016
(CCIP-016). The principles and basics of each method are discussed in the
following sub-sections.
2.3.1. AISC DG11 MethodThis method is described in AISC Steel Design Guide Series 11 Floor Vibration
Due to Human Activity (AISC DG11) (Murray et al. 1997). This method is based
on calculating the peak acceleration response of a floor system and compares it
with acceptable levels of acceleration. The acceptable levels of acceleration are
dependent on the floor function such as an office floor, floor for shopping centers
etc.
17
2.3.1.1. Acceptance Criteria for Human ComfortThe acceptance criteria for floor vibrations are based on human comfort. The
influence of vibration frequency and amplitude on human comfort was initially
studied by Reiher and Meister in 1931 and extended further later by Lenzen in
1966 (Alvis 2001 ; Boice 2003 ; Lenzen 1966). The Lenzen modified Reiher-
Meister scale for vibration acceptance is shown in Figure 2-7.
Figure 2-7 Modified Reiher-Meister Scale (Alvis 2001)
The reaction of people who feel vibration depends very strongly on what they are
doing. People in offices or residences are disturbed at peak acceleration of about
0.5% of the acceleration of gravity ( g ) whereas people taking part in an activity
will accept acceleration levels 10 times greater (5% g or more) (Allen 1990a ;
Murray et al. 1997). People’s perception is also affected by the characteristics of
the vibration response including frequency, amplitude and duration (Hanagan &
Murray 1997). Figure 2-8 demonstrates the recommended acceptable peak
acceleration response for different environments and their variation with
18
frequency. From comfort studies for automobiles and aircrafts it was found that in
the frequency of 4 to 8 Hz humans are especially sensitive to the vibration. This is
explained by the fact that many organs in the human body resonate at these
frequencies (Alvis 2001) whilst outside this frequency range, people accept higher
vibration acceleration levels (Murray et al. 1997).
The usual assessment is based on the measurement of acceleration. However,
acceptable vibration levels vary with the frequency of the motion, hence it is
necessary to filter the acceleration (ISO10137 2007). The multiplying factors for
rms (root mean square) acceleration in Table 2-3 are applied to the base curves
presented in Figure 2-8. The ISO 10137 acceptance criteria will be discussed
further in Section 2.3.2.
Figure 2-8 Acceptability criteria for vertical floor accelerations (Ebrahimpour & Sack 2005)
19
Table 2-3 Multiplying factors for low probability of adverse comment (ISO10137 2007)
Place TimeMultiplying factor forexposure to continuous
vibration 16 h day 8 h night
Impulsive vibrationexcitation with up to 3
occurrencesCritical working areas(e.g., hospitaloperating theatres
Day 1 1
Night 1 1
ResidentialDay 2 to 4 60 to 90
Night 1.4 20
OfficeDay 4 128
Night 4 128
WorkshopsDay 8 128
Night 8 128
People’s perceptions of vibration effects in buildings are very subjective, vary
widely and can include annoyance, loss of mental concentration and even
apprehension about the safety of the structure. None of these effects, however, are
considered especially harmful. In general, the effects on the structure from floor
vibrations due to human excitation are not critical to its integrity as the vibration
amplitudes are well below the critical stress limits for strength and fatigue.
2.3.1.2. Evaluation of Floor’s Peak Response to WalkingThe time dependent walking force component that matches the fundamental
frequency of a floor can be represented in Equation (2-2). It should be noted that
only one harmonic of Equation (2-1) is included since all other harmonics
vibration are small in comparison to the harmonic associated with resonance
(Murray et al. 1997);
)2(0 tifCosrFF stepi 2-2
where
0F = person’s weight;
ir = dynamic load factor (DLF);
i = the order of harmonic number;
stepf = the walking frequency in Hz; and
20
t = the time variable in seconds.
The dynamic load factor (DLF), )( ir , for which the lowest harmonic ).( stepfi of
the walking frequency can match a natural frequency of the floor structure )( 1f ,
was calibrated to form Equation (2-3) (Murray et al. 1997);
)35.0exp(83.0 1fri 2-3
where 1f is the fundamental frequency of floor structure.
In general, the vibration response reduces as (the ratio of a floor modal
frequency to footfall rate) increases. If is close to 1.0, the footfall rate equals
the natural frequency of the mode, and the mode responds strongly in the
resonance to the first harmonic of the footfall force. If is close to 2.0, the mode
responds resonantly to the second harmonic of the footfall force, and so on. These
resonances lead to significantly greater response when is close to 1.0, 2.0, 3.0
or 4.0 than for intermediate values of . For 4 there is much less sensitivity
to the exact value of (i.e. the response does not build up over time). The
response is characterised by an initial peak response followed by a decaying
sinusoid (Willford & Young 2006). Consequently, this method of floor
assessment can lead to conservative predictions for low pace frequencies, and un-
conservative predictions for higher pace frequencies.
2.3.1.3. Idealisation of Single Degree of Freedom SystemsA floor can be represented by a single degree of freedom (SDOF) system with an
equivalent mass ( 1m ), stiffness ( 1k ) and damping ( 1c ) as shown in Figure 2-9.
The equation of motion that governs a SDOF system is expressed in Equation
(2-4) (Chopra 1981);
)()()()( 111 tFtaktactam 2-4
21
where
)(ta = acceleration;
)(ta = velocity;
)(ta = displacement;
)(tF = excitation force; and
t = time.
Figure 2-9 Schematic of single degree of freedom system
The steady state acceleration response of a SDOF system to a harmonic excitation
force, )(tF , can be calculated using Equation (2-5) (Inman 1996);
2211
221
2
mkc
Fa
2-5
where
a = peak acceleration;
F = excitation force amplitude; and
= excitation force frequency in radians/sec.
22
The damping coefficient of the system ( 1c ) can be calculated using Equation
(2-6) (Thorby 2008);
1111 2 mkc 2-6
where 1 is the damping ratio of the system.
The fundamental circular frequency of the SDOF can be calculated using
Equation (2-7) (Inman 1996);
1
11 m
k 2-7
The maximum acceleration response ( pa ) occurs when the frequency of the
excitation force coincides with the natural frequency of the system, i.e. 1 ,
hence Equation (2-5) can be reduced to Equation (2-8) to obtain the maximum
acceleration;
112 mFap
2-8
Substitution of Equation (2-3) for walking excitation into Equation (2-8) to
introduce the walking force and multiplying the resulting equation by a constant
( R ) yields the resonance response acceleration of the floor which can be written
as;
11
10
2)35.0(83.0
mfExpRFap
2-9
where R is a reduction factor which takes into account the fact that the floor may
not reach the full steady state resonant motion due to walking. Moreover, the
23
walking person and annoyed person are not simultaneously at the location of
maximum response. It is recommended by Murray et al. (1997) that the reduction
factor )(R to be taken as 0.5 for floor structures with two way mode shape
configuration.
The peak acceleration due to walking excitation for an idealised floor represented
as a SDOF floor structure can then be estimated using Equation (2-10) (Murray et
al. 1997);
WfPExp
gap
1
1)35.0(
2-10
where
gap = estimated peak acceleration in units of gravity acceleration ( g );
1f = the fundamental frequency of floor structure;
1 = the damping ratio of the floor;
W = the effective weight of the floor; and
P = constant force (0.29 kN) due to person’s weight defined by Equation (2-11);
kNRFP 29.083.0 0 2-11
where
0F = 0.7 kN; and
R = 0.5.
The effective weight (W ) and the natural frequency ( 1f ) of the floor in Equation
(2-10) are to be determined using the procedure described by Murray et al. (1997).
24
2.3.1.4. Floor Damping Value for AISC DG11 MethodDamping in floor systems is a very complex quantity and it affects the level and
duration of vibrators. Damping in floor systems is composed of three primary
components:
a) Material damping, which is the ability of the construction material themselves
to dissipate the excitation energy. The type of material used in construction of
the floor system is a factor that can determine the overall damping of the floor,
for example, timber floors have higher damping than concrete floors. Table
2-4 demonstrates damping ratios for various type of bare floor systems such as
composite floors, prestress floors, wood joist floors etc.
b) Damping from the floor fit-out and furniture, which is highly dependent on the
height of partitions and the type of furniture used. For example, traditional
office floors with full-height partitions and filing cabinets have higher
damping than modern paperless office without full-height partitions.
c) Damping due to attached mechanical services (e.g. ducts) and the availability
of false ceilings and false floors. For example, a floor with a false ceiling has
higher damping than a floor without a false ceiling.
Table 2-4 Examples of damping values for the fundamental mode of floors in buildings(ISO10137 2007)
Type of floorRange of spans for
damping ratiosgiven (m)
Values of dampingfor preliminary
design of bare floorsSteel joist/concrete slab simply supported 9 to 15 1.3%
Steel joist/concrete slab, continuous slab
construction across walls4 to 8 1.5%
Fully composite steel beams with shear
connectors to concrete slab6 to 20 1.8%
Prestressed concrete, precast 2 to 15 1.3%
Reinforced concrete, monolithic 5 to 15 1.5%
Wood joist floors 2 to 9 2.0%
25
Damping in a vibrating structure is associated with dissipation of mechanical
energy, generally by conversion into thermal and sound energy. In most cases, the
structural mass and stiffness can be evaluated rather easily, either by simple
physical consideration or by generalised expressions. On the other hand, the basic
energy loss mechanism (damping) in practical structures is seldom fully
understood. Consequently, it usually is not feasible to determine the damping
coefficient by means of corresponding generalised damping expression. For this
reason, the damping in most structural systems is either estimated during the
design stage or evaluated by measurements for existing floors.
The damping ratio of the floor is required to be assumed prior to calculating the
peak acceleration of the floor. It is obvious from Equation (2-10) that the
maximum floor acceleration is very sensitive to the assumed damping value. For
example, peak acceleration response using Equation (2-10) for a floor system with
dynamic properties of W = 275 kN , 1f = 6.0 Hz and 1 = 3% is about 0.43%
gravity (g) which is considered to be acceptable for office floors (compared to
Figure 2-8). The peak acceleration response for this floor is significantly increased
to 0.52% g when the damping ratio ( 1 ) reduced to the 2.5%, which is considered
to be unacceptable. Therefore, overestimation of damping during the design phase
can lead to excessive vibrations for floors in service.
Engineers have to guess the damping values according to the fit-out of the floor
systems. The function or the fit-out of the floor could be altered during the design
lifespan of the floor system. This alteration may affect the damping ratio of the
floor and hence the level of acceleration. For example, the damping ratio of floor
system originally designed as a traditional office floor is significantly decreased
when the floor is modified to a modern electronic office.
Hewitt & Murray (2004) and Murray et al. (1997) suggested damping ratio values
for different type of floors. Figure 2-10 illustrates the traditional and modern floor
systems with different fit-out and attached services. The damping ratio of each
type of a floor system is illustrated in Table 2-5. These damping ratio values of
26
floor systems were determined according to the partition height, fit-out and the
availability of suspended ceilings and services. It is clear from Table 2-5 that the
transformation from traditional floors to modern floors results in a decrease in the
damping ratio of the floor system. Consequently, the peak acceleration response
of floor systems due to human excitation is significantly increased as the floor
response is damping dependent.
Figure 2-10 Types of office floors (Hewitt & Murray 2004)
27
Table 2-5 Damping ratios for different type floors (Hewitt & Murray 2004)
Type of floor systemDamping
ratio
Traditional office: full-height partition with (or without) suspended ceiling and
ductwork attached below the slab. Full-height partitions running perpendicular to the
beam span will provide sufficient damping to eliminate floor-vibration problems.
5%
Electronic office: Nearly no paperwork with limited number of filing cabinets. No
full-height partitions, with suspended ceilings and ductwork attached below the slab.2-2.5%
Electronic office: Nearly no paperwork with limited number of filing cabinets. No
full-height partitions, no suspended ceilings or ductwork attached below the slab.2%
Open office plan: Cubicles and no full-height partitions, with suspended ceiling and
ductwork attached below the slab.2.5-3%
Open office plan: Cubicles with no full-height partitions, suspended ceiling or
ductwork below the slab.2-2.5%
Office Library: Full-height bookshelves in heavily loaded room with suspended
ceiling and ductwork attached below the slab.3-4%
Office Library: Full-height bookshelves in heavily loaded room with no suspended
ceiling and ductwork attached below the slab.3%
2.3.1.5. Summary of AISC DG11 ProcedureThe procedure to assess the dynamic performance of a floor system using AISC
DG11 can be summarised as follows:
i) calculate the natural frequency ( 1f ) and effective weight (W ) of the floor
system using the procedure described by Murray et al. (1997);
ii) estimate the damping ratio of the floor using Table 2-5;
iii) calculate the peak acceleration response of the floor using Equation (2-10);
and
iv) compare the peak acceleration response with acceptable level of acceleration
in Figure 2-8. For example, the maximum acceptable level of acceleration
response of an office floor is 0.5% of gravity ( g ).
28
2.3.2. SCI P354 MethodThis procedure is a European method was written to enable engineers to
determine the vibration response of floors. This method was originally described
in SCI P076 which was published in 1987 and then with improved accuracy in
SCI P356 which was published in 2007. It is based on the calculation of natural
frequency and modal mass of the floor and its predicted damping. The rms (root
mean square) acceleration ( rmsa ) for floors with fundamental frequencies between
3 Hz and 10 Hz can be calculated (assuming a resonant response to one of the
harmonic of walking frequency) using Equation (2-12) (Smith et al. 2007);
bffo
sxyexyrms RWmFa
221.0
,, 2-12
where
exy , = mode shape factor at the point of excitation;
sxy , = mode shape factor at the point of response;
= damping ratio obtained from Table 2-6;
oF = weight of person (746 N);
m = modal mass of the floor;
fW = weighting factor obtained for human perception of vibrations based on the
fundamental frequency of the floor obtained from Figure 2-11; and
bfR = resonance build up factor obtained from Equation (2-13) (Smith et al. 2007);
step
stepp
vfL
bf eR2
1 2-13
where
stepf = step frequency in Hz;
pL = length of the walking path in meters; and
29
stepv = velocity of walking in ( sm ) obtained from Equation (2-14) (Bachmann &
Ammann 1987 ; Smith et al. 2007);
5.483.467.1 2 stepstep ffv HzfHz step 4.27.1 2-14
Figure 2-11 Weighting factor for human perception of vibrations (Smith et al. 2007)
Table 2-6 Damping ratios for various floor types (Smith et al. 2007)
Floor FinishesDamping
Ratio ( )
For completely bare floors or floors where only a small amount of furnishings
are present.1.1%
For fully fitted out and furnished floors in normal use. 3.0%
For a floor where the designer is confident that partitions will be appropriately
located to interrupt the relevant mode of vibration, (i.e. the partition lines are
perpendicular to the main vibrating elements of the critical mode shape).
4.5%
30
Once the weighted rms acceleration ( rmsa ) of the floor has been weighted, the
Response Factor ( RF ) can be calculated using Equation (2-15) (Smith et al.
2007);
005.0rmsaRF 2-15
If the response factor ( RF ) values are within the limits of the multiplying factors
for continuous vibrations in Tables (2-3 and 2-7), the floor is considered to be
acceptable. Whereas for the situation where the floor has a higher response than
would be acceptable under the conservative limits for continuous vibration, the
method permits the use of Vibration Dose Value (VDV). The VDV analysis is
effectively allowing the response to be greater than those specified for continuous
vibrations, but only for small periods of time. The VDV of a walking activity of
duration ( aT ) that occurs ( an ) times in an exposure period is calculated Equation
(2-16) (Smith et al. 2007):
468.0 aarms TnaVDV 2-16
where
rmsa = the rms acceleration ( 2sm ) obtained from Equation (2-12).
an = the number of times the activity will take place in an exposure period; and
aT = the duration of an activity (for example, the time taken to walk along a
corridor) in seconds obtained from Equation (2-17);
step
pa v
LT 2-17
31
Table 2-7 Recommended multiplying factors based on single person excitation (Smith et al.2007)
Place Multiplying factor for exposure tocontinuous vibration
Office 8
Shopping mall 4
Dealing floor 4
Stairs – Light use (e.g. offices) 32
Stairs – Heavy use (e.g. public buildings, stadia) 24
The VDV value can be directly compared to the limits given in Table 2-8.
Alternatively, this equation can be rearranged to give the number of times an
activity can occur in an exposure period and still correspond to ‘a low probability
of adverse comment’ as expressed in Equation (2-18) (Smith et al. 2007);
4
68.01
rmsaa a
VDVT
n 2-18
The exposure periods that should be considered are a 16 h day and an 8 h night,
and a VDV analysis can be considered to be satisfactory if the floor will be
traversed fewer than times in the exposure period.
Table 2-8 Vibration dose value limits (Smith et al. 2007)
PlaceLow probability of
adverse comment
Adverse comment
possible
Adverse comment
probable
Buildings 16 h day 0.2 to 0.4 0.4 to 0.8 0.8 to 1.6
Buildings 8 h night 0.13 0.26 0.51
32
2.3.3. Hivoss MethodAnother method for calculating the floor peak response and assessing its
performance include the recently published European Commission guide
described in Human-induced vibration of steel structures (Hivoss) (Feldmann et
al. 2009 ; Hechler et al. 2008 ; Hivoss 2010).
Similar to the AISC DG11 and SCI P354, this method directly relates the peak
response to the total damping which has to be assumed during the design phase.
The other dynamic properties of the floor, i.e. floor natural frequency and modal
mass are required to be determined using a prescribed procedure in order to
calculate the peak response.
2.3.3.1. Damping Estimation Using Hivoss MethodIn this guideline, the total damping of the floor for the design and assessment of
an existing floor purposes is taken as the sum of contributions from structural
damping, furnishing and finishes as shown in Table 2-9. For example, the
damping ratio of a composite floor with a suspended ceiling used as an open plan
office can be determined as a contribution of 1 = 1% for structural damping, 1 =
1% for furniture damping and 1 = 1% for finishes damping which yields the
overall estimated damping ratio of the floor to be 1 = 3%.
33
Table 2-9 Estimation of floor’s damping design values (Hechler et al. 2008)
Type of floor Damping ratio
Structural damping 1
Wood 6%
Concrete 2%
Steel 1%
Composite (Steel-Concrete) 1%
Damping due to furniture 1
Traditional office for 1 to 3 persons with separation wall 2%
Paperless office 0%
Open plan office 1%
Library 1%
Residential 1%
School 0%
Gymnastic rooms 0%
Damping due to the finishes 1
Ceiling under the floor 1%
Free floating floor 0%
Total damping 1111
2.3.3.2. Determination of Natural Frequency and Mass ofFloor Using Hivoss MethodThe natural frequency and modal mass of a simple plate can be calculated using
equations presented in Figure 2-12. These equations are applied according to
fixity conditions (type of supports) and length ( L ) to width ( B ) ratio ( ) of the
floor system. It should be noted that these equations are applied only for isotropic
plates but not for orthotropic plate. An example of such isotropic plates is given in
Figure 2-12.
34
Figure 2-12 A procedure to calculate the natural frequency and modal mass of isotropic
plates (Hechler et al. 2008)
Orthotropic floors such as composite floors with beams in longitudinal direction
and a concrete plate in transverse direction have different stiffness in length
( y)EI( ) and width ( x)EI( ) where xy )EI()EI( . A typical composite floor
(orthotropic) is shown in Figure 2-13. The first natural frequency of the
orthotropic plate being simply supported at all four edges can be determined using
Equation (2-19);
35
y
xy
)EI()EI(
lb
lb
ml)EI(
f
42
41 212
2-19
where
l = the length of the floor (in x-direction) in metres;
b = the width of the floor (in y-direction) in metres;
E = the Young's modulus ( 2/ mN ) to be 10% higher than the static modulus;
xI = the moment of inertia for bending about the x-axis ( 4m );
yI = the moment of inertia for bending about the y-axis ( 4m ); and
m = the mass of floor including finishes and a representative amount of imposed
live load ( 3/ mkg ). Expected values for residential and office buildings are 10% to
20% of the imposed live load.
Figure 2-13 Typical composite floor consists of a slab and beams (Hivoss 2010)
The modal mass of a composite floor system can be calculated according to the
mode shape of the floor. For a composite floor shown in Figure 2-13, the modal
mass ( 1m ) can be calculated using Equation (2-20);
222
22
18
2
yxyxMm 2-20
36
where
M = total mass of the floor;
x = deflection of the beam;
y = deflection of the slab assuming the deflection of the supports (i.e. the
deflection of the beam) is zero; and
yx .
2.3.3.3. Hivoss Acceptance CriteriaThe design value to calculate the response of a floor system is One Step Root
Mean Square 90% (OS-RMS90) response velocity as illustrated in Figure 2-14.
This value covers the response velocity of the floor for a significant step with
intensity of 90% of person’s steps walking.
Figure 2-14 The OS-RMS90-values as a function of step frequency and body mass (Hechler et
al. 2008)
The Root Mean Square (RMS) of the response velocity can be determined using
Equation (2-21) (Hechler et al. 2008);
2)(1
0
2 PeakT
RMSvdttv
Tv 2-21
37
where
T = time;
)t(v = velocity function;
RMSv = RMS response velocity; and
Peakv = peak response velocity.
Once the floor damping ratio, natural frequency and modal mass were determined
using Table 2-9 and Equations (2-19) and (2-20), the class of the floor can then be
obtained using the OS-RMS90 diagram as shown in Figure 2-15.
The diagram is used by entering the modal mass on the mass-axis and the
corresponding frequency on the frequency-axis. The OS-RMS90 value and the
acceptance class can be read off at the intersection of extensions at both entry
points. The OS-RMS90 diagrams are introduced for floor systems with damping
ratios ranging from 1% to 9%. Hence, an appropriate OS-RMS90 diagram
matching the damping ratio of the floor has to be selected in order to obtain the
class of the floor and its OS-RMS90 value. It should be noted that the OS-RMS90
velocity values are in millimetres per second.
The acceptance or otherwise of the floor can be assessed using Figure 2-16
according to the class and the function of the floor. For example, class D floor is
considered to be acceptable for office floors whereas class F is not recommended.
38
Figure 2-15 OS-RMS90 application for floors with 3% damping ratio (Hechler et al. 2008)
39
Figure 2-16 Classification of floor response and recommendation for the application of
classes (Hivoss 2010)
2.3.4. CCIP-016 MethodThis design guide is described in “A design guide for footfall induced vibration of
structures: A tool for designers to engineer the footfall vibration characteristics of
buildings or bridges” which was published by the Cement and Concrete Industry
Publication no. 016 (CCIP-016) (Willford & Young 2006). For the AISC DG11,
SCI P354 and Hivoss guidelines, the response of the floor to walking excitation is
calculated as result of a harmonic force coinciding with the natural frequency of
the floor. In this assessment method, the floor response is calculated for each
walking harmonic and the total response of the floor is a combination of all
walking harmonic forces up to the fourth harmonic.
The procedure to evaluate the performance of a floor system using CCIP-016
method is as follows:
1. calculate the response for each mode of the floor to each of the first four
harmonics ( if ), from i = 1 to i = 4, for a particular footfall rate ( stepf ).
40
2. identify the vertical modes of the floor up to 15 Hz. All vertical modes with
frequencies up to 15 Hz can potentially contribute significant response and
should be included in the calculation. For complex and composite floor
systems, FE analysis is essential to identify the mode shapes and hence the
corresponding frequencies and modal masses.
3. Calculate the harmonic frequency ( if ) using Equation (2-22);
stepi fif . 2-22
where
i = harmonic number from i = 1 to i = 4; and
stepf = walking frequency in Hz.
4. Calculate the harmonic force, iF , for this harmonic frequency using Table 2-2
and Equation (2-23) for each mode of the floor;
0.FDLFFi 2-23
where
DLF = dynamic load factor obtained from Table 2-2; and
0F = static weight of the walker.
5. Calculate the real and imaginary acceleration (jireala ,, ,
jiimaga ,, ) for each
mode using Equations (2-24) and (2-25);
22
2
jj
j
j
j
j
j BAA
mF
ffa ,is,xye,xyii
,i,real
2-24
41
22
2
jj
j
j
j
j
j BAB
mF
ffa ,is,xye,xyii
,i,imag
2-25
where
= mode shape;
j = mode shape number;
jf = frequency of mode j ;
e,xy = mode shape value at the point of excitation;
s,xy = mode shape value at the point of response;
2
1
j
j ffA i
;
j
jj ffB i
2 ;
j,i = 1 when the number of steps to cross the footpath exceeds 10 steps
j = damping ratio of mode j and it can be estimated from Table 2-10.
Table 2-10 Suggested damping values for different structures (Willford & Young 2006)
Structural typeDamping
ratio
Bare steel composite or post tensioned concrete floors with little or no fit-out 0.8% to 1.5%
Bare reinforced concrete floors 1% to 2%
Completed steel composite or post tensioned floors with low fit-out 1.5% to 2.5%
Completed steel composite or post tensioned floors with typical fit-out 2% to 3%
Completed reinforced concrete floors with typical fit-out 2.2 to 3.5%
Completed steel composite, post tensioned or reinforced concrete floors with
extensive fit-out and full-height partitions3% to 4.5%
6. sweep the footfall frequency ( stepf ) and hence the walking harmonic force
using Table 2-2. The footfall frequency varies from slow walking of 1.0 Hz to
fast walking of 2.8 Hz. A convenient frequency increment (e.g., 0.01 or 0.1
42
Hz) for this range of step rate (1 – 2.8 Hz) is required to obtain the floor
response for each harmonic frequency.
7. sum the real and imaginary responses for all modes of the floor (included in
the calculations) for this harmonic frequency to obtain the total real ( ireala , )
and imaginary ( iimaga , ) accelerations for each harmonic force using Equation
(2-26);
J
j,i,reali,real j
aa ; J
j,i,imagi,imag j
aa 2-26
where J is the number of modes included in the calculations.
8. find the magnitude of this acceleration ( ia ), where i is the harmonic of the
footfall, which is the total response in all modes of the floor to this harmonic
(at this footfall frequency) as illustrated in Equation (2-27);
22i,imagi,reali aaa 2-27
9. convert this acceleration magnitude ( ia ) to a response factor ( i)RF( ) using
Figure 2-17. First, calculate the baseline peak acceleration for a response
factor of 1 at this harmonic frequency ( iRFa ,1 ).
If if < 4 Hz,i
iRF fa 0141.0
,1 2sm
If 4Hz < if < 8 Hz, 0071.0,1 iRFa 2sm
If if >8 Hz, 4,1 1082.2
iiRF fa 2sm
43
Divide this response factor into the total acceleration response for this
harmonic as illustrated in Equation (2-28);
i,RF
ii a
a)RF(
1
2-28
Figure 2-17 Baseline RMS acceleration (Willford & Young 2006)
10. find the total response factor ( RF ) for this particular footfall rate using
Equation (2-29), which is the 'square root sum of the squares' combination of
the response factor for each of the four harmonics;
24
23
22
21 ))(())(())(())(( RFRFRFRFRF 2-29
11. repeat this calculation for other footfall rates according to the specified
frequency increment and find the critical rate that generates the maximum
response.
44
An example to demonstrate the procedure to calculate the response factor (RF) for
a step frequency of 2.18 Hz is shown in Table 2-11. This calculation is for two
span simply supported footbridge with dynamic properties of 37,000 kg modal
mass, 1.5% damping ratio and natural frequencies of vertical modes are mode 1 =
4.2 Hz, mode 2 = 6.6 Hz and mode 3 = 16.9 Hz. It should be noted from Table
2-11 that only the first two vertical modes are included in the calculation as the
third mode is above 15 Hz.
Table 2-11 Response factor calculation for walking at 2.18 Hz (Willford & Young 2006)
Harmonic number 1 2 3 4
Harmonic frequency (Hz) (Eq. 2-22) 2.18 4.36 6.54 8.52
Real and Imag. Responses (m/s2)
(Eqs. 2-24 & 2-25)Real Imaginary Real Imaginary Real Imaginary Real Imaginary
Mode 1 2.78E-03 5.09E-05 1.41E-02 3.83E-03 -2.82E-03 1.19E-04 -1.84E-03 4.05E-05
Mode 2 1.13E-03 1.28E-05 1.33E-03 4.88E-05 1.53E-02 3.78E-02 -2.75E-03 1.38E-04
Mode 3 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Total (Eq. 2-26) 3.91E-03 6.37E-05 1.55E-02 3.88E-03 -1.82E-02 3.79E-02 -4.59E-03 1.79E-04
Magnitude of response (m/s2)
(Eq. 2-27)3.91E-03 7.10E-02 7.10E-03 7.80E-03
Base curve acceleration (m/s2) 9.20E-03 7.10E-03 7.10E-03 8.33E-03
Response factor (Eq. 2-28) 0.41 2.25 5.93 0.59
Total RF 2.35 Hz (Eq. 2-29) 6.38
The response factors for all walking frequencies for the range of 1-2.8 Hz are
illustrated in Figure 2-18. From this figure, the maximum response will be due to
a pedestrian walking at 2.32Hz and the response factor (RF ) value is about 8.5.
The maximum value of the total response factor ( RF ) is compared with the
acceptance criteria. If the footfall vibration level is below RF = 8, this is almost
satisfactory for commercial buildings such as offices, retail, restaurants, airports
and the like where some people are seated. However, there are instances in which
some people complain at this vibration level, particularly when it occurs regularly
(Willford & Young 2006).
45
Figure 2-18 Response for all walking frequencies (Willford & Young 2006)
2.4. Damping Estimation from Measured DataThere are different methods that are used in damping estimation from recorded
data. For example, Logarithmic Decrement Method (LDM) is used in the time
domain analysis and Half-Power Bandwidth (HPB) is used in the frequency
domain analysis. The selection of damping analysis method depends on the
characteristics of the vibration data including level of noise and number of modes
included in the vibration analysis. The following sub-sections provide a brief
description for the methods commonly used to obtain the damping ratio from
recorded data.
2.4.1. Logarithmic Decrement Analysis (LDM)The LDM probably is the most popular method that is used to determine damping
of an oscillator. When a SDOF oscillator is excited by an impulse or an initial
displacement, its response takes the form of a time decay (De Silva 2007). In the
logarithmic decrement analysis, the decay in vibration amplitude ( ) which is
defined as the natural log of the ratio of the size of two peaks with i cycles apart,
can be estimated using Equation (2-30) (Blanchard et al. 1977 ; Clough 1975 ;
Thorby 2008);
46
in
n
xx
i
ln1 2-30
where
nx = the amplitude of nth cycle; and
inx is the amplitude of the thin cycle.
The damping ratio can then be found from Equation (2-31);
2
2-31
It should be noted that this procedure assumes SDOF system response behaviour.
For multi-degree of freedom (MDOF) systems, the modal damping ratio for each
mode can be determined using this method if the initial excitation is such that the
decay takes place primarily in one mode of vibration. In other words, substantial
modal separation and the presence of real modes (not complex modes with non-
proportional damping) are assumed (De Silva 2007).
2.4.2. Half-Power Bandwidth (HPB)The half-power bandwidth method as illustrated in Figure 2-19 is commonly used
in estimating damping in the frequency domain. The response of an excited
oscillator is usually recorded in time domain and it can be converted to frequency
domain using Fast Fourier Transform (FFT). The results obtained from this
method are very good for single degree of freedom systems with small values of
damping. This method is also used extensively for MDOF and continuous systems
both in the laboratory and in the field, including systems that do not have linear
viscous damping and even systems that do not have normal modes in the classical
sense (Olmos & Roesset 2010). The maximum amplitude )( max related to a
particular natural frequency )( nf is determined from the frequency domain trace
and then the maximum amplitude divided by the square root of two )2( is
47
calculated to obtain the values of )&( 21 ff as shown in Figure 2-19. The damping
ratio )( can then be calculated using Equation (2-32);
nfff
212 2-32
Frequency
Am
plitu
de
or 3 dB in Log scale
f2fnf1
max2
max
Figure 2-19 Half-Power Bandwidth
The use of this method for damping estimation has a clear meaning only to the
degree to which a structure can be successfully modelled by a SDOF system or by
a series of decoupled SDOF systems. Nevertheless, the use of the HPB method
can be extended to MDOF structures based on the assumption that each peak in
the frequency response function is affected only by the mode under study. On the
other hand, the method is challenged in cases of MDOF structures with modes
having nearly the same natural frequency (closely spaced modes) leading to
possible mode coupling. The degree of this mode coupling in a structure depends
on the interplay among its damping distribution, its geometric characteristics
(from which its natural frequencies can be found) and its type of excitation
(Papagiannopoulos & Hatzigeorgiou 2011). This method can be used when the
frequency trace for a structure response is available regardless of knowing the
input data.
48
Haritos (1993) investigated an alternative optimised method to obtain the damping
level. The “equivalent area” was tested and compared to the “peak value” in the
frequency domain and half-power bandwidth. The basic concept of the
“equivalent area method” is to equate the area under the measured transfer
function trace. The reason behind the use of this concept is that by conducting
such integration, the influence of “noisiness” is minimised because the integration
is a form of smoothing operation. The area can be determined by using standard
numerical integration such as Simpson’s rule. Haritos used a Monte Carlo style
simulation to identify the statistical characteristics of predicted damping levels of
a SDOF. The equivalent area method is considered more than satisfactory for
determination of damping levels below about 8%.
The accuracy of the estimated level of damping may vary depending on the
prediction method. The accuracy is influenced by a number of factors in particular
the “noisiness” of the data. It is reported that the equivalent area methods
produces sufficiently accurate estimates for system with low damping (Haritos
1993).
2.4.3. Circle-Fit MethodSometimes the measured modes are closely spaced and interfere with each other
and hence the half-power bandwidth can not accurately estimate the damping as it
is difficult to identify the peak and frequency band (i.e. 21 & ff ) for each mode.
The equation of motion for a SDOF illustrated in Figure 2-9 under forced
excitation ( tiFe ) can be expressed as (Beards 1996);
tiFekxxcxm 2-33
where
x = response acceleration;
x = response velocity;
x = response displacement;
49
F = force amplitude;
= excitation frequency in radian; and
t = time.
A solution tiXex can be assumed, yields Equation (2-34) (Beards 1996);
222222
2
)()()()(
cmkci
cmkmk
FX
2-34
The solution consists of two components; the real part ( )Re( FX ) in phase with
the force and imaginary part ( )Im( FX ) in quadrature with force.
Nyquist plots of imaginary and real part were introduced into vibration analysis
by Kennedy and Pancu in 1947 to provide a simple and fairly accurate method for
analysing vibration test data (Kennedy & Pancu 1947 ; Thorby 2008). An
example of the Nyqusit plot for a given value of ( c ) is illustrated in Figure 2-20.
Experimentally the curve in Figure 2-20 can be obtained by plotting the measured
amplitude and phase of )( FX for each excitation frequency ( f ). The phase
angle can be obtained using Equation (2-35) (Beards 1996);
cmk 2
)tan( 2-35
The method of estimating the damping from Half-Power Bandwidth of the
amplitude response described in Section 2.4.2 can be applied. The half-power
points occurs when 45 and 135 and the frequencies ( 21 & ff ) can be
identified as:
1f = excitation frequency at the phase angle 45o below the phase of the resonant
frequency ( nf ); and
50
2f = excitation frequency at the phase angle 45o above the phase of the resonant
frequency ( nf ).
Figure 2-20 Nyquist plot for a system with viscous damping (Beards 1996)
Imregun (1991) addressed that the circle-fit method gives reliable results when
there are enough data points around resonance and that damping is not too low.
On other hand, this method should not be used when the data contains noise
around resonance (Iglesias 2000 ; Imregun 1991)
2.4.4. Random Decrement Technique (Randec)This method was first introduced by H.A. Cole at NASA during the late 1960s
and early 1970s (Asmussen et al. 1999). The concept of Randec is based on the
principle that the response of a system due to the random excitation consists of
three components, the response to an initial displacement, response to an initial
velocity and the response to a random input load. Averaging large number of
51
segments with the same initial condition will eliminate the forced vibration
component. Thus, a typical free vibration decay curve can be obtained.
Consequently, the damping ratio )( and natural frequency )( nf can be extracted
from the free vibration decay trace (Brincker et al. 1991 ; Rodrigues & Brincker
2005).
Considering two response time histories )(tx and )(ty , simultaneously measured,
the auto )(XXD and cross )(XYD Randec functions can be mathematically
defined by the following expressions (Rodrigues & Brincker 2005):
)()(1)(1 i
N
i iXX tTxtxN
D
2-36
)()(1)(1 i
N
i iXY tTxtyN
D
2-37
where
N = the number of averaged time segment; and
)( itTx = the triggering condition applied to time history.
In order to obtain the free vibration curve, a triggering level and segment length
are required to be defined as shown Figure 2-21. One important aspect of the
application of Randec technique is the definition of the triggering level ( )( itTx ) in
Equations (2-36) and (2-37) and the length of time segments that are extracted
from a time history. In general, it is advantageous to use a large number of
triggering points. However, in order to eliminate the effect of noise, a large value
of triggering level ( )( itTx ) should be specified since the lower values of triggering
level in the time history are more contaminated by the noise than the larger
values. Therefore, some balance must be achieved between the large number of
triggering points and having a high value of triggering level ( )( itTx ). A good
option is to consider the standard deviation of the response time history ( x ) to
52
specify the triggering level ( )( itTx ) with optimum value of xitTx 2)(
(Rodrigues & Brincker 2005).
Figure 2-21 Basic concept of the Random Decrement Technique
(Rodrigues & Brincker 2005)
2.5. Remedial Measures to Suppress Floor VibrationsAlthough structural engineers have design guidelines for evaluating the floor
vibration response during the design phase, there is still a large number of
constructed floors that exhibit excessive vibrations. These problematic floors arise
from inappropriate assumptions made in prediction of damping during the design
stage or due to change of use or alterations. Few options are available to rectify a
floor with excessive levels of vibration.
The relocation of the vibration source is the cheapest corrective method such as
placing the vibration source (e.g. a gym) on the ground slab or placing sensitive
equipment near columns or walls where the vibrations are less severe than at mid-
bay (Koo 2003).
Floor damping depends primarily on the presence of non-structural components
such as partitions, ceilings, mechanical services and furnishing. Full-height
partitions are most effective in adding damping to the floor system (Allen &
Pernica 1998). Adding non-structural elements such as full-height partitions with
53
the aim of increasing damping in most cases is not possible due to architectural
and functional requirements (Setareh et al. 2006).
Increasing the floor stiffness can reduce human induced vibration because it
increases the natural frequency of the floor and hence shifting the resonance to
higher harmonics. Stiffening methods include increasing floor member depths and
ensuring that there is a composite action between the beams and the concrete slab.
Introducing new columns between existing columns from the affected floor down
to the foundations is very effective in the case of flexible floor structures, but is
often unacceptable to the owner. Adding members, as shown in Figure 2-22, can
stiffen light-frame floors. However, it is important to determine whether the
vibration is caused by flexible supports (poor seating, flexible beams, etc.) before
proceeding with stiffening the joists (Allen & Pernica 1998). In many instances
there is physically not enough space to introduce new structural elements and also
this solution is not conservative as it does not take the account of damping.
Figure 2-22 Stiffening technique for steel joists and beams (Allen & Pernica 1998)
Adding mass can reduce the vibration level but in most cases it is not practical as
it may create overstress in structural members or it could reduce the natural
frequency of the floor, which makes it more susceptible to vibration.
Another option to reduce floor vibrations is to employ dampers. This technique is
discussed in details in the following sections.
54
2.6. DampersPassive, semiactive and active dampers can be used effectively to reduce
excessive floor vibrations. This type of rectification measure increases the
damping of floor systems and hence decreases the response vibrations.
Mechanical dampers or viscoelastic material can often be installed more cheaply
than structural stiffening and are usually the only practical means of vibration
control in existing structures (Webster & Vaicaitis 1992).
2.6.1. Passive Tuned Mass Dampers (TMD)The principle of a tuned mass damper (TMD) was initially utilised when Den
Hartog in 1947 reintroduced the dynamic absorber invented by Frahm in 1909 (De
Silva 2005 ; Hartog 1956). Generally, a TMD consists of a mass ( 2m ), spring ( 2k )
and dashpot ( 2c ), and is tuned to the natural frequency of the primary system ( 1f ).
When the primary system as illustrated in Figure 2-23 begins to oscillate it sets
the TMD into motion and hence the TMD absorbs energy from the vibrating floor.
The TMD inertia forces produced by this motion are anti-phase to the excitation
force. The first use of a TMD for floor vibration application was reported by
Lenzen (1966) who used small TMDs with a total mass of about 2% of the floor
mass. The TMDs were made of steel hung by springs and dashpots from the floor
beams. Lenzen reported floors with annoying vibration characteristics became
satisfactory by tuning the TMDs to a natural frequency of about 1.0 Hz less than
that of the floor and using a damping ratio of 7.5% (Setareh 2002).
TMDs are typically effective over a narrow frequency band and must be tuned to
a particular natural frequency. They are not effective if the structure has several
closely spaced frequencies and they can potentially increase the vibration if they
are off-tuned (Webster & Vaicaitis 1992). One TMD can only damp one mode of
vibration and if damping of several modes is necessary the arrangement becomes
quite complex (Bachmann 1995).
55
Figure 2-23 Typical Representation of Two Degree of Freedom Tuned Mass Damper-
structure system
A TMD splits the natural frequency of the primary system into a lower )'( 1f and
higher frequency '2f as shown in Figure 2-24. If there is zero damping then
resonance occurs at the two undamped resonant frequencies of the combined
system ( ''& 21 ff ). The other extreme case occurs when there is infinite damping,
which has the effect of locking the spring ( 2k ). In this case the system fitted with
a TMD becomes one degree of freedom with stiffness of ( 1k ) and a mass of
( 21 mm ). Using an intermediate value of damping such as optimum damping
( opt ), it is possible to control the vibration of the primary system over a wider
frequency range (Smith 1988). For the optimum damper, the values of the
damper’s natural frequency and damping ratio ( opt ) are set to obtain minimum
and equal height peaks at '1f & '2f (Puksand 1975).
56
3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
Frequency (Hz)
Res
pons
e
Without TMD
With TMD
f '2f '1
f1
Figure 2-24 Example showing the effects of attaching a TMD to a SDOF system
The first step in the design of a TMD is to determine the desired mass ratio ( ) as
defined by Equation (2-38);
1
2
mm
2-38
The larger the mass of the damper )( 2m the larger the separation between the two
new frequencies ( ''& 21 ff in Figure 2-24) which are created by the damper. This
would normally increase the effectiveness of the damper over a broader range of
frequencies and also decreases the vibration level of the primary system.
However, there are normally some structural and physical limitations on the size
of the damper and its mass. For most practical cases a mass ratio ( ) of 0.01 to
0.02 is recommended.
The optimum natural frequency of a TMD ( 2f ) can be obtained from Equation
(2-39) (Al-Hulwah 2005 ; Setra 2006);
57
11
2ff 2-39
where 1f is the natural frequency of the primary system.
The optimum damping ratio ( opt ) of the vibration absorber (TMD) corresponds to
the mass ratio of the coupled system as described in Equation (2-40) (Setra 2006);
3)1(83
opt 2-40
An example of a recent TMD proposed for floor vibrations is a Pendulum Tuned
Mass Damper (PTMD) shown in Figure 2-25. Experiments were undertaken on a
testing floor to evaluate the performance of the PMTD in reducing floor vibrations
and it is reported that the damper reduced the floor vibration in the range of 50-
70% (Setareh et al. 2006).
Figure 2-25 Pendulum Tuned Mass Damper (Setareh et al. 2006)
58
2.6.1.1. Damping ElementsThe conventional TMD consists of mass, spring and damping elements. This
damping element can be a viscous damper, viscoelastic damper or liquid damper.
The following sub-sections describe the characteristics of each damping element
and its components.
Viscous Damper
A dashpot as illustrated in Figure 2-26 can act as damping element of a passive
TMD. The dissipation in a dashpot takes place by the conversion of the
mechanical energy into heat, using a piston that deforms a very viscous substance,
such as silicone, and displaces it. Another family of viscous dampers is based on
the flow of a fluid in a closed container. The piston is not limited to deforming the
viscous substance, but forces the fluid to pass through calibrated orifices (Setra
2006).
Figure 2-26 Example of viscous damper (Setra 2006)
The main difference between these two technologies is as follows. In the case of a
pot or wall damper, the dissipative force will depend on the viscosity of the fluid,
59
whereas, in the case of an orifice damper, this force is due mainly to the density of
the fluid. Orifice dampers will therefore be more stable with respect to variations
in temperature than pot dampers or wall dampers (Setra 2006).
Viscoelastic Damper
Rather than using a viscous material, the viscoelastic materials such as polymers
can be used to dissipate energy by working in shear. Figure 2-27 shows a
viscoelastic damper formed from layers of viscoelastic materials between metal
plates. When this type of device is excited, the relative displacement of the outer
plates with respect to the central plate produces shear stresses in the viscoelastic
layer, which dissipates the energy (Setra 2006).
Figure 2-27 Viscoelastic damper (Setra 2006)
Liquid Damper
The first liquid damper prototype was proposed in the 1900s by Frahm to control
rolling in ships. Since the 1970s, these dampers are installed on satellites to
reduce long period vibrations and have been used in buildings since 1980s. The
damper consists of a container filled with a liquid as illustrated in Figure 2-28.
The liquid acts as the secondary mass and the damping is provided by friction
with the walls of the container (Setra 2006).
60
Figure 2-28 Liquid damper (Setra 2006)
2.6.1.2. Application of Passive TMDs on Floor SystemsAn example of a floor system which was retrofitted by a conventional viscous
TMD is Terrace on the Park building. The floor is a reinforced concrete slab
supported by steel beams. The floor was symmetrically partitioned into four
dining/dancing halls (Webster & Vaicaitis 1992).
Preliminary free vibration tests of the floor found that the first natural frequency
to be about 2.3 Hz and the second one was 3.9 Hz. This very low frequency is
well below the recommended levels for floors and corresponds closely to the beat
of many dances. The measured damping ratio of floor was about 3.1%. Guests of
the dining halls complained about the vibrations with observations of sloshing
waves in cocktail glasses and chandeliers that bounced to the beat of the band
gave credence to these concerns. Preliminary vibration tests performed during
dance events showed that the floor accelerations and displacements sometimes
reached 7% g and 3 mm, respectively (Webster & Vaicaitis 1992). It should be
noted that the measured value of acceleration response of this floor was about 14
times greater than the acceptable level of an office floor of 0.5% g.
One TMD of 4% mass ratio was installed in the corner of a ballroom extending
from the floor beam to the roof beam as shown in Figure 2-29a. Calculations
61
showed that the floor beams supporting the TMD would be overstressed with a
damper mass ratio greater than 4%. The typical TMD as depicted in Figure 2-29b
was tuned for optimum frequency and damping and the measurements of TMD
performance during dance events revealed that the TMDs reduced ballroom floor
vibrations by at least 60 percent.
Figure 2-29 Ballroom floor long section with a TMD (Webster & Vaicaitis 1992)
The example of using viscous damper for retrofitting was for retrofitting a
structure with very high acceleration response compared with office floors
acceleration response due to the walking excitation. The maximum displacement
of Terrace Park building was 3 mm due to dance excitation while the maximum
displacements of office floors due to walking excitation is in the order of 0.1 mm.
This very small displacement makes it difficult to develop a conventional viscous
TMD to retrofit office floor systems. Setra (2006) reported that because of the
compressibility of the fluid, friction in the joints, and tolerance in the fixings, it is
not easy to achieve a viscous damper that can tolerate such small displacements.
A liquid damper was used as a damping element to retrofit an office floor. This
damper consists of a steel plate as the spring, and two stacks of steel plates, acting
as additional mass to adjust the TMD frequency as illustrated in Figure 2-30.
Damping is provided by liquid filled bladders confined in two rigid containers
62
instead of conventional dashpot or damping elements connecting the additional
mass to the original structure (Hanagan et al. 2003 ; Shope & Murray 1995).
Figure 2-30 Liquid TMD (Hanagan et al. 2003)
This type of liquid TMD was installed to retrofit a composite floor system with
joists spaning 15.85 m and girders spacing 4.88 m. Heel drop impact tests
identified a significant dynamic response at two natural frequencies of 5.1 Hz and
6.5 Hz. The two modes were controlled by fourteen liquid TMDs. The dampers
were hung from the bottom chords of the existing floor joists to control the two
modes. The acceleration histories for a person walking perpendicular to the joist
span before and after the installation of the dampers are shown in Figure 2-31
(Hanagan et al. 2003 ; Koo 2003 ; Shope & Murray 1995).
The using of liquid dampers attempt provided limited details on the damper
design and its physical limitations.
63
Figure 2-31 Acceleration responses of the floor due to walking without and with liquid TMDs
(Hanagan et al. 2003)
A TMD with viscoelastic damper element were used to reduce the level of
vibration of an office floor. The dampers were designed by the 3M Company,
Minnesota, and consist of an outer frame that rests on the floor, connecting
elements, and an inner frame as shown in Figure 2-32. Four springs and a
viscoelastic damping element connect the outer frame to the inner mass carrying
frame. The inner frame can hold a number of steel plates, which provide the mass
for the TMD and allow for tuning (Hanagan et al. 2003).
64
Figure 2-32 TMD with viscoelastic damping element (Hanagan et al. 2003)
Two dominant modes of vibration with frequencies of 5 and 6 Hz were identified.
One TMD was used to control the 5 Hz mode and two TMDs were used to control
the 6 Hz mode. The TMDs were successfully reduced the level of vibration and
the response of the floor without and with damper due to walking is illustrated in
Figure 2-33.
The using of viscoelastic damping element attempt provided limited details on the
damper design and its performance specifications. Further, they were costly and
complex.
65
Figure 2-33 Walking induced response of the office floor without and with TMDs (Hanagan
et al. 2003)
2.6.2. Semiactive Control DampersDuring the 1980s, the automotive industry researched, developed and tested
various types of semiactive shock absorbers. That research produced a new type
of control actuator that has applications in civil, mechanical, and aerospace
engineering. The term semiactive describes a system that consists of a variable
actuator that requires little power to operate.
Setareh (2002), Koo (2003) and Koo et al. (2004) reported the use of a class of
semiactive tuned mass dampers called ground-hook tuned mass dampers
(GHTMD) as shown Figure 2-34. This comprises a TMD with the semiactive
damping element. A magnetically responsive fluid can be used as a damping
element, which is based on a suspension of micron-sized, magnetisable particles
66
in a carrier fluid as illustrated in Figure 2-35.
Figure 2-34 schematic of semiactive tuned mass damper (Koo 2003)
Altering the strength through the application of a magnetic field precisely controls
the yield stress of the fluid. The alteration of the inter-particle attraction, by
increasing or decreasing the strength of the field, permits continuous control of
the fluid’s rheological properties (Koo et al. 2004). In other words, the magneto-
rheological (MR) fluid mixture thickens, and even becomes solid, when it meets a
magnetic field. As the magnetic field strength increases, the resistance to fluid
flow at the activation regions also increases. This mechanism is similar to that of
hydraulic dampers, in which resistance is caused by the fluid passage through an
orifice. This variable resistance to fluid flow allows the use of MR fluid in
electrically controlled viscous dampers and other devices (Koo 2003).
Figure 2-35 Typical magneto- rheological damper (Koo 2003)
67
The semiactive damper shown in Figure 2-36 was mounted on a primary structure
to form two DOF system in order to evaluate the effectiveness of a semiactive
TMD that uses an MR damper. The physical representation of the test rig with
mounted semiactive TMD is shown in Figure 2-37.
Figure 2-36 Semiactive tuned mass damper (Koo 2003)
Based on analytical and experimental studies, it was found that the semiactive
TMD is more effective than its equivalent passive TMD (for the same mass), in
reducing the level of displacement when subjected to harmonic force excitation.
However, in the case of passive TMDs, their effectiveness depends largely on the
existing floor damping level. Specifically, it was found that GHTMD can
outperform its equivalent passive TMD by about 14%. Even though semiactive
TMD have limited advantage over their passive counterparts in reducing the
maximum levels of vibration, it is reported that they perform in a much more
robust manner when subjected to off-tuning of floor mass and natural frequency
(Koo 2003 ; Koo et al. 2004 ; Setareh 2002).
68
Figure 2-37 Test rig and primary components (Koo 2003)
Although the semiactive TMD is reported to be more robust than the passive
TMD, it is expected that their costs would be significant and ongoing maintenance
would be more demanding due to sophisticated control system. Furthermore, there
has been no effort made using such dampers on floor systems.
2.6.3. Active Control DampersHanagan & Murray (1995) developed and tested an active electro-magnetic
actuator that uses a piezoelectric velocity sensor and a feedback loop to generate
control forces effectively adding damping to the supporting structure. An actively
controlled mass provides a larger degree of control compared with a passive
device with an equivalent reactive mass. Given the high level of control, the
active system could be less disruptive to the building function than most other
rectification measures. The active device can be more compact and can be
installed with relative speed and ease. There are also disadvantages to the active
69
control scheme. The cost of the components to provide a single control circuit was
reported to be high. Furthermore, maintenance and reliability issues also detract
from the attractiveness of an active system, however as the technology advances,
the cost may reduce (Hanagan & Murray 1995, 1997).
Hanagan et al. (2003) investigated an active damper to correct floor systems. It
was reported that the active damper reduced the peak amplitude of walking
excitation by approximately 12% of the peak amplitude for the uncontrolled floor
system. More recently Reynolds et al. (2009) implemented an active control
system to reduce human induced vibrations of an in service floor which had a
damping ratio of 3% and a modal mass of approximately 20 tonnes with
approximately 9 modes between 4 and 10 Hz. Reductions of up to 50 % were
observed experimentally.
2.7. Concluding RemarksRecent changes in building construction have included the use of light composite
and long span floor systems. Although these changes have many advantages, such
floor systems can suffer from excessive vibration due to human activities. This
problem is exacerbated in office buildings due to the reduction in inherent
damping associated with modern fit-outs. Excessive floor vibrations are often
realised after the completion of construction or following structural modifications
and normally arise due to inadequate knowledge of the damping values in the
design process.
The human activities such as walking, dancing, aerobics and running can create
annoying problems to the in service floors. The force produced from walking is
dominated by the pacing rate and it becomes a problem when the floor natural
frequency coincides with one of the walking harmonics of the walking force.
The most commonly used floor assessment methods for floor vibrations have been
discussed in this chapter. The response of a floor subjected to human excitation is
related to the frequency, mass and damping of the floor. The dynamic response of
70
floor systems due to footfall excitations obtained from these assessment methods
is sensitive to the assumed damping value. As it is difficult to know how a
particular floor will be fitted out, the damping value could be overestimated
resulting in excessive floor vibrations. The assessment process also involves a
comparison of the floor response with human comfort which is subjective.
Methods to estimate the damping of a floor system using recorded data were
discussed in this chapter. The damping of floor systems can be obtained from time
history or frequency domain. The accuracy of the measured damping from the
recorded data depends on the floor system behavior in terms of frequencies,
modes of vibration, excitation force and the characteristics of the utilised
measurement method. Consequently, the method used to measure a floor damping
must be carefully considered as each method has limitations and may be suitable
for a particular floor rather than another.
There are few options available to correct a floor with excessive levels of
vibration. The relocation of the vibration source is the cheapest corrective method
such as placing the vibration source on the ground slab or placing sensitive
equipment near columns or walls where the vibrations are less severe than at mid-
bay. Increasing the floor stiffness can reduce human induced vibration because it
increases the natural frequency of the floor and hence shifting the resonance to a
higher walking harmonics. However, in many cases there is physically not enough
space to introduce new structural elements. Adding mass can reduce the vibration
level but in most cases it is not practical as it may create overstress in structural
members or it could reduce the natural frequency of the floor, which makes it
more susceptible. Adding non-structural elements such as full-height partitions
with the aim of increasing damping and stiffness in most cases is not possible due
to architectural and functional requirements.
The use of mechanical dampers is a possible option to reduce excessive
vibrations. Dampers to rectify problematic office floor systems experiencing small
displacement response are not practically available yet. Past attempts reported in
71
the literature to treat office floor vibrations using other types of dampers provide
limited information about the construction of dampers and their performance
specifications. Floor vibrations due to walking excitation typically produce
dynamic displacements that are generally very small. In reality, it is difficult to
produce a practical viscous damper that provides a reasonable level of damping
given the small displacements. While viscous dampers were used in some floor
applications, these had large displacements compared with what would occur in
office floors.
For floor retrofitting application, there are usually physical limitations associated
with access and presence of mechanical services attached to the soffit of the slab
and beams. Hence for a floor damper to be practical, it needs to be sufficiently
small to be accommodated in the available ceiling space and it should also allow
for easy adjustment of frequency for tuning.
Given the above mentioned limitations, an alternative concept for a simple and
cost effective TMD specifically for floor vibrations with small displacements is
required.
72
3. Viscoelastic Damper
3.1. IntroductionThis chapter discusses the development of a new viscoelastic tuned mass damper
and the prediction of its dynamic properties (i.e stiffness, modal mass and
damping ratio) using an approximate analytical model. The analytical model is to
take into account the factors which may affect the behaviour of the damper
including boundary conditions, material properties and geometric parameters.
In order to assess the concept of the viscoelastic damper, prototypes were
developed and tested. The results from the tests were used to validate the
analytical model and were also used for benchmarking against Finite Element
(FE) analysis.
3.2. Damping Using Viscoelastic MaterialsThe use of viscoelastic materials in reducing the effect of vibrations is common in
mechanical engineering applications especially in machine vibrations. Recently it
also became a solution for floor vibrations. Indeed an effective way to increase
damping and reduce transient and steady state vibration is to add a layer of
viscoelastic material, such as rubber as shown in Figure 3-1. The combined
system would have a higher damping level and thus reduces unwanted vibration
(Inman 1996).
Embedded viscoelastic materials (VEM) offer the advantage of reducing
vibrations over a broad range of frequencies compared with TMDs which work
optimally only for a specific mode of vibration. Use of VEMs is a cheap method
of increasing the damping if incorporated during construction (Ljunggren 2006 ;
Ljunggren & Ågren 2002).
73
An example of viscoelastic damping is the Resotec system, which is illustrated in
Figure 3-1. This product comprises a thin layer of high damping viscoelastic
material with an overall thickness of about 3 mm. Resotec is sandwiched between
the top flange of the floor steel beams and concrete slab for a proportion of the
beam near each end where the shear stresses are the greatest. It is reported as
shown in Figure 3-2 that the damping of a fitted out floor is typically doubled by
the incorporation of Resotec (Willford et al. 2006). However, this product needs
to be incorporated within the floor during construction and is not suitable as a
rectification measure.
Figure 3-1 Resotec product installation (Willford et al. 2006)
74
Figure 3-2 Performance of resotec product (Willford et al. 2006)
3.3. The Concept of New Viscoelastic DamperThe mass-spring conventional viscous TMD system consists of three elements; a
mass (m), a spring (k) and a dashpot (c). The proposed TMD to replace the
conventional viscous TMD is in a cantilever beam form as illustrated in Figure
3-3. It consists of a viscoelastic material sandwiched between two constraining
layers. The dynamic properties of the sandwich beam (i.e. damping coefficient,
modal stiffness and modal mass) are required to be calculated in order to optimise
it for a given floor. The three key elements (m, k and c) of the viscoelastic damper
are highly dependent on the geometry of the damper, the dynamic properties of
the rubber and the properties of constraining layers. The overall damping mainly
comes from the viscoelastic materials, which replaces the damping element
(dashpot) in viscous dampers. The damping coefficient of the viscoelastic damper
depends on the dissipation loss factor of the rubber used. In this composite
sandwich beam, the viscoelastic material experiences considerable shear strain as
it bends, dissipating energy and attenuating vibration response (Mace 1994). The
flexural stiffness of the cantilever beam is acting as the spring element in the
typical viscous damper and it is highly dependent on the geometry of the
75
sandwich beam, constraining layer material and the stiffness of the rubber. The
end mass and the modal mass of the cantilever beam represent the mass of the
damper.
Figure 3-3 Viscoelastic damper compared with viscous damper
The proposed damper can be installed within false ceilings as illustrated in Figure
3-4a. The proposed damper can also be housed within false floors as shown in
Figure 3-4b. To install the proposed damper within a very tight space it can be
scaled down to the required size and used in a multiple arrangement.
Figure 3-4 Proposed viscoelastic damper installed within false ceilings and false floors
76
3.4. Development of Analytical ModelThe simplest form of a viscoelastic mechanical damper as described earlier is a
constrained viscoelastic layer in a beam. There are many factors which affect the
damping performance of viscoelastic materials in sandwich beams including
material type, thickness, temperature and bonding. The viscoelastic damper
proposed in this research is for internal use so variation in the temperature is not
significant. Consequently, the variation in the dynamic properties of the
viscoleastic damper due to the variation in the temperature is negligible. The resin
used for bonding the layers can be easily selected so that it has minimal slip at the
interfaces of the layers. Hence the two main remaining factors to be taken into
account for the design of the damper are the viscoelastic material type and
thickness.
In order for the viscoelastic sandwich beam (as shown in Figure 3-5) to be used as
a damper, its natural frequency and damping ratio need to be estimated. There are
two methods for obtaining a solution, namely, an exact solution and an
approximate method as discussed below.
Figure 3-5 Typical sandwich beam
77
The equation of motion for the sandwich beams was considered by a number of
researchers. Kerwin (1959) analysed the three layer system and derived an
expression for flexural stiffness of the sandwich beams. DiTaranto (1965) derived
a sixth order differential equation governing the motion of the sandwich beams. In
contrast, Equation (3-1) was derived by Mead & Markus (1969) for a sandwich
beam with arbitrary boundary conditions subjected to forced vibration. The
solution of Equation (3-1) is complex as it involves solving a sixth order
differential equation. This solution can be complicated further by other loading
configurations such as the addition of an end mass to the tip of a sandwich
cantilever. Classical exact solutions are discussed by Mead (2007) and Mead &
Markus (1969);
01111 2
jg
DAjYjg n
iin
tnn
ivn
vin 3-1
where
= mode shape function;
n = mode number;
g= shear parameter;
= viscoelatic material loss factor;
Y = geometric parameter;
= circular natural frequency;
= overall disipation loss factor;
= mass density;
A = cross-sectional area of the beam;
tD = flexural rigidity of the constraining layers; and
1j .
Equation (3-1) is based on the following realistic assumptions related to the
behaviour of the damper:
78
(i) the viscoelastic core resists shear stress but not direct flexural stress;
(ii) shear strains in the constraining plates are negligible;
(iii) transverse strains in the core and constraining plates are negligible; and
(iv) no slip occurs at the interfaces of the core and constraining plates.
3.5. Approximate Analytical MethodAn approximate analytical method developed by Mead (1982) is utilised to solve
Equation (3-1) for the viscoelastic sandwich beam. The flexural rigidity ( totalEI )( )
and the overall dissipation loss factor of the viscoelastic damper ( ) are estimated
based on the dissipation loss factor of the viscoelastic material ( ), thickness of
viscoelastic layer, geometric parameter (Y) and Young’s moduli of the top and
bottom plates constraining the viscoelastic material. This method can be applied
to any sandwich beam configuration such as simply supported or cantilever
beams.
The flexural rigidity ( totalEI )( ) of the viscoelastic sandwich beam can be
calculated using Equation (3-2) (Mead 1982);
331122
2
)1('21))1(1(1)( IEIE
gggYgEI total
3-2
where
= the dissipation loss factor of the rubber;
Y = the geometric parameter to be calculated using Equation (3-3);
g = the shear parameter to be calculated using Equation (3-4);
1E = the modulus of elasticity of the top constraining plate;
3E = the modulus of elasticity of the bottom constraining plate;
1I = the moment of inertia of the top constraining plate about its neutral axis; and
3I = the moment of inertia of the bottom constraining plate about its neutral axis.
79
The geometric parameter (Y ) is calculated using Equation (3-3);
))(())((
33113311
23311
IEIEAEAEdAEAEY
3-3
where
1A = the cross-sectional area of the top constraining plate;
3A = the cross-sectional area of the bottom constraining plate; and
d = the distance between top and bottom constraining plate centroids.
The shear parameter ( g ) is calculated using Equation (3-4);
33112
2
11AEAEKh
GbgB
3-4
where
G = the shear modulus of the rubber;
b = the width of the sandwich beam;
2h = the thickness of the viscoelastic core; and
BK = the wave number of the sandwich beam.
The performance of a tuned mass damper (TMD) is highly dependent on damper
properties in terms of the damper modal mass ( 2m ), damper natural frequency
( 2f ) and the damping ratio of the damper ( 2 ). Therefore, the damper should
satisfy the optimum properties as discussed in Section 2.6.1 to attain the
maximum reduction of vibration of the primary system.
The properties of conventional viscous dampers can be controlled by selecting
separate mass, spring and dashpot. This approach is not applicable for the
proposed viscoelastic damper as the three properties are interrelated. The stiffness
and the overall damping of the proposed viscoelastic damper need to be
80
determined in order to satisfy Equations (2-39) and (2-40). The closed form
solution for Equation (3-1) to obtain the stiffness and the damping ratio of the
proposed damper is quite complex. As the alternative method is approximate,
therefore an error is expected in the frequency and the damping of the viscoelastic
damper. The accuracy of this approximate method in calculating the natural
frequency and the damping ratio of the viscoelastic damper are to be assessed
experimentally using prototype dampers. In addition to the experimental values,
the approximate method values of the natural frequency and damping ratio are
compared with FE results of a model of the viscoelastic damper.
3.6. Design of Proposed Viscoelastic DamperThe proposed viscoelastic damper consists of two constraining metal plates
bonded together with high damping rubber as illustrated in Figure 3-5. A simple
procedure for designing an optimum damper is outlined below. This procedure
determines the dynamic properties of a viscoelastic damper using the approximate
mathematical solution. The procedure is summarised as follows;
(i) Determine the basic dynamic properties of the primary system (floor) to be
retrofitted with a damper (i.e. 1f , 1 and 1m for an equivalent SDOF
system). These properties are to be based on the as built conditions of the
floor. This assures that the developed damper will suppress a specific
mode of vibration of the floor.
(ii) Determine a suitable mass ratio ( ) (Equation (2-38)) for the damper
based on given physical limitations and the required reduction in the
response of the primary system. A mass ratio of 1% to 2% is practical in
most cases.
(iii) The optimum natural frequency of the TMD, 2f , can be calculated
according to the specified mass ratio using Equation (2-39).
(iv) The optimum damping ratio of the TMD ( 2 ) can now be calculated
81
according to the specified mass ratio using Equation (2-40).
(v) Trial dimensions for the damper can be proposed to suit any physical
limitations (i.e. 1h , 2h , 3h , L and b as illustrated in Figure 3-5). In
addition, the material properties for the constraining layers and
viscoelastic material (rubber) need to be specified ( 1E & 3E for the
constraining layers and & G for the viscoelastic material). For
simplicity the top and bottom layers can be of the same material (i.e. 1E =
3E ).
(vi) Calculate the overall flexural rigidity totalEI )( of the sandwich beam
without an end mass using Equations (3-2 - 3-4).
The wave number ( BK ) for a cantilever sandwich beam without an end
mass can be calculated using Equation (3-5) (Inman 1996);
LKB
875.1 3-5
where L is the length of the viscoelastic cantilever beam.
(vii) Calculate the natural frequency of the damper with an end mass using
Equation (3-6);
2
22 2
1mkf
3-6
where
2k = the modal stiffness of the cantilever beam; and
2m = the modal mass of the cantilever beam.
82
For the cantilever beam, the modal stiffness of the beam ( 2k ) can be
obtained from Equation (3-7);
32)(3
LEIk total 3-7
where
totalEI )( = the flexural rigidity obtained from Equation (3-2); and
L = the length of the viscoelastic cantilever beam.
The modal mass of a uniform viscoelastic cantilever beam with an end
mass can be calculated from Equation (3-8) (Buchholdt 1997);
endmALm 14033
2 3-8
where
= mass density of the sandwich beam;
A = overall cross-sectional area of the sandwich beam; and
endm = end mass at the tip of the cantilever sandwich beam.
As the correct damper frequency is critical in reducing the floor vibrations,
the end mass can be used to fine tune the damper frequency.
(viii) Calculate the overall dissipation loss factor and then the estimated
damping ratio ( 2 ) of the damper using Equation (3-9) (Mead 1982);
)1)(1()2(12 222
YgYgYg
ee
e 3-9
where
Y = geometric parameter obtained from Equation (3-3); and
83
eg= shear parameter for the cantilever sandwich beam with end mass.
Equation (3-4) needs to be modified to consider the effect of the added end
mass on the natural frequency of the system. Consequently, the wave
number value of the system is altered because it is proportional to the
frequency as shown in Equation (3-10) (Mead 1982 ; Nashif 1985);
totalB EI
AfK)(
2 22 3-10
Substitution of Equation (3-10) into the Equation (3-4) yields the
expression in Equation (3-11);
)11()(2 331122 AEAEA
EIfh
Gbg totale
3-11
This procedure is required to be repeated by altering the material and dimensions
of the viscoelastic damper until the optimum damper properties are achieved.
One of the main factors that affect the performance of a viscoelastic damper is the
rubber dissipation loss factor ( ). This material property needs to be established
for the type of rubber to be used. The determination of this property is discussed
in the following section.
3.7. Determination of Viscoelastic Material PropertiesIn order to design a viscoelastic TMD, the material properties for the viscoelastic
material (rubber) need to be identified ( & G ). Many types of commercially
available rubbers do not have adequate technical specifications concerning their
material properties and consequently it becomes necessary to undertake specific
tests on the acquired rubber to determine the shear modulus ( G ) and the
dissipation loss factor ( ). This can be achieved using one of two types of tests:
84
(i) direct measurements using a Dynamic Mechanical Analyser (DMA)
(DMA2980 2002 ; Menard 2008); and
(ii) back calculation from experimental results performed on a prototype
damper.
These two methods were both utilised to find the properties of the rubber used in
developing the viscoelastic cantilever damper in this research and are discussed
below.
3.7.1. Dynamic Mechanical Analyser (DMA)Dynamic mechanical analysis is a testing technique that measures the mechanical
properties of materials as a function of time, temperature and frequency. In DMA
testing, a small deformation is applied to a sample in a cyclic manner with the
measured response providing information on the stiffness and damping properties
for the material. The setting used for this research is the dual cantilever clamp
mode (as shown in Figure 3-6) which is used to analyse weak to moderately stiff
samples such as rubber. In this setup, the sample is rigidly clamped using the
clamping jaws with the exterior jaws acting as fixed supports to the ends of
sample while the centre clamping jaw moves up and down to excite the sample to
the desired amplitude and frequency. Two methods of DMA tests can be used to
determine the properties of rubber. They are a frequency sweep mode and strain
sweep mode (DMA2980 2002).
The testing performed by this machine separates the viscoelastic response of
material into the two components of the complex value of modulus ( *E ): the real
part corresponds to the elastic modulus ( E ) and the imaginary part refers to the
damping or loss component ( E ). The standard complex variable notation is
defined by Equation (3-12) (Nashif 1985);
EEE * 3-12
85
The separation of the measurement into the two components describes the two
independent processes within the materials-elasticity (energy storage) and
damping (energy dissipation). This is the fundamental feature of dynamic
mechanical analysis that distinguishes it from other mechanical testing techniques.
The loss tangent ( )(Tan ) represents the dissipation loss factor of the rubber ( )
and is calculated using Equation (3-15).
EE)(Tan
3-13
Figure 3-6 The DMA machine with dual cantilever clamp mode (DMA2980 2002)
86
For the rubber used in developing the damper, three samples measuring 35 x 10.8
x 5.2 mm were tested. The DMA uses the term “Tan Delta” for the dissipation
loss factor ( ) while the term “Storage Modulus” refers to the stress/strain ratio
( E ) of the rubber. The average measured loss factor ( ) was found to be 0.12
and the average measured shear modulus ( G ) was 690 kPa based on the
assumption that ( GE 3 ) for elastomeric materials (Nashif 1985).
In the frequency sweep test as illustrated in Figure 3-7, the dissipation loss factor
and the storage modulus of the sample do not significantly change due to the
variation in the frequency although a slight increase in the storage modulus when
the frequency increases can be observed. On the other hand, the dissipation loss
factor and storage modulus are significantly changed due to the variation in the
strain amplitude for the strain sweep test with a constant excitation frequency (10
Hz for this test) as illustrated in Figure 3-8. The dissipation loss factor increases
when the strain amplitude increases while the storage modulus decreases when the
strain amplitude increases.
Figure 3-7 Example of the DMA test result for a rubber sample using the frequency sweep
87
Figure 3-8 Example of the DMA test results for rubber samples using the strain sweep
Generally, the dissipation loss factor and storage modulus of rubber depend on the
variation in the excitation frequency and the corresponding strain amplitude. The
proposed viscoelastic damper would be tuned to a particular frequency hence the
variation in the dissipation loss factor and storage modulus of the rubber due to
the variation in the frequency is negligible. Since the energy transmitted to the
damper from a floor is relatively low, the variation in the rubber strain at
resonance is also low and hence the dissipation loss factor and storage modulus do
not significantly change.
3.7.2. Back Calculation from Prototype TestingThe rubber dissipation loss factor ( ) can be back calculated from vibration tests
if access to a Dynamic Mechanical Analyser is not available. This method
requires construction of a prototype sandwich beam damper with the selected
rubber. The prototype is then tested to obtain the overall damping ratio of the
damper ( 2 ) and the total flexural rigidity ( totalEI )( ) from basic vibration testing.
The damper needs to be fixed at one end and is dynamically excited by either
mechanical exciter or using a pluck test. The resulting vibration response needs to
88
be measured using either attached displacement transducer (LVDT) or an
accelerometer. A better alternative if available, could be a non-contact sensor to
measure the vibration of damper. These two measured properties along with the
geometric parameters and other material properties of the damper are substituted
into Equations (3-2 - 3-4) to back calculate the ( ) and ( G ) values of the rubber.
The flexural rigidity ( totalEI )( ) can be obtained from the measured natural
frequency using Equations (3-6 - 3-8).
The damping ratio ( 2 ) of the viscoelastic beam can be estimated from the time
domain of the excited viscoelastic beam using the Logarithmic Decrement
Method (LDM) or Half-Power Bandwidth (HPB) method in the frequency
domain. Once the ( totalEI )( ) and ( 2 ) of the viscoelastic beam are obtained, the
procedure described in Section 3.6 is used to back calculate the ( G ) and ( ) of
the rubber.
The accuracy of back calculation method compared to the DMA test in
determining the shear modulus and the loss factor is investigated using prototype
dampers as presented in the following section.
3.8. Validation of Analytical Model Using Prototype DampersAlthough the natural frequency of the viscoelastic damper can be fine tuned by
the added end mass, the mass ratio has to be maintained in order to satisfy the
required mass ratio in Equation (2-38) and the corresponding optimum damping
ratio (Equation (2-40)). The mass of the damper ( 2m ) determines the separation
between the two new frequencies ( ''& 21 ff in Figure 2-24) which are created by
the damper. This would normally increase the effectiveness of the damper over a
broader range of frequencies and also decreases the vibration level of the primary
system. In addition, sometimes there is a structural or space limitations with added
mass, so the tuning of the damper by the added mass becomes impractical.
Therefore, it is important to calculate the stiffness of damper and hence the
corresponding natural frequency as accurately as possible to avoid the need for
89
extensive testing. It should be noted that a large end mass may interfere with
space limitations whilst a low end mass will decrease the performance of the
damper. Hence, it is necessary to know the accuracy of the values obtained from
the approximate method and their effect on the performance of the viscoelastic
damper.
3.8.1. Validation of Analytical Model Using Physical TestingTwo prototypes of cantilever sandwich beams were constructed to examine the
damper concept and to experimentally validate the approximate analytical method
presented earlier. A prototype of viscoelastic cantilever damper referred here as
“Prototype 1” of length 500 mm was constructed for two configurations, normally
without and with an end mass as shown in Figure 3-9. The constraining layers of
the Prototype 1 were 1 mm thick steel plates, the rubber core of 12 mm thick and
the prototype width of 25 mm. The rubber dissipation loss factor and shear
modulus were obtained using a Dynamic Mechanical Analyser (DMA) machine
discussed in Section 3.7. The damper was fixed at one end to a rigid support and
subjected to pluck tests. A non-contact sensor was set to measure the time domain
response of the excited viscoelastic cantilever beam.
Figure 3-9 Prototype 1 vibrational test using non-contact accelerometer
Table 3-1 shows the details of dampers along with their predicted fundamental
frequency and damping ratio. The natural frequencies for the Prototype 1
configurations without and with an end mass were extracted from recorded time
90
histories from the pluck tests using Fast Fourier Transform (FFT) analysis. The
values of natural frequencies for the Prototype 1 without and with an end mass
were about 11.4 Hz and 4.2 Hz, respectively. These values of natural frequencies
were in good agreement with the values obtained from the analytical model
particularly for the damper with an end mass as it can be seen in Table 3-1. The
error between experimental and predicted natural frequency values were about
10% for the prototype without an end mass while the frequency values were in
excellent agreement for the prototype with an end mass.
The time histories obtained from pluck tests for Prototype 1 with and without an
end mass were filtered and normalised to unity accelerations as illustrated in
Figure 3-10. The normalised trace makes it easy to compare the response decay of
both cases (i.e. damper without and with an end mass). The damping ratios were
calculated using the log decay method. The plotted data revealed that damping
ratio for the Prototype 1 without an end mass was about 5.6% and it was about
5.7% for the damper with an end mass.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.5
0
0.5
1
Time (s)
Acc
eler
atio
n R
espo
nse
(m/s
2 )
Prototype 1 without end massPrototype 1 with end mass
= 5.6%
= 5.7%
Figure 3-10 Time history for the Prototype 1 without and with an end mass
The experimental values of damping ratio of the damper configurations without
and with an end mass were in good agreement with the values obtained from the
analytical model. The error between experimental and predicted damping ratio
91
values were about 2% for the prototype without an end mass while the error was
about 9% for the prototype with an end mass.
In order to increase the confidence level in the predicted values of the natural
frequency and damping ratio from the analytical model, another prototype damper
was built and tested. “Prototype 2” was of a length 750 mm with constraining
layers of 1 mm thick steel plates, a rubber core of 32 mm thick and a damper width
of 25 mm as summarised in Table 3-1. The rubber used in this prototype was the
same as for prototype 1. For the damper configuration without and with end mass,
Prototype 2 was subjected to pluck tests to obtain its natural frequency and
damping ratio.
The measured values of the frequency for the damper without and with an end
mass were 10.1 Hz and 5.6 Hz, respectively. The normalised time domain to unity
acceleration responses due to the pluck test for the prototype damper without and
with end mass are shown in Figure 3-11. The damping ratio values for the
prototype damper without and with an end mass were 5.6% and 6.7%,
respectively.
The experimental values of natural frequencies and damping ratios were in good
agreement with predicted values obtained from approximate analytical method.
The difference between the analytical method and experimental values for the
natural frequencies of the Prototype 2 without an end was about 5% while it was
about 11% for the prototype with an end mass. For the damping ratio values, the
difference between the experimental and analytical method was about 3% for the
prototype without an end mass whereas the value was about 16% for the prototype
with an end mass.
92
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.5
0
0.5
1
Time(s)
Acc
eler
atio
n R
espo
nse
(m/s
2 )
Prototype 2 without end massPrototype 2 with end mass
= 6.7%
= 5.6%
Figure 3-11 Time history for the Prototype 2 without and with an end mass
The values of natural frequencies and damping ratios of the prototypes without an
end mass were also utilised to calculate the shear modulus ( G ) and dissipation
loss factor ( ) of the rubber using the back calculation method described in
Section 3.7.2. As a comparison, based on this back calculation method the
estimated average values of (G ) and ( ) of the rubber were found to be 640 kPa
and 0.10 respectively which were in very good agreement with the values of 690
kPa and 0.12 obtained from the DMA test.
It should be noted that the dissipation loss factor of the rubber ( ) is the factor
that determines the upper limit of overall dissipation loss factor ( ) of the
composite system. In other words, the ( ) value ( 2 ) of the composite system
can not exceed the ( ) value of the rubber (Mead 1982 ; Nashif 1985).
3.8.2. Validation of Analytical Model Using FE AnalysisThe prototype dampers detailed in Table 3-1 were modelled in finite element (FE)
program using commercially available software package ANSYS. The aim of this
analysis was to further validate the values of natural frequency and damping ratio
obtained from the analytical model and demonstrate the applicability of the FE
analysis for modelling the new viscoelastic damper.
93
The constraining layers and rubber core of the prototype dampers were modelled
with solid elements. The constraining layers were assumed to be linear elastic,
while the rubber layer was assumed to be hyperelastic. The rubber was modelled
as an elastomer (Neo-Hookean) with rubber dynamic properties of shear modulus
(G = 690 kPa), a dissipation loss factor ( = 0.12) and a rubber Poisson’s ratio
( = 0.3). The added end mass to the tip of the cantilever viscoelastic damper was
taken to be structural mass element (Mass21). The damping was modelled in
ANSYS for each material as a constant stiffness multiplier (Damp) using
Equation (3-14) (Thorby 2008);
kmm 212 3-14
where
= damping ratio of the material;
= natural frequency of the prototype damper in radians/sec;
m = modal mass of the prototype damper;
k = modal stiffness of the prototype damper;
1 = mass multiplier; and
2 = stiffness multiplier.
For simplicity, and to comply with ANSYS input requirements the mass
multiplier can be eliminated from Equation (3-14) and the resulting expression
would be in the form of Equation (3-15);
f 2 3-15
where f is the fundamental frequency of the damper in Hz.
The constraining layers and rubber core dimensions of the counterpart viscoelastic
damper were modelled in ANSYS according to the damper geometry
94
demonstrated in Table 3-1. The geometry, meshing and fixity of the prototype
dampers are illustrated in Figure 3-12. In this FE model of the viscoelastic
damper, the rubber layer was assumed to be fully bonded to the steel constraining
layers. This means, as in the analytical model assumptions, there is no slip
between the interfaces of the viscoelastic and constraining layers. This can be
achieved in ANSYS by preventing all the relative translations and rotations
between the interfaces of the rubber and constraining layers (at the coincident
nodes).
Figure 3-12 FE model of the prototype damper
A harmonic analysis was conducted to obtain the natural frequency plot of
Prototype 1 and its damping ratio. The prototype was subjected to a unit harmonic
force of 1 N applied at the tip of the damper. The natural frequency of the damper
was found to decrease from 11.8 Hz for the prototype without an end mass to 4.6
Hz for the prototype with an end mass as illustrated in Figure 3-13. The
corresponding damping ratio obtained from harmonic analysis using the Half-
Power Bandwidth (HPB) method were about 5% for the prototype without an end
mass and 5.3% for the prototype with an end mass as shown in Figure 3-13.
95
2 4 6 8 10 12 14 16 180
2
4
6
8
10
12
Frequency (Hz)
Res
pons
e (m
m)
Prototype 1 without end massPrototype 1 with end mass
= 5.0%
= 5.3%
Figure 3-13 Response of the Prototype 1 without and with an end mass using FE harmonic
analysis
In order to obtain the response of the prototype damper in the time domain, a
transient dynamic analysis was performed. The prototype damper was subjected
to an initial displacement of 5 mm at the tip of the damper and then released to
vibrate freely. The resulting values of damping ratios of the prototype damper for
the configuration of without and with an end mass using the log decay method
were 5.2% and 5.5%, respectively, as illustrated in Figure 3-14 and summarised in
Table 3-1.
The values of damping ratios for the Prototype 1 obtained from transient analysis
were in good agreement with the values obtained from harmonic analysis with an
error of about 4% for both cases without and with an end mass.
96
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
Time (s)
Dis
plac
emen
t (m
m)
Prototype 1 without end massPrototype 1 with end mass = 5.2%
= 5.5%
Figure 3-14 Time history for Prototype 1 without and with an end mass using FE transient
analysis
Similarly, Prototype 2 was modelled using FE and the results for the harmonic
and transient analyses for the damper without and with an end mass are plotted in
Figures 3-15 and 3-16) and summarised in Table 3-1. The values of damping
ratios obtained from harmonic analysis were in excellent agreement with values
obtained from transient analysis with a percentage error of about 2%.
2 4 6 8 10 12 14 16 180
1
2
3
4
5
6
Frequency (Hz)
Res
pons
e (m
m)
Prototype 2 without end massPrototype 2 with end mass
= 5.6%
= 5.4%
Figure 3-15 Response of Prototype 2 without and with an end mass using FE harmonic
analysis
97
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Dis
plac
emen
t (m
m)
Prototype 2 without end massPrototype 2 with end mass
= 5.7%
= 5.5%
Figure 3-16 Time history for Prototype 2 without and with an end mass using FE transient
analysis
Table 3-1 summarises the values of natural frequencies and damping ratios
obtained from the analytical model, FE analyses and experimental tests for the
two prototype dampers. The FE results for both frequency and damping were in
good agreement with the analytical and experimental results. This indicates that
the approximate analytical method for determining the damper properties could be
used with an acceptable level of confidence. Furthermore, FE models can also be
utilised for determining damper properties with good accuracy.
It can be seen from Table 3-1 that the predicted values of the natural frequency
and damping ratio obtained from approximate analytical model agree well with
the experimental and FE values. The difference between the analytical model and
FE values for the natural frequencies was in the range of 5% - 11% while it is in
the range of 1% - 12% with experimental values. The difference between the
analytical model values and FE values for the damping ratios was in the range of
2% - 9% while it is in the range of 2% - 16% with experimental values. The
difference is larger for the cases with end mass due to the greater approximation
for the wave number ( BK ) in Equation (3-10).
98
Table 3-1 Analytical model, FE analysis and experimental results for prototype dampers 1 &
2 without and with an end mass
Prototype
damper
No.
Dimensions (mm)End
mass
Analytical
modelFE analysis
Experimental
results
1h 2h 3h b L kgf
(Hz)
(%)
f
(Hz)
(%) f
(Hz)
(%)LDM HPB
1 1 12 1 25 5000 12.7 5.5 11.8 5.2 5.0 11.4 5.6
0.545 4.2 5.2 4.6 5.5 5.2 4.2 5.7
2 1 32 1 25 7500 10.6 5.8 9.3 5.5 5.4 10.1 5.6
0.545 4.9 5.6 5.2 5.7 5.6 5.5 6.7
Refer to Figure 3-5 for damper dimensions.
f = damper natural frequency.
= damper damping ratio
LDM = logarithmic decrement method to calculate damping ratio in time domain.
HBP = half-power bandwidth method to calculate damping ratio in frequency domain.
The analytical and FE models are generally overestimating the natural frequency
values compared with the experimental results while the values of damping ratio
obtained from the three methods were comparable. On the other hand, the
analytical and FE models are generally underestimating the natural frequency and
damping ratio values for the prototype dampers with an end mass compared with
experimental results.
3.9. Concluding RemarksA new viscoelastic damper based on a sandwich beam concept was presented.
This damper would take the form of a cantilever beam and can be attached to the
top or soffit of vibrating floors. When the damper is used as a TMD, the cantilever
would vibrate with the floor and the vibration energy would be dissipated through
stressing of the sandwiched rubber layer.
An analytical model based on an approximate solution of the equation of motion
of a sandwich beam is presented. This method allows the dynamic properties of a
sandwich beam (i.e. natural frequency and damping ratio) to be calculated based
on basic input of material properties and geometric configuration.
99
Two of the key properties for the analytical model are the shear modulus ( G ) and
dissipation loss factor ( ) for the rubber. As there are many types of rubbers
available commercially which do not normally come with adequate technical data,
the G and need to be obtained experimentally. These values can be obtained
using a dynamic mechanical analyser (DMA) or using back calculation from a
prototype testing. Three identical samples of a commercial rubber were used to
obtain average values of the shear modulus and dissipation loss factor of the
rubber used in this research. Two prototypes of cantilever sandwich beams with
same constraining layers and rubber material properties but different dimensions
were developed for the back calculation method. The dampers were tested for the
configurations of without and with an end mass. The values of shear modulus and
dissipation loss factor obtained from the back calculation method were in good
agreement with values obtained from DMA tests. This indicates that the back
calculation method can be used to obtain the shear modulus and dissipation loss
factor of the rubber with a reasonable confidence if a DMA testing machine is not
available.
The two prototype dampers were also utilised to experimentally validate the
analytical model. The prototypes were fixed at one end to a rigid support and
subjected to pluck tests. A non-contact sensor was set to measure the time domain
response of the excited viscoelastic cantilever beam. Using the recorded response
time history, the damping ratio was calculated using the log decay method while
the natural frequency was obtained from Fast Fourier Transform (FFT) analysis.
The prototype dampers were also modelled in ANSYS software to obtain the
natural frequency and overall damping ratio of the dampers and compare them
with the analytical and experimental values. It was found that the values of natural
frequencies and damping ratios obtained from analytical models, FE analyses and
experimental tested were in good agreement. The difference between the
analytical model and FE values for the natural frequencies was in the range of 5%
- 11% and in the range of 1% - 12% with experimental values. The difference
between the analytical model values and FE values for the damping ratios was in
100
the range of 2% - 9% and in the range of 2% - 16% with experimental values.
Given the reasonable level of accuracy of the prediction from the analytical model
combined with its simplicity, it is considered to be sufficient and reliable for the
design of the new viscoelastic damper. Therefore, this analytical model will be
used for the remainder of this thesis to predict the damping ratio and natural
frequency of the various viscoelastic dampers developed throughout the research.
The application of the new viscoelastic damper in reducing the level of vibration
in retrofitted structures will be discussed in the next chapter. The sensitivity of the
primary system response to variations in the natural frequency, damping ratio and
location of the new damper are also covered in the following chapter.
101
4. Performance of Viscoelastic Dampers
4.1. IntroductionThis chapter discusses the effectiveness of the viscoelastic damper in reducing the
level of vibration of floors. Two case studies to investigate the performance of the
new TMD are presented in this chapter. A case study of a small size viscoelastic
damper prototype was developed to reduce the level of vibration of a steel beam.
The second case study was to examine the new TMD further on a larger size
structure with higher damping ratio subjected to human induced excitation such as
heel drop and walking excitations. Furthermore, the predicted values of reduction
in the level of vibration for the two beams were compared with FE and
experimental test results. A sensitivity analysis covering factors such as variation
in the damper natural frequency, damper damping ratio and the departure of
damper from the location of maximum response (anti-node) was also performed
with the results presented herein.
4.2. Case Study 1 – Steel BeamA steel beam measuring 3000 mm long, 100 mm wide and 25 mm thick was used
to assess the effectiveness of the proposed viscoelastic tuned mass damper in
reducing vibrations. In order to predict the response of the steel beam, its dynamic
properties including modal mass ( 1m ), modal stiffness ( 1k ) and damping ratio
( 1 ) are required. The predicted modal mass ( 1m ) for the simply supported beam
can be calculated from Equation (4-1) (Buchholdt 1997);
21ALm
4-1
where
= mass density of the beam;
102
A = cross-sectional area of the beam; and
L = length of the beam.
The stiffness ( 1k ) of the simply supported beam can be calculated using Equation
(4-2) (Buchholdt 1997);
3148
LEIk 4-2
where
E = Young modulus of the steel; and
I = moment of inertia of the beam.
The natural frequency of the beam can then be calculated from Equation (4-3);
1
11 2
1mkf
4-3
Using Equations (4-1 - 4-3) with a mass density of 7,850 3m/kg and Young’s
modulus of 200,000 MPa for the steel beam, yields 1m = 29.67 kg , 1k = 47.4
m/kN and 1f = 6.4 Hz.
4.2.1. Measurement of the Steel Beam Dynamic PropertiesTo obtain the dynamic properties of the steel beam experimentally, it was
subjected to pluck tests. The middle of the beam was initially displaced and then
released with free vibration recorded using an accelerometer. The normalised time
history of the steel beam response due to the pluck test is shown in Figure 4-1.
The estimated value of damping ratio obtained from the time history using the log
decay method was about 0.3%.
103
0 5 10 15 20 25 30-1
-0.5
0
0.5
1
Time (s)
Res
pons
e (m
m)
= 0.3%
Figure 4-1 Normalised steel beam response due to the pluck test
The natural frequency of the steel beam was also measured experimentally by
transferring the time history into the frequency domain using the Fast Fourier
Transform (FFT) technique. It was found that the measured natural frequency of
the steel beam was about 6.3 Hz as illustrated in Figure 4-2, which is in excellent
agreement with the predicted value of about 6.4 Hz as described in Section 4.2.
5 5.5 6 6.5 7 7.5Frequency (Hz)
Mag
nitu
de fn= 6.3 Hz
Figure 4-2 The measured natural frequency of the steel beam
104
4.2.2. Prediction of Steel Beam Response without and withDamperIn order to assess the effectiveness of the new viscoelastic damper, the maximum
response of the beam without the damper was assessed. The beam was excited
using a harmonic force with a frequency range covering the natural frequency of
the beam. The damper was then attached to the beam and the response of the
retrofitted beam due to harmonic force obtained. The effectiveness of the damper
can be obtained from comparing the response of the retrofitted beam with the
response of the beam without the damper (i.e. the response of a SDOF compared
with the response of a two DOF system).
The idealised SDOF of the steel beam was subjected to a unit harmonic force
expressed in Equation (4-4);
)tsin(F)t(F 0 4-4
where
)t(F = excitation force;
0F = excitation force amplitude (1 N);
= excitation force frequency in radians/sec; and
t = time in seconds.
In order to match the SDOF frequency, the frequency of harmonic force was
swept for a frequency range of 5.0-7.5 Hz with frequency increment of 0.001 Hz.
The maximum response acceleration for the bare steel beam with 1m = 29.67 kg ,
1f = 6.4 Hz and 1 = 0.3% was found to be 5.6 2sm as shown in Figure 4-3.
The addition of the damper converts the bare steel beam into a two degree of
freedom system. To predict the effectiveness of the optimum damper in reducing
the level of vibration of the steel beam, the response of the two degree of freedom
system is required to be calculated. The equation of motion for a two degree of
105
freedom system shown in Figure 2-23 can be expressed in Equation (4-5) (Thorby
2008);
00
0 1
2
1
22
221
2
1
22
221
2
1
2
1 )t(Fxx
kkkkk
xx
ccccc
xx
mm
4-5
where
1111 2 mkc ;
2222 2 mkc ;
1 = damping ratio of the primary system; and
2 = damping ratio of the damper.
To determine the response of a two DOF system, the equation of motion as
expressed in Equation (4-5) needs to be solved. The response acceleration of the
primary system 1X can be found by solving Equation (4-5) using the Mechanical
Impedance Method ( tieXx 11 , tieXx
22 and tieF)t(F 01 ), resulting in the
expression presented by Equation (4-6):
24
212
21221212122
2122121122
222
2222
202
1mmm4mkmkcckkmmcmckckc
c)mk(FX
4-6
where
0F = the excitation force amplitude; and
= the excitation force circular frequency and other parameters of the equation
are defined in Figure 2-23.
The addition of the damper to the primary structure creates two splitting
frequencies )'f'&f( 21 as shown in Figure 2-24. The two splitting frequencies can
be calculated using Equation (4-7) (Irvine 1986);
106
21
21212
21221212212,1 2
4)()(21'
mmmmkkkkmkmkkmkm
f
4-7
The two DOF system resonates at both splitting frequencies and the maximum
acceleration of the primary system can be calculated using Equation (4-6) by
substituting either 'f12 or 'f22 as obtained from Equation (4-7)
whatever value of acceleration response is greater.
The two splitting frequencies were found to be 1f = 6 Hz and 2f = 6.6 Hz using
Equation (4-7). The retrofitted beam was subjected to the unit harmonic force
defined by Equation (4-4) to predict the maximum response acceleration of the
steel beam with the added damper. The frequency of harmonic force was swept
for the frequency range of 5.0-7.5 Hz with frequency increment of 0.001 Hz in
order to match the splitting frequencies. The maximum acceleration response for
the steel beam with attached damper was obtained using Equations (4-6) and (4-7)
and it was about 0.5 2sm as shown in Figure 4-3. The reduction factor in the steel
beam response due to the addition of the damper is the ratio of the beam without
the damper response to the retrofitted beam response and it was about 12.5.
5 5.5 6 6.5 7 7.50
1
2
3
4
5
6
Frequency (Hz)
Bea
m A
ccel
erat
ion
Res
pons
e (m
/s2 )
Steel beam without damperSteel beam with damper
Figure 4-3 Prediction of steel beam response without and with damper due to 1 N harmonic
force using Equations (2-5) & (4-6)
107
4.2.3. Development of Viscoelastic Damper for Steel BeamGiven that the dynamic properties of the steel beam were determined, the
viscoelastic damper could then be developed using the procedure outlined in
Section 3.6 to satisfy the optimum damper properties. The geometry and the
predicted dynamic properties of the viscoelastic damper for a 1% mass ratio
damper are listed in Table 4-1. The predicted properties of the viscoelastic
damper, i.e. natural frequency and damping ratio are based on the properties of the
available rubber with = 0.12 and G = 690 kPa. It should be noted that the
dissipation loss factor of the rubber used in the development of the damper is not
sufficient to provide the optimum damping ratio for the viscoelastic damper with
the given thickness, width and length of the rubber and constraining plates. The
optimum damping ratio of a damper with the 1% mass ratio using Equation (2-40)
was 6% whereas the calculated value using the analytical model to develop the
damper was 5.4%. A rubber with a higher dissipation loss factor would be needed
to increase the damping ratio of the damper.
Table 4-1 Viscoelastic damper properties for steel beam based on available rubber
Length ( L ) 500 mm
Width ( b ) 25 mm
Thickness of steel top constraining layer ( 1h ) 1 mm
Thickness of rubber ( 2h ) 12 mm
Thickness of steel bottom constraining layer ( 3h ) 1 mm
Mass density of rubber ( ) 550 3mkg
Dissipation loss factor of rubber ( ) 0.12
Rubber shear modulus 690 kPa
End mass ( endm ) 220 g
Natural frequency of damper ( 2f ) 6.3 Hz
Damping ratio ( 2 ) of damper 5.4 %
108
4.2.4. Measurement of Steel Beam ResponseThe bare beam was experimentally tested with a harmonic excitation using a
rotating unbalanced mechanical shaker located at a distance of about one third of
the span from one of the ends as shown in Figure 4-4. A lightweight
accelerometer was attached to the beam at a distance of one third of the span from
the shaker to record the steel beam response. To obtain the maximum response of
the steel beam, the shaker frequency was tuned to match the fundamental
frequency of the steel beam. The maximum measured acceleration response of the
steel beam due to the shaker excitation was about 5.3 2sm as shown in Figure
4-6.
Figure 4-4 Viscoelastic damper attached to a vibrating steel beam
The damper was attached at the mid-span of the simply supported beam using a
rigid bracket as illustrated in Figure 4-5. Under normal conditions, the extra
bending moment due to the addition of the damper at the mid-span of the beam is
negligible because it just produces an additional moment of about 1% of the
bending moment produced by the primary system.
The retrofitted beam was experimentally tested with harmonic excitation using the
same mechanical shaker. The maximum response of the beam with the damper
due to the shaker excitation was about 0.46 2s/m . Figure 4-6 shows the
acceleration responses in the time domain for the bare and the retrofitted beam.
The response of the beam was reduced by a factor of 11.5, which is in good
agreement with predicted value of 12.5.
109
Figure 4-5 Viscoelastic damper attached to the steel beam
The overall damping of the retrofitted system was estimated using a curve fitting
method in the frequency domain described by Haritos (2008) for both splitting
frequencies and the average damping value was found to be about 3%, which is a
significant increase from the original 0.3% damping.
0 10 20 30 40 50 60
-5
-2.5
0
2.5
5
Acc
eler
atio
n (m
/s2 )
Response of steel beam without damper
0 10 20 30 40 50 60
-2.5
-0.50.5
2.5
Time (s)
Response of steel beam with damper
Figure 4-6 Steel beam response without and with damper attached
110
4.2.5. FE Modeling for the Steel BeamThe bare steel beam and retrofitted beam with the TMD were both modelled using
FE program (ANSYS). The steel beam was modelled in ANSYS using solid
elements and assumed to be linear elastic and the damping of the beam was
modelled using a constant stiffness multiplier as described in Section 3.8.2. Modal
analysis was performed for the bare beam in order to obtain the fundamental
natural frequency ( 1f ) and the corresponding mode shape ( 1 ). The fundamental
natural frequency of the simply supported steel beam obtained from the FE modal
analysis was about 6.4 Hz, which is in very good agreement with the predicted
and measured values. In addition, the modal mass ( 1m ) and the stiffness ( 1k ) of
the steel beam can be extracted from the modal analysis to compare them with
predicted values.
In ANSYS, the kinetic energy can be used to extract the modal mass from the
modal analysis. ANSYS normalises the mass, so the modal mass matrix is an
identity matrix and the modal stiffness matrix is a diagonal matrix of the 21
values, where is the circular natural frequency of each mode in radians/sec. In
FE, the model is meshed into an appropriate number of elements ( N ), and each
element has its own physical mass ( iM ) and modal mass ( im ). The modal mass of
each element ( im ) is a function of the mode shape and the physical mass of the
element as presented in Equation (4-8);
j,ij,iT
j,ij,i Mm 4-8
whereT
j,i = transpose mode shape value of the element;
j,i = mode shape value of the element;
i = element number; and
j = mode number.
111
The total modal mass of the mode ( jm ) is the summation of all element modal
masses as expressed in Equation (4-9);
N
iij mm
14-9
With the mode shape normalised to unity, the kinetic energy ( kE ) of the steel
beam is a function of the modal mass ( jm ) and the mode circular frequency ( j )
and it can be calculated using Equation (4-10);
2
21
jjk mE 4-10
The kinetic energy for the fundamental mode of the steel beam ( 461 .f Hz)
obtained from the modal analysis using ANSYS was about 23,475 Joules.
Substitution of the kinetic energy value and the circular frequency of the
fundamental mode into Equation (4-10) yields the value of the modal mass of the
steel beam to be about 29 kg. This value of modal mass obtained from FE model
was in excellent agreement with the predicted value of (29 kg).
For simple beams and floor plates, the prediction of modal mass is straightforward
but for complex floors with irregular geometry where the general expressions to
calculate the modal mass do not exist, the FE methodology described above can
be easily adopted to obtain the modal mass. For simple structures such as the steel
beam under consideration, it is more convenient to use the general expressions to
calculate the modal mass.
In order to obtain the response of the steel beam using the FE model, the beam
with 0.3% damping ratio was subjected to a unit harmonic force at mid-span. The
maximum acceleration response obtained from a FE harmonic analysis for the
beam without the damper in the frequency domain was about 5.5 2sm as
112
illustrated in Figure 4-7. A damper with properties demonstrated in Table 4-1 was
attached to the beam. The viscoelastic damper was modelled as an equivalent
viscous damper using a linear combination element spring-damper14
(COMBIN14), which is defined by a damper stiffness ( 2k ) and damping
coefficient ( 2c ). The mass of the damper was assumed to be a structural mass
element (Mass21). The FE harmonic analysis was performed using the same
harmonic force utilised for the beam without the damper. The FE harmonic
analysis revealed that the maximum acceleration response for the beam with
damper attached was about 0.5 2sm as shown in Figure 4-7. The addition of the
TMD reduced the response of the beam by a factor of 11. This reduction factor is
in good agreement with experimental results (11.5) and the predicted value (12.5).
5 5.5 6 6.5 7 7.50
1
2
3
4
5
6
Frequency (Hz)
Acc
eler
atio
n (m
/s2 )
Steel beam without damperSteel beam with damper
Figure 4-7 Response of the steel beam with and without damper attached in the frequency
domain using FE analysis
The response in the time domain of the steel beam without and with the damper
attached was also investigated using FE transient analysis. The maximum
acceleration response for the beam without the damper due to the force defined by
Equation (4-4) applied at the mid-span was about 5 2sm as illustrated in Figure
4-8. It can be seen from Figure 4-8 that the steel beam requires a long time to
reach the maximum acceleration response. This long time to reach the maximum
113
response is because of the low damping ratio of the steel beam.
The time required to reach the steady state response ( aT ) can also be calculated by
substitution of Equation (2-17) into Equation (2-13) which forms Equation (4-11);
an Tfbf eR 121 4-11
For nf = 6.4 Hz, 1 = 0.003 and response build up factor ( bfR )= 0.999, the
calculated time to reach the steady state response ( aT ) obtained from Equation
(4-11) was about 60 s which agrees well with the experimental time shown in
Figure 4-6 for the beam response without damper.
The retrofitted beam was also subjected to the same unit force as defined by
Equation (4-4) but with the excitation frequencies matching each of the splitting
frequencies of 1f = 6 Hz and 2f = 6.6 Hz which were created due to the addition
of the damper. The maximum acceleration response of the retrofitted steel beam
was about 0.4 2sm . The reduction in the acceleration response of the retrofitted
steel beam was about 12.5. This value of reduction in the beam response is in
excellent agreement with the experimental and predicted values. This further
demonstrates that a FE analysis can be used for predicting the performance of the
proposed damper, which would be particularly useful for complex floor systems.
114
0 5 10 15 20 25 30 35 40 45 50-5
0
5
Time (s)
Acc
eler
atio
n (m
/s2 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.4
-0.2
0
0.2
0.4
Steel beam response without damper
Steel beam response with damper
Figure 4-8 Response of the steel beam with and without damper in time domain using
transient analysis in FE analysis
4.2.6. Sensitivity of Steel Beam Response to TMD NaturalFrequencyThe sensitivity of the steel beam response to variations in damper natural
frequency from its optimum value was conducted using the analytical model, FE
modelling and experimental tests. The analytical solution was performed using
Equations (2-5) and (4-6). In the three methods, the damping ratio of the damper
of 5.4% was kept constant whilst the frequency of the damper was varied from
-18% to 41% from the value of optimum frequency of 6.3 Hz. The change in the
damper frequency was achieved by altering the mass of the damper. The reduction
in retrofitted beam response due to the variation in the damper natural frequency
using the three methods is illustrated in Figure 4-9. The results of sensitivity
analyses obtained from the three methods clearly show that the performance of the
damper in reducing the vibration is sensitive to its tuning. However, for this case,
the viscoelastic damper was still able to reduce the vibration by a factor of 9 when
its frequency was out of tune by 2.5%. This indicates that the system has some
leeway in terms of damper tuning before significant loss of performance occurs.
This issue will be examined further in this thesis. It should be noted that this range
of variation ( 2.5%) from the optimum frequency due to the change of the
damper end mass causes a small variation in the mass ratio (in the range of 5%)
115
which has negligible effect on the damper performance.
-18% -15% -12% -9% -5% 0 5% 12% 20% 29% 41%0
2
4
6
8
10
12
14
Variation in Damper Frequency
Red
uctio
n Fa
ctor
in B
eam
Res
pons
e
AnalyticalFE modelExperimental
Figure 4-9 Steel beam response due to the variation in damper natural frequency using
analytical, FE and experimental results
4.2.7. Sensitivity of TMD Performance to Damping Ratio of TMDand BeamIn order to assess the effectiveness of the damper when it does not have the
optimum damping value, FE analysis and the developed analytical model (based
on Equations (2-5) and (4-6)) were utilised to investigate the sensitivity of the
steel beam response to variations in the damping ratio of the damper. In this
analysis the damper damping ratio ( 2 ) was varied from 1% to 10%.
The retrofitted beam was subjected to a harmonic force and the reduction in the
response for this range of damper damping ratios was obtained and plotted in
Figure 4-10.
116
Figure 4-10 Steel beam response due to the variation in the damping ratio of the damper
using Equations (2-5) & (4-6) and FE analysis
The steel beam response for this range of variation in the damping ratio of the
damper was also investigated using the FE model. The stiffness and the mass of
the damper and hence the damper frequency were kept constant while the
damping ratio of the damper was varied from 1% to 10%. A harmonic analysis
was conducted for each damping ratio using a 0.5% increment in the damping.
The reduction in the steel beam response for each damper damping ratio is shown
in Figure 4-10. It can be seen from Figure 4-10 that the results from the FE model
are in a good agreement with analytical predictions. In both cases i.e. the
analytical and FE analysis, the damper can be quite effective over the range of 5-
7% for 2 (i.e. 15% from the optimum value of 2 ). Within the 15%
variation range in the damper damping ratio, the damper can still achieve about
95% of its reduction performance compared with optimum performance.
The effectiveness of the damper to variations in the steel beam damping ratio was
also investigated using Equations (2-5) and (4-6) for different damping ratios of
the steel beam. The response of the steel beam was obtained for each damping
ratio of the steel beam within the range of 1% to 10%. It was found, as shown in
Figure 4-11, that the damper performance was sensitive to variations in the
117
damping ratio of the steel beam. This study indicates that if the steel beam has a
damping ratio higher than 1%, the reduction in the response is approximately
constant when the damper damping ratio ranges from 5 to 7%.
From the above analyses to examine the sensitivity of the damper performance to
the departure from the optimum damping ratio ( 2 = 6%), it was found that this
performance was not highly influenced if the change is within 15% of the
optimum value. Furthermore, the damping ratio of the damper has less effect if
the steel beam damping ratio is greater than 1%. Therefore, if the damper does not
utilise rubber at the optimum damping value or there is an error in prediction of
damping of up to 15%, the effectiveness of the damper will not be significantly
affected particularly when used with floors having a damping ratio of 1-3%.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
2
4
6
8
10
12
14
Damper Damping Ratio
Red
ucio
n Fa
ctor
0.3% beam damping ratio1% beam damping ratio2% beam damping ratio3% beam damping ratio4% beam damping ratio5% beam damping ratio
Figure 4-11 Reduction in the steel beam response for different damping ratio of the beam
and damper using Equations (2-5) & (4-6)
4.2.8. Sensitivity of Steel Beam Response to TMD Point ofAttachmentThe sensitivity of the damper performance to variations in the damper’s point of
attachment along the length of the beam was investigated using three methods,
namely analytical, FE modelling and via experiment. For the analytical analysis,
118
the steel beam was idealised as an equivalent SDOF system while the retrofitted
beam was idealised as a two DOF system. The damping ratio of the damper was
kept constant throughout this sensitivity analysis. On the other hand, the modal
mass of the damper for each location was calculated according to the typical
fundamental mode shape of the steel beam using Equation (4-8). Because of the
variation in the damper modal mass, the stiffness of the damper was modified for
each damper location to attain the optimum frequency. The reduction factors in
the retrofitted steel beam response due to the departure from the point of
maximum response (anti-node) were obtained using Equation (2-5) for the SDOF
system and Equation (4-6) for the beam with the damper as shown in Figure 4-12.
The damper was also experimentally relocated along the steel beam and the
response of the beam compared with the predicted values. Figure 4-12 shows that
the measured reduction values were comparable with those obtained from the
analytical model. In addition, the same process of relocation of the damper along
the length of the beam was repeated using the FE model. In this FE modelling, all
parameters of the damper, i.e. stiffness, mass and damping ratio, were kept
constant. The FE model results as shown in Figure 4-12 revealed a good
agreement with the predicted and measured values. The study indicates that the
damper would remain effective in reducing the vibration if it is located within the
central one third of the length of the beam. This analysis was based on the steel
beam with a damping ratio of 0.3%. The effect on the performance of damper due
to its relocation from the point of maximum response when it is attached to a
structure with a higher damping ratio will be investigated in Chapter 6.
119
Figure 4-12 Variation in damper point of attachment along the length of the steel beam
4.3. Case Study 2 – Concrete T beamIn order that the effectiveness of the new viscoelastic damper was fully assessed,
another prototype was developed for an experimental floor. This floor is
essentially a segment of a reinforced concrete floor system with a reinforced
concrete beam and a composite slab. The cross section of the experimental floor is
shown in Figure 4-13 and is referred to herein as “T beam”.
Figure 4-13 Cross-section of T beam floor used in Case Study 2
The T beam has a span of 9.5 m, a total weight of 6,000 kg and is simply
supported at the ends using roller supports, which lave small contribution to the
120
overall damping of the T beam, as shown in Figure 4-14. The long span of the T
beam and its geometry make it relatively flexible and easily excited by footfall
excitation. Hence, the T beam was a prime candidate for retrofitting using the
newly developed viscoelastic damper. Prior to the design of the damper, the T
beam was tested using various forms of excitation including heel drop, walking
and impulse loading using a modal impact hammer. From these tests, the natural
frequencies, mode shapes and damping ratios were determined using experimental
modal analysis. The results from the modal analysis are presented in the following
section.
Figure 4-14 T beam supports
In addition, the T beam was subjected to heel drop excitation and the response of
the beam was recorded to obtain its dynamic properties. Simple assessment
techniques were utilised, which would normally be used in the field such as
extracting the natural frequency of the T beam from a time history record by
converting it to the frequency domain using FFT and utilising the half-power
bandwidth (HPB) method to estimate the apparent overall damping. The natural
frequency of the T beam was found to be 4.2 Hz and the measured damping ratio
using HPB was about 2.3% as illustrated in Figure 4-15.
121
3 3.5 4 4.5 5 5.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (Hz)
Acc
eler
atio
n (m
/s2 ) = 4.2 Hz = 2.3%f
Figure 4-15 T beam response due to heel drop excitation
In addition to the value of damping ratio obtained from the half-power bandwidth
(HPB) method, the Random Decrement (Randec) technique was utilised to
estimate the damping ratio of the T beam. This technique to estimate the damping
in the time domain was discussed in Section 2.4.4. The damping ratio obtained
from this method was about 2.9% as illustrated in Figure 4-16.
0 0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
T be
am R
espo
nse
= 2.9%
Figure 4-16 Normalised T beam response to heel drop using Randec technique
122
4.3.1. Experimental Modal AnalysisTo determine the mode shapes for simple systems such as beams and plates as
shown in Figures (4-17 and 4-18), analytical or experimental methods can be
utilised. The mode shapes are useful for determining the locations on a floor
system that are prone to maximum excitation with the least amount of effort. It is
also important to know the mode shapes to properly determine a retrofit for a
problem floor (Alvis 2001).
The mode shapes and corresponding frequencies can be predicted analytically for
simple floor system or using Finite Element (FE) analysis for complex floor
systems. The actual mode shapes can also be obtained using experimental modal
analysis. The fundamental mode shape for the simply supported T beam is a half-
sine shape as shown in Figure 4-17 and is given by Equation (4-12) (Inman 1996);
Lxx sin)( 4-12
0 1 2 3 4 5 6 7 8 90
0.5
1
T Beam length (m)
Nor
mal
ised
def
lect
ion
Mode shape
(x)
x
Figure 4-17 Fundamental mode shape for a simply supported T beam
123
Figure 4-18 Typical mode shapes for simply supported floor system (Murray et al. 1997)
In order to perform experimental modal analysis, grid points must be created to
set the locations of the accelerometers to be used in recording the response. In this
case, the top surface of the T beam was divided into 3 rows and 11 columns to
form 33 grid points as shown in Figure 4-19.
124
Figure 4-19 T beam grid points
Thirteen accelerometers and one displacement transducer were used in this
experiment. The displacement transducer was installed at the expected anti-node
of the fundamental mode shape, which is normally the mid-span of the T beam.
The accelerometers were distributed on the T beam in three patterns in order to
record the acceleration response of all grid points. Ten out of the thirteen
accelerometers were relocated in the three patterns while three of them were kept
at the same grid points as reference points as shown in Figure 4-20.
The T beam was excited using heel drop and impact hammer excitations. In order
to minimise the effect of noise, the heel drop excitation was repeated and recorded
eight times for each pattern location. The recorded acceleration data obtained
from the heel drop excitations was analysed using a commercial modal analysis
program “ARTeMIS”. This program can average the recorded data for each
accelerometer to minimise the effect of noise. The recorded data from heel drop
excitation was then used to estimate the natural frequencies, the mode shapes and
damping ratios. The fundamental mode shape obtained from the experimental
modal analysis using heel drop excitation is illustrated in Figure 4-21, which is in
very good agreement with the typical first mode shape of a simply supported
beam.
125
Figure 4-20 Distribution of accelerometers for three rounds
Figure 4-21 First mode shape of the T beam obtained from experimental modal analysis
For the fundamental mode of the T beam, ARTeMIS can estimate the values of
the natural frequency and damping ratio as illustrated in Figure 4-22. The
ARTeMIS values of fundamental natural frequency and damping ratio were 4.2
Hz and 1.9% respectively. Other mode shapes, frequencies and damping ratios
obtained from experimental modal analysis are illustrated in Figures (4-22 and
4-23).
126
Figure 4-22 Estimation of the T beam natural frequency and damping ratio using ARTeMIS
Figure 4-23 T beam mode shapes obtained from experimental modal analysis
Mode 1 of 4.2 Hz can be affected by the second harmonic of walking while mode
2 of 7.2 Hz can be affected by the third or fourth harmonic of walking excitation.
Mode 3 of 15.6 Hz and higher modes are not critical for walking excitation.
127
Based on the experimental modal analysis and FFT results it was found that the T
beam had a fundamental natural frequency of 4.2 Hz. The values of the damping
ratio obtained from HPB, Randec and experimental modal analysis were 2.3%,
2.9% and 1.8% respectively. In general, the higher than expected damping for this
bare RC beam is simply due to the presence of cracks in the beam because of
earlier load tests. This earlier load test degraded the stiffness of the beam and
reduced the natural frequency from 5 Hz (before the load test) to 4.2 Hz (after the
load test).
The HPB and Randec are ideally suited for systems with well separated natural
frequencies. When the natural frequencies are relatively close, the accuracy of
these methods reduces. Further, the HPB method relies on identification of the
peak amplitude in the frequency domain; hence, if the resolution is coarse, the
maximum peak may not be fully captured. If this is the case, then HPB would
overestimate the damping. The high value obtained using the Randec method
could be attributed to contribution from the higher modes. Given the T beam did
not have any furniture on it or services suspended from it, a damping value of
2.9% would be considered high. A damping value of the range of 1.8% to 2.3% as
predicted by the modal analysis and HBP would be closer to expectations. Given
this uncertainty, a sensitivity analysis will be performed to examine the influence
of various T beam damping ratios on its performance when fitted with the
viscoelastic TMD.
As earlier discussed for a simply supported beam with a uniformly distributed
mass, the modal mass is approximately half of the total mass (Equation (4-1)).
Hence, for the equivalent SDOF system, the modal mass of the T beam can be
taken as 3,000 kg. For a tuned mass optimum damper with a mass ratio of 1%, the
required damper mass is 302 m kg. Using Equations (2-38 - 2-40), the optimum
damper is required to have a natural frequency of 242 .f Hz and damping ratio
of 2 = 6%.
128
The predicted maximum acceleration responses of the T beam without damper
and with optimum damper for the three measured damping ratios of the beam
using Equations (2-5) and 4-6) are shown in Figure 4-24. For each measured
damping ratio of the beam, the beam was subjected to the harmonic force defined
by Equation (4-4) but with an amplitude ( 0F ) of 1 kN and the excitation frequency
was swept from 3 to 6 Hz. It is obvious from Figure 4-24 that the lower value of
damping ratio of 1.8% obtained from experimental modal analysis can
overestimate the effectiveness of the damper in reducing the vibration. Therefore,
for conservative damper design, the value of damping ratio obtained from the
Randec technique (2.9%) was adopted to predict the effectiveness of the damper
in reducing floor vibrations.
1.8% 2.3% 2.9%2
3
4
5
6
7
8
9
10
Damping ratio of the T beam
Acc
eler
atio
n (m
/s2 )
T beam without damperT beam with damper
ReductionFactor = 2.1
ReductionFactor = 2.4
ReductionFactor = 2.8
Figure 4-24 Acceleration response of the T beam without and with optimum damper for
different beam damping due to 1 kN harmonic excitation force using Equations (2-5) and
(4-6)
The evaluation of the reduction factor in the frequency domain in the T beam
response with a damping ratio of 2.9% was performed as depicted in Figure 4-25.
The reduction factor in the retrofitted T beam response was around a factor of 2.1.
It should be noted that this reduction factor is based on the optimum damper but
not on the as built damper.
129
3 3.5 4 4.5 5 5.50
1
2
3
4
5
6
Frequency (Hz)
T be
am R
espo
nse
(m/s
2 )
T beam without damperT beam with optimum damper
Figure 4-25 Acceleration response of the T beam with and without optimum damper in
frequency domain due to 1 kN harmonic force using Equations (2-5) and (4-6)
4.3.2. Viscoelastic Damper Design for T BeamA commercial rubber with a dissipation loss factor ( ) of 0.15 was utilised to
develop the viscoelastic damper. The geometry, damping ratio and natural
frequency of the damper were obtained using the procedure described in Section
3.6. The resulting damper properties are listed in Table 4-2. It should be noted that
the dissipation loss factor of this rubber is not sufficient to provide the optimum
damping ratio of 6% for the TMD with the given thickness, width and length of
the rubber and plates. A rubber with a higher dissipation loss factor would be
needed to increase the damping (such rubbers can be sourced from specialist
suppliers but are not readily available). However, the sensitivity of damper
performance to the damping ratio was previously investigated for the steel beam
in Section 4.2.7. It was found that the retrofitted primary system response is not
very sensitive to the damping ratio of the damper when it is in the range of 15%
from the optimum damping ratio of 6% (i.e. damper can still provide a major
improvement).
130
Table 4-2 Predicted viscoelastic damper properties for the T beam
Length (L) 510 mm
Width (b) 100 mm
Thickness of top constraining layer (h1 ) (steel) 6 mm
Thickness of rubber (h2) 38 mm
Thickness of bottom constraining layer (h3) (steel) 6 mm
Dissipation loss factor of rubber (β) 0.15
Rubber shear modulus 637 kPa
End mass (mend) 29 kg
Natural frequency of damper 4.2 Hz
Damping ratio (ζ2) of damper 4.5%
The dynamic properties of the constructed damper were obtained experimentally
by exciting the prototype damper using a pluck test. The pluck test value for the
natural frequency ( 2f ) of the 30 kg damper was 4.2 Hz with a damping ratio ( 2 )
of 4.6% as shown in Figure 4-26. The natural frequency was extracted from the
frequency domain while the damping ratio was obtained using the log decay
method. These values of damping ratio and natural frequency of the constructed
viscoelastic damper were in very good agreement with the predicted values
obtained from the approximate analytical method used for the design of
viscoelastic damper that are listed in Table 4-2.
0 1 2 3 4 5 6-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
Acc
eler
atio
n (m
/s2 )
2 = 4.6%
Figure 4-26 Response of damper developed for the T beam due to a pluck test
131
4.3.3. Application of TMD to T BeamThe viscoelastic damper was attached at the mid-span of the T beam using a
bracket shown in Figure 4-27. The damper was tuned by adjusting the end mass to
achieve the optimum damper frequency of 4.2 Hz in order to attain the maximum
damper performance. The effectiveness of the damper was then evaluated
according to the response of the T beam due to heel drop and walking excitations.
Figure 4-27 Tuned mass viscoelastic damper attached to the experimental T beam
4.3.4. Performance of the Damper Due to Heel Drop ExcitationThe T beam was subjected to repeated heel drop excitations with a total of eight
tests recorded. The time histories were transferred to the frequency domain using
FFT and then averaged to assess the effectiveness of the viscoelastic damper in
reducing the T beam vibrations. Figure 4-28 shows the response of the T beam in
the frequency domain due to heel drop excitations for both cases with and without
the damper. The value of reduction in the acceleration was about 2.0, which
agreed well with the predicted value for an optimum damper of 2.1. This
predicted value of the reduction factor of 2.1 due to the addition of the damper
was based on the optimum damper properties. The predicted reduction factor
according to the as built damper with 30 kg, 2f = 4.2 Hz and 2 = 4.6% was 1.9
which is still in good agreement with experimental value.
132
3 3.5 4 4.5 5 5.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (Hz)
Acc
eler
atio
n (m
/s2 )
T beam without damperT beam with damper
Figure 4-28 T beam response due to heel drop excitation
The overall damping of the retrofitted beam was estimated using the log decay
method in time domain. Figure 4-29 clearly shows the increase of damping in the
time history response of the T beam due to heel drop excitations. The overall
apparent damping ratio of the T beam with damper attached was found to be
6.1%.
0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1
0 0.5 1 1.5 2 2.5 3-1
-0.5
0
0.5
1
Time (s)
Acc
eler
atio
n (m
/s2 )
1 = 2.9%
= 6.1%
T beam with damper
T beam without damper
1
Figure 4-29 Response of the T beam without and with damper due to heel drop excitation
with measured damping ratios using log decay method
133
The bare T beam and the T beam with the attached damper were also modelled
using ANSYS. The beam was modelled in ANSYS using two node beam
elements (Beam188) while the slab was modelled using shell elements (Shell181).
The viscoelastic damper was modelled as an equivalent viscous damper. As
earlier load tests on the T beam degraded its strength, the Young’s modulus of the
concrete was adjusted in the FE model so that the measured natural frequency of
the first mode matched that obtained experimentally. The T beam was excited by
1 kN harmonic force to evaluate the effectiveness of the damper in reducing the
vibration level. Based on the FE results the reduction factor in the acceleration
response was about 1.9 (as shown in Figure 4-30), which is in excellent agreement
with the experimental and predicted reduction factors of 2.0 and 1.9, respectively.
3 3.5 4 4.5 5 5.50
0.5
1
1.5
2
2.5
3
3.5
Frequency (Hz)
Acc
eler
atio
n (m
/s2 )
T beam with damperT beam without damper
Figure 4-30 T beam response in frequency domain for cases without and with as built
damper due to 1 kN harmonic force based on FE analyses
4.3.5. Performance of the Damper Due to Walking ExcitationThe effectiveness of the damper in reducing the T beam response due to walking
along the length of beam was also investigated. In order to match the natural
frequency of the T beam, as much as possible, the walker practiced the correct
step frequency and attempted to maintain this step frequency throughout the
testing. The number of time history samples for each case of beam without and
with the damper was 24 records.
134
It was found that the reduction factor in frequency domain was only 1.4 as shown
in Figure 4-31. This is attributed to the second harmonic of the walking rate not
matching the first mode frequency of the beam and the length of the beam was not
long enough to reach the full steady state response. This exact coincidence
between the step rate or one of its harmonics and the natural frequency of the
floor can be difficult to constantly achieve because humans tend to have a natural
self selecting walking rate which can be difficult to alter and then maintain to
achieve a certain walking rate.
3 3.5 4 4.5 5 5.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Frequency (Hz)
Acc
eler
atio
n (m
/s2 )
T Beam without damperT Beam with damper
Figure 4-31 T beam measured acceleration response due to walking excitation based on
averaging of 24 records for cases with and without damper
The time history records for both cases with and without dampers were filtered for
the frequency range of (2-20 Hz) to obtain the maximum acceleration in the time
domain. The peak acceleration responses of the T beam without and with damper
were plotted in a column chart as shown in Figure 4-32. The maximum
acceleration response of the T beam without the damper attached was about 2.7%
g, which reoccurs in four records as clearly seen in Figure 4-32. This recurrence
of the maximum value of the acceleration response indicates that the second
harmonic of the step frequency may have coincided with the natural frequency of
the T beam. Thus, the value of 2.7% g can be adopted as a maximum response of
the T beam. The average value of the 24 peak acceleration responses due to the
135
walking excitation for the T beam without the damper was about 2% g. The
maximum acceleration response for the retrofitted T beam in the time domain was
about 2% g while the average of the 24 peaks was about 1.4% g. For both cases
i.e. the maximum response and the average response of the 24 records, the
reduction in the acceleration response was about 1.4. The response of the T beam
without and with damper was also compared using the root mean square (rms)
acceleration. It was found that the rms acceleration for the beam without damper
was about 2.1% g while it was about 1.5% g for the retrofitted beam, which both
agrees well with the average values.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240
0.05
0.1
0.15
0.2
0.25
Record Number
Acc
eler
atio
n (m
/s2 )
1 2 3 4 5 6 7 8 9 10 11 1213 14 15 1617 18 19 2021 22 23 240
0.05
0.1
0.15
0.2
0.25
Record Number
T beam without damper T beam with damper
Figure 4-32 Peak accelerations of T beam with and without the damper from 24 walking
excitation records
Clearly, the damper could perform better if the T beam was excited to full steady
state resonance. However, the damper was still effective in producing a
reasonable reduction in the T beam response. An option to further reduce the T
beam response would be to adopt a higher mass ratio for the damper.
4.3.6. FE Model for Walking ExcitationThe T beam response due to walking excitations was investigated using ANSYS.
An equivalent concentrated time-dependent walking force )t(F was imposed at
the mid-span of the T beam, which is the anti-node for the fundamental modes.
136
The maximum acceleration response of the T beam was also collected from this
location. To account for the moving characteristic of the walking force along the
length of the T beam, Equation (4-12) is merged with the forcing function
(Equation (2-1)). In addition, the static weight can be subtracted from the equation
so that only the harmonic applied force is used for analysis to form the equation of
a moving force as expressed in Equation (4-13) (Heinemeyer et al. 2009).
tLv
SintifSinrFtF stepN
iistepi
10 2 4-13
where
0F = person’s weight;
ir = walking dynamic load factors are given in Table 2-1;
i = harmonic number;
N = number of walking harmonics included in the forcing function;
stepf = step frequency;
t = time in seconds;
i = phase angle for each harmonics ( 01 , 22 and 23 );
stepv = walking speed defined by Equation (2-14); and
L = length of walking path as illustrated in Figure 4-33.
137
Figure 4-33 Equivalent walking force function according to the mode shape
Substitution of Equation (2-14) into Equation (4-13) yields the elimination of the
walking speed and the walking force function expressed in terms of step
frequency only as defined by Equation (4-14);
Lt.f.f.SintifSinrFtF stepstep
N
iistepi
5483334666712 2
10 4-14
For the T beam with a fundamental frequency of 4.2 Hz, the second harmonic of
the step rate can match its fundamental frequency. Therefore, the equivalent time
dependent walking force at the mid-span of the T beam without damper as
generated from Equation (4-14) is shown in Figure 4-34.
In order to allow for reaching the steady state response, the mode shape
component was removed from Equation (4-14). Thus, the forcing function for on-
the-spot walking applied at the mid-span of the T beam is given by Equation
(4-15);
138
N
instepi tifSinrFtF
10 2 4-15
0 1 2 3 4 5 6-300
-200
-100
0
100
200
300
400
Time (s)
Wal
king
For
ce (N
)
Figure 4-34 Equivalent time dependent walking force for the bare T beam using Equation
(4-14)
The on-the-spot walking function was applied at the mid-span and the maximum
acceleration response obtained from transient analysis was about 4.4% g as shown
in Figure 4-35. The walking force along the length of the beam presented by
Equation (4-14) was also applied at the mid-span of the beam to determine its
response. The maximum acceleration response of the T beam without damper due
to walking along the length of the beam was about 3.7% g as depicted in Figure
4-35. The higher response due to on-the-spot walking compared with walking
along the length of the beam using FE analysis confirms that the T beam did not
reach the steady state response due to walking along the length of the beam. This
was also confirmed using the resonance build up factor obtained from Equation
(2-13) for the given values of damping, step rate and length of the beam, which
yield a resonance build up factor of about 0.8. The difference between the
responses of the T beam without damper due to on-the-spot walking was about
16% greater than the response due to walking along the length of the beam. This
value of acceleration response due to walking along the length of the beam
without the damper obtained from FE modelling is quite comparable with
predicted values using Equation (2-10).
139
0 2 4 6-400
-200
0
200
400
0 2 4 6-0.5
-0.25
0
0.25
0.5
0 2 4 6-400
-200
0
200
400
Time (s)
Wal
king
For
ce (N
)
0 2 4 6-0.5
-0.25
0
0.25
0.5
Time (s)
Acc
eler
atio
n (m
/s2 )
On-the-spot walking force Response due to on-the-spot walking
Walking along the length of beam force Response due to walking along the length of beam
Figure 4-35 T beam loading and response without damper due to on-the-spot walking and
walking along the length of the beam using FE models
The addition of the damper to the beam creates two resonating frequencies, which
are 4.0 and 4.4 Hz. The frequency of the walking force was adjusted in Equation
(4-14) in order that the second harmonic matches the new resonating frequencies
of the retrofitted T beam. A transient analysis was performed for each walking
force and the maximum acceleration response of the retrofitted beam was taken as
the greatest absolute value obtained from both walking forces. On-the-spot
walking was applied at mid-span of the retrofitted beam to reach the steady state
response using FE transient analysis and maximum acceleration response was
predicted as 2.4% g as shown in Figure 4-36. The walking along the length of the
beam was defined for each splitting frequency using Equation (4-14). The
maximum response acceleration for the T beam with attached damper due to
walking along the length of the beam was about 2.4% g as shown in Figure 4-36.
These response results indicate that the retrofitted beam can reach the steady state
for walking along the length of the beam. This value of maximum acceleration
response due to walking along the length of the beam for the beam with damper
attached obtained from the FE transient analysis was in good agreement with the
maximum acceleration value of 2% g obtained from the experimental work.
140
0 2 4 6-400
-200
0
200
400
0 2 4 6-0.3
-0.2
-0.1
0
0.1
0.2
0 2 4 6-400
-200
0
200
400
Time (s)
Wal
king
For
ce (N
)
0 2 4 6-0.3
-0.2
-0.1
0
0.1
0.2
Time (s)
Acc
eler
atio
n (m
/s2 )
On-the-spot walking force Response due to on-the-spot walking
Response due to walking along the length of beamWalking along the length of beam force
Figure 4-36 T beam loading and response with damper due to on-the-spot walking and
walking along the length of the beam using FE models
The reduction factor in the T beam response for both cases with and without
damper using walking excitation in the FE model was about 1.6 as illustrated in
Figure 4-37. This value of reduction in T beam response is in good agreement
with the 1.4 value obtained from the experimental tests.
0 1 2 3 4 5 6-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time (s)
Res
pons
e A
ccel
erat
ion
(m/s
2 )
T beam without damperT beam with damper
Figure 4-37 T beam response without and with damper due to walking along the length of
the beam using FE models
141
4.3.7. Sensitivity of the T Beam Response to TMD NaturalFrequencyThe sensitivity of the T beam response to variations in the frequency of the
damper was analytically examined using Equations (2-5) and (4-6) to compare it
with the previous steel beam results. In this sensitivity analysis, the damping ratio
of the as built damper of 4.6 % was kept constant whilst the change in the damper
frequency was achieved by changing the mass of the damper. The end mass of the
damper was varied from -18% to 41% from the optimum mass of 30 kg associated
with the optimum frequency of 4.2 Hz. This range of variation in the damper mass
from the optimum mass yields a variation in the frequency of the damper from
optimum frequency (4.2 Hz) that ranges from 10% to -15% (4.6 Hz to 3.6 Hz). It
was found, as shown in Figure 4-38, that the T beam response was very sensitive
to variations in the natural frequency of the damper. To maintain 90% of the
maximum reduction factor for the beam with attached damper, the variation in the
frequency of the damper should be in the range of 2.5% from the optimum
frequency of the damper. On the other hand, to achieve 80% of the maximum
reduction factor, the variation in the frequency of the damper should be in the
range of 4% from the optimum frequency of the damper.
-14.0 -9.5 -4.0 -2.5 0 2.5 4.0 9.5 14.350
60
70
80
90
100
Damper Frequency (Percentage)
Red
uctio
n Fa
ctor
(per
cent
age) Optimum damper
natural frequencyf2 = 4.2 Hz
Figure 4-38 Reduction factor of the T-beam response with attached TMD due to variations
in the natural frequency of the damper using Equations (2-5) and (4-6)
142
The sensitivity of the T beam response to variations in the damper natural
frequency was also experimentally investigated by modifying the end mass and
hence the natural frequency of the damper. The mass of damper was modified by
20% from the optimum damper mass of 30 kg to achieve modification in
frequency by -7% and +13% from the optimum value of 4.2 Hz. Eight heel drop
excitations were recorded for each modified damper frequency and then converted
to frequency domain for comparison. Figure 4-39 shows that the efficiency of the
damper was considerably affected by increasing the frequency by 10% and this
highlights the importance of tuning the damper.
3.5 4 4.5 5
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Acc
eler
atio
n (m
/s2 )
T beam without damperOptimum frequency damper (4.2 Hz)Lower frequency damper (7% from optimum)Higher frequency damper (13% from optimum)
Figure 4-39 Sensitivity of the damper due to variation in the natural frequency
The performance of the damper to variations in the frequency of the damper was
also investigated using FE analysis. This variation in the damper frequency was
achieved by modifying the end mass of the damper while the stiffness and
damping ratio of the damper were kept constant to the as built damper values of
2k = 20,892 N/m and 2 = 4.6%. Similar to the analytical and experimental results,
the FE harmonic analysis results as illustrated in Figure 4-40 indicated that the
response of the beam is sensitive to the variation of the damper natural frequency
from its optimum value.
143
-10% -5% 0 5% 10%1
1.2
1.4
1.6
1.8
2
Frequency Variation
Red
uctio
n Fa
ctor
Figure 4-40 Sensitivity of damper performance to the variation in its natural frequency using
FE model
4.3.8. Sensitivity of the T Beam Response to TMD DampingRatioThe sensitivity of the T beam response to variations in the damping ratio of the
damper was investigated using the analytical model and FE analysis. The natural
frequency and mass of the damper were kept constant while the damping ratio of
the damper was varied from 0.5% to 10% with an increment value of 0.5%.
1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Damper Damping Ratio
Red
uctio
n Fa
ctor
AnalyticalFE model
Optimum damping ratio
= 6%2
Figure 4-41 Reduction factor in the T beam response with damper attached for different
damping ratios of the damper using Equations (2-5) & (4-6) and FE model
144
It was found from the analytical and FE harmonic analysis methods, that the
damper performs well when variations in the damping ratio of the damper are
within 25% from the optimum damping ratio, i.e. in the range of 4.5-7.5%. The
corresponding decrease in the reduction factor was about 5% of the maximum
reduction factor for the analytical modal as shown in Figure 4-41. The values of
reduction factors obtained from analytical model and FE harmonic analysis of the
retrofitted T beam response due to variations in the damping ratio of the damper
were in good agreement.
The analyses of the steel beam and T beam responses due to variations in the
damping ratio of the damper from its optimum value revealed that the response of
the beams was not sensitive to the damping ratio of the damper. The damper
performs well for both case studies when the variation in the damping ratio of the
damper is within 25% from the optimum damping value of 6% for the 1% mass
ratio damper i.e. when the damping ratio is within the range of 4.5% to 7.5%.
4.4. Concluding RemarksTwo prototypes of viscoelastic damper were developed and tested to investigate
the performance of the viscoelastic damper in reducing the levels of vibration.
The TMDs were designed using the approximate analytical model presented in
Chapter 3. The properties of the dampers were validated using FE analysis and
experimental tests. The values of natural frequency and damping ratio of the
proposed dampers obtained from analytical, FE model and experiments were in
good agreement.
One of the prototype dampers was developed to retrofit a simply supported beam.
The natural frequency of the steel beam ( 1f ) was obtained from three methods
namely closed form solution, FE analysis and experiment. The values obtained
from the three methods were in excellent agreement and in the order of 1f = 6.3
Hz. Similarly, the modal mass ( 1m ) of the steel beam was found to be 29 kg. The
damping ratio ( 1 ) of the steel beam was measured experimentally and found to
be 0.3%. A prototype viscoelastic damper of 1% mass ratio was developed using
145
the approximate analytical model to retrofit the steel beam. A commercial rubber
was used in the development of the damper with measured shear modulus ( G ) of
about 690 kPa and a dissipation loss factor ( ) of about 0.12. The dynamic
properties of the as built viscoelastic damper were a natural frequency ( 2f ) of 6.3
Hz, a modal mass ( 2m ) of 0.29 kg and a damping ratio ( 2 ) of 5.4%. The
dissipation loss factor of the rubber used in the development of the damper was
not sufficient to provide the optimum damping ratio of 6% for the viscoelastic
damper. A rubber with a higher dissipation loss factor would be needed to
increase the damping ratio of the damper and bring it closer to the optimum value.
The response of the original steel beam to mechanical harmonic excitation was
measured and found to be 5.3 2sm . The response of the retrofitted steel beam to
the same excitation was found to be 0.46 2sm . The response of the retrofitted
steel beam was reduced by a factor of 11.5, which is in good agreement with the
predicted value of 12.5 and the FE analysis value of 11. The overall damping of
the retrofitted system was found to be about 3%, which is a significant increase
from the original 0.3% damping.
In order that the effectiveness of the new viscoelastic damper could be fully
assessed, a large scale concrete T beam was retrofitted with the second prototype
viscoelastic damper. The T beam was simply supported and had a span of 9.5 m
and a modal mass of 3,000 kg. The mode shapes and the corresponding
frequencies were predicted using experimental modal analysis. The first
fundamental natural frequency and damping ratio were measured and found to be
4.2 Hz and 1.8%, respectively.
The performance of the damper was investigated using an analytical model, FE
analysis and experimental tests. For the experimental tests, the T beam without
and with the damper was subjected to heel drop and walking excitations. For the
heel drop, the reduction in the response of the beam was about 2.0, which is in
good agreement with the analytical and FE harmonic results. For walking
146
excitation, the maximum measured acceleration obtained from 24 time history
records was about 2.7% g whereas for the retrofitted T beam, the maximum
acceleration response was about 2% g. The reduction in the acceleration response
due to the walking excitations was about 1.4. This reduction factor was consistent
with the FE prediction of 1.6. The reason that the walking reduction factor being
smaller than that with heel drop tests is due to the fact that the walking excitation
of the beam without the damper was not sufficient to reach the full steady state
response. This was confirmed by FE analysis.
The sensitivity of the retrofitted structure response to variations in the natural
frequency of the new damper was experimentally and numerically investigated by
modifying the end mass and hence the natural frequency of the damper. The
efficiency of the damper was considerably affected by altering the frequency by
more than 2.5% and this highlights the importance of tuning the damper to the
optimum natural frequency.
It was found from the two case studies using the analytical model and FE analyses
that the response of the primary system with the damper is sensitive to the
damping ratio of the primary system. The damper is very efficient when the
damping ratio of the primary system is less than 1% where a reduction factor in
excess of 10 being achievable. The reduction factor decreases when the damping
ratio of the primary system increases. For common floors with damping of 2-3%,
the expected reduction factor in vibration due to the addition of the viscoelastic
TMD is in the order of 2.0.
The sensitivity of the primary structure (floor) response to variations in the
damping ratio of the damper was also investigated using the analytical model and
FE analysis. It was found for both case studies that for 1% damper mass ratio, the
damper performs well when variations in the damping ratio of the damper are
within 25% from the optimum.
147
The sensitivity of the steel beam response to the relocation of the damper
attachment from the point of the maximum response was investigated using the
analytical model, FE analysis and experimental tests. The study revealed that the
damper would remain effective in reducing the level of vibration if it is located
within the central one third of the length of the beam. This indicates that when the
installation of the damper at the point of maximum response is not possible, the
damper can be relocated within a certain distance according to the mode shape of
the retrofitted system.
The results presented in this chapter confirm that the new damper can achieve
significant reduction in floor vibrations particularly when the damping ratio of the
retrofitted structure is low.
The two case studies of prototype viscoelastic TMDs discussed in this chapter
were for a single viscoelastic damper. The options of utilising the viscoelastic
damper in a multiple form at one location and distributed on a structure are
discussed in the following chapter.
148
5. Application of Multiple Dampers
5.1. IntroductionFor a typical floor, a single TMD could be quite large to offer the typical 1% mass
ratio. To avoid the limitations of large size of single dampers, a multiple tuned
mass damper (MTMD) system is proposed to replace the single damper. The
effectiveness and robustness of multiple dampers will be investigated in this
Chapter.
A MTMD system of four identical dampers with a mass ratio of 0.25% each was
developed to replace the single damper with a 1% mass ratio for the T beam that
was previously described in Chapter 4. The four dampers were arranged in a
crucifix form and attached to the T beam at the same point of attachment used for
the single damper (i.e. mid-span of the beam). The T beam with the multiple
dampers at the mid-span was excited using heel drop and walking excitations to
compare the effectiveness of the MTMD system with the single damper. The
dampers were then evenly distributed along the T beam to test the effectiveness of
the MTMD in reducing the levels of vibration when attached as a distributed
system.
Given the success of the laboratory based trials, the new viscoelastic damper was
developed as MTMD configurations on a real office floor. The office floor
response due to walking excitation was determined using an FE model and field
measurements. Once the dynamic properties of the office floor were determined,
multiple viscoelastic dampers were designed and constructed. The response of the
office floor with and without a MTMD system is discussed in this chapter.
149
5.2. Multiple Tuned Mass Dampers (MTMD)For a single TMD to be effective in an office floor, it needs to be of significant
size and finely tuned to the natural frequency of the floor system. Either mistuning
the frequency or departing from the optimum damping ratio by 25% can reduce
the effectiveness of a TMD (Bakre & Jangid 2004). Furthermore, there could be
size limitations, which restrict the dimensions of the damper for real floors due to
the presence of ducts or services in the false ceiling. A multiple tuned mass
damper (MTMD) system can be utilised to replace the single damper approach to
overcome these limitations. Furthermore, malfunction of a single damper in a
MTMD system will not normally cause potentially detrimental effects on the
structural response as may be the case for the single TMD solution, so that the
MTMD system can be a more robust solution (Chen & Wu 2003).
The mass ratio ( ) of the MTMD system can be specified according to the
required spacing of the splitting frequency and the desirable reduction in the
response of the primary system. Li & Liu (2002) proposed that the optimum
frequency spacing increases approximately linearly with the increase of the total
mass ratio.
For a typical case of a MTMD system made of identical dampers attached to a
primary structure shown in Figure 5-1, the mass ratio ( ) is defined by Equation
(5-1);
Figure 5-1 Schematic five degrees of freedom system
150
1
2.mmn
5-1
where
n = number of individual dampers;
2m = mass of each individual damper; and
1m = mass of primary system.
The equation of motion that governs the multi-degree of freedom (MDOF) system
shown in Figure 5-1 can be written in the form of Equation (5-2) (Li & Ni 2007 ;
Park & Reed 2001) ;
0000
000000000000
4
000000000000
4
00000000000000000000
5
4
3
2
1
22
22
22
22
322221
5
4
3
2
1
22
22
22
22
222221
5
4
3
2
1
2
2
2
2
1 )t(F
xxxxx
kkkk
kkkk
kkkkkk
xxxxx
cccc
cccc
cccccc
xxxxx
mm
mm
m
5-2
The response acceleration 1X can be found by solving Equation (5-2) using the
Mechanical Impedance Method ( tiXex and tieF)t(F 0 ), where 0F , and t
are excitation force amplitude, excitation frequency in radians/sec and time.
Solving Equation (5-2) yields the expression of the response acceleration of the
primary system as presented by Equation (5-3);
2
2
1
21221
1
22
1
212
2
2
1
212
1
21
1
212
2
2
22
22
22
1
021
44
21
kmm
mmkk
mkcc
kk
mmckmc
kk
cc
k
kF
X 5-3
where
2 = the natural circular frequency of an individual damper; and
2 = damping ratio of an individual damper.
151
The value of maximum acceleration (for a retrofitted system) from Equation (5-3)
can be compared with the value calculated from Equation (2-5) (for SDOF
system) to obtain the reduction in acceleration response.
5.3. Multiple Dampers at the Mid-span of the T beamThe effectiveness of MTMD system in reducing floor vibrations was investigated
experimentally using the T beam described in Section 4.3. Four dampers with a
mass ratio of 0.25% each were developed for the T beam to replace the single
damper of 1% mass ratio. The T beam with four dampers can be represented using
an equivalent mass, stiffness and damping by a multiple degree of freedom
MDOF model configuration as shown in Figure 5-1. The required and predicted
properties of each of the multiple dampers as determined using the approximate
analytical method described in Section 3.6 are listed in Table 5-1 with a typical
damper shown in Figure 5-2.
Table 5-1 Properties of each viscoelastic damper in the MTMD configuration
Length (L) 520 mm
Width (b) 50 mm
Thickness of top constraining layer (steel) 6 mm
Thickness of rubber (h2) 20 mm
Thickness of bottom constraining layer (steel) 1 mm
Dissipation loss factor of rubber (β) 0.18
Mass density of rubber () 550 kg/m3
Rubber shear modulus (G) 680 kPa
End mass (mend) 7.1 kg
Natural frequency of damper 4.2 Hz
Damping ratio ( 2 ) of damper 4.8 %
The properties of the viscoelastic damper shown in Figure 5-2 were
experimentally determined using pluck tests to obtain the damping ratio and
natural frequency of the damper. It was found from the experimental tests that the
damping ratio of the viscoelastic damper was about 4.5%. The required end mass
to achieve the optimum natural frequency of 4.2 Hz was 8 kg. These differences
152
between the values of natural frequency and damping ratio of the damper obtained
from the experimental tests and the values of Table 5-1 are simply due to the
approximate nature of the model used to produce the results in the table and
construction tolerances.
Figure 5-2 Prototype viscoelastic damper developed for multiple damper system to replace
the single large damper
All four dampers were almost identical and arranged in a crucifix form and
attached to the T beam at the same point of attachment used for the single damper
as shown in Figure 5-3 (i.e. at mid-span). The total mass of the MTMD system
was closer to the mass of the single large damper with its properties described in
Section 4.3.2. The retrofitted T beam was excited using heel drop excitation to
measure the effectiveness of dampers in reducing the acceleration response.
Figure 5-4 shows the response of the T beam in the frequency domain with an
associated reduction factor in acceleration response of about 2.1.
153
Figure 5-3 Four viscoelastic tuned mass dampers at the centre of the T beam
3 3.5 4 4.5 5 5.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (Hz)
Acc
eler
atio
n (m
/s2 )
T beam without damperSingle damper at mid-spanFour dampers at mid-span
Figure 5-4 Response due to heel drop with four dampers attached at the centre of T beam
The damping ratio of the retrofitted T beam was estimated using the log decay
method. The average damping ratio of the retrofitted T beam using the MTMD
system obtained from 24 time history records due to the heel drop excitation was
about 6.2% as shown in Figure 5-5. Figure 5-4 illustrates that the response
reduction in the T beam with four dampers was in good agreement with the
response reduction of the T beam with the single large damper.
154
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Time (s)
Acc
eler
atio
n (m
/s2 )
= 6.2%
Figure 5-5 Response of the T beam with four dampers at the centre due to heel drop
excitation
5.3.1. Distributed Damper SystemsIn some cases, there may be limitations on the space available for placing a single
large damper or a set of smaller multiple dampers at the location of maximum
response of the floor. In these situations, multiple dampers could be distributed on
floor systems. In such distributions, the dampers are not located at the location of
maximum response (anti-node) but spatially distributed along the length of the T
beam as shown in Figure 5-6. To maintain the ultimate performance of the damper
system, the dampers are required to be redesigned to compensate for the loss in
the effectiveness of their mass due to the relocation of the dampers away from the
centre (where the peak response occurs). Consequently, the mass of the dampers
need to be increased and hence the stiffness needs to be increased to satisfy the
optimum frequency.
155
Figure 5-6 T beam with four spatially distributed dampers
To compensate for the loss in the damper modal mass due to the departure from
the point of maximum response, the fundamental vibration mode shape needs to
be determined. Experimental modal analysis or FE modal analysis can be used to
obtain the mode shape for floor systems especially for complex systems. For
simple plates (shown in Figure 4-18), the mode shape ( )y,x( ) is two directional
and for plates simply supported along all edges, the mode shape can be calculated
using Equation (5-4) (Hivoss 2010);
yx Lysin
Lxsin)y,x( 5-4
where
x = the location coordinate at which the mode shape value is required on x axis;
y = the location coordinate at which the mode shape value is required on y axis;
xL = the dimensions of the plate in x direction; and
yL = the dimensions of the plate in y direction.
The contribution to the modal mass at a position )y,x( would be given by )m( xy
via Equation (5-5) (Buchholdt 1997);
156
2),(. yxmmxy 5-5
Component Mode Synthesis (CMS) can also be used to evaluate multi-degree of
freedom (MDOF) system response for the purpose of finding the optimum
parameters of the Tuned Mass Damper (TMD). Hurty (1965) developed CMS as a
method specifically designed for analysis of structures consisting of an
assemblage of sub-structures. The advantage of using this method is that the
response of a system with TMDs can be found using a reduced number of mode
shapes of the original structure. It derives the system equations of motion by first
deriving the equations of motion for each individual substructure separately and
then relating the displacement and force conditions at their junction points
(Setareh & Hanson 1992a, 1992b).
5.3.2. Multiple Dampers Distributed on T beamIn order to assess the effectiveness of the distributed dampers, the T beam was
tested again for the distributed damper configuration shown in Figure 5-6. In this
experiment the same identical dampers as described in Section 5.3.1 were used
without modification. Therefore, a lower damper performance in reducing the
floor vibrations was expected.
Figure 5-7 shows the response of the T beam in the frequency domain due to heel
drop excitation with the four dampers evenly distributed along the T beam as
shown in Figure 5-6. The figure clearly shows that the dampers were still effective
and provided a reduction in the acceleration response of approximately 1.8.
157
3 3.5 4 4.5 5 5.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Freqeuecy (Hz)
Acc
eler
atio
n (m
/s2 )
T beam without damperSingle damper at mid-spanFour dampers at mid-spanFour distributed dampers
Figure 5-7 T beam response with four spatially distributed dampers
The average damping ratio of the T beam with four dampers spatially distributed
along the length of the T beam obtained from 24 time history records was about
6% as shown in Figure 5-8.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
Acc
eler
atio
n (m
/s2 )
= 6%
Figure 5-8 T beam response with four distributed dampers attached
To investigate the sensitivity of the T beam response when one or more dampers
are not functioning, some dampers were deactivated by preventing their vertical
movement using a timber stud. An example of a deactivated damper is shown in
158
Figure 5-9. The T beam was then excited either by heel drop or walking excitation
to measure the effectiveness of the modified system with fewer active dampers.
Figure 5-9 Deactivated damper
The effectiveness of the distributed MTMD system was investigated for different
possible cases of deactivated dampers. The reduction factors for seven possible
cases of active and deactivated dampers are listed in Table 5-2 for heel drop and
walking excitations.
The T beam response with distributed dampers was also investigated using FE
analysis using the modelling method described in Sections 4.2.5 and 4.3.4. All
seven possible cases of active and deactivated dampers were investigated using
the FE analysis. The T beam was subjected to harmonic forces and walking along
the length of the beam to compare the results with the experimental values. The
results of FE harmonic and transient analyses are shown in Table 5-2. It is quite
evident that the extreme dampers (No.1 & 4) were not very effective in reducing
159
the levels of vibration particularly for walking because of the loss in the modal
mass of dampers. Furthermore, in the case of walking excitation the dampers are
less effective because the system does not reach steady state response in contrast
to the harmonic excitation. According to Equations (4-12) and (5-5), the mass of
dampers No. 1 & 4 need to be increased by three times while for dampers No. 2 &
3 the mass needs to be increased by 10%. It was found that the dampers have a
reasonable effectiveness when they are distributed at the middle fifth of the T
beam (i.e. 10% span ratio from the point of maximum response). Hence the
system can be located away from the point of maximum response of the T beam
and distributed in the centre fifth of the beam to overcome any possible
limitations to the size and place of the dampers due to the interference with
services in the false ceiling or the false floor space.
Table 5-2 Sensitivity analysis of distributed dampers
Active damper Reduction factor
Heel drop Walking
1 2 3 4 Experiment FEM Experiment FEM
x x x x 1.8 1.8 1.4 1.4
x x x - 1.7 1.7 1.3 1.4
x x - x 1.6 1.6 1.2 1.3
- x x - 1.5 1.6 1.3 1.3
- x - - 1.3 1.2 1.2 1.2
x - - x 1.3 1.3 1.1 1.1
x - - - 1.1 1.1 1.0 1.1
x = active damper
- = deactive damper
Sensitivity analyses for modifying the damper end mass and hence stiffness to
compensate the loss of modal mass of the damper due to the departure from the
anti-node was investigated using FE modelling. The compensations in the modal
mass of each damper and the corresponding stiffness to achieve the optimum
frequency of 4.2 Hz for the distributed damper system that was illustrated in
Figure 5-6 are listed in Table 5-3.
160
The maximum acceleration response obtained from walking along the length of
the T beam with the four modified distributed dampers was about 0.23 2sm .
This corresponds to a reduction factor of 1.6 compared with the beam without
dampers. This reduction factor is equal to that obtained for identical dampers
located at mid-span, which is consistent with expectation since both cases have
the same total modal mass and damping ratio. Given that the additional mass to
dampers 1 & 4 was rather large, another alternative case was investigated. In this
case, all identical dampers were located at a distance of 1 m from the mid-span as
shown in Figure 5-10 in order to investigate the effectiveness of the identical
dampers. It was found that the response acceleration of the T beam due to walking
was about 0.24 2s/m . Although the identical as built dampers described in Table
5-1 were not modified to compensate for the loss in the modal mass because of
the departure from the point of maximum response (anti-node), the dampers were
still very efficient in reducing the T beam vibration.
Figure 5-10 Identical dampers located at a distance of 1 m away from the point of maximum
response
Table 5-3 Dynamic properties of modified distributed dampers
Damper
No.
Mass (kg)Adjusted stiffness
(N/m)
Damping
ratioRequired total mass
for modified damper
Modal mass of
modified damper
1 23.2 8 15,807 4.5%
2 8.8 8 6,038 4.5%
3 8.8 8 6,038 4.5%
4 23.2 8 15,807 4.5%
161
The maximum response accelerations of the T beam for the four cases of damper
setups obtained from walking along the T beam are summarised in Table 5-4. The
FE transient analysis indicated that the acceleration response of the T beam due to
the departure of the dampers from the point of maximum response by a distance
of 1 m does not significantly increase. The acceleration response of the T beam
due to walking excitation for distributed dampers offset by 1 m from the centre
was in excellent agreement with four dampers at mid-span and the modified
distributed damper setup studied.
Table 5-4 Acceleration response of T beam to walking excitations for different damper setup
Damper configurationsAcceleration
m/s2
Reduction
factor
Four identical dampers evenly distributed 0.27 1.37
Four identical dampers 1 m away from mid-span 0.24 1.54
Four identical dampers at mid-span 0.23 1.60
Four modified dampers evenly distributed 0.23 1.60
T beam without dampers 0.37 -
5.4. Multiple Viscoelastic Dampers on an Office FloorThe new viscoelastic damper was thoroughly tested and found to be successful in
reducing the levels of vibration for the steel beam and the T beam as discussed
earlier. The MTMD system was also successfully tested on the T beam in two
configurations, one concentrated at the mid-span and the other in distributed form.
In order to assess the scaling of the dampers to the size necessary for real floors
and to examine their effectiveness under normal operating conditions, a real office
floor was investigated. The office floor is located in a building highrise tower in
the Melbourne CBD. The floor plan of this office is shown in Figure 5-11. The
floor comprises a 120 mm thick slab acting compositely with steel beams. The
concrete used in the construction of this floor was a lightweight concrete with 30
MPa strength and 1900 3mkg mass density. The problematic bay is 12.7 m long
and 9 m wide with a 3 m secondary beam spacing. The floor was designed for an
imposed live load of 5 kPa with a false floor, ceiling and other services
162
accounting for an additional load of 1.17 kPa. The floor has a typical modern
office fit-out with low height partitions as shown in Figures 5-12 and 5-13). A
specific problematic bay was identified by the tenants as a lively bay and
uncomfortable for deskwork.
Figure 5-11 Plan of the floor and problematic bay
The problematic floor bay has two long corridors A & B in Figure 5-11, which are
perpendicular to each other. The walking paths are thus long enough for the
vibration energy to build up. Moreover, the distance from the intersection of these
corridors to the closest working station is just about 1 m, which is too close to
avoid the troublesome vibration effects. In this location, occupants at their
163
workstation feel the vibration, with noticeable shaking of plants in pots when
people walk along the two corridors ‘A’ and ‘B’ that are shown in Figures 5-12
and 5-13).
Figure 5-12 Pathway ‘A’ of the problematic bay along the secondary beams
Figure 5-13 Pathway ‘B’ of the problematic bay crossing the secondary beams
164
The building manager sought assistance from the building designer to assess the
floor dynamic performance. The designer investigated the floor and carried out a
numerical analysis to compare the floor response acceleration with the acceptance
criteria for walking excitation. The numerical analysis was carried out in
accordance with the AISC DG11 (Murray et al. 1997), which is widely used in
assessing floors in Australia. As discussed in Section 2.3.1.1, for office floor
systems with natural frequencies in the range of 4-8 Hz, the maximum acceptable
acceleration is 0.5% g.
The preliminary numerical analysis estimated the natural frequency of the floor
system to be about 4.9 Hz. Assuming that the damping ratio of the floor was 3%,
the peak acceleration was calculated using AISC DG11 and found to be 0.44% g,
which is within the acceptability range. This value of peak acceleration is
marginal and based on the assumption of a 3% damping ratio. If the damping ratio
is assumed to be 2.5%, the peak calculated acceleration would be 0.53% g, which
exceeds the acceptance criteria.
The assumptions and subjectiveness of the assessment can cause large
discrepancies. Therefore, to reduce the vibration of the floor, several solutions
were considered by the designer including the stiffening of the floor. Based on
careful consideration, the designer suggested the installation of a TMD to increase
the overall damping of the office floor. The designer suggested to their client to
pursue this option with the research team at Swinburne University of Technology.
Accordingly, the client contacted the research team in order to rectify the floor
using the viscoelastic dampers.
Before the dampers were designed, the problematic floor was required to be
assessed experimentally to measure the level of acceleration under walking
excitation as well as to determine its dynamic properties.
165
5.4.1. Determination of Floor Natural FrequencyThe natural frequency of the floor was extracted from the heel drop excitations by
converting the time domain into the frequency using Fast Fourier Transform
(FFT). The measured natural frequency of the floor was about 6.2 Hz as
illustrated in Figure 5-14, which is higher than the predicted value of 4.9 Hz.
0 2 4 6 8 10 12 14 16 18 200
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Frequency (Hz)
Mag
nitu
de
= 6.2 Hzfn
Figure 5-14 Natural frequency of the bare floor obtained from heel drop excitation
5.4.2. Determination of the Peak Acceleration Due to WalkingIn order to assess the peak acceleration response of the original floor
experimentally, eight time histories for walking along the critical path (Pathway
‘A’) were recorded. The reason for considering Pathway ‘A’ as a critical path was
because of the high chance to approach the maximum response due to walking
along this longer path. An example of the response to walking along Pathway ‘A’
is illustrated in Figure 5-15 with a maximum acceleration value of around 0.6% g.
166
0 1 2 3 4 5 6 7 8 9 10-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time (s)
Acc
eler
atio
n (m
/s2 )
Figure 5-15 Response acceleration of the bare floor due to walking along Pathway ‘A’
The peak accelerations from eight time histories of the floor due to walking along
the critical path are shown in Figure 5-16. While there were some variations from
one record to another, the average maximum response was 0.52% g with the
maximum of the eight records of 0.67% g. It is obvious from Figure 5-16 that five
out of the eight records exceeded the acceptable limit of the peak acceleration.
The variation in the acceleration of the floor is due to the difficulty in controlling
the step frequency of walking so that the third harmonic matches the natural
frequency of the floor. The root mean square (rms) acceleration response was
about 0.53% g, which also exceeded the acceptable limit and agrees well with the
average maximum response of 0.52% g.
1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Test Number
Peak
Acc
eler
atio
n (%
g)
Threshold = 0.5%g
Figure 5-16 Measured peak accelerations of the floor for the eight walking records
167
An on-the-spot walking was also carried out to measure the steady state
acceleration response of the floor. This on-the-spot walking was conducted very
close to the area of maximum response (approximately the centre of the
problematic bay). In order to achieve the maximum response, the walking was
attempted at a frequency of about 2 Hz, which is one third of the measured natural
frequency of the floor. An example of the floor response acceleration due to on-
the-spot walking is illustrated in Figure 5-17. The average value of maximum
response acceleration of the floor for several time history records was about 0.7%
g. It is clear from the on-the-spot walking tests that if the walking force is
maintained at the right frequency for a sufficient time the peak acceleration of the
floor would far exceed the recommended maximum limit.
0 1 2 3 4 5 6 7 8 9 10-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Res
pons
e A
ccel
erat
ion
(m/s
2 )
Figure 5-17 Measured original floor response acceleration due to on-the-spot walking
According to the field tests, it was concluded that the office floor often fails to
comply with the acceptable level for human comfort. Therefore, remedial work
was required for this office floor to reduce the excessive floor vibration due to
walking excitation.
168
5.4.3. Determination of Floor Damping RatioSeveral time histories were recorded from the heel drop excitations to obtain the
value of the damping ratio using the logarithmic decrement method (LDM) and
Randec method.
In order to eliminate the noise from the measured signal, the signals were filtered
in the frequency domain for the frequency band of 2-20 Hz. The filtered signals
were transformed back to the time domain using Inverse Fast Fourier Transform
(IFFT). It appeared from the response of the floor due to heel drop excitation as
shown in Figure 5-18 that the damping ratio of the floor is not constant. The
damping values for every five cycles starting from the second cycle with an
overlap of two peaks were 3.8%, 2.4% and 2.3% as shown Figure 5-18. The
average value of damping ratio obtained from the three laps was about 2.8%.
Alternatively, the damping was also calculated for eleven peaks and it was found
that the damping ratio was about 2.9% as illustrated in Figure 5-18.
1 1.5 2 2.5 3 3.5 4-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Acc
eler
atio
n (m
/s2 )
2.4%
3.8%
2.3%
2.9%
Figure 5-18 Response of the floor due to the heel drop excitations using LDM
The damping ratio was also determined using the Randec method. Two damping
values were obtained from this method as illustrated in Figure 5-19, which
represent two distinct slopes for the response cycles. The first part provided a
damping value of 4.3%, which was constant over the first three peaks while the
169
second part produced a damping value of 2.9%, which was constant over the
remaining peaks.
0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Time (s)
Res
pons
e
= 2.9%
= 4.3%
Figure 5-19 Damping ratio of the floor obtained from heel drop excitation using Randec
Both the LDM and Randec produced relatively high damping values for the first
few cycles. These values of damping are higher than expected for such a floor
where DG11, SCI P354, Hivoss and CCIP-016 would suggest a damping ratio of
3%. It is acknowledged that the measurement of damping using the heel drop
excitation can overestimate the damping because it measures not only energy
dissipation (the true damping ( )) but also the transmission of vibrational energy
to the structural components. Murray (1998) suggests that the modal damping of
the floor to be approximately two third to one-half of the LDM damping.
Consequently, the average value of damping for the initial cycles could be taken
as 2.7% (based on two third factor as suggested). This is closer to the damping
values, which were obtained from the cycles following the initial response. In
designing the TMD for this floor, it is more conservative to overestimate the
damping value of the floor rather than underestimate it. Therefore, for the
development of TMDs for this specific floor, a damping value of 3% was adopted
because it is consistent with the typical design value.
170
To fully assess the retrofitted floor response, a sensitivity analysis will be
performed which includes a change in the damping ratio of the floor as presented
in Chapter 6.
5.4.3.1. Dynamic Testing of Building FloorsSCI P354 described two types of floor modal testing methods to estimate the
damping of existing building floors in Appendix C: when the excitation force
creating these responses is not measured and when it is measured (Smith et al.
2007).
5.4.3.1.1. Modal Testing of Floors without Measuring theExcitation ForceThere are three methods to estimate the damping without measurement of the
excitation force. These are ambient vibration survey (AVS), heel drop and
rotating mass shaker. The AVS method was discussed in Section 4.3.1 and its
accuracy depends on the grid points covering floor area of interest. The grid needs
to be dense enough to describe floor mode shapes of interest in sufficient detail,
otherwise a problem of so called “spatial aliasing” could occur, leading to
incorrect identification of mode shapes. The second method is the heel drop
excitation, which overestimates the damping and the obtained value of damping
needs to be adjusted as discussed earlier. The third method (rotating mass shaker)
estimates the damping using the half-power bandwidth. This process is slow and
prone to errors due to low frequency resolution obtainable particularly for the case
of closely spaced modes of vibration.
A common feature in the above methods where the excitation force is not
measured and hence the modal properties tend to be less complete and reliable.
This is because the lack of measurement of forces requires a number of
assumptions to be made to enable extraction of modal properties, and some of
them may not be correct, leading to considerable errors (Smith et al. 2007).
171
5.4.3.1.2. Modal Testing of Floors with Measurements Made ofthe Excitation ForceTwo types of tests are commonly performed for the modal testing with measured
excitation force: impact testing (using an instrumented hammer or a heel-drop on
an instrumented force plate) and shaker testing (using a single shaker or an array
of shakers distributed over the floor area). In the case of an instrumented hammer,
the force is measured by a load cell installed at the tip of the hammer. A problem
with impact testing of floors using measured impulses is the fact that the
excitation energy is supplied over a very short period of time (relative to the
natural periods of modes of vibration being excited) and the response decays
quickly as well, within a second or two. This leads to poor signal-to-noise ratios
which leading to errors in the damping estimation. The problem of poor signal-to-
noise ratios in the instrumented impact testing can be resolved by employing an
instrumented shaker excitation (Smith et al. 2007).
5.4.4. FE Model to Determine Floor Dynamic PropertiesThe experimental work that was conducted to determine the natural frequency,
damping ratio and the peak acceleration response was necessary to assess the
existing condition of the floor. This was also necessary for obtaining data to
validate a FE model of the floor, which was later used for assessing the
performance of the TMDs to be fitted. One significant factor in designing the
damper is to estimate the modal mass of the floor in order to determine the
required optimum mass of the damper. General expressions are available for some
simple structures, such as simply supported beams, to calculate the modal mass of
the structure for dynamic analysis but no such general expressions are available
for complex and irregular floors. Therefore, FE modelling is an alternative tool to
be used to predict the modal mass of the fundamental mode for such complex
floors as well as to predict the floor response.
The floor was modelled using ANSYS to obtain the dynamic properties such as
the natural frequency and modal mass of the floor and then to predict the floor
acceleration response due to walking excitations. The slab was modelled using
172
shell elements (Shell181) while the secondary and primary beams were modelled
as beam elements (Beam188). The beam end connections were assumed to be pin
joints. The steel and concrete were assumed to be linear elastic and the floor
damping was modelled as a stiffness multiplier. Standard steel properties were
used in the analysis with a Young’s modulus of 200 GPa and density of 7,8503mkg . For dynamic analysis, the modulus of elasticity of the concrete was
multiplied by a factor of 1.35. The dynamic Young’s modulus of the concrete was
taken as 26 GPa and its density was 1,900 3mkg . In addition to the self-weight
and services, 10% of the live load was used in the analysis.
5.4.4.1. FE Model for Problematic BayThe process of the FE analysis began with modelling the problematic bay only to
determine the mode shapes of the floor and their corresponding frequencies.
Modal analysis was conducted for this model and it was found that the
fundamental natural frequency of the problematic bay was about 5.1 Hz with a
mode shape as shown in Figure 5-20. The obtained natural frequency of the bay of
5.1 Hz did not agree with the measured natural frequency of 6.2 Hz.
Consequently, the model was extended to include the adjoining bays in order to
increase the stiffness and hence the natural frequency due to the continuity of the
concrete slab.
173
Figure 5-20 Fundamental mode shape for the model of the problematic bay only
5.4.4.2. FE Model for Problematic Bay with Adjoining BaysThe adjacent bays in the vicinity of the problematic bay were included in this
revised model. From the modal analysis of this revised model, it was found that
the natural frequency of the first mode was about 6.1 Hz, which is very close to
the measured natural frequency of 6.2 Hz. The mode shapes of the corresponding
natural frequencies are shown in Figure 5-21 where it is obvious that the anti-node
of mode 1 (fundamental mode) is located around the centre of the problematic
bay.
174
Figure 5-21 First eight mode shapes for the model of the problematic bay with adjoined bays
The modal mass of the problematic bay for the model was calculated from the
kinetic energy value of the mode using Equations (4-8 - 4-10) as discussed in
Section 4.2. The modal mass of the fundamental mode was about 15 tonnes.
In order to have a higher level of confidence on the values of natural frequency
and modal mass of the problematic bay and adjacent bays, another model of the
entire floor was created and analysed as described in the following section.
175
5.4.4.3. FE Model for Entire FloorModal analysis for a model of the entire floor was conducted in order to obtain the
natural frequencies and their corresponding mode shapes which are shown in
Figure 5-22. It is obvious from this figure that the first mode is not critical for the
problematic floor bay, as it does not yield any anti-nodes in that bay. On the other
hand, the fourth mode with natural frequency of 6.2 Hz seems to be the resonant
mode for the problematic bay with an anti-node located around the centre of the
bay. The modal mass obtained from this model was found to be 16.5 tonnes.
It is obvious from Figures 5-21 and 5-22) that up to a frequency of 8.4 Hz, the
critical mode for the problematic bay is the mode with a natural frequency of 6.2
Hz. Generally, apart from mode No. 1 in Figure 5-21, which corresponds to mode
No. 4 in Figure 5-22, the other modes do not have significant impact on the
response of the problematic bay, as the problematic bay does not extensively
oscillate and the motion mainly occurs in the other bays. Hence, the critical
frequency of the excitation force for the problematic bay would be 6.2 Hz and the
maximum response takes place when the third harmonic of the walking step
frequency coincides with the frequency of this mode.
As the variation in both values of natural frequency and the modal mass for the
two models (entire floor and floor with adjoining bays) was only about 1% and
7% respectively, the model of the problematic bay with adjoining bays can be
adopted for further FE analyses such as the maximum acceleration response of the
floor due to walking excitation.
176
Figure 5-22 First eight mode shapes and corresponding frequencies for the model of entire
floor
177
5.4.5. Peak Acceleration of the Floor Using Analytical and FEModelsThe peak acceleration response of the problematic bay due to walking excitation
was investigated using analytical and FE models. The values obtained from the
analytical and FE models were compared with the values obtained from the field
tests.
5.4.5.1. Peak Acceleration Obtained From Analytical SolutionThe equivalent SDOF model of the floor for mode No. 4 in Figure 5-22 in terms
of the modal mass ( 1m ), frequency ( 1f ) and modal damping ( 1 ) were taken to be
16,500 kg , 6.2 Hz and 3% respectively. The value of modal mass was obtained
from the FE model for the entire floor while the values of frequency and modal
damping were based on the field tests.
For a quick simplified analytical estimation of the floor response to walking, the
SDOF system was subjected to only the third harmonic component of the walking
force that matches the floor natural frequency. The steady state acceleration of the
floor bay ( pa ) can thus be computed by manipulating Equation (2-8) to produce
Equation (5-6);
1111 2.
2 mPr
mFap
5-6
where
r = Fourier coefficient of walking excitation;
P = walker’s weight;
F = applied force;
1 = damping ratio of the original floor; and
1m = modal mass of the floor.
178
The substitution of r = 0.1, P = 800 N, 1 = 3% and 1m = 16,500 kg into Equation
5-6) yields pa = 0.081 2sm or pa = 0.82% g. It should be noted that the person’s
weight was taken as 800 N to match the weight of the person who conducted the
walking test in the experimental work.
The peak acceleration of the floor due to the third harmonic force of the walking
excitation (0.82% g) exceeds the accepted comfort limit of 0.5% g. It should be
noted that the value obtained from analytical model is for steady state response of
the floor. This value of peak acceleration agrees well with the measured value
particularly with on-the-spot walking floor response. Using the adjusted values of
natural frequency, damping ratio and mass of 6.2 Hz, 3% and 16.5 tonnes of the
floor respectively, DG11 gives a peak acceleration response of 0.28% g. This
reduction in the peak acceleration response from 0.44% g to 0.28% g was because
of the increase in the natural frequency of the floor from 4.9 Hz to 6.2 Hz. Both
the measured and analytical values do not agree with value of peak acceleration
calculated by the procedure described in AISC DG11.
5.4.5.2. Peak Acceleration Obtained from FE ModelThe peak acceleration was also predicted using FE modelling. The force for
walking along the problematic bay was modelled in FE using Equation (4-14). A
transient dynamic analysis was conducted to extract the peak acceleration
response due to walking along the critical path. The peak acceleration obtained
from the FE model was about 0.87% g as shown in Figure 5-23. Similar to the
field tests and analytical results, the peak acceleration obtained from FE model
exceeded the acceptable value of comfortable acceleration limit of 0.5% g.
An on-the-spot walking test at the point of maximum response (anti-node) was
also carried out using the same FE model. The peak acceleration obtained from
the transient analysis was about 0.9% g as shown in Figure 5-24. It can be noted
from this analysis that the value of peak acceleration was very close to that
obtained from walking along the critical path of the problematic bay. This
indicates that the floor can reach the steady state response by walking along the
179
critical path. It is most likely that the floor did not reach the steady state response
during the in-situ tests due to lack of consistency in the step frequency throughout
the walking path.
0 1 2 3 4 5 6 7 8-400
-200
0
200
400
600
Inpu
t For
ceat
Mid
-spa
n (N
)
0 1 2 3 4 5 6 7 8-0.1
-0.05
0
0.05
0.1
Res
pons
eA
ccel
erat
ion
(m/s
2 )
Time (s)
Figure 5-23 Floor acceleration response due to equivalent walk along the critical path and
response at mid-span using FE analysis
0 1 2 3 4 5 6 7 8-500
0
500
Forc
e (N
)
0 1 2 3 4 5 6 7 8-0.1
-0.05
0
0.05
0.1
Time (s)
Res
pons
eA
ccel
erat
ion
(m/s2 )
Figure 5-24 Floor response from FE analysis due to on-the-spot walking at the centre of
problematic bay
180
5.4.6. Development of Multiple Viscoelastic DampersA system of tuned mass dampers were designed and fabricated to attenuate the
excessive floor vibration level. One of the most challenging design requirements
was that the dampers had to be installed within the limited space of the false floor
with a cavity height of 150 mm.
For a total floor modal mass of 16,500 kg, a TMD with mass ratio of 1% would
have a mass of 165 kg. While this could be sufficient for an effective TMD
design, there were two uncertainties, which needed to be considered. The first, the
modal mass of the floor was based on 10% of the nominal live load. In reality, this
could be as high as 20% of the nominal live load as suggested by Hivoss (2010).
If this is the case, then the mass of the TMD needed to be increased. Furthermore,
during the installation of the MTMD system not all of the dampers will be
installed at the point of maximum response. Hence, the modal mass of the MTMD
system would reduce. Therefore, for a conservative design, it was decided to
adopt a higher mass ratio for the MTMD system to account for these two possible
uncertainties, which could potentially be reducing the effectiveness of the MTMD
system. Accordingly, the MTMD system was designed with a total mass of 276
kg. In order to assess the effect of change of floor mass on the performance of the
MTMD system a sensitivity analysis was performed as presented in Chapter 6.
A single viscoelastic damper with a configuration as shown in Figure 5-25a would
be too large to be housed within the 150 mm false floor. Consequently, a four
armed MTMD system in a crucifix configuration with 62.5 kg masses on each arm
as shown in Figure 5-25b was considered. This configuration was still too large to
be accommodated within the false floor. Alternatively, twelve distributed dampers
each with a relatively small mass were proposed to replace the single large and
four dampers options to satisfy the false floor space constraints. The MTMD
system consisted of 12 cantilever dampers with a total mass of 276 kg arranged in
3 sets (4 dampers per set) as shown in Figure 5-25c with an overall height of
about 140 mm.
181
Figure 5-25 Plan view of TMD configurations
One appropriate type of commercially available rubber was selected for
developing the viscoelastic damper. A Dynamic Mechanical Analyser (DMA) was
utilised to obtain the mechanical properties of the rubber as a function of
frequency, amplitude and temperature. Once the mechanical properties of the
material were determined, the dimensions of the damper such as length, width,
thickness, end mass, etc., were estimated using the procedure outlined in Section
3.6 with the aim of satisfying the target optimal frequency and damping ratio of
the TMD. The dampers were then manufactured and tested to confirm their
dynamic properties. The thickness of the steel constraining layers was 6 mm and
the rubber core was 20 mm thick and 50 mm wide for each damper. The damper’s
arm had a length of 400 mm with an end mass of 22.5 kg. Pluck tests were
performed on each damper to confirm their dynamic properties with a typical time
trace and the Fourier transformation for the damper response shown in Figure
5-26.
Due to construction tolerances between dampers, minor variations in natural
frequency and damping ratio could be expected. From the pluck tests, each
damper had a natural frequency of approximately 6.2 Hz and a damping ratio of
approximately 5%. Whilst the damper could be tuned to the floor natural
frequency, it could not be designed with the optimum damping ratio of about 6%
using Equation (2-40), due to the limited dissipation loss factor of the available
rubber.
182
1 2 3 4 5-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Time (s)
Acc
eler
atio
n (m
/s2 )
0 2 4 6.2 8 10 120
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Frequency (Hz)
Mag
nitu
deFigure 5-26 Damper response to pluck test
5.4.7. Preliminary Estimation of the Retrofitted Floor ResponseAn estimation of the response of the floor with dampers could be made by
adopting the equivalent single damper concept. For twelve dampers with a total
mass of 276 kg supported by three units 1.2 m apart, the computed modal mass for
the equivalent single damper, 2m , is approximately 250 kg. This takes into
account the departure of the damper units from the anti-node of the floor’s
fundamental mode shape. The total mass ratio based on the modal mass obtained
from the FE model of the entire floor of 16.5 tonnes, , is thus about 1.5% for
the whole damper system. Based on this mass ratio and damping values of the
dampers, the maximum acceleration response of the retrofitted floor could be
determined using Equations (2-8) and (5-3). Due to the addition of the dampers, a
reduction of 50% in peak response was expected compared with the original floor.
If this could be confirmed experimentally, the peak response of the retrofitted
floor would be reduced to be within the acceptable range.
183
5.4.8. Installation of Viscoelastic Dampers and TestingThe twelve dampers were divided into three sets with each set consisting of four
viscoelastic dampers bolted to a steel base as shown in Figure 5-27. Each steel
base consists of 1001006 mm steel square hollow section welded to 10 mm
thick steel base plate. The steel plate was anchored to the floor using 10 mm
diameter Dynabolts to form a rigid connection, which could transmit the energy
from the floor to the dampers. One set of viscoelastic dampers was installed at the
centre of the problematic bay while the other sets were installed along the line of
the critical path at a distance of 1.2 m from the centre as illustrated in Figure 5-27.
Figure 5-27 Viscoelastic dampers installed within false floor
184
5.4.9. In-situ Vibration MeasurementsThe floor was subjected to walking and excitation using an electrodynamic
shaker. The floor vibration was measured using accelerometers located at the
centre of the bay.
5.4.9.1. Retrofitted Floor Response to Walking ExcitationA large number of walking induced vibration tests were undertaken in which the
walking speed and step length were adjusted in an attempt to closely match the
third harmonic of the step frequency with the floor natural frequency so as to
attain the maximum floor response. Measurements taken before the installation of
the dampers revealed a peak acceleration floor response of about 0.67% g, which
exceeded the recommended level for human comfort of 0.5% g. After the dampers
were installed, the maximum acceleration level from all walking tests was found
to decrease to around 0.3%0.4% g, which is well within the acceptable range. It
was also observed that the floor felt less lively after installation of the dampers in
both standing and sitting positions of an observer. Samples of the peak floor
response to walking are shown in Figure 5-28 whilst typical time traces for the
walking induced response are shown in Figure 5-29.
1 2 3 4 5 6 7 80
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.7%
Record Number
Peak
Acc
lera
tion
(g)
1 2 3 4 5 6 7 80
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.7%
Record Number
a) Original Floor b) Floor with Dampers
Threshold = 0.5% g
Figure 5-28 Peak acceleration responses for the floor without and with the MTMD system
due to walking excitation
185
0 2 4 6 8 10
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time (s)
Acc
eler
atio
n R
espo
nse
(m/s
2 )
0 2 4 6 8 10
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Time (s)
b) Floor with Dampersa) Original Floor
Figure 5-29 Typical traces for acceleration responses of the floor without and with the
MTMD system due to walking excitation
5.4.9.2. Retrofitted Floor Response to Shaker ExcitationGiven the possible variation in the walking frequency and hence response, it was
decided to use a mechanical shaker to obtain the response of the floor without and
with the TMDs. A series of tests were also conducted using a shaker (shown in
Figure 5-27) which excited the floor with a defined dynamic force and a
frequency range covering the resonant frequencies of the floor without and with
the dampers. The response of the floor was measured before and after the
installation of the dampers with resulting time histories as depicted in Figure 5-30.
Peak values obtained from acceleration time traces were collected and plotted in
Figure 5-31 for different tests. Generally, there was a reduction of 40% in peak
floor response to the shaker excitation, from 1.59% g for the original floor (Figure
5-30a) to 0.96% g for the floor with dampers (Figure 5-30b).
186
0 10 20 30 40 50 60-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Acc
eler
atio
n R
espo
nse
(m/s
2 )
0 10 20 30 40 50 60-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time (s)
b) Floor with Dampersa) Original Floor
Figure 5-30 Typical acceleration responses for the floor without and with the MTMD system
due to shaker excitation
1 2 3 4 5 6 7 80
0.5%
1.0%
1.5%
2.0%
Record Number
Peak
Acc
eler
atio
n (g
)
1 2 3 4 5 6 7 80
0.5
1.0%
1.5%
2.0%
Record Number
b) Floor with Dampersa) Original Floor
Figure 5-31 Peak acceleration response for the floor without and with the MTMD system due
to shaker excitation
5.4.10. FE Model for Floor with DampersThe office floor with viscoelastic dampers shown in Figure 5-25c was modelled
via FE to evaluate the performance of the dampers. The dampers were modelled
using FE elements as described in Section 4.3.4. The excitation force for walking
along the problematic bay was modelled in FE using Equation (4-14). The
187
acceleration response of the floor with the attached dampers was investigated for
all splitting frequencies, which were created due to the addition of the dampers.
The waking frequency in Equation (4-14) was adjusted so that the third harmonic
of walking matches each splitting frequency. The maximum acceleration response
obtained from a transient analysis for all possible third harmonic walking
frequencies was found to be 0.4% g as shown in Figure 5-32. This confirms that
the effectiveness of the dampers in reducing the level of vibration by
approximately 50% (i.e. from 0.87% g (as discussed in Section 5.4.5.2) to 0.4%
g). The addition of the dampers reduced the acceleration response of the floor to
below the acceptable level to be in a good agreement with the experimental
measurements.
0 2 4 6 8
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Acc
eler
atio
n R
espo
nse
(m/s
2 )
0 2 4 6 8
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
a) Original Floor b) Floor with dampers
Figure 5-32 Acceleration responses of the floor without and with dampers due to walking
excitation based on FE analyses
5.5. Concluding RemarksThis chapter discussed the development of an analytical solution for natural
frequencies and steady state response for a system consisting of a primary
structure combined with multiple identical tuned mass dampers. The steady state
response of a primary structure fitted with n identical TMDs would be similar to
that when it is connected to an equivalent single TMD. Consequently, in the event
that a single damper can not be used because of any restrictions in fabrication or
188
installation then several smaller dampers with the same frequency and damping
ratio could be used instead.
For the T beam previously retrofitted with a single damper, four viscoelastic
dampers with a total mass ratio of 1% were developed to replace the single
damper. The retrofitted T beam was excited using heel drop excitation. The
overall response of the retrofitted T beam with multiple dampers was halved with
a corresponding increase in the damping ratio to 6.2% from the maximum original
value of 2.9%. The four dampers were then evenly distributed along the length of
the T beam to test the sensitivity of the beam response to the position of dampers
in relation to the location of maximum response. It was found that a reduction
factor of 1.8 in the response of the T beam could still be achieved. Although the
modal mass ratio reduced because of the departure from the anti-node position,
the dampers performed well when they were all located inside 10% of the beam
span from the anti-node.
A bay of a real office floor experiencing excessive floor vibrations was also
investigated. The building designer recommended the installation of viscoelastic
dampers to reduce the annoying vibrations.
Field measurements were performed to obtain the natural frequency, damping and
maximum response of the problematic bay due to walking. The in-situ tests
revealed that the dynamic properties of the bay were a fundamental natural
frequency of 6.2 Hz, damping ratio of 3% and a maximum acceleration response
due to walking excitation along the length of the bay of 0.67% g, which exceeded
the acceptable level for human comfort of 0.5% g.
A model of the entire floor was created in ANSYS to obtain the natural frequency,
modal mass of the fundamental mode of the bay and the maximum acceleration
response due to walking. The natural frequency obtained from the FE analysis
was in good agreement with the measured value. Further, the maximum
acceleration response due to walking was about 0.87% g and a modal mass of the
189
fundamental mode was about 16.5 tonnes.
For a damper system to fit in the available false floor spacing of only 150 mm, a
distributed multiple viscoelastic damper system was developed. The damper
system consisted of 12 individual dampers grouped in 3 sets.
After the dampers were installed, the maximum measured acceleration obtained
from several walking tests was found to decrease to around 0.3%0.4% g, which
is well within the acceptable limit of 0.5% g. It was also observed that the floor
felt less lively after installation of the dampers in both standing and sitting
positions. The FE analysis for the floor with dampers revealed that the floor
acceleration response due to walking was reduced by at least 40%.
A series of tests were also conducted using a shaker, which excited the floor with
a defined dynamic force and a frequency range covering the resonant frequencies
of the floor without and with dampers. Using the shaker excitation, there was a
reduction of 40% in peak floor response as a result of the dampers. The response
was reduced from 1.59% g for the original floor to 0.96% g for the floor with
dampers.
The sensitivity of the retrofitted floor response to variations in damping ratio of
the floor will be investigated in the next chapter. Further, the effectiveness of the
MTMD system in reducing the vibration of the retrofitted office floor to variations
in damping ratio, damper mass ratio, damper frequency and location of the
dampers will also be investigated in the next chapter.
190
6. Parametric Analyses
6.1. IntroductionThe successful development, installation and testing of multiple viscoelastic
TMDs were presented in Chapter 5. In this chapter, parametric studies are
presented to investigate the influence of several parameters on the performance of
MTMD. These parameters are variations in the floor damping ratio, damper
damping ratio, damper natural frequency, damper mass ratio and damper point of
attachment. These studies will be performed on the validated FE model, which
was developed in Chapter 5 to represent the real office floor that was retrofitted
with the MTMD system.
6.2. Performance of MTMD to Variations in Floor Damping RatioIn the design of a floor system, it is difficult to determine the damping ratio with a
high degree of accuracy. Even for an existing floor, the damping value may
change from one location to another depending on the fit-out in a specific area.
Further, the measurement of damping is performed indirectly and can be
complicated by the presence of multiple modes and signal quality. Hence, the
sensitivity of a retrofitted floor system with TMDs to variations in floor damping
ratio is investigated using FE analysis.
The damping ratio of the same office floor presented in Section 5.4 was varied
from 1% to 5%. The location of as built dampers and their properties in terms of
natural frequency, modal mass and damping ratio of 5% were kept constant. A
transient dynamic analysis was performed for the original floor to each assumed
damping ratio in order to obtain the maximum acceleration response due to
walking from one end of the floor to another using the walking force defined by
Equation (4-14). The retrofitted floor was also subjected to the same walking
force to obtain its maximum response with distributed damper for each assumed
damping ratio of the floor. The addition of distributed MTMD system can alter
191
some of the original vibration modes of the floor and create new modes.
Consequently, the step rate was adjusted in order that the third harmonic of
walking matches each relevant frequency. A transient dynamic analysis was
performed for all these step rates in order to obtain the maximum response of the
retrofitted office floor.
The maximum floor acceleration response obtained from the FE transient analysis
was reduced to below the acceptable level for all damping ratios of the floor as
illustrated in Figure 6-1. The dampers were very effective in reducing the
acceleration response when the floor damping was low, while as expected the
effectiveness of the dampers degraded when the floor damping was high. These
analyses clearly show that if the damping ratio of the floor is underestimated or
overestimated it would have significant effect on the floor response. However,
when the floor is fitted with a MTMD system, such variations in floor damping
would have been much less significant. Indeed, within 20%, the variation in
damping for the floor under consideration, the resulting change in the retrofitted
floor response is almost negligible with the MTMD system reducing the
maximum vibration by a factor of 2.0.
This FE analysis investigating variations in floor damping indicated that
retrofitted floor response does not significantly change for a floor damping ratio
ranging from 1% to 5% although the response of the floor without dampers
significantly changes for this range of damping ratio.
192
1% 1.5% 2% 2.5% 3% 3.5% 4% 4.5% 5%0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Damping Ratio
Acc
eler
atio
n R
espo
nse
(% g
)
Original floorFloor with MTMDs
Threshold = 0.5% g
Figure 6-1 Acceleration response of the office floor without and with as built MTMD system
to variations in the floor damping ratio using FE analysis
6.3. Performance of MTMD to Variations in Damper DampingRatioA series of FE analyses were conducted to investigate the influence of variation in
the damping ratio of the individual dampers of the MTMD system on the
retrofitted floor response described in Section 5.4. The natural frequency and
modal mass of the as built MTMD system were kept constant throughout the
analysis. The damping ratio of the floor was assumed to be constant with a value
of 3% while the damping ratio of the damper system was varied from 3% to 8.5%.
The office floor retrofitted with the as built MTMD system but with varying
damping ratio was subjected to walking excitation for each splitting frequency
that were created from the addition of the MTMD system to the floor. The
walking was simulated using Equation (4-14) for a walk from one end of the floor
span to another. The step rate was adjusted to match the new frequencies of the
problematic bay.
From the FE analyses, it was found that the response of the retrofitted floor was
not very sensitive to the damping ratio of the MTMD system as shown in Figure
6-2. The maximum floor acceleration response reduced to the acceptable level for
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the entire range of variations in the damping ratio of the damper system (3% -
8.5%). It was also found that the floor response almost linearly decreased when
the damping ratio increased for this range of variations in the damper damping
ratio. The result from this analysis indicates that if the MTMD system does not
utilise rubber with the optimum properties, such system would still be effective in
reducing the maximum floor acceleration. In other words, the consequence for not
adopting optimum damping ratio for the dampers is minor.
3% 4% 5% 6% 7% 8%0.38
0.39
0.4
0.41
0.42
0.43
0.44
Damper Damping Ratio
Acc
eler
atio
n R
espo
nse
(% g
)
Figure 6-2 Maximum acceleration response of the office floor to variations in the damping
ratio of the MTMD system using FE analysis for floor with 3% damping ratio
6.4. Performance of MTMD to the Variation in Mass RatioOne of the design considerations for the MTMD system is the mass ratio. To
determine the mass ratio, knowledge of the modal mass of the floor is required.
This in turn requires knowledge of two items, firstly, mode shape of the
fundamental frequency, which can be determined from an FE analysis for
complex or irregular floors, and secondly, the dead and live loads need to be
determined. While the dead load can be determined with high degree of accuracy,
the live load is estimated with a lower degree of confidence. Murray et al (1997)
suggests a value of 10% of the nominal live load to be used for floor vibration
analysis while Hivoss (2010) suggested the live load percentage of up to 20%.
Given that the designer of the MTMD system has the flexibility of nominating the
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mass ratio of the system and there could also be some uncertainty in relation to
the actual modal mass of the floor, a sensitivity analysis is performed to
investigate the effect of variation in the mass ratio of the MTMD system on the
response of the retrofitted floor.
The damping ratio of the floor was kept constant at 3%. Further, the MTMD
system natural frequency and damping ratio were kept constant to the values of
6.1 Hz and 5%, respectively. To achieve the damper frequency of 6.1 Hz for all
damper mass ratios, the stiffness of the damper was adjusted according to the
specified mass ratio. The mass ratio in this FE analyses was varied from 0.75% to
1.6%. The retrofitted floor was subjected to walking excitations with step
harmonics matching all splitting frequencies, which were created due to the
addition of the MTMD system. The peak acceleration responses of the retrofitted
floor for each mass ratio of the MTMD system are shown in Figure 6-3.
It was found that the floor response did not dramatically change due to the
variation in the mass ratio of the damper system. For a benchmark value of 1% for
mass ratio, the response of the retrofitted floor with 0.75% mass ratio system was
higher by 2.4%. For a retrofitted floor with MTMD system with a mass ratio of
1.6%, the acceleration response decreased by 10.7%.
0.8% 0.9% 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6%0.39
0.4
0.41
0.42
0.43
0.44
0.45
0.46
MTMD Mass Ratio
Acc
eler
atio
n R
espo
nse
(% g
)
Figure 6-3 Acceleration response of the 3% damping ratio office floor to variations in the
MTMD system mass ratio using FE analysis
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It was also found that the acceleration response of the floor due to walking
excitation is acceptable for all damper mass ratios in this range despite the
difference in the frequency band of the splitting frequencies produced by the
addition of MTMD. The frequency band for a MTMD system mass ratio of 0.75%
is slightly narrower when it compared with frequency band of 1.6% damper mass
ratio as illustrated in Table 6-1.
Table 6-1 Comparison floor acceleration response to variations in the MTMD mass ratio
MTMD system
mass ratio
MTMD system
modal mass (kg)
Frequency
band (Hz)
Acceleration
response (% g)
0.75% 123 5.9 – 6.4 0.46
1.00% 165 5.8 – 6.4 0.44
1.25% 206 5.8 – 6.5 0.42
1.60% 264 5.7 – 6.5 0.39
It can be concluded from this study that small change in the mass ratio would not
have a significant change on the performance of MTMD system. However,
improvements in MTMD system performance can be achieved by increasing the
mass ratio if there is no space or practical limitations.
6.5. Performance of MTMD to Variations in Damper’s FrequencyThe natural frequency of the damper is an important factor that affects the
performance of the MTMD system in reducing the levels of vibration. The
optimum natural frequency of the damper system for a retrofitted floor with 6.2
Hz floor natural frequency and 1.5% damper mass ratio was calculated to be 6.1
Hz. The effectiveness of the damper system in reducing the floor vibrations is
expected to be reduced when the damper natural frequency departs from the
optimal value. Mistuning of dampers can occur due to variations or errors in
estimating the floor frequency and damper frequency.
As the purpose of this sensitivity study was to investigate the performance of the
floor response to the variation in the damper natural frequency, the floor natural
frequency of 6.2 Hz was assumed to be constant and the variation was imposed on
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the natural frequency of the damper system. The damper natural frequency was
changed from 5 Hz to 7 Hz. The stiffness of the dampers in the FE analysis was
adjusted to achieve the required natural frequency while the damper mass and the
damping ratio of the damper remained constant. The retrofitted floor was
subjected to a walking force and a transient analysis was performed as described
in Section 6.2.
The FE result for the retrofitted floor response due to walking excitation revealed
that the response was not significantly affected by the variation in the natural
frequency of the damper in the range of %5.2 (i.e. from 5.95 to 6.25 Hz) from
the optimum value (6.1 Hz) as shown in Figure 6-4. In this frequency range, the
dampers can achieve up to 90% of their ultimate performance. For variation of
natural frequency of damper in the range of %5 (i.e. from 5.8 to 6.4 Hz), the
damper can still reduce the floor vibration to an acceptable level but the damper
system loses about 23% of its effectiveness.
5 5.2 5.4 5.6 5.8 5.9 6 6.1 6.2 6.3 6.4 6.6 6.8 70.3
0.4
0.5
0.6
0.7
0.8
Damper Frequency (Hz)
Acc
eler
atio
n R
espo
nse
(% g
)
Optimum dampernatural frequencyof 6.1 Hz
Figure 6-4 Maximum acceleration response of the retrofitted office floor to variations in the
natural frequency of MTMD system using FE analysis
The above analysis was for the configuration that all dampers are tuned to the
same frequency. Another sensitivity analysis was undertaken to the variation of
damper frequency that each set of dampers has different frequency from the
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optimum frequency within the frequency band of 6 Hz to 6.5 Hz (i.e. -1.6% to
6.6% variation from the optimum frequency of 6.1 Hz). The adjustment of the
damper frequency and the properties of the floor and dampers were described in
this sub-section earlier. The retrofitted floor was subjected to walking excitation
and the maximum response for each case was obtained as detailed in Table 6-2. It
was found that the dampers for the cases of having different frequencies were
very effective in reducing the floor response compared to when all were tuned to
the optimum frequency as clearly seen in Table 6-2. This is due to the fact that
multi-modes of this particular floor are controlled and hence better damper
performance. There are about 4 modes as shown in Figure 5-22 which can be
controlled for the range of damper frequency of 6.0 Hz to 6.5 Hz. It can be
concluded from this analysis that for floors with the closely spaced modes, the
dampers should be tuned within a small frequency band around the optimum
frequency for better damper performance.
Table 6-2 Floor response due to the tuning of damper sets to different frequencies
CaseDamper Frequency (Hz)
Floor Response (%g)Set 1 Set 2 Set 3
1 6.10 6.10 6.10 0.40
2 6.00 6.20 6.40 0.28
3 6.10 6.30 6.50 0.33
4 6.05 6.40 6.00 0.29 Set 2 is the central set at the point of the maximum response.
Set 1 is located to the left of Set 2 and is at the distance of 1.2m along the critical path (See Figure 5-11
for directions and critical path).
Set 3 is located to the right of Set 2 and is at the distance of 1.2m along the critical path (See Figure 5-11
for directions and critical path).
Damper frequency (Hz) 6.00 6.05 6.10 6.20 6.30 6.40 6.50
Variation from optimum frequency of 6.1 Hz -1.6% -0.8% 0 1.6% 3.3% 4.9% 6.6%
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6.6. Performance of MTMD to Departure from Location ofMaximum ResponseThe departure of the dampers from the location of maximum response was
investigated using FE analysis. This departure from the location of maximum
response may decrease the effectiveness of the damper system in reducing the
level of vibration as the modal mass decreases for the same total MTMD system
mass. To investigate the sensitivity of the floor response to the damper location,
the three groups of identical dampers (which were described in Section 5.4) were
moved along the floor.
The MTMD system were distributed along the X and Y axes on a circumference
of a circle originated at the point of maximum response (approximately the centre
of the problematic bay) in a distributed form as illustrated in Figure 6-5. The three
sets of MTMD were relocated five times. The radius of the circle began at 1 m
and increased by 0.5 m up to 3 m from the point of maximum response. Modal
analysis for the first 10 modes was performed for each MTMD location to
determine the new natural frequencies of the problematic bay and the
corresponding mode shapes.
The floor with distributed MTMD system was subjected to walking excitation to
obtain the maximum acceleration response. In order to obtain the maximum
acceleration response for each location of MTMD, the FE transient analysis due to
walking excitation was repeated to match all relevant frequencies of the
problematic bay. The maximum acceleration response for each MTMD system
location is shown in Figure 6-6. The analyses revealed that the dampers remained
very effective in reducing the floor vibration for the circular area with 1.5 m
radius originated at the point of maximum response. Beyond this circular area, the
effectiveness of the dampers was decreased and when the radius of the circular
area exceeded 2.5 m the dampers could not reduce the floor acceleration to the
acceptable limit. In terms of span ratios, the MTMDs were very effective when
located up to 25% of the span from the point of maximum response, while the
MTMDs could not reduce the floor acceleration to the acceptable limit when they
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were located up to the 35% of the span from the point of maximum response.
Figure 6-5 MTMD system distributed apart from the point of maximum response
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30.4
0.5
0.6
0.7
0.8
0.9
1
Location of MTMDs from Point of Maximum Response (m)
Acc
eler
atio
n R
espo
nse
(m/s
2 )
Original floorFloor with MTMDs
Figure 6-6 Acceleration response of the 3% damping ratio office floor to different MTMD
system locations using FE analysis
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Modal analysis can be used to identify the best area of a floor to position the
distributed dampers. Since the shape of the problematic floor shown in Figure 6-7
was rectangular, the area of maximum response was taken as elliptical shape. The
contour plot of the first mode of the problematic bay is useful in determining the
most effective area for distribution of the MTMD system. This type of sensitivity
analysis is essential for the distributed damper system particularly when it is
difficult to install all dampers at the point of maximum response.
Figure 6-7 First mode shape of the problematic bay without damper
6.7. Concluding RemarksThe sensitivity of the retrofitted floor acceleration response to variations in the
floor damping ratio, damper damping ratio, damper natural frequency, damper
mass ratio and damper point of attachment was performed using FE analyses.
From the investigation of the specific floor, it was found that the addition of the
MTMD system could reduce the floor response to the acceptable level when the
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damping ratio of the floor was varied in the range of 1% to 5%. However, the
efficiency of dampers decreased for the situation when the floor had a high
damping ratio.
The FE analysis investigating variations in the damper damping ratio revealed that
the response of the retrofitted floor was not very sensitive to the damping ratio of
the damper. The damper system can reduce the floor acceleration response to the
acceptable level when the damping ratio of the damper was above 3% and the
MTMD system can reduce the maximum vibration by a factor of 2.
Based on the sensitivity analysis related to the mass ratio of the MTMD, it was
found that the floor acceleration response did not significantly change due to
variations in the mass ratio of the MTMD system. The MTMD system can reduce
the maximum acceleration response of the floor to 0.46% g when the mass ratio
was 0.75% while the maximum acceleration response of the floor was reduced to
0.4% g when the mass ratio was 1.6%.
The FE analysis for variations in the natural frequency for the individual dampers
of MTMD system revealed that the MTMD systems can achieve 90% of its
optimal performance when the variation in the damper frequency is in the range of
2.5% from the optimum value. When the variation in the natural frequency of
the individual dampers is in the range of 5%, the MTMD system can reduce the
floor acceleration response to the acceptable level but the MTMD system loses
about 23% of its effectiveness.
FE analyses were performed to investigate the sensitivity of the floor response to
the departure of the MTMD system from the location of maximum response. It
was found that the MTMD system remained very effective in reducing the floor
vibration when it was distributed within a span ratio of 25% from the point of
maximum response. Beyond the 35% of the span from the point of maximum
response, the MTMD system can not reduce the floor acceleration to below the
acceptable limit. On the other hand, the dampers can be more robust in reducing
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the floor vibration when damper sets are differently tuned within a small
frequency band around the optimum frequency to control several modes of the
floor or to account for variation in floor frequency over time.
The use of distributed MTMD system offers a significant advantage in reducing
the size and space required for a single TMD. Based on the sensitivity analyses
presented in this chapter, it is clear that the developed viscoelastic MTMD system
provides a robust solution in reducing floor vibrations. Such a MTMD system is
tolerant to some variations in the mass ratio of the system and damping ratios of
the floor and dampers. On the other hand, dampers remain efficient with some
changes in damper frequency and location of attachment to the floor.
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7. Conclusions
This thesis investigated the reduction of excessive floor vibrations in existing
office floors using a new viscoelastic damper. The new damper can be used as a
single tuned mass damper or in multiple damper configurations. The new damper
is simple in form, compact and cost efficient. Using this new damper, a significant
reduction of floor vibration can be achieved. This achievement has been
demonstrated using analytical models, experimental results and FE analyses.
The major findings and conclusions from this research are summarised below.
7.1. Floor Assessment and Rectification MethodsMost common methods to assess the dynamic performance of floor systems were
reviewed. The floor peak response due to human excitation depends largely on the
dynamic properties of the floor system. The designer can calculate, with a high
degree of accuracy, the natural frequency and mass of the floor using available
expressions or FE analysis. However, the damping has to be assumed based on
construction materials, fit-out of the floor and furnishing. The engineers may not
know the details of the fit-out or the furnishing during the structural design phase
and hence there could be a high degree of discrepancy between the estimated and
the actual damping values.
Commonly used remedial measures to reduce the levels of vibration were
reviewed. Increasing the floor stiffness can reduce human induced vibration
because it increases the natural frequency of the floor and hence shifts the
resonance to a higher walking harmonic, but is often unacceptable due to practical
limitations. Adding full-height partitions with the aim of increasing damping in
most cases is not possible due to architectural and functional requirements. Few
case studies of floor systems with excessive levels of vibration rectified by
increasing the damping using dampers have been reported. Most of the case
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studies were based on TMDs with viscous dampers applied to floor systems that
experiencing large displacements. While few concepts for other types of dampers
for office floors have been discussed in the literature review, there is little or no
information available on their design limitations and physical properties. Further,
there are no specific TMDs commercially available for floor applications.
7.2. Concept of the New DamperA new viscoelastic TMD based on a sandwich beam concept was developed. This
damper takes the form of a cantilever beam and can be attached to a vibrating
floor within the space of the false ceiling or false floor. The damper can be in the
form of a single damper, multiple dampers at one location or distributed dampers.
The damper can be easily constructed from commercially available materials.
With attachment of an end mass, the damper can be easily tuned to achieve the
optimum frequency, which is a key parameter of TMD performance.
7.3. Development of Analytical ModelAn analytical model based on an approximate solution of the equation of motion
of a sandwich beam was presented. This method allows the dynamic properties of
a sandwich beam (i.e. natural frequency and damping ratio) to be calculated based
on basic input of the damper material properties and geometric configuration.
The viscoelastic (rubber) layer represents the damping element of the new damper
and its dynamic properties can be determined using a DMA machine or by back
calculation using measured data obtained from a prototype testing if a DMA
testing machine is not available. Both methods were utilised in this research and
were presented in this thesis.
7.4. Validation of Analytical ModelTwo prototype dampers were developed to validate the analytical model and
determine its accuracy in calculating the natural frequency and damping ratio of
the proposed damper. Each damper was tested for the configurations of without
and with an end mass. The difference between the experimental and predicted
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values was in the range of 1% - 12% for the frequency, while it was in the range
of 2%-19% for the damping ratio.
The viscoelastic dampers were also modelled using FE to obtain the overall
damping ratio and the natural frequency of the viscoelastic dampers. It was found
that the values of damping ratios and natural frequencies obtained from the FE
models were in good agreement with the values obtained from the analytical
model and experimental tests. The difference between the analytical model and
FE values for the natural frequencies was in the range of 5% - 11% while it was in
the range of 1% - 12% with the experimental values. The difference between the
analytical model values and FE values for the damping ratios was in the range of
2% - 9% while it was in the range of 2% - 16% with the experimental values.
Given the reasonable level of accuracy of the predictions from the analytical
model combined with its simplicity, it is considered to be sufficient and reliable
for the design of the new viscoelastic damper. Therefore, this analytical model
was used for the remainder of this thesis to predict the damping ratio and natural
frequency of the various viscoelastic dampers developed throughout the research.
7.5. Application of Viscoelastic TMD on Simple BeamsTwo prototype dampers were used to reduce the vibration of two simple beams.
The effectiveness of each damper in reducing the level of vibration was
investigated using the analytical model, FE analysis and experimental tests. The
sensitivity of the retrofitted beams response to variations in the damper natural
frequency, damping ratio and mass was also investigated.
One of the prototype viscoelastic dampers was developed to retrofit a simply
supported steel beam with dimensions of 3000 mm long, 100 mm wide and 25 mm
thick. The natural frequency of the steel beam ( 1f ) was 6.3 Hz and its modal mass
( 1m ) was 29 kg. One other important aspect affecting the steel beam response was
its damping ratio. The damping ratio ( 1 ) of the steel beam was determined
experimentally and found to be 0.3%. In order to obtain the response of the bare
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beam, it was experimentally tested with a harmonic excitation using a rotating
unbalanced mechanical shaker. To obtain the maximum response of the steel
beam, the shaker frequency was tuned to match the fundamental frequency of the
steel beam. The maximum measured acceleration response of the steel beam due
to the shaker excitation was about 5.3 2s/m .
A prototype viscoelastic damper of 1% mass ratio was developed using the
approximate analytical model to retrofit the steel beam. A commercial rubber was
used in the development of the damper with measured shear modulus (G ) of
about 690 kPa and a dissipation loss factor ( ) of about 0.12. The viscoelastic
damper had a natural frequency ( 2f ) of 6.3 Hz, a modal mass ( 2m ) of 0.29 kg and
a damping ratio ( 2 ) of 5.4%. The dissipation loss factor of the rubber used in the
development of the damper was not sufficient to provide the optimum damping
ratio of 6% for the viscoelastic damper with the given thickness, width and length
of the rubber and constraining plates. A rubber with a higher dissipation loss
factor would be needed to increase the damping ratio of the damper.
The viscoelastic damper was attached at mid-span of the steel beam and tested
using the mechanical shaker. The maximum acceleration responses of the
retrofitted steel beam was about 0.46 2s/m . The response of the beam was
reduced by a factor of 11.5, which was in good agreement with the predicted
reduction factor of 12.5. The steel beam without and with damper was also
modelled using FE and the reduction factor in the beam response was about 11
which is in good agreement with experimental and analytical values. The overall
damping of the retrofitted system was found to be about 3%, which is a significant
increase from the original 0.3% damping.
Using the analytical and validated FE analyses, it was found that the response of
the retrofitted beam was sensitive to variations in the damping ratio of the steel
beam. The damper was very efficient when the damping ratio of the beam was
less than 1% whereas the response reduction factor was significantly reduced
when the damping ratio of the beam was above 1%. On the other hand, the
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reduction in the response of the steel beam was less sensitive to the damping ratio
of the damper. For 1% damper mass ratio, the damper performed well when the
damping ratio of the damper was in the range of 15% from the optimum value
of 6%. Within this range, the damper can maintain about 90% of its optimum
performance.
The sensitivity of the steel beam response to variations in the natural frequency of
the damper from the optimum value was investigated using the analytical model
and FE analysis as well as experimental tests. The change in the damper
frequency was achieved by changing the mass of the damper. The results of
sensitivity analyses obtained from the three methods revealed that a variation in
the damper frequency of up to 2.5% from the optimum value has little effect on
the efficiency of the viscoelastic damper.
The response reduction factor obtained from the FE analysis and experimental
tests was found to be insensitive to the variations in position of damper location
within the middle fifth of the span for the steel beam.
In order that the effectiveness of the new viscoelastic damper could be fully
assessed, a second prototype damper was developed to rectify a large scale
experimental concrete T beam. The T beam had a span of 9.5 m, a total weight of
6,000 kg and was simply supported at the ends. The mode shapes and the
corresponding frequencies were predicted using experimental modal analysis. The
value of the fundamental frequency was about 4.2 Hz. The corresponding modal
mass of this beam was estimated to be 3,000 kg.
The viscoelastic TMD with a mass ratio of 1% was developed to retrofit this T
beam. The developed damper had a mass ( 2m ) of 30 kg, frequency ( 2f ) of 4.2 Hz
and damping ratio ( 2 ) of 4.5%. The material properties of the rubber used in the
development of the damper were dissipation loss factor ( ) of 0.15 and shear
modulus (G ) of 637 kPa. It should be noted that the dissipation loss factor of this
rubber was not sufficient to provide the optimum damping ratio of 6% for the
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viscoelastic TMD with the given thickness, width and length of the rubber and
constraining layers. A rubber with a higher dissipation loss factor would be
needed to increase the damping (such rubbers can be sourced from specialist
suppliers but were not readily available). The T beam without and with damper
was excited using heel drop and the measured value of reduction in the T beam
response was found to be about 2.0, which was in very good agreement with the
predicted analytical and FE harmonic analysis values of 1.9.
The measured maximum acceleration in the time domain of the bare T beam for
24 time history records due to walking excitation was about 2.7% g whereas the
maximum acceleration response for the retrofitted T beam was about 2% g.
Therefore, the reduction in the acceleration response due to the walking excitation
was about 1.4. The maximum response acceleration using the FE model without
damper due to the walking excitation was about 3.7% g while the maximum
acceleration response with attached damper was about 2.3% g. This translates to a
reduction factor of 1.6 for walking excitation, which is again in very good
agreement with the experimental results.
The difference in the apparent efficiency (reduction factor) from the harmonic
excitation and walking is attributed to the fact that under walking excitation, the T
beam did not reach steady state response. Therefore, the damper did not reach its
maximum potential. This is simply because of the limited span of the specimen.
The sensitivity of the retrofitted beam response to variations in the natural
frequency of the damper was investigated using analytical and FE analyses as well
as experimental tests by modifying the end mass of the damper. It was found from
the three methods that the response of the T beam was sensitive to variations in
the natural frequency of the damper. To maintain 90% of the maximum reduction
factor for the system with the attached damper, the variation in the frequency of
the damper should not be greater than 2.5% of the optimum frequency of the
damper. On the other hand, to achieve 80% of the maximum reduction factor, the
variation in the frequency of the damper should not be more than 4% from the
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optimum frequency of the damper. These findings are consistent with the results
for the steel beam.
Based on the detailed laboratory experiments, analytical and FE investigations it
can be concluded that the developed damper can provide significant reductions in
floor vibrations. Furthermore, the analytical model developed for this viscoelastic
damper can predict the dynamic behaviour with a good degree of accuracy.
7.6. Application of New TMD in Multiple FormThe concept of multiple viscoelastic tuned mass dampers was presented in this
research to overcome possible shortcomings of a single large damper. An
analytical model was introduced to calculate the natural frequencies and steady
state response of a system consisting of a primary structure (floor) which is
retrofitted using multiple identical tuned mass dampers. The steady state response
of the primary structure with n identical TMDs would be similar to the case when
it is retrofitted by an equivalent single TMD.
Four individual dampers with a mass ratio of 0.25% each were developed for the
T beam to replace the 1% mass ratio single damper. The properties of the
developed damper were mass ( 2m ) of 8 kg, frequency ( 2f ) of 4.2 Hz and
damping ratio ( 2 ) of 4.8%. The dampers were set up in a crucifix form and
attached at the mid-span of the T beam, which was the same point of attachment
for the single damper. The T beam was excited using heel drop excitation to
measure the effectiveness of dampers in reducing the acceleration response. It was
found from several heel drop records that the average reduction factor in
acceleration response of the T beam was about 2.1, which is in very good
agreement with the analytical and FE results of 1.9.
In some cases, there may be limitations on the space available for placing a single
large damper or a set of smaller size dampers at the location of maximum
response of the floor. In these situations, the multiple tuned mass damper
(MTMD) system could be spatially distributed along the length of floor beams. To
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demonstrate and evaluate this arrangement, the multiple dampers described above
were evenly distributed along the length of the T beam, which was tested again.
The average reduction factor in the T beam response due to several heel drop
excitations was in the order of 1.8. The four spatially distributed dampers were
also investigated using FE harmonic analysis and the response reduction factor
was 1.8, which is in excellent agreement with the measured value. The floor was
also excited by walking and the average reduction factor obtained from several
walking records was about 1.4. This value was in good agreement with value of
response reduction for the floor with the single large damper. Again, the
difference in the damper performance from the harmonic and walking excitations
is attributed to the fact that under walking excitation on the T beam, it did not
reach steady state response because of the limited span of the specimen.
7.7. Application of Multiple TMD on a Real FloorA distributed MTMD system was developed and installed on a real office floor to
reduce the level of vibration due to walking excitation. The office floor is in a
highrise building located in Melbourne and comprises a 120 mm thick slab acting
compositely with steel beams. The problematic bay is 12.7 m long and 9 m wide
with 3 m secondary beam spacing. This floor bay has two long corridors, which
are perpendicular to each other. The walking paths are thus long enough for the
vibration energy to build up and walking to induce excessive vibrations to occur.
It was reported by occupants that they could feel the vibration at their
workstations when people walked along the adjacent corridors.
The preliminary numerical analysis estimated the natural frequency of the floor
system to be about 4.9 Hz and the peak acceleration response using AISC DG11
was about 0.44% g based on the assumption of 3% damping ratio, which is within
the acceptable range. The in-site measurement of the office floor vibration
revealed that the natural frequency of the floor was about 6.2 Hz with an average
damping ratio of about 3%.
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A large number of walking tests were undertaken in which the walking speed and
step length were adjusted in an attempt to closely match the third harmonic of the
step frequency with the floor natural frequency in order to achieve maximum floor
response. The maximum acceleration response of the floor due to walking
excitations was about 0.67% g with an average value of 0.52% g. This average
value of 0.52% g exceeded the acceptable peak acceleration (0.5% g) due to
walking excitation for office floors.
A distributed MTMD system was developed and attached to the office floor to
attenuate the excessive vibrations. One of the most challenging aspects of the
design requirements was that the dampers had to be installed within the limited
space of a false floor with a cavity height of 150 mm. Twelve dampers each with a
relatively small mass were developed instead of a single large damper to fit within
the false floor space. The 12 damper system had a total mass of 276 kg (22.5 kg
for each damper) and was arranged in 3 crucifix sets (with 4 dampers each) .
After the MTMD system was installed, the maximum acceleration level from
several walking tests was found to decrease to around 0.3%0.4% g, which is well
within the acceptable level of human comfort. It was also observed that the floor
felt less lively after installation of the dampers in both standing and sitting
positions.
A series of tests were also conducted using an electrodynamic shaker, which
excited the floor with a defined dynamic force and a frequency range covering the
resonant frequencies of the floor without and with dampers. There was a reduction
of 40% in peak floor response to the shaker excitation. The response was reduced
from 1.59% g for the original floor to 0.96% g for the floor with dampers.
The performance of this floor was also assessed using FE analysis. A model of the
floor was built and tuned using the natural frequency measured. The same model
was then subjected to simulated walking excitation without dampers and with
dampers. The peak acceleration of the office floor without dampers was found to
212
be 0.87% g. The acceleration response of the floor with attached distributed
dampers was about 0.4% g. The addition of the dampers reduced the acceleration
response of the floor to the acceptable level with a reduction factor of 2.0, which
was in a good agreement with the experimental value.
7.8. Sensitivity of Floor Response to variations in Floor andTMD PropertiesThe sensitivity of the retrofitted office floor acceleration response to variations in
floor damping ratio, damper damping ratio, damper natural frequency, damper
mass ratio and damper point of attachment was investigated using FE analyses.
For the office floor discussed above, it was found that the addition of the
distributed MTMD could reduce the floor response to an acceptable level when
the damping ratio of the floor varied in the range of 1% to 5% although the
efficiency of dampers decreased when the floor was assigned a high damping
ratio. For 1% floor damping, the response of the retrofitted floor was reduced
from 1.8% g to 0.49% g with a reduction factor in the floor response of 3.7. On
the other hand, for 5% floor damping, the response of the retrofitted floor was
reduced from 0.57% g to 0.36% g with a reduction factor in the floor response of
1.6. The analysis investigating the variation in the damper damping ratio revealed
that the response of the retrofitted floor was not sensitive to the damping ratio of
the damper. The damper system could reduce the floor acceleration response to
the acceptable level when the damping ratio of the damper was above 3%.
Based on the sensitivity analysis related to variation in the modal mass of the floor
and hence the resulting change in the mass ratio of the MTMD, it was found that
the floor acceleration response does not significantly change due to the variation
in the mass ratio of the MTMD system. When the mass ratio was 0.75%, the
MTMD system could reduce the maximum acceleration response of the floor to
0.46% g while the maximum acceleration response of the floor was reduced to
0.4% g when the mass ratio was 1.6%.
213
The analysis for variations in the natural frequency for the individual dampers of
MTMD system revealed that the MTMD systems could achieve 90% of its
optimal performance when the variation in the damper frequency in the range of
2.5% from the optimum value. For a 5% variation in the natural frequency of
the individual dampers, the MTMD system could reduce the floor acceleration
response to the acceptable level but the MTMD system loses about 23% of its
maximum effectiveness.
It was found that the MTMD system remained very effective in reducing the floor
vibration when all individual dampers were distributed within a span ratio of 25%
from the point of maximum response. When all individual dampers were located
beyond this span ratio, the effectiveness of the MTMD system was significantly
decreased and the system could not reduce the floor response to the acceptable
level when all dampers were positioned away from the point of maximum
response by a distance of 35% of the span length. The relevant mode shape of the
floor could be used to position the dampers in the most efficient locations.
The use of distributed MTMD system offers a significant advantage in reducing
the size of each damper and hence the space required for retrofitting. Based on the
sensitivity analyses presented in this research, it is clear that the developed
viscoelastic MTMD system provides a robust solution in reducing floor
vibrations. Such a MTMD system is tolerant to some variation in floor mass and
damping and it remains efficient with some changes in damper frequency,
damping and location of attachment to the floor.
7.9. Recommendations for Future WorkThe damper was developed to control one mode of vibration of floor systems. The
control of more than one mode using multiple damper systems requires further
study. This could be possibly achieved by tuning the several TMDs in a
distributed system to modes which require suppression.
214
The viscoelastic material was bonded to the constraining layers using a rubber
adhesive. The durability of adhesives and changes in the rubber properties with
time are required to be investigated. The rubber may harden with time and may
deviate from its original properties. A long-term creep testing under continuous
cyclic loading could be performed.
The natural frequency of the damper can be adjusted through the end mass to
achieve the optimum frequency. The tuning of the damper could be enhanced by a
mechanism that allows the end mass to be translated thereby altering the
cantilever stiffness and hence the damping frequency.
215
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