Tuning TMDs to Fix Floors in MDOF ShearBuildings

Jennifer M. RinkerHenri P. Gavin

Duke UniversityCivil and Environmental Engineering

February 11, 2013

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Outline

Background

Theory

Results

Conclusions

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Outline

Background

Theory

Results

Conclusions

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Enforcing nodes in harmonically driven continuous systems

Methods have been derived in the past to tune TMDs to enforce nodesin harmonically driven continuous systems (Cha and Rinker 2012)

I Node amplitude of vibration is zero

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Enforcing nodes in harmonically driven continuous systems

Methods have been derived in the past to tune TMDs to enforce nodesin harmonically driven continuous systems (Cha and Rinker 2012)

I Node amplitude of vibration is zero

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Application to discrete systems

Objective: given the NDOF shear buildingshown at right, choose the parameters of theTMD(s) to enforce nodes at desired floors

System characteristics:

I N degrees of freedom

I Nf external harmonic forces

I Na attachment locations

I Na nodes, i.e., fixed floors

I 3Na TMD parameters (mi , ci , ki )

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Discretized model of shear building

Model inter-story connections as springs and dampers

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Solution overview

Given

I a set of forcing locations (xf) and amplitudes (f)

I a forcing frequency ()

I a set of desired node locations (xn)

I a set of attachment locations (xa)

the TMDs may be tuned with the following procedure:

1. Solve for the required steady-state forces (fa) that the TMDs mustexert upon the system to create the nodes

2. Use these forces to determine the steady-state system behavior withattachments

3. Verify that the required forces can be delivered passively

4. Calculate the TMD parameters

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Outline

Background

Theory

Results

Conclusions

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Step 1: solving for fa

Consider an arbitrary discrete system

[M]x + [C ]x + [K ]x = F (1)

in which F contains harmonic components from both the external forcing(Fe) and the attached TMDs (Fa).

At steady-state, x(t) = xejt , Fe(t) = Feejt , and Fa(t) = Faejt , andthe system may be written(

2[M] + [K ] + j[C ])

x = Fe + Fa. (2)

Isolating x yieldsx = [Z ]1

(Fe + Fa

)(3)

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Step 1: solving for fa (cont.)

For convenience, define selection matrices of 0s and 1s such that

Fa = [Sa ]fa, 0 = [Sn]x. (4)

Multiplying Eq. 3 by [Sn] and using Eq. 4 yields

0 = [Sn][Z ]1 (Fe + [Sa ]fa) , (5)

which may be solved for fa:

fa = ([Sn][Z ]

1[Sa])1

[Sn][Z ]1Fe. (6)

When does Eq. 6 not have a solution?

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Step 1: solving for fa (cont.)

fa = ([Sn][Z ]

1[Sa])1

[Sn][Z ]1Fe. (7)

A solution exists if all of the following conditions are satisfied:

I [Sn] has as many rows as [Sa] has columns same number of nodesand attachments

I the columns of [Sa] are linearly independent TMD locations aredistinct

I the rows of [Sn] are distinct node locations are distinctI Na is not greater than N cannot have more TMDs/nodes than

degrees of freedom

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Steps 2 & 3: solving for the steady-state displacement andverifying passivity

Once fa has been determined, the steady-state deflection with attachmentscan be calculated from

x = [Z ]1(Fe + [Sa ]fa

). (8)

For a system to be passive, the required force must lag behind thedeflection at the attachment point by no more than , i.e.,

0 xi fa,i < (9)

for all TMDs.

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Step 4: choosing the TMD parameters

Two complex equations, 5 unknowns (mi , ci ,ki , and complex TMD displacement) prescribe one TMD parameter and solve forthe other unknowns (Cha and Rinker, 2012)

Prescribing the steady-state absolute amplitude of each TMD |zi |,

mi =|fa,i |2|zi |

(10)

ki =(a2i + b

2i aimi2)mi2

b2i + (ai mi2)2(11)

ci =bim2i 3

b2i + (ai mi2)2(12)

where ai = R(fa,i/xi ) and bi = I(fa,i/xi ).

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Outline

Background

Theory

Results

Conclusions

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Example: 3DOF, one node

Base excitation: xb = 10 cm, =20 rad/sSystem parameters:

Parameter Floor 1 Floor 2 Floor 3

Mi (kg) 1001 999 1000

Ki (kN/m) 450 441 467

Ci (kg/s) 54 63 79

Passive arrangements:

xn = 1 xn = 2 xn = 3

xa = 1 Y Y Yxa = 2 N Y Yxa = 3 N N Y

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Example: 3DOF, one node

Fixing the first floor: m = 375.0 kg, c = 0 kg/s, k = 150.0 kN/m

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3DOFanim.aviMedia File (video/avi)

Effects of mistuning

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Example: 5DOF, one node

Trangular loading: Femax = 100 kNSystem parameters:Mi = 1000 kg, Ci = 100 kg/s,Ki = 450 kN/m

Passive arrangements:

xn = 1 2 3 4 5

xa = 1 Y N N N Nxa = 2 Y Y Y N Nxa = 3 Y Y Y N Nxa = 4 N N N N Nxa = 5 N N N N N

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Example: 5DOF, one nodeFixing the first floor with TMD on third floor:m = 61.2 kg, c = 16.9 kg/s, k = 15.3 kN/m

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5DOFanim.aviMedia File (video/avi)

Outline

Background

Theory

Results

Conclusions

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Conclusions

I It is possible to tune TMDs to enforce nodes in discrete systemsthrough the following procedure:

I There is considerable freedom in choosing the TMD parameters tosuit experimental conditions

I If a system is banded and base excited, fixing a given floor will alsoisolate all higher floors

I The effects of mistuning can be examined in the case of unknownforcing frequencies

I It can be difficult to find a node-TMD arrangement that satisfiespassivity

I Possible extension to multiple-frequency excitation

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Questions?

Reference: Cha, P.D. and Rinker, J.M. Enforcing nodes to suppressvibration along a harmonically forced damped Euler-Bernoulli beam.Journal of Vibration and Acoustics. Volume 134, issue 5 (2012).

This material is based on work supported by the National ScienceFoundation Graduate Research Fellowship.

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BackgroundTheoryResultsConclusions