dynamic torsional modeling and analysis of a fluid mixer
TRANSCRIPT
Rochester Institute of TechnologyRIT Scholar Works
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8-1-2001
Dynamic torsional modeling and analysis of a fluidmixerJoel Berg
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Recommended CitationBerg, Joel, "Dynamic torsional modeling and analysis of a fluid mixer" (2001). Thesis. Rochester Institute of Technology. Accessedfrom
Dynamic Torsional Modeling andAnalysis of a Fluid Mixer
by
Joel S. Berg
A Thesis Submittedin
Partial Fulfillmentof the
Requirements for the
MASTER OF SCIENCEIn
Mechanical Engineering
Approved by:
Dr. Mark H. KempskiThesis Advisor
Dr. Josef TorokThesis Committee
Dr. Hany GhoneimThesis Committee
Dr. Edward HenselDepartment Head
DEPARTMENT OF MECHANICAL ENGINEERINGCOLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
August 2001
I Joel S. Berg hereby grant permission to the Wallace Memorial Library ofthe Rochester Institute of Technology to reproduce my thesis entitledDynamic Torsional Modeling and Analysis of a Fluid Mixer in whole orin part. Any reproduction will not be for commercial use or profit.
August 1, 2001
Joel S. Berg
Acknowledgements
I would like to thank the following people for their contribution and support
in this endeavor:
Dr. Mark Kempski for his patience and guidance as an instructor, an
advisor, and as a friend over the long road which has become this work;
My thesis committee, Dr. Josef Torok and Dr. Hany Ghoneim, for takingtime out of their schedules to review this thesis;
Craig Bahr for doing me a tremendous favor;
My'sponsors'
for providing the means and subject matter to undertake and
complete this endeavor;
All of the people that I have had the privilege to learn academic,
professional, and practical lessons from (there are far too many of you to
list here);
My children Rachel and Andrew for giving me a perspective they cannot
yet begin to realize;
Most of all I would like to thank my wife Kellie for her patience and
understanding in enduring several years of sacrificing some of our time
together so that I could pursue my MS Degree.
Abstract
Mixers and agitators are used in a variety of processing industries. Each
application has its own uniqueness requiring a high degree of customization in
process design and mechanical design. Many of the processing and mechanical
performance characteristics of mixers are derived from torque cell and tachometer
measurements usually located between the motor and speed reducer.
This thesis deals with the development of a dynamic modeling and analysis
procedure to simulate the torsional response of mixers. This procedure will allow
for the characterization of the torsional response at any point within the system, as
well as relate the response as observed at the measurement location on full scale
tests to any point of interest within the system.
Various modeling options were developed for each of the mixing subsystems
and compared to determine which configurations more accurately describe the
system torsional dynamics. The developed modeling options were simulated
usingSimulink
and MATLAB. For torsional frequency verification of the
simulation model, a finite element model was constructed, analyzed, and
compared to the simulation model. Also, the results of a full scale test were
obtained and compared to the simulation model. Recommendations for usage,
further study, and model development are also discussed.
TABLE OF CONTENTS
Abstract i
1. INTRODUCTION 1
1.1 Background 7
1.1.1 Mixing Overview 1
1.1.2 Impeller Overview 9
1.2 Reason for Thesis 10
1.3 Current Knowledge 11
1.3.1 Torsional Natural Frequencies 12
1.3.2 Load Fluctuations 13
1.4 Thesis Overview 14
2. METHODOLOGY 15
2.1 Modeling Overview 15
2.1.1 Basic Theory 16
2.2 Simulation Overview 20
2.2.1 Simulink Block Diagram Models 20
2.2.2 Model Simulation 22
2.2.3 Output Data Format 23
2.2.4 Simulation Example 23
2.3 Model Specifics 26
2.3.1 Three-Phase AC Induction Motor Model 27
2.3.2 Torque Cell Model and Flexible Coupling Model 41
2.3.3 Speed Reducer Model 44
2.3.4 Mixer Shaft Model 47
2.3.5 Impeller Modeling 48
2.4 Simulation Specifics 51
2.4.1 Motor Simulation Subsystem 54
2.4.2 Torque Cell/Flexible Coupling Simulation Subsystem 57
2.4.3 Speed Reducer Simulation Subsystem 60
2.4.4 Shaft Simulation Subsystem 62
2.4.5 Impeller Simulation Subsystem 63
2.4.6 Simulation Model Studies 67
2.4.6.1 Motor Modeling Study 70
2.4.6.2 Torque Cell Modeling Study 73
2.4.6.3 Speed Reducer Modeling Study 75
2.4.6.4 Shaft Discretization Study 76
2.4.6.5 Impeller Load Modeling Study 77
2.5 Model Verification 79
2.5.1 Torsional Frequency Analysis Using Finite Element Techniques 79
2.5.2 Full Scale Testing 81
3. RESULTS 84
3. 1 Simulation Model Studies 84
3.1.1 Motor Modeling 84
3.1.2 Torque Cell Modeling 90
3.1.3 Speed Reducer Modeling 96
3.1.4 Shaft Discretization Study 100
3.1.5 Impeller Load Modeling 104
3.2 Model Verification: 1 12
3.2.1 Finite Element Analysis Results 112
3.2.2 Full Scale Test Results 116
4. DISCUSSION OF RESULTS 128
4.1 Model Studies 128
4.1.1 Motor Modeling 128
4.1.2 Torque Cell & Flexible Coupling Modeling 131
4.1.3 Speed Reducer Modeling 133
4.1.4 Shaft Modeling 134
4.1.5 Impeller Load Modeling 135
4.2 Model Verification 137
4.2.1 FE Model 137
4.2.2 Full Scale Testing 139
5. CONCLUSIONS AND RECOMMENDATIONS 143
5.1 Conclusions 143
5.2 Usage Recommendations 146
5.3 Recommendations for Further Study 148
5.3.1 Refined Impeller Modeling 148
5.3.2 Refined Motor Subsystem Incorporating Electrical Subsystem 149
5.3.3 Lateral Subsystem Modeling and Analysis 150
5.3.4 Torsional Subsystem Interaction with Lateral Subsystem 150
5.3.5 Load Monitoring 151
5.3.6 Mixing Application Effect on Loading and System Dynamics 152
5.3.7 Non-linear Coupling Stiffness 152
5.3.8 Refined Speed Reducer Modeling 152
5.3.9 Modal Damping & Modal Resonance Study 153
6. APPENDIX A 154
6. 1 Test Equipment Specifications 154
6.2 ManufacturerData Sheets 155
6.2.1 Motor Performance Sheets 155
6.3 Equation Development: Damped Free Response (adapted from Ref 9) 157
7. APPENDIX B 158
7.1 Bond Graph Theory 158
7.2 Block Diagram Modeling 172
8. REFERENCES 175
IV
LIST OF TABLES
Table 1-1 Application Classes 3
Table 2-1 Full Scale Mixer Test Parameters 82
Table 3-1 Torsional Modal Frequencies: Motor Study 87
Table 3-2 Torsional Modal Frequencies: Torque Cell vs Flexible Coupling 91
Table 3-3 Torsional Modal Frequencies: Reducer Study 96
Table 3-4 Torsional Modal Frequencies: Shaft Study 100
Table 3-5 Torsional Modal Frequencies: FEA vs Simulation Model 113
Table 3-6 Torque Cell Measured Frequencies 122
Table 3-7 Simulated Torque Cell Measured Frequencies 126
Table 7-1 Energy and Power Variables 159
Table 7-2 Causal Forms 169
LIST OF FIGURES
Figure 1-1 Typical Mixer Arrangement 2
Figure 1-2 Swirl vs. Baffled Flow Conditions 6
Figure 2-1 Spring-Mass-Dashpot System 16
Figure 2-2 Example System Simulink Model 21
Figure 2-3a Example System: Velocity vs Time 24
Figure 2-3b Example System: Velocity Magnitude vs Frequency 25
Figure 2-4 Typical Mixer Arrangement 26
Figure 2-5 Word Graph of Typical Mixer Arrangement 27
Figure 2-6 Typical NEMA B Motor Performance Curve 29
Figure 2-7 Motor Performance Curve, Current and Torque 32
Figure 2-8 Motor Performance Curve for 1600-1 800 RPM Range 34
Figure 2-9 Thevenin Equivalent Torque Source 35
Figure 2-10 Approximated Motor Performance Curve 37
Figure 2-1 1 Motor Subsystem Bond Graph 38
Figure 2-12 Flexible Coupling Stiffness vs Torque 42
Figure 2-13 Torque Cell Bond Graph Model 43
Figure 2-14 Flexible Coupling Bond Graph Model 44
Figure 2-15 Double Reduction Speed Reducer Bond Graph Model 45
Figure 2-16 Single Reduction Speed Reducer Bond Graph Model 46
Figure 2-17 Lower Shaft Element Bond Graph Model 48
Figure 2-18 Resistive Impeller Load Bond Graph Model 49
Figure 2-19 Effort Source Impeller Load Bond Graph Model 49
Figure 2-20 Simulink Model of Typical Mixer Configuration 51
Figure 2-21 Simulink Model of System with Torque Cell and Two-Element Lower Shaft 52
Figure 2-22 Simulink Model of Motor Configuration One 54
Figure 2-23 Dual-Step and Ramp-Step System Input Torque 55
Figure 2-24 Simulink Model of System with Ramp-Step Input 55
Figure 2-25 Simulink Model of Motor Incorporating Speed-Torque Curve 56
Figure 2-26 Simulink Model of Flexible Coupling Configuration One 57
Figure 2-27 Simulink Model of Flexible Coupling Configuration Two 58
Figure 2-28 Simulink Model of Torque Cell Configuration One 58
Figure 2-29 Simulink Model of Torque Cell Configuration Two 59
Figure 2-30 Simulink Model of Speed Reducer Configuration One 60
Figure 2-31 Simulink Model of Speed Reducer Configuration Two 61
Figure 2-32 Simulink Model of Speed Reducer Configuration Three 61
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e 2-33 Simulink Model of Lower Shaft Element 62
e 2-34 Simulink Model of Resistive Impeller Load 63
e 2-35 Simulink Model of Effort Source Impeller Load 64
e 2-36 Simulink Model of Effort Source and Alternating Effort Impeller Load 65
e 2-37 Simulink Model of Resistive and Alternating Effort Impeller Load 66
e 2-38 System Model Version One: Simulink Model 68
e 2-39 System Model Version Two: Simulink Model 69
e 2-40 Motor Modeling Study: Simulink Models 72
e 2-41 Torque Cell Modeling Study: Simulink Models 74
e 2-42 Speed Reducer Modeling Study: Simulink Models 75
e 2-43 Shaft Discretization Modeling Study: Simulink Models 76
e 2-44 Impeller Load Modeling Study: Simulink Models 78
e 2-45 Finite Element Model Diagram 79
e 3-1 Motor No-Load Model Validation 85
e 3-2 Motor No-Load Torque vs Speed 86
e 3-3 Output Torque vs Time: Motor Study 87
e 3-4 Output Torque Frequency Response: Motor Study 88
e 3-5 Impeller Speed vs Time: Motor Study 88
e 3-6 Reducer Output Speed vs Time: Motor Study 89
e 3-7 Reducer Output Speed vs Time: Motor Study (Zoomed) 89
e 3-8 Output Torque vs Time: Torque Cell Study 91
e 3-9 Output Torque Frequency Response: Torque Cell Study 92
e 3-10 Cell Torque vs Time: Torque Cell Study 92
e 3-1 1 Cell Torque Frequency Response: Torque Cell Study 93
e 3-12 Impeller Speed vs Time: Torque Cell Study 93
e 3-13 Comparison of Tachometer Speed and Impeller Speed 94
e 3-14 Comparison of Cell Torque and Output Torque 94
e 3-15 Comparison of Cell Torque and Output Torque (Zoomed) 95
e 3-16 Output Torque vs Time: Reducer Study 97
e 3-17 Output Torque Frequency Response: Reducer Study 97
e 3-18 Cell Torque vs Time: Reducer Study 98
e 3-19 Cell Torque vs Time: Reducer Study (Zoomed) 98
e 3-20 Cell Torque Frequency Response: Torque Cell Study 99
e 3-21 Impeller Speed vs Time: Reducer Study 99
e 3-22 Output Torque vs Time: Shaft Study 101
e 3-23 Output Torque vs Time: Shaft Study (Zoomed) 101
e 3-24 Output Torque Frequency Response: Shaft Study 102
e 3-25 Cell Torque vs Time: Shaft Study 102
e 3-26 Cell Torque Frequency Response: Shaft Study 103
e 3-27 Impeller Speed vs Time: Shaft Study 103
e 3-28 Output Torque vs Time: Resistive vs Effort Model 105
e 3-29 Output Torque Frequency Response: Resistive vs Effort Model 106
e 3-30 Cell Torque vs Time: Resistive vs Effort Model 106
e 3-31 Cell Torque Frequency Response: Resistive vs Effort Model 107
e 3-32 Impeller Speed vs Time: Resistive vs Effort Model 107
e 3-33 Output Torque vs Time: Alternating Impeller Load 108
e 3-34 Cell Torque vs Time: Alternating Impeller Load 108
e 3-35 Cell Torque Frequency Response: Alternating Impeller Load 109
e 3-36 Impeller Speed vs Time: Alternating Impeller Load 109
e 3-37 Output Torque vs Time: Resistive Model with Alternating Effort 110
e 3-38 Cell Torque vs Time: Resistive Model with Alternating Effort 110
e 3-39 Impeller Speed vs Time: Resistive Model with Alternating Effort 11 1
e 3-40 Cell Torque Frequency Response: Simulation Model 11 3
e3-41a Torsional Mode Shapes: Modes 1,2 114
VI
Figure 3-41 b Torsional Mode Shapes: Modes 3, 4, 5 115
Figure 3-42 System Response at Torque Cell with 10 sec Ramp Up (Trial 1) 118
Figure 3-43 Steady State System Response at Torque Cell (Trial 2) 121
Figure 3-44 Torque Cell Power Spectrum 123
Figure 3-45 Tachometer Speed vs Time: Simulation Model 124
Figure 3-46 Tachometer Speed vs Time: 22 to 23 sec 124
Figure 3-47 Cell Torque vs Time: Simulation Model 125
Figure 3-48 Cell Torque vs Time: 22 to 23 sec 125
Figure 3-49 Simulation Model Power Spectrum: 0 to 500 Hz 126
Figure 3-50 Simulation Model Power Spectrum: 0 to 50 Hz 127
Figure 7-1 One-Port Component Elements 161
Figure 7-2 One-Port Source Elements 163
Figure 7-3 Transformer Element 164
Figure 7-4 Gyrator Element 164
Figure 7-5 Three-Port Junction Elements 165
Figure 7-6 Causal Strokes 166
Figure 7-7 Example System Bond Graph Development 171
Figure 7-8 One-Port Element Block Diagrams 171
Figure 7-9 Two-Port Element Block Diagrams 171
Figure 7-10 Three-Port Junction Block Diagrams 173
Figure 7-11 Example System Block Diagram 174
VII
1. INTRODUCTION
1.1 Background
1.1.1 Mixing Overview
GENERAL
Mixing is an important and integral part of the processes of various industries.
Chemical processing, food processing, waste treatment, pulp and paper, minerals
processing, and pharmaceuticals are just a few of the industries that employ
mixing or agitation processes. In many chemical processes, agitation serves a
key role in the overall success and quality of the end product(s). Typically,
agitation is achieved through the use of an impeller rotating in a fluid medium.
The fluid could be any combination of liquid, gas, or solid constituents to which
work is done to achieve the desired results. Power is delivered to the impeller
through a shaft which is driven by a prime mover. Usually the prime mover is a
motor (electric, hydraulic, or air) which is coupled to a speed reducer (torque
amplifier) which then drives the agitator shaft. A representation of a typical
arrangement can be seen in Figure 1-1 . Mixers generally consist of the same
mechanical components as pumps and compressors but are subjected to
significantly different design parameters and operating conditions. There are
various mounting configurations that can be employed: open tank, closed tank,
single unit per tank, multiple units, and floating platforms in lagoons. Mounting
locations can be top-entering, bottom entering, and side entering with the location
indicating where the shaft enters the tank. There are many mechanical design
parameters to consider for shaft and impeller configurations, as well as other
components, which greatly affect the system dynamics.
SpeedTorque
ReducerCell t
HMotor
i
Impeller
i
Mixer
Shaft
Figure 1-1 Typical Mixer Arrangement
PROCESS DESIGN PARAMETERS
Before an investigation into the mechanical behavior of mixers can be initiated,
there must first be a general understanding of mixer applications in the process
industry, and some of the parameters which go into the sizing and specification of
a mixer. For a much more complete description of the process related information
which will be discussed refer to [Ref 4] and [Ref 5]. The two basic processing
categories are physical processing and chemical processing with each having a
number of unique mixing process subsets. The most common process subsets
are blending fluids of differing viscosity, suspending solids in liquids, dispersing
gases or solids in liquids, mass transfer, and heat transfer. The general
application classes relating to both physical and chemical processes are listed
below in Table 1-1 [Refs 3,4]
Physical
Processing
Application
Class
Chemical
Processing
Suspension
Dispersion
Emulsification
BlendingPumping
Liquid-Solid
Liquid-Gas
Immiscible
Liquids
Miscible Liquids
Fluid Motion
DissolvingAbsorption
Extraction
Reactions
Heat Transfer
Table 1-1 Application Classes
There are a multitude of process dependent mixing parameters and variables
which are considered in the process design. The process designer must first
determine which fluid regime (laminar, turbulent or transitional) will be required for
the process. Once the required fluid regime has been established the proper flow,
fluid shear, and pressure head requirements must be determined. With fluid
regime and required fluid mechanics established, the proper impeller type can
then be selected to meet these requirements. Proper selection and sizing
depends heavily on basic turbomachinery dimensionless groups, with empirically
determined scaling and modifying factors. Mixers, pumps, compressors, and
turbines all follow some of the same basic laws with respect to power draw, flow,
and pressure head. Therefore performance curves based on specific speed,
pressure head, flow, and power similar to those used for pumps can also be
developed for mixer impellers. Some of these relationships with respect to mixers
will be presented below, as well as definitions and descriptions of the empirical
constants used.
IMPELLER DIMENSIONLESS GROUPS
The primary impeller parameters used in the development of dimensionless
groups are the rotational speed (N) and diameter (D). There are other important
impeller and process related modifying factors whose effects will also be
discussed. The dimensionless groups of concern for this discussion are pressure
head, flow and power. Other parameters such as Reynolds Number, Froude
Number, and Weber Number are very important in the process design but are not
of direct importance to the torsional system analysis for reasons which will be
made clear below. For some of the more advanced impeller designs, lift and drag
coefficients also play an important role in sizing and design (but will not be
considered herein).
The two relationships of importance from a process design standpoint to be
considered are pressure head (H), and volumetric flow rate (Q) per the following:
[Refs 1,7]
HocN2 D2
Eqn 1.1
QocND3
Eqn 1.2
These relationships are not directly important to the torsional system, however by
the following relationship to the power (P) their importance becomes clear:
PocQ-h Eqn 1.3
Substitution of 1 . 1 and 1 .2 into 1 .3 results in the following relationship:
PocN3 D5
Eqn 1.4
4
Therefore the power draw will be proportional to the cube of the rotational speed
and the fifth power of impeller diameter. Development of usable equations from
the proportionality equations has been achieved through years of testing and
characterization of parameters to the following generic form:
P =
pN3D5NM Eqn 1.5
where p is the density of the mixing medium and NM represents the impeller
mixing power characteristics number. In actuality NM is a combination of a many
factors dependent on the process conditions and the type of impeller used (shown
below in Eqn 1 .6). The three basic groups these modifying factors fall into are:
1 . The hydrodynamic behavior of the impeller type
2. Relative geometric parameters of the impeller and the tank
3. Fluid regime characteristics
The hydrodynamic behavior of an impeller type is established by operating at
different speeds and impeller diameters at well defined process conditions. This
is done to determine the power number (NP) of a particular type of impeller and is
used to compare power draw characteristics of different impeller types.
The geometric parameters which affect power draw (in addition to the
diameter) are relative parameters which describe the boundary conditions. The
impeller relative distance to tank components such as the tank walls, baffles, coils
and tank bottom, as well as relative distance to other impellers are characterized
in the Proximity Modifier (Mp). Typically, the Proximity Modifier is derived through
empirical relationships and tables proportional to impeller diameter.
Fluid regime characteristics which affect power draw are viscosity, swirl, and
the presence of gas. The Viscosity Modifier (Mv) is dependent upon the viscosity
of the fluid as well as flow type (i.e. laminar, turbulent or transitional). Therefore
relationships have been developed relating Mv to the Reynolds Number. A Swirl
Modifier (Ms) is applied due to the loss of power experienced when the fluid swirls
in a vessel. Swirl can be caused by insufficient (or non-existent) tank baffling, low
liquid coverage, or a high power to volume ratio. A visual comparison of swirl vs.
baffled conditions can be seen in Figure 1-2.
Swirling Baffled
Figure 1-2 Swirl vs. Baffled Flow Conditions
The Gas Modifier (MG) is applied when either gas is supplied to the system, or
gas is evolved (due to chemical reactions) within the system. The power loss due
to the presence of gas is a function of impeller parameters such as type, speed,
diameter, and location and of gas parameters such as flow rate, and type of
sparge.1
The modifying factors can be related back to the overall mixing power
characteristic in the following form:
NM =NP-MP-MV-MS-MG Eqn 1.6
However, as complicated as the derivation of some of the above factors can get,
once the process design has been performed they are all constant and along with
the density, can be collapsed into a single power factor (FP) and the power can be
calculated by the following modification to Eqn 1 .5:
P =FP-N3
Eqn 1.7
MECHANICAL DESIGN PARAMETERS
Once the power has been determined, sizing of the mechanical components
can be undertaken. Based on the calculated power requirement and desired
operating speed, the resulting torque requirement can be used to size the motor
and speed reducer and establish preliminary minimum requirements for shafting
and impeller blade thickness. The torque transmitted by the shaft to the
impeller(s) can be found by dividing the impeller power draw by the rotational
speed. Using the power relationship presented in Eqn 1 .7 the following equation
is developed:
T=-=FpN*'D
=
FpN2D5
Eqn 1.8N N
P
1A sparge is a device inside a vessel which delivers gas.
7
When D is constant, as it would be after process design is complete, the Fp
and rf terms can be combined into a single term and Eqn 1 .8 reduces to the
following:
T =RN2
Eqn 1.9
R is a proportionality constant relating the torque to the rotational speed for a
given set of mixing parameters and fixed impeller selection.
Along with torsional loading, mixers have to be designed to withstand
significant bending loads which are due to asymmetric fluid conditions at the
impeller(s) and long overhangs of shafting into the vessel. Mechanical sizing
procedures may vary slightly for each mixer vendor and the exact methods are
usually proprietary. In general, final sizing is determined by a combination of
stress analysis, deflection analysis, and shaft lateral natural frequency
determination.
8
1.1.2 Impeller Overview
There are various different types of mixing impellers, each with its own power,
pumping and shear characteristics. The two main classifications for impellers are
radial flow and axial flow (with radial and axial indicating the primary direction of
discharge from the impeller). Another variable important in understanding the
dynamics of mixers is the number of blades an impeller possesses. The blade
number not only affects the hydrodynamic performance of an impeller from power
draw and flow perspective, it also affects the dynamics of the mechanical system
in both the rotary and lateral reference frames. When observing the dynamic
response of rotational systems, torsional and lateral frequencies which lie near the
blade passing frequency can have a significant effect on force amplification of the
lateral and torsional sub-systems. Therefore the blade passing frequency
becomes the second most important system frequency after the shaft first lateral
frequency. Along with blade passing frequency, the impeller operating frequency
(rotational speed) is also an important system forcing function in both reference
frames. The forced response of a mixer is dominated by both of these
frequencies.
1.2 Reason for Thesis
As in most industries, the process industry is constantly seeking to improve
process results. The impact on mixer manufacturers is to push the envelope of
current mixer process and mechanical technologies. In many cases this forces
equipment design into one of two categories:
1 . trying to get more output from the same size equipment thereby increasing
the yield of the process.
2. trying to do the same job with smaller equipment to reduce initial capital
costs.
Historically mixers, like most mature products, have been developed by
general engineering methods as well as trial and error methods employing
empirical relations arrived at through years of testing and redesign. A greater
understanding of the dynamics involved is needed to develop mathematical
relationships that have a more complete physical meaning than some of the
empirical relationships used currently, while at the same time validate the
empirical relations. The purpose of this thesis is to develop a modeling
methodology which can be used as a design or analysis tool to simulate the
torsional system dynamics of a fluid mixer. The resulting modeling methodology
will allow for greater insight into the overall system dynamics of mixers and form a
solid basis for further study.
10
1.3 Current Knowledge
As discussed in Section 1.1, mixing impellers follow the same physical
relationships as all other groups of turbomachinery. One major difference with
mixing impellers versus other types of impellers and propellers are the boundary
conditions. Unlike the well defined, tightly controlled boundary conditions of
pumps and compressors, and the open, semi-infinite regime of marine propellers,
mixing impellers operate in a region that is somewhere in between. The varied
boundary conditions for mixing impellers results in a significant variation in power
draw characteristics when compared to pump or marine propeller applications.
Through laboratory testing, and full scale testing, parametric relationships have
been developed to account for the boundary conditions, and reasonably accurate
values for loading have been established. Using static equivalents of loading
values, many analysis methods can and have been employed to calculate
stresses, evaluate frequencies, and determine component and subsystem size
requirements. Many mixing system components can be sized based on the
knowledge of a few loads and operating parameters and by using guidelines
established by equipment suppliers or industry standards (such asAGMA2
ratings
for gear boxes). However in general, most analysis work performed is static and
linear in application.
2American Gear Manufacturers Association
11
1.3.1 Torsional Natural Frequencies
In its simplest mechanical form, a mixer is both a lateral spring and mass
system as well as a torsional spring and mass (rotational inertia) system.
Therefore, lateral and torsional natural frequencies have always been a design
concern. However, since the majority of mixers are designed with overhung shafts
subjected to high bending loads, stress and lateral natural frequencies (critical
speed) usually limit the design. Typically this results in a mixer design that is, by
comparison, torsionally stiff and has torsional frequencies that are far removed
from any potential system operating frequencies. Based on this, torsional critical
speed evaluations are rarely performed. Exceptions to this occur when mixing
systems possessing lower or intermediate journal bearings (steady bearings) are
used. The extra bearing(s) reduce the bending loads and add lateral stiffness
thus leading to designs that have"slender"
shafts with high L/D (length to
diameter) ratios. This design results in a much more compliant torsional system
with torsional frequencies that lie closer to operating frequencies. Another
exception is the recent trend in mixer design towards larger mixers with higher
solidity impellers which have higher rotary inertia thus lower torsional system
frequencies.3
Higher solidity in this sense indicates a large blade area with respect to impeller diameter.
12
1.3.2 Load Fluctuations
As discussed in Section 1.1 process conditions play an important role in
initially determining the proper sizing of an agitator. Aside from affecting the
expected power draw, process conditions can have a significant effect on the
lateral loading, impeller blade loading, and torsional load fluctuations an agitator
will experience. It has been observed and measured that alternating blade loads
increase with the severity of the mixing application. Along with the increase in
blade loading is an increase in the alternating component of the torque. Mixers
operating in fluids (particularly those in a defined space such as a tank) have fairly
noisy torque signals. However the two predominant excitation frequencies that
the dynamic components exhibit are the impeller lower shaft rotational speed and
the blade passing frequency (shaft speed times number of impeller blades).
Application severity can be categorized for various process types with each
category possessing its own assumed level of blade load fluctuations. In addition,
torque fluctuations are used in mechanical design procedures as a peak load in
static analyses or as load range in fatigue analyses. The assumed torque
variations are typical values based on measured fluctuations in laboratory tests as
well as field measurements.
13
1.4 Thesis Overview
There are five major sections in this thesis: Chapter 1 introduction, Chapter 2:
Methodology, Chapter 3: Results, Chapter 4: Discussion of Results, Chapter 5:
Conclusions and Recommendations, and there are also two Appendices. Chapter
1, while serving as a basic background and introduction to the topic, establishes
some of the key relationships between the process design of a mixer and the
mechanical subsystems, and defines the key parameters needed in the torsional
modeling and analysis. Chapter 2 discusses the development of the torsional
system model and simulation parameters. Chapter 3 presents the results of the
simulations as well as the test data results. Chapter 4 is the discussion related to
the results and a comparison of the simulation to the test data. Chapter 5
discusses the overall conclusions, recommendations for further model
refinements, and usage recommendations. Contained within the Appendices are
the test equipment data sheets, model component calculations, appropriate
manufacturer specification sheets for some of the components which were
modeled, and overview of bond graph theory and block diagram development.
14
2. METHODOLOGY
2. 1 Modeling Overview
To study the dynamics of real systems, a representation or model of that
system needs to be developed. In an engineering sense, a model can be a
scaled physical representation of the real system or a mathematical
representation which captures the time histories and relationships between
system variables. A"good"
model is one that will accurately predict the behavior
of a system to the satisfaction of the analyst without creating superfluous
information which unnecessarily overcomplicates the situation. For a
mathematical model this means including only those variables which are
necessary based on the design or problem at hand. The model can be a simple
or complex representation of the real system with the level of detail being
dependent upon many factors, therefore there must be some understanding of the
system behavior to know what the important variables are. For many complex
systems this can be a very difficult task. However, it may be possible to divide the
system into several subsystems and/or components which are easier to
understand and predict both physically and mathematically. Once the behavior of
the subsystems and components has been established, the interrelationships
between them can be explored leading to an understanding of the overall system.
Thus the system behavior or output(s) can be predicted for nearly any given input
or series of inputs to the system. This method by which a system is divided into
smaller parts for analysis can be referred to as a system modeling approach.
15
2.1.1 Basic Theory
For dynamic mechanical systems one of the most well known and simplest
models is the spring, mass, dashpot system per Figure 2-1 . This example system
has three components (the spring, the mass, and the dashpot) each with its own
behavioral properties which can be described by how that component handles
energy.
Mass
Spring
X
T77XDashpot
Figure 2-1 Spring-Mass-Dashpot System
The spring stores potential energy which in the simplest linear case is
proportional to the square of the displacement of the free-end. The mass (inertia)
stores energy kinetically and is proportional to the square of the rate of
displacement. The dashpot acts as an energy dissipater which (in the linear case)
is proportional to the square of the rate of displacement.
If the mass is given some initial displacement (causing the spring to be
displaced thus storing potential energy) and is then released, the system will then
try to return to its"free"
state. In doing so, the potential energy stored in the spring
16
will be transformed into kinetic energy in the moving mass. Simultaneously, the
dashpot will be dissipating some of the energy due to the velocity of the mass.
The problem then becomes a study of viscously damped free vibration for which
the dynamic behavior of the system can be easily predicted and understood
knowing the mass, spring rate, and damping coefficient of the system. The
differential equation of motion for this system can be represented in the following
form:
M'x + Cx + Kx = F(t) Eqn 2. 1
whereM is the mass, C is the damping coefficient of the dashpot, K is the spring
constant of the spring, F(t) is an applied forcing function to the mass and x is the
mass displacement. In the free vibration case F(t)=0which leads to assuming a
solution of the form:
x =est
Eqn 2.2
where e is the base of the natural log and s is a constant. The general solution to
this problem can be found to be:
x =A-eSi'
Eqn 2.3
where A and B are constants which are evaluated from initial conditions and s7
and s2 are the characteristic roots. In this example the initial conditions would be
the initial displacement of the spring, and the system at rest (initial velocity of
mass equal to 0). Development of Eqn 2.3 from Eqns 2. 1 and 2.2 can be found in
the Appendix. The resulting motion of the system is dependent upon the values of
17
M, C, and K and can be overdamped, underdamped (oscillatory) or critically
damped.
The simple linear translation system model can become a basis for a system in
the rotary reference frame. A simplified representation of the agitator system
would be the classical spring, mass, dashpot system with slight alteration to reflect
the rotational reference frame. The torsional spring would represent the torsional
compliance of the agitator shaft, the mass (rotary inertia) would be a lumped
parameter representing the shaft and impeller inertia, and the rotary dashpot
would represent the resistance and energy dissipated by the impeller rotating in
the fluid. The primary shortcoming of the simplified model is that when rotating in
a fluid, an impeller dissipates energy such that the power absorbed is proportional
to the square of the rotational speed (see Eqn 1.9)
instead of the linear power
relationship as is the case with the simple system presented earlier in this section.
Another problem with the simple torsional spring-mass-dashpot analog is that it
doesn't capture some key components of the real system. The motor and speed
reducer also play important roles in the overall dynamics of the system therefore a
more intricately defined system model needs to be employed.
As systems get more complex so do the mathematical relationships. If the
simple system is connected to two other similar systems, a free body diagram can
be constructed and equations of motion can be developed. However, adding
more systems starts to become overly complicated and analytical solutions start to
become laborious. A matrix approach can be used but that also gets to be too
complicated for hand solution. In the current thesis, a matrix system modeling
18
approach should be used for the torsional modeling of the agitator to capture the
physical behavior and interrelationships between system parameters. One
approach that works very well is the application of bond graph theory. Some of
the key concepts relating to bond graph theory as well as procedures for
developing a bond graph of a physical system are presented in Appendix B. The
methods for transforming a bond graph model into a block diagram model are also
presented in Appendix B.
19
2.2 Simulation Overview
As indicated in the previous section, a simple spring-mass-dashpot system has
a fairly straight forward closed form solution. However, the mathematical
complexity of such an evaluation increases significantly when more system
components are added or a more complete representation of the system is
required. In such cases numerical methods, finite element methods, or some
other form of computational methods need to be used. The speed, power, and
popularity of personal computers has resulted in a large number of software
packages for dynamic modeling and analysis of systems being available. The two
packages used in computer simulation and analysis of the system dynamics for
this thesis wereMATLAB
andSimulink
by Mathworks, Inc. A finite element
model of the system was constructed and analyzed in theAnsys
software
package to verify the system frequencies obtained from the model simulation.
2.2.1 Simulink Block Diagram Models
As discussed and demonstrated in Appendix B, a bond graph can be easily
converted into a block diagram model. The Simulink Toolset in Matlab is a
dynamic simulation tool which analyzes block diagram representations of systems.
For the simple spring-mass-dashpot system block diagram of Figure 7-1 1
(developed in Appendix B) subjected to a sinusoidal input force, the Simulink
model is as follows:
20
Spring-Mass-Dashpot System \
w,e 1
t
e2
IVW
1
Is
f2
Sine Waver-fc* -
Sum1
f3
e3 1
Cs
e4
^<
Capacitance Term
f4
^^
V1
Resistance Term
Figure 2-2 Example System Simulink Model
In this representation, integration is represented by the frequency (Laplacian)
domain representation Vs. The / and C terms in the denominators of the
integration blocks represent the values for inertia and capacitance respectively.
The R term in the amplifier gain block represents the resistance. Within Simulink
it is possible to "tapoff"
of any signal flow line in the diagram and output the
information to a scope (graph) or to the Matlab workspace for further post
processing. Simulink is not limited to the simple integrator blocks presented in
Figure 2-2. Both continuous and discrete system transfer functions can be
modeled in a single block to represent a subsystem.
21
2.2.2 Model Simulation
Once the system parameters have been identified, the state of the system
needs to be determined. The system can either start from rest or at some non
zero initial conditions for some or all of the state variables. The next step is to
choose a solver. There are several different solvers to chose from for fixed time-
step or variable time-step integration. In the case of numerically stiff systems,
such as the one being investigated here, the fixed time-step solvers cannot
capture the behavior of the high frequency transients within the system and often
will not converge to a solution. The ODE23s solver (which is based on a modified
Rosenbrock formula of order 2 [Ref 10]) was the solver of choice used in this
analysis. A complete description of the solvers can be found in the both the
Matlab and Simulink software help files or in the User Guides.
All of the system parameters (stiffness terms, inertia terms, resistance etc) can
be entered directly within the Simulink model. However it is much easier to make
all of the parameters variables that can be initialized within a script file. The script
file can also be used to set the Simulation parameters, retrieve system outputs,
and create the plots of the output data. All of the simulations performed for this
report were executed in this manner.
22
2.2.3 Output Data Format
All of the data presented within this report will be in the form of torque vs time
plots, rotational speed vs time plots, torque vs frequency plots, and tabulated
modal frequency values. The measurement locations of the torque and speed
response vary depending upon the focus of each study but typically include the
impeller torque and speed, lower shaft speed, torque cell (flexible coupling) torque
and motor torque.
The response magnitude vs frequency plots are based on using the Matlab
FFT function. The torque quantity of interest (whether it be output torque or
torque cell torque) is divided by the motor input signal and a Hanning window is
used to eliminate the transient effects at the beginning and end of the time
sequence. In addition to the frequency plots, a table containing the frequencies
derived from the graph is also presented.
2.2.4 Simulation Example
The following exercise will illustrate the method by which data will be presented
in the Results Section of this thesis. The example system (Figure 2-2) discussed
previously and developedthroughout this chapter and in Appendix B (Section 7) is
the system used. The values for inertia (mass), capacitance (1 /stiffness), and
resistance used in this example are 10, 20, and 2 respectively. The sinusoidal
input source has an amplitude of 10 and a frequency of 10 rad/sec. The
simulation was run with the ODE23 solver over a time span of 40.96 seconds.
The sampling period used was 0.01 sec and a 4096 point FFT was performed on
23
the input and output signals. Plots for the time response and frequency response
are given in Figure 2-3a and b. The transient response is evident in Figure 2-3a
as it slowly decays leaving the forced response at steady state. The only system
natural frequency as obtained from Figure 2-3b is approximately 0.22Hz. This
agrees with the hand calculation for the damped natural frequency of this2nd
order
system:
6)d=
K C 20
M AM 104-102
1.4107rad/sec = 02245Hz
where cod is the damped natural frequency, K is the stiffness, M is the mass, and C
is the damping coefficient.
Velocity of Mass: Step Input
V
e
I
o
c
i
t
y
Figure 2-3a Example System: Velocity vs Time
24
V
e
I
o
c
i
t
y
M
a
g
n
i
t
u
d
e
Frequency Response
1600
! ; | { | | | ! ! Ii
1400
1200
1000
800
600
400
200
0 \) 1 : ^ !"
^ ^ r ^ : !T
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Frequency, Hz
Figure 2-3b Example System: Velocity Magnitude vs Frequency
25
2.3 Model Specifics
In this section the overall system model will be developed as well as the base
components and subsystems. A typical arrangement for a mixer is illustrated in
Figure 2-4.
Torque
Cell
Parallel
Gear Set
Motor
Mixer
Shaft
Ll=Q=
3
Figure 2-4 Typical Mixer Arrangement
The system displayed in Figure 2-4 is referred to as a right angle mixer since the
configuration utilizes a right angle reducer. A slight rearrangement of the system
with all of the components arranged vertically is also presented in Figure 2-4. For
the latter case, each gear set has been lumped into a single component (keeping
in mind that there is still a reduction present) for a simpler representation.
Combining the gear sets and gear shafting into a speed reducer subsystem, the
following system word graph representation can be developed.
26
Input I *'1^ Motor I 1nl_^ Torque I
e
tc ^Speed I
e
srshaft I *SH fc> Impeller
1/
' / Cell i i Reducer ' / /r' i *
m tc*sr
7sh
Figure 2-5 Word Graph of Typical Mixer Arrangement
The identification and development of important physical characteristics of each of
the subsystems identified in Figure 2-5, as well as the modeling considerations for
each, will be the focus of this section.
2.3.1 Three-Phase AC Induction Motor Model
The rotation of the impeller in the mixing medium requires torque as discussed
in previous sections. The purpose of the motor is to deliver the required torque to
the agitator assembly at a nominal speed. Most mixing applications utilize a
three-phase AC induction motor to supply torque. Some applications use air
motors or hydraulic motors but usage is relatively infrequent. Due to their
overwhelming presence in industrial applications as compared with other types of
prime movers, only three-phase AC induction motors are being included in this
investigation. AC motors follow the same basic principles as all other types of
motors in that current is used to generate magnetic fields which then can be used
to produce torque. The motor has a magnetic rotor and a pair of poles for each
phase wound around an electromagnetic stator. When alternating three-phase
power is supplied to the stator it creates a rotating magnetic field. This in turn
causes the rotor to try and match the rotation of the magnetic field. When the
rotor rotation is perfectly synchronized (i.e. same speed) as the stator, this is
referred to as the synchronous speed of the motor and is dependent upon the
27
electrical frequency and the number of pairs of poles. For instance, if the
electrical frequency is 60Hz and the motor has two pairs of poles (4 poles), the
synchronous speed would be 1800rpm per the following:
60#zx60sec/min,_
,_
= 1 ZOOrpm Eqn 2.42pairs
From a torsional standpoint, the rotor inertia is the main physical parameter
which will be incorporated into the model. However, the torque vs speed
characteristics of the motor also need to be considered since their relationship to
one another determines what torque is available and at what speed. Figure 2-6 is
a characteristic representation of a typical performance curve for an AC induction
motor (the exact type is a NEMA design B). When this type of curve is supplied
by motor manufacturers, data is typically presented in relative terms with respect
to synchronous speed and full load torque. For example, in Figure 2-6, the
ordinate is the operating speed with respect to the synchronous speed and the
abscissa is the torque with respect to the motor's rated full load torque.
28
300%
250%
200%
3<r>-i
o
H
73B
O
"3
150%
100%
50%
0%
locked
Breakdown Torque
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
% Synchronous Speed
Figure 2-6 Typical NEMA B Motor Performance Curve
An important observation of this relationship (for this type of motor) is that
rated levels do not necessarily indicate a physical limit but an optimum operating
level due to electrical and thermal implications (which will not be discussed here).
It is very possible that operationabove the full load torque will occur at some point
in the operation of the motor. In fact it is inevitable in most cases as indicated by
the curve. If a motor is started "across theline"
(full voltage applied with motor at
rest) the motor willfollow this curve exactly from rest to full speed and produce
torque well over the full load value (and in this case it can overload to nearly
250%). Full speed will be whatever speed corresponds to the torque required by
the load. The relationship between torque and rotational speed for the motor does
29
not change with the physical characteristics of the load connected to the motor
(only where the motor operates on the curve). However, what does change is the
time required to reach the operational point on the curve. A higher load inertia will
result in a longer startup period as long as there is enough available torque
throughout the motor's acceleration from rest to operating speed.
There are several terms within Figure 2-6 that correspond to important AC
motor performance descriptors and their definitions are as follows:
Slip: The difference between the actual rotor speed and the synchronous speed
of the rotating magnetic field in the stator. It is an indirect measurement of the
torque since a higher load torque will cause more slip in the motor.
Locked Rotor Torque: Also referred to as the starting torque, it is the zero speed
torque developed by a motor, or put another way, the torque available at rest to
initially accelerate the system inertia.
Pull-up Torque: It is the minimum torque supplied by the motor between locked
rotor torque and breakdown torque. This torque must be greater than the
combination of the system rotary inertia and driven torsional loads or the motor will
stall (returning the motor to the locked rotor torque).
Breakdown Torque: The maximum momentary torque a motor can supply at
overload conditions. In a practical sense, extended operation at this point would
result in the motor "trippingout"
if equipped with the proper protection or
overheating and burning the windings if not protected.
30
Full load torque: This is the torque supplied by the motor at the rated power and
full load speed. For example, the full load torque of a 40hp motor with a full load
speed of 1775rpm would be:
A^-l r = USft-lbs Eqn 2.51115rpm x 5252hp /{ft lbs rpm)
Also within Figure-2-6 is the theoretical impeller load curve. It is the torque
load developed by the impeller (according to the relationship developed in
Section 1.1.1 and presented in Eqn 1.9)
from rest to full speed. Comparison of
the impeller load curve to the motor performance curve demonstrates the
suitability of this design of motor for mixing applications. There is a fairly high
starting torque available to initially accelerate the inertia loads. The torque then
drops off where the extra torque is not needed and increases again as the
impeller load increases.
One of the benefits of this type of motor from an operational standpoint is the
relative stability of the output speed. There is little degradation of motor speed
even at a breakdown torque of 250% of full load torque. The down side of this is
that small fluctuations in speed can represent very large fluctuations in torque
leading to speed measurement being a less than ideal means of measuring
torque.
In Section 2.1.3,
an electric motor was cited as a common example of a
gyrator. From a modeling standpoint, there are some key motor parameters which
must be considered, one of which is the motor torque constant, KM. The motor
torque constant defines the relationship between the torque and the current.
31
Another motor parameter, the motor generator constant Kg, relates the voltage to
the motor output speed. Both constants are equal to one another but are
represented differently based on the units of the different physical quantities
involved. For the motor being discussed, the relationships of current to speed,
torque to speed, and the relationship between torque and current (torque
constant) are charted in the Figure 2-7. The torque constant, KM, was derived by
dividing the torque by the current throughout the speed range.
Motor Performace Curves
0-1800 RPM
Q.
6
i.5 a
600 800 1000 1200 1400
MotorOutput Speed (RPM)
1600 1800
Figure 2-7 Motor Performance Curve, Current and Torque
For Figure 2-7, the y-axis represents both the torque (ft-lbs) and current (amps) at
the indicated speed and the2nd
y-axis represents torque vs current at the
32
indicated motor speed. It is apparent from the graph that a linear torque constant
would not be particularly accurate for the entire range of operation.
In the sizing procedure for an agitator, a relatively accurate estimate of the
operating power requirements is usually developed. Although unique
circumstances can occur and result in severe overload cases, mixers usually
operate well below the breakdown torque and are sized such that, even in the
case of severe load fluctuations, the peak load will be no more than 20 to 50%
over the motor full load rating. Due to this, the motor will operate the majority of
the time in a fairly narrow band about the design speed and torque. With this in
mind, attempts to model the"startup"
portion of the motor speed-torque curve will
be abandoned and the focus of the modeling efforts will be the operating range
from breakdown torque to no load (synchronous speed). Concentrating on just
this portion of the operating range results in the following modification to Figure 2-
7.
33
Motor Performace Curves
1700-1800 RPM
1600 1620 1640 1660 1680 1700 1720 1740 1760 1780 1800
Motor Output Speed (RPM)
Figure 2-8 Motor Performance Curve for 1600-1800 RPM Range
All of the axes and relationships of Figure 2-8 are the same as that of Figure 2-7.
Even in this operating range, which is the"linear"
portion of the speed torque
curve, KM can vary by a factor of 2.5 throughout. The zone where the majority of
the operation would take place is an even narrower band between 1740 and 1790
rpm. In this range KM is nearly constant at a value of 2.5 in-lb/amp so a linear
value would be fairly accurate for steady state operation in a well defined
operating range.
As Figures 2-7 and 2-8 indicate, the current for an AC motor is not constant
and cannot accurately be approximatedas such leading to the decision that a flow
source will not be considered as the system input. However the voltage input to a
motor can be considered to be relatively constant (460 volts in this case) leading
34
to an effort source being the appropriate choice for a driver of the system. If
voltage is used as the effort into the gyrator, rotational speed would be the forward
output. Since determining the torsional frequencies is of major importance, the
rotor inertia needs to be included in the model. From a modeling standpoint, flow
into an inductive element results in a derivative causality relationship. Subsequent
causal assignment results in derivative causality propagating throughout the
model. Since this would be undesirable, for reasons already discussed, another
approach was used.
In modeling DC motors an ideal source is combined with a motor resistance
value to closer approximate the behavior of a real source. In a simple case, the
resistance is constant and the resulting relationship between speed and torque is
linear with the slope of the line being the resistance (see Figure 2-9).
Torq
Thevenin 1
ue vs Speed
iquivalent Source
- \
Ol
3T-n
O
H
i*r~ "
Rm
Equivalent Source
Ideal Source
Motor Output Speed
Figure 2-9 Thevenin Equivalent Torque Source
35
This configuration is referred to as a Thevenin-Equivalent source. However, for
the AC motor this type of a representation is a very poor approximation of the
relationship between the torque and speed. A source is needed that remains
relatively constant for most of the speed range with a sharp drop-off beyond
1675rpm. A closer approximation would be to assume an exponential relationship
between speed and torque of the following form:
'"synch a
r r
T{co) =T0-T0ey T T)
Eqn 2.6
In this relationship T0 is an original starting torque, (Osynch is the synchronous
speed of the motor, co is the output speed of the rotor, and t is parameter
determined through a curve fit of the speed torque curve. The only portion of the
motor curve which was fit was the section from the breakdown torque to the
synchronous speed. A least squares fit was used and the results are presented in
Figure 2-10:
36
324T Motor, 40hp, 1775 rpm
Speed-Torque Relationship
Approximated Speed-
Torque Curve
Actual Speed-
Torque Curve
200 400 600 800 1000 1200 1400 1600
MotorOutput Speed (RPM)
Figure 2-10 Approximated Motor Performance Curve
The relationship was next modified to quantify the slip torque with respect to
speed per the following:
TSUP(Q))=T0e Eqn 2.7
Tslip(o)) serves as a pseudo-resistance which continually tracks and adjusts the
supply torque. The amount of slip torqueassociated with the current operating
speed is subtracted from the reference torque leaving the speed-torque curve as
presented in Figure 2-10 and per the following equation.
'
MOTOR~ *
REFERENCE*SLIP
Eqn 2.8
37
The impact this modeling approach will have on the response of the simulated
system will be to accelerate the system to operating speed at a faster rate than
would be exhibited by the actual speed-torque curve.
Incorporation of the slip vs torque relationship as a nonlinear resistance results
in the following bond graph model for the AC motor.
Impeller
1
Input!"' ^ Motor 1 'M|^ Torque
e
TC
I*TC
^ Speed 1 gSR^ shaft
I
Reducer / '
frSR
e
SH ^
I'i \ Cell
^SH
"^-^_
Im
k
"3 h
. Input
Slk| -
t|- m i ^. Torque
1il
l_ I i Cell"m 1
e3 hf
Rm
Figure 2-11 Motor Subsystem Bond Graph
In this bond graph"load"
represents the remainder of the system which the motor
will be driving, Se is the reference torque, RM is the resistive term representing slip,
and / is the rotor inertia. Effort is being imposed upon the 1 -junction by the
reference torque. Assigning integral causality to the rotor inertia term results in
flow being imposed upon the load. Conversely, the load is imposing an effort on
the motor which makes sense since the rotating impeller is trying to draw torque
from the motor to maintain the operating speed which is being imposed upon it.
38
Effects of Variable Frequency Inverter
It is fairly common that, based on the processing requirements of the
Customer, a mixer will need to be operated at more than one fixed output speed.
If just two operating speeds are required then a two-speed motor can be used.
This motor type has separate windings with different pole numbers to achieve
different output speeds (1800/1200, 1800/900, 1200/900 etc). However, when
more than two speeds are required or the optimal output speed for the process is
not known ahead of time, a variable speed drive will be required. The two major
groupings for variable speed drives are mechanical and electrical. Electrical
drives are much more common in the mixing industry and will be the focus herein.
The most common method of adjusting the output speed of an AC motor is to
vary the frequency of the electrical power supplied to it (leading to the name
variable frequency drives, VFDs). For example, if power is delivered at 60 Hz, a
VFD can control the speed of the driven equipment to half of its normal speed by
adjusting the frequency to 30hz. Similarly, if the frequency is increased to 90Hz
the speed of the driven equipment would increase by a factor of 1 .5.
There are some important characteristics of AC motor response to variable
frequency which need to be addressed. Most VFDs are assumed to operate in
the linear range of the motor speed-torque curve which is typically less than 150%
of full motor torque. In this range both torque and current follow nearly the same
curve resulting in the torque and slip varying linearly with the current. Since slip
varies linearly with the current, speed will not. If we assume constant torque in the
frequency range being observed, the relationship between slip and torque will be a
39
constant. There are other factors such as the available voltage which can effect
motor performance at frequencies beyond 60Hz (but will not be discussed here).
Therefore, if the frequency is halved the synchronous speed of the motor is halved
but the actual speed of the rotor is not. The difference in speeds is due to the slip.
For example: if the motor operates at 60Hz, the synchronous speed is 1800 rpm
and the full load speed is 1775 rpm resulting in 25 rpm of slip. If the frequency is
reduced to 30Hz, the synchronous speed is 900 rpm and the full load speed would
be 875 rpm since the slip remains constant at 25 rpm.
From a modeling standpoint provisions need to be made to insure the proper
speed-torque relationship is used if operating frequencies other than 60Hz are
used. The form of the approximated speed-torque curve developed in Eqn 2.8
already accounts for this in the region of interest. This holds since the slip torque
in the equation is based on the difference between synchronous and operating
speed. Therefore, by adjusting only the synchronous speed in the motor slip
parameter, variable frequency can be handled easily by the model.
The most common VFD types used in industrial mixing applications produce
pulse-width-modulated (PWM) signals. There are various electrical and thermal
implications associated with PWM drives. The thermal and most of the electrical
implications of VFDs will not be considered herein as they do not directly affect
the dynamics of the agitator system. One important electrical characteristic of a
PWM VFD to consider is the effect the switching frequency can have on the
motor. Some PWM VFDs can produce electrical spikes at the frequency of the
output signal and harmonics of the output signal. This electrical spike will be
40
transformed into a torsional spike by the motor. Depending on the torsional
frequency characteristics of the driven system, serious consequences can arise
such as mechanical resonance. This effect will not be incorporated into the
simulation but is an important point to consider in designing or analyzing a system
which utilizes variable frequency drives.
2.3.2 Torque Cell Model and Flexible Coupling Model
A torque cell is a torque measuring device that is inline with the system.
Typically, torque cells are not a normal offering on units built for Customers but
are used extensively on test units and dynamometers. When used, usually it is
placed either between the motor and speed reducer or between the speed
reducer output shaft and the in-tank shaft. Since most mixers utilize speed
reducers, high output torque values make it impractical and expensive to place the
torque cell on the low speed shaft assembly. Connection to the motor shaft and
the reducer input shaft is achieved through the use of two flexible couplings.
When a torque cell is not used, just one flexible coupling is used between the
motor and the speed reducer.
A flexible coupling is used due to the motor shaft and reducer input shaft
typically being mounted on separate bearings (this is not the case with gearmotors
which have the speed reducer pinion mounted to the motor output shaft). From a
torsional standpoint, a flexible coupling is nothing more than a torsional spring.
There is a large variety of different coupling types available and with many
different material options. The most prevalent type used in mixing applications is
41
the flexible grid type due to its high load capacity and misalignment allowances.
The torsional stiffness of the exact coupling type used for the analysis is
presented in Figure 2-12:
Flexible Grid CouplingTorsional Stiffness vs Torque
0.5 -
tl 0.4
a
*c 0.3
CouplingStiffness y
8
g0.2-
N /0.1 -
Design Breakdown
0 - 1 1 11 1 1
Torque
' ' 1
Torque
1 1 . 1 11 l
2000
Torque, in'lbs
Figure 2-12 Flexible Coupling Stiffness vs Torque
It is apparent from the figure that the stiffness of the coupling varies with load.
Considering the fact that the normal operating range of the coupling will be
somewhat centered about the design torque, a constant coupling stiffness is a
reasonable approximation.
The torque cell can have two effects on the system. First, it requires that an
additional flexible coupling be used in the system (one coupling to connect to the
motor shaft and another to connect to the reducer shaft). The second effect it has
is added flexibility due to its own spring rate. Therefore, instead of having one
spring term with a single flexible coupling, there are three spring terms with the
torque cell. As indicated, torque cells are usually only used on test equipment.
However, an important point to consider is that the speed, torque, and power
42
characteristics of an impeller type (or process) are usually determined on test
equipment which incorporates a torque cell. Therefore, the measurement device
may effect impeller characterizations by being an integral part of the system. A
modeling effort which incorporates the torque cell can be compared to a model
without the torque cell to determine whether the test data needs to be modified to
negate the"colorizing"
effect of having the measuring device inline with the
system.
The bond graph model for the torque cell is as follows:
Input
,1'
Motor| ( Torque i tc I ^ Speed
KCell I i Reducer < /
"tc
' 'SR
Shaft - Impeller
Motor
cp^l
(, (n
h
L t i
Speed
Reducer
Figure 2-13 Torque Cell Bond Graph Model
Kcpgi and Kcpg2 are the flexible coupling spring constants, Icpgl and Icpg2 are the
flexible coupling inertia values, KceU is the torque cell stiffness, and IceU is the
torque cell inertia. In the case where only a flexible coupling is being evaluated,
the torque cell terms and the second set of coupling terms vanishes leading to the
following bond graph:
43
Input
tl
,I
Motor \\Flexible i
'
fc | ^_ Speed ie
sr ^ Shaft Impeller
Motor Speed
Reducer
Figure 2-14 Flexible Coupling Bond Graph Model
2.3.3 Speed Reducer Model
A speed reducer (or gearbox) can come in several different configurations
with single or multiple gear reductions and a multitude of gear ratios. In mixing
applications, most heavy size units are double reduction. Triple reduction
gearboxes are also fairly common at low output speeds or for extremely large
units. The orientation is typically right-angle with the input shaft horizontal and the
output shaft down although parallel shaft reducers are also extensively used.
The main components of a speed reducer from a torsional standpoint are
rotating shafting (input, intermediate, & output) and gearing. The shafts act as
torsional springs and the gears each have rotary inertia. Also, each gear set acts
as a transformer which reduces speed thus increasing torque (by conservation of
energy). Based on bond graph theory, the rotational speed would be the flow
44
variable and the torque would be the effort variable. From a systems standpoint,
an important feature of this type of transformer is that inertia and stiffness values
get reflected through at the square of the gear ratio. For instance, a rotational
inertia value of 5 in*lb*sec at the input end of a 10:1 ratio speed reducer would
appear to be 500 in*lb*sec at the output end (5*102=500). Similarly, 5 in*lb/rad of
stiffness at the input would reflect to 500 in*lb/sec at the output. Therefore small
changes in stiffness and inertia values of all equipment at the high speed end can
have a more significant effect on system frequencies than changes at the low
speed end.
The bond graph model for the speed reducer (Figure 2-15) includes terms for
the gear ratios in the from of transducer elements, gearbox shafting stiffness
values, and gearing inertia values.
Input MotorTorque I |
e
tc ^.Speed i
e
sr j ^ haftCell '
, I Reducer ' / II * TC
*SR
|
Impeller
Torque
Cell
TF_
CRl t< I, tg
{7
TF
i;R2 i
1_J j iD*.
K
W Shaft
Figure 2-15 Double Reduction Speed Reducer Bond Graph Model
In this figure, Kin is the input shaft stiffness, GR1 is the gear ratio of the high speed
set, GR2 is the gear ratio of the low speed set, IHs is the rotary inertia of the high
45
speed set, ILS is the rotary inertia of the low speed set, KBp is the stiffness of the
low speed bevel pinion, Input is the input signal, and Load represents the
"downstream"
subsystems and components. A simpler representation of the
reducer would be to lump the inertia terms (properly reflected) into a single inertia
value, combine both gear ratios into a single transformer, and combine the spring
terms into a single value. The simplified system would be as follows:
Input Motor
Torque
Cell
Impeller
Shaft
Figure 2-16 Single Reduction Speed Reducer Bond Graph Model
Another potentially significant parameter to consider in modeling a speed
reducer is power transmission efficiency. Inefficiencies can affect the torsional
system by introducing additional energy dissipation components. Typically,
gearboxes of the type employed in the mixing industry introduce a decrease in
efficiency of roughly 2.5% per gear reduction. It is somewhat of a composite
value that includes gear friction, bearing losses, oil seal losses and churning
losses. Hence, using this relationship, a double reduction gearbox would have an
46
efficiency of 95%, or put another way, 5% of the energy delivered by the prime
mover would be dissipated in the gearbox as generated heat. Some types of
gearboxes, such as worm gear reducers, have significant losses due to gearbox
efficiencies. Since the expected energy dissipation of the gearbox type
considered in the modeling is not that significant as compared to the energy
delivered to the mixing medium by the impeller, its losses will not be included in
the model.
2.3.4 Mixer Shaft Model
The mixer shaft can be discretized as finely as desired. The level of
discretization is dependent on the number of vibration modes to be considered
and the complexity of the system. This requires some knowledge of the system in
question and can lead to iterations in the analysis model to capture all frequencies
of concern. Each torsional shaft section consists of a compliance (spring)
component, and an inertial component. In the model employed here (Figure 2-
17), the impeller rotational inertia was also lumped into the inertia term of the shaft
section preceding it.
47
Input
re
1"l
e
^ Torque I
Cell rt
TC
TC
^. Speed
ReducerM SR ^ Shaft
1 ^SR
SH '
tl
Motor'
tMi 1
r~
--
'shaft
1J
k
e5 /5
Speed
Reducer
,
KSR fc.
e
2 ^J , I SHlk. T
\ i SR
)
t^ 1 |
2
.
^Iiiipeller
1 *3 if
| "Shaft
1
Impeller
Figure 2-17 Lower Shaft Element Bond Graph Model
Kshaji is the stiffness of the shaft section and Ishaft is the inertia of the shaft section
and any lumped inertia at the end of the shaft section (such as an impeller).
2.3.5 Impeller Modeling
The most important component on an agitator is the impeller since it is the
component which performs the mixing and delivers the energy supplied by the
prime mover to the fluid. As discussed earlier, the selection of the proper impeller
to obtain the desired process results can, in some cases, be a very complicated
endeavor with many variables. Many of these complexities do not need to be
considered in this analysis since the configuration and application being explored
has been fixed. This allows for modeling the impeller as a torsional damper which
dissipates energy according tohydrodynamic relationships developed for
turbomachinery as stated in the development of Eqn. 1 .9. In the stated
relationship, the torque draw of the impeller does not vary linearly with the
48
rotational speed, it varies with the square of the speed. The bond graph model of
the impeller is presented in Figure 2-18.
\~e'
Input I *'h. Motor I "M
fc Torqe I 'tc ^Speed I
*sr ^ shaft
l . 'sh fc. Impeller|
I i Cel1 'i Reducer I /
'/ l
I frM <>
TC*SR |
'SH
'
^J ^J
LShaft | | SHfc.
o | -^ Rimp |I / SH t
2
I J
Figure 2-18 Resistive Impeller Load Bond Graph Model
Rimp represents the resistance function relating torque to rotational speed. The
impeller rotational inertia is not in the impeller model but instead has been lumped
into the shaft subsystem inertia term. Not included in the impeller inertia term are
any virtual mass effects due to entrained fluid.
An alternate form to the resistance model for impeller loading is to assume that
the torque draw of the impeller is an effort source (sink actually) imposed on the
system. This configuration is modeled in Figure 2-19.
_
!Torque
|Irc ^ JJpeed^ |_ *sr ^ shaft [._, ^ Impeller|
-J
i
Input |_^_^ Motor [> > | ^
R/ducer
| ^ ^att
rrjl\ *
Mi
TC*SR |
"SH I
Shaft \-\-s*_k> n I 2_fc. s. I
I t SH "2 I
l '
Figure 2-19 Effort Source Impeller Load Bond Graph Model
49
In the arrangement, Se has been deliberately modeled in a non-conventional
manner to stress the fact that the power is being absorbed at this location. The
advantage in modeling the impeller as an effort source is the ability to observe the
response of different impeller power draw scenarios such as alternating torsional
loads and impact loads. One complication of this model is ensuring energy
balance between source and sink efforts. Also, if an alternating sinusoidal effort is
employed, the frequency of the alternating component will be a constant in the
model whereas the"actual"
frequency of oscillation (shaft rotational speed and/or
blade passing frequency) varies.
50
2.4 Simulation Specifics
In this section, details of the Simulink model development will be discussed as
well as any assumptions or simplifications made in characterizing components.
Following the development of the subsystems and components will be a section
outlining the various analyses performed and reasons for each.
Several different system configurations will be discussed in this section. One
such configuration is shown in Figure 2-20.
motor
full
load
torque
3 phase AC motor
40hp, 324T
1800RPM
n
flex_couphng
input
Outputn~
1 7.4 Ratio
double reductionT imp
lower_shaft
lmp_1
Figure 2-20 Simulink Model of Typical Mixer Configuration
The system is driven by a step input to the motor (which represents a base torque)
and the motor output signal represents the rotational speed of the motor. The
motor is also subjected to a feedback torque from the next capacitive element in
the system (for this system that would be the flexible coupling compliance). The
torque cell/flexible coupling, speed reducer, and torsional shaft subsystems all
have a rotational speed input and output, and a torque input and output (there are
several instances presented later in this section where the torque cell and reducer
subsystems are modeled such that the input and output variables don't follow this
convention).
Organization of the diagrams is based upon the bond graph model developed
in Section 2.1. In Figure 2-20, the flexible coupling is indicated in the model
51
instead of the torque cell, however the two are interchangeable from a model
functionality standpoint. The subsystems in Figure 2-20 do not appear to be in the
same form as that of Figure 2-2. This is due to a Simulink feature called sub-
masking. It allows the user to group components into subsystems and
diagrammatically represent them in a more user friendly format. A more familiar
form for each of the subsystem diagrams will follow. The configuration being
presented in Figure 2-20 contains a step torque input and submodels for the
motor, flexible coupling, speed reducer, lower shaft, and impeller.
Variations of simulation model options will be presented for the subsystems as
well as for complete systems. One such variation includes an extra shaft section
and utilizes a torque cell (instead of the flexible coupling model) and is shown in
Figure 2-21 .
motor
full
load
torque
3 phase AC motor
40hp, 324T
1800RPM
torque cell
Lebow 1 1 05H-2K
input
Output
~~n~
speed
reducer
COsr K
?Wshl K COsM
cvWE ^w^\
jtlTCWX,
lmp_1
Figure 2-21 Simulink Model of System with Torque Cell and Two-Element
Lower Shaft
The interchangeability of components makes it easy to investigate different
configurations and modeling options and observe the effects it can have on overall
system dynamics.
The base parameters for the system being simulated is as follows:
Motor: 40hp, 1800rpm, NEMA Design B, 3-phase AC induction motor
Torque Cell: Eaton-Lebow, 1105H-2K
52
Flexible Couplings: Falk 1060T flexible grid type
Speed Reducer: 17.4 ratio double reduction right-angle drive
Lower Shaft: 3.5 inch diameter by 209 inches long 1020 steel
Impeller: 3-bladed high solidity impeller
A more complete description of the base component specifications can be found
in Appendix A.
53
2.4.1 Motor Simulation Subsystem
The performance characteristics of the motor and its response to variable
frequency drives were discussed in Section 2.3. Four different modeling
scenarios for the motor were considered in the analysis to determine the effect the
motor speed torque relationship has on system response.
Constant Torque Model: The first model (Figure 2-22) incorporates just the
inertia of the motor and assumes a step input equal to the motor full load torque.
The inputs to the subsystem are the reference torque (in this case motor full load)
and the feedback torque from the subsequent subsystem (flex coupling). The
single output of this submodel is the motor output speed.
3-PhaseACMotor
Subsystem
(DTref
0r>
1
Jm.s -CD
Figure 2-22 Simulink Model of Motor Configuration One
Dual Step Model and Ramp Step Model: Two variations of this scenario to
pursue are a) to assume a dual-step input and b) to assume a ramp-step input to
the motor. For both cases, the input torque initiates at a value equal to the
breakdown torque (or the starting torque could be used) and has a final value
equal to the full load torque of the motor within a specified time interval.
Representations for each of the two input options can be seen in Figure 2-23.
54
Input TorqueModeling Options
S
1-1
o
| !_4-- Starting Torque
1
/Ramp-StepModel-i-i iy-
. l j
V i /iii
\i/ Full Load
L J i j
i i
i i
-1-
r -\r-!- -
"'1/Torque
i r--
tntttttrrL
Dual-Step Model ' L j L _l l J ~-
Time
Figure 2-23 Dual-Step and Ramp-Step System Input Torque
The full system model incorporating the ramp-step input can be seen Figure 2-24:
Mixer Torsional System Model
40hp @ 102 RPM
Ramp
Rampl
->
->
3 phase AC motor
40hp, 324T
1800RPM
flex_couplmg
Input
Outpu1
17.4 Ratio
double reduction
lower_shaft
lmp_1
Figure 2-24 Simulink Model of System with Ramp-Step Input
This configuration is the same as that of Figure 2-20 except for the motor input
torque. The dual-step model is similar in form to Figure 2-24 with the exception
that the two ramp inputs are replaced with a single step input that results in an
input of the form of Figure 2-23 (dashed curve). The base torque in the diagram
represents the motor breakdown torque. One of the ramps has a negative slope
55
and initializes at the same instant as the step. The second ramp is used to "shut
off"
the first ramp at the desired time and torque level.
Speed-Torque Model: The third scenario is to incorporate the speed torque
curve as discussed in Section 2.3. 1 . The Simulink model for this arrangement is
as follows:
3-Phase ACMotor
Subsystem
CD-
f(u) <
Slip Torque
--KD
Figure 2-25 Simulink Model of Motor Incorporating Speed-Torque Curve
In this approach, the system is driven by the breakdown torque, which is
incorporated as a step input. The slip torque is then fed back (subtracted) from
the reference torque to simulate the speed-torque curve. The other input to the
subsystem is the feedback torque and motor output speed is the subsystem
output. In the "SlipTorque"
block/fa) represents the slip function developed in
Eqn 2.7. A no load analysis of just the motor subsystem driven by the reference
torque will be used to verify that the motor speed-torque characteristics are
incorporated properly. This does however include the resistance load attributed to
the slip torque which defines thespeed-torque curve.
56
2.4.2 Torque Cell/Flexible Coupling Simulation Subsystem
The torque cell and flexible coupling models are used interchangeably
throughout.
Flexible Coupling Model: In the simplest model of the flexible coupling, the
coupling inertia is either ignored or lumped elsewhere and only the coupling
stiffness is considered (Figure 26).
o-
w in
d>
Sum
Kcpg
Transfer Fen
-KDT out
w infb ->T fb
Figure 2-26 Simulink Model of Flexible Coupling Configuration One
The subsystem inputs are the motor rotational speed and the speed reducer
feedback speed. The outputs are both coupling torque: one line feeds into the
reducer while the other feeds back to the motor. To keep the energy variables in
the system causal, this coupling model requires that the next energy storage
component (in the reducer) be an inertance. This configuration will be discussed
further in the speed reducer simulation section.
A coupling model which includes coupling inertia is seen in Figure 2-27:
57
OfSum
T in
Kcpg
s
-KD
Transfer Fen
J*.
T fb
1
Jcpg.s
Sum1 Transfer Fcn1
-KDw out
Figure 2-27 Simulink Model of Flexible Coupling Configuration Two
In this configuration, the inputs are motor speed and speed reducer feedback
torque. The outputs are coupling rotational speed and coupling feedback torque.
Torque Cell Model: The torque cell model is very similar to the flexible coupling
model. From a torsional standpoint it can be viewed as three flexible couplings in
series, two of which are flexible couplings and the other would be the stiffness and
inertia of the torque cell. The Simulink model is as follows: (Figure 2-28)
Torque Cell Subsystem
with Flexible Couplings
w in I~
Kcpgl
-KD
Jcpgl .s
Sum4Transfer Fcn4
Transfer Fcn5
-*
T in
Kcpg2
Transfer Fcn1
J?
Transfer Fcn2
Sum3
Jcpg2.s
rT,3Transfer Fcn3
Figure 2-28 Simulink Model of Torque Cell Configuration One
In this representation, the subsystem inputs are motor rotational speed and speed
reducer feedback torque. The outputs are the rotational speed of the second
58
coupling and the feedback torque to the motor. An additional output (Tcell) has
been added to capture the torque in the torque cell and export it to the Matlab
workspace.
A variation of the torque cell model presented in Figure 2-28 is to lump the
torque cell and coupling inertia into a single value. This results in the same inputs
and outputs as was presented for the flexible coupling model of Figure 2-26. This
representation is as follows: (Figure 2-29)
Torque Cell Subsystem
zvith Flexible Couplings
o-r KCPQ1
-KD
Jcpgl .s
Sum4Transfer Fcn4
Transfer Fcn5
-KDTcell
d>Kcpg2
Transfer Fcn2
Transfer Fcn1
-KD
Figure 2-29 Simulink Model of Torque Cell Configuration Two
The arrangement depicted in Figure 2-29 would be used with a simple speed
reducer model to maintain integral causality within a system model.
59
2.4.3 Speed Reducer Simulation Subsystem
There are three speed reducer configurations to be considered within the
scope of the analyses to be performed. The simplest of the models is one that
includes a transformer for the total gear ratio and total inertia for the reducer per
Figure 2-30:
CD-
T in1
TFratio
gear ratio
Fratio
gear ratio
T infb
Jred.s
Sum1 reducer J
-KD
-KDw fb
Figure 2-30 Simulink Model of Speed Reducer Configuration One
Subsystem input values are the coupling torque and the lower shaft feedback
torque. The outputs are reducer output speed and reducer feedback speed
(scaled by the gear ratio).
A slight variation on the configuration of the model in Figure 2-30 is to include
a term for the stiffness of the reducer (which is predominated by the high speed
shaft stiffness value) as noted in Figure 2-31 .
60
CD-
Sumequiv K
Fratio
T in
gear ratio
Sum1 reducer Jw out
-KDT fb
Figure 2-31 Simulink Model of Speed Reducer Configuration Two
In this arrangement, inputs are flexible coupling speed and lower shaft feedback
torque while outputs are reducer output speed and reducer feedback torque. The
speed reducer model of Figure 2-31 can be further discretized to represent each
individual gear set as separate transformers per Figure 2-32.
(D-
w in I ^
Jhsgr.s
CD-
TFHS
HS ratio 2
Kbp
-KDT fb
Jlsgr.s
TFLS <
-KD
Figure 2-32 Simulink Model of Speed Reducer Configuration Three
Here, subsystem inputs and outputs are the same as that of Figure 2-31 . With
two transformers there will be three distinct sub-groupings to consider when
reflecting inertia and stiffness terms:
1 . High Speed/Low Torque
2. Intermediate Speed/Intermediate Torque
3. Low Speed/High Torque
61
The subsystem inertia terms have been grouped such that the inertia for the high
speed shaft and pinion have been reflected and lumped in with the high speed
gear. The inertia for the bevel pinion (intermediate) shaft and bevel pinion have
been reflected and lumped in with the low speed gear inertia. The low speed gear
inertia also includes the inertia for the reducer output shaft.
2.4.4 Shaft Simulation Subsystem
The shaft subsystem model includes a stiffness term and an inertia term. In all
of the analyses employed herein, impeller inertia is lumped into the shaft inertia
term. For any analysis that includes more than one shaft lump, the impeller inertia
is only lumped into the shaft term that the impeller connects to. The Simulink
subsystem model of the shaft is shown in Figure 2-33.
o>
Sum
Kshaft
-KD
Transfer Fen
d>T in
J Sum1
T fb
1
Jshaft.s
Transfer Fcn1
<D
Figure 2-33 Simulink Model of Lower Shaft Element
The inputs to the shaft element model are the rotational speed imparted on the
subsystem and the feed back torque of the impeller and/or torque from the
following subsystem. The outputs are shaftrotational speed and shaft feedback
torque.
62
2.4.5 Impeller Simulation Subsystem
Four distinct modeling scenarios were considered for the impeller load.
Impeller Model 1: The model corresponding to the equation development
presented in previous sections is the resistance model with impeller torque
proportional to the square of the rotational speed (see Figure 2-34).
Impeller Subsystem
Lumped Model
d> T
Product
Rimp-KD
T impR imp
Figure 2-34 Simulink Model of Resistive Impeller Load
In this arrangement the input is lower shaft speed and the output is the
corresponding impeller torque. The impeller model is used in conjunction with a
lower shaft model and the impeller torque feeds back to the shaft section.
Impeller Model 2: To determine the effect that the modeling approach of
Impeller Model 1 has on the system dynamics, a comparison can be made to a
simplified version of the impeller. The simple model would be to assume that the
impeller torque is an effort source (or sink) as discussed in the impeller modeling
specifics of Section 2.3.5. This can be accomplished by placing an effort source
in the overall system model at the lower shaft or by incorporating it into an impeller
model per Figure 2-35.
63
Impeller Subsystem
Constant Effort Model
(D HI P KDTerminator
StepT imp
Figure 2-35 Simulink Model of Effort Source Impeller Load
With regards to the system dynamics, this model is the same as using an effort
source. This convention was used solely for the purpose of continuity in the
system models. The input is lower shaft speed which ends in a terminator block.
In Simulink, a terminator block is used to avoid having unconnected system signal
lines. The output is the effort source which in this case is presented as a step
input.
Impeller Model 3: As discussed in previous sections, a significant amount of
alternating load is present in most mixing systems. The primary frequency content
of the alternating loads are output shaft rotational speed and impeller blade
passing frequency (recall that blade passing is the shaft rotational speed
multiplied by the number of impeller blades). The percent of torque fluctuation
can be between 1 0% and 50% with 35% being the most typical value for a
rigorous application. The value considered herein will be a 25% fluctuation at
shaft rotational speed and an additional 10% at the blade passing frequency for a
35% total fluctuation (as a percentage of the nominal full load torque). These
fluctuation percentages are approximate values based on observations. The
64
fluctuation can be incorporated into the modeling as a pair of sinusoids which are
added to the nominal torque per Figure 2-36.
Impeller Subsystem
AlternatingEffort Model
(D ?! Step
w in Terminator
P^Sine Wave
FV
Sum
-KDT imp
Sine Wavel
Figure 2-36 Simulink Model of Effort Source and Alternating Effort Impeller
Load
This arrangement is similar to the constant effort load model with the exception of
the alternating component. In this modeling effort, it was assumed that the two
sinusoidal signals initiate at a 60deg phase angle to one another (at the midpoint
of adjacent blades spaced 120deg apart).
Impeller Model 4: An alternate approach to that of Figure 2-36 is to incorporate
the alternating components as discussed with the2nd
order resistive impeller load
instead of the constant effort model. This configuration can be seen in Figure 2-
37.
65
Impeller Subsystem
Resistancewith Alternating Effort Model
(^^*VJ w
Pw in w
Product
Rimp
R imp
aSine Wave
P^Sum
->T imp
Sine Wavel
Figure 2-37 Simulink Model of Resistive and Alternating Effort Impeller Load
All four of the impeller models presented have the shaft rotational velocity as an
input and the impeller torque as the output.
66
2.4.6 Simulation Model Studies
The simulation model studies are intended to form a basis for determining the
effect different modeling options can have on the simulated system response and
the system sensitivity to some of the model parameters. The model studies being
considered are as follows:
1 . Motor Modeling Study
2. Torque Cell Modeling Study
3. Speed Reducer Modeling Study
4. Shaft Discretization Study
5. Impeller Load Modeling Study
For each study, the different parameters investigated will be compared to one
another using the time domain and frequency domain response plots of torque
and rotational velocity at several different locations throughout the system model.
The frequency response plots are obtained through the use of the Matlab FFT
function. There are two base model configurations which are used throughout the
model studies. The first is System Model Version One (see Figure 2-38) which is
used for the Motor Modeling Study and Torque Cell Modeling Study. All
components (except for the specific component which is the focus of each study)
remains unchanged. Due to causal considerations, a different system model was
used for the Speed Reducer, Shaft Discretization, and Impeller Load Modeling
Studies (see System Model Version Two, Figure 2-39).
67
System Model Version 1
motor
full
load
torque
3 phase AC motor
40hp, 324T
1800RPM
flex_coupling
Motor:
Speed-Torque Model
o>
Flexible Coupling
(D-
T,c
-
(D-
f(u)
Slip Torque
input
Output
*-* I["^
lower_shaft
seed reducer
I l_OA*lOsrtb Timp VV
Kcpg
lmp_1
-KD
-KD
Speed Reducer: Qff
Single-Reduction *
o
Lower Shaft:
Single Lump
TFratio
gear ratio
gear ratio
Tsh
T
Sum!Transfer Fcn1
-KD
Impeller:
2nd Order Resistance Model
KD O fm sh CO sh
**" Rimp
R imp
^D OT imp T imp
Figure 2-38 System Model Version One: Simulink Model
68
System Model Version 2
motor
full
load
lorque
3 phase AC motor
40hp, 324T
1800RPM
torque cell
Lobow1105H-2K
input
Oulpul
Motor:
Speed-Torgue Model
mxSH'' i y lower_shafi
17.4 Ratio |double reduction /\f\~"~~~
~~
T imp j")
lmp_1
o
f(ui 4
Slip Torque
Torgue Cell
Tic
T,c
-KD
O-HE>Kcpg!
Jcpgl.s
Transfer Fcn4
~KDTcell
measurement
-
Kcpg2
Sum3
Jcpg2.s
Sum3 Transfer Fcn3
Speed Reducer:
Double-Reduction
O-H
Single Lump
03dr r^
Impeller:
2nd Order Resistance Model
Transfer Fcnl
I KD 0 [ Rimp
R imp
KD OT imp T imp
<d
Figure 2-39 System Model Version Two: Simulink Model
69
2.4.6.1 Motor Modeling Study
The first motor analysis to be performed is the validation of the Simulink model
incorporating the resistance function to obtain the proper speed-torque
relationship. This is accomplished by running the simulation of the motor
subsystem only with a reference input equal to the breakdown torque (300 ft-lbs)
as discussed in Sections 2.3.1 and 2.4.1 . With no load applied to the model, the
steady state motor output speed should reach the synchronous speed which in
this case is 1800 rpm. The slip torque should start at 300 ft-lbs and reach a
steady state value of 0 ft-lbs.
Once the model had been validated, simulations were run to determine the
significance of the speed-torque curve to the system model through comparison
with the constant torque model, and ramp-step model. A complete representation
of the system model, subsystem components, and motor model options used can
be seen in Figures 2-38 and 2-40. The subsystem models used to create the
complete system model (System Model Version One) were the flexible coupling of
Figure 2-26, the speed reducer of Figure 2-30, a single element lower shaft, and
the impeller of Figure 2-34. For the constant effort model, the motor full load
torque (1 18 ft-lbs) was applied to the motor model of Figure 2-22. The dual-step
and ramp-step models (see Figure 2-24) used the same subsystem models as
that of the constant effort model and had input signals as presented in Figure 2-
23. The starting torque was 300 ft-lbs and thefull load torque was 118 ft-lbs. The
decline time was set to 0.125 sec for the dual-step and 0.25 sec for the ramp-step.
The fourth configuration incorporating the motor curve had an input signal of 300
70
ft-lbs and had the same subsystems as the previous two configurations. Initially,
calculation of the dual-step decline time was based on the rise time of a step
response of a2nd
order system (as found in [Ref 1]). The relationship developed
is based on the rise time to reach 90% of the steady state value and is equivalent
to 1 .8 divided by the natural frequency of the primary vibration mode. In this case
(as will be shown in the Results Section) the first natural frequency of the system
was found to be 19.5 Hz (3.1 rad sec) resulting in an expected rise time of 0.58
sec. Subsequent comparison to the results of the speed-torque model indicated
that a value of 0.125 sec should be used instead. The decline time for the ramp
step model was based on doubling the decline time of the dual step model to
result in the same total angular momentum being imparted on the system.
71
Motor Model Study Utilizing System Model Version 1
motor
full
load
torque
3 phase AC motor I
40hp, 324T'
1800RPM I
flex_coupling
Input
Output
speed reducer
lower_shaft
lmp_l
L
Motor Model Configuration One:
Constant Effort Model
motor
full
load
torque
OTin
Motor Model Configuration Two:
Dual-Step Model
base torquel
(D-
Tin
"
Tfc
Motor Model Configuration Three:
Ramp-Step Model
ehbase torque
o-^
T in
(D-
Tfc
Motor Model Configuration Four:
Speed-Torque Model
TiT"*(D-
motor
lull
load
torque
f(u)
Slip Torque
(f>
1
Jm.s
-KD
-KD
-KD
-KD
Figure 2-40 Motor Modeling Study: Simulink Models
72
2.4.6.2 Torque Cell Modeling Study
The purpose of the torque cell study is to determine the effect the additional
stiffness (flexibility) terms of the cell and extra flexible coupling have on the
system as compared with a system with a single flexible coupling. Also of
importance is determining the differences between the torque being drawn by the
impeller and that being measured in the torque cell. The system models (System
Model Version One) for both the flexible coupling configuration and the torque cell
configuration included the motor with speed-torque curve, speed reducer of Figure
2-30, a single element lower shaft, and the impeller of Figure 2-34. The flexible
coupling model used was that of Figure 2-26 ,and the torque cell model used was
that of Figure 2-29. The torque cell and flexible coupling model options can be
seen in Figure 2-41.
73
Torque Cell Model Study Utilizing System Model Version 1
motor
full
load
torque
3 phase AC motof
40hp, 324T
1800RPM
flex_coupling/torque cell
input
Output
speed reducer
03 srfc
K
mm.lower_shafl
TlfBp X.+-
lmp_1
Flexible Coupling Model Configuration One:
Tfc
(D-
-
Kcpg
<D
Torgue Cell Model Configuration Two
(L)T,c
Dm *H I
Kcpgl
Jcpgl .s
Sum4Transfer Fcn4
->
oisrtb
Kcpg2
Transfer Fcn2
Transfer Fcn1
XD
Figure 2-41 Torque Cell Modeling Study: Simulink Models
74
2.4.6.3 Speed Reducer Modeling Study
For the speed reducer modeling study, a simple single reduction model was
compared to a double reduction, more highly refined model. This was done to
determine the effect a coarse model has on simulation results and whether a more
refined model is needed. The system model (System Model Version Two)
included the motor with speed-torque curve, torque cell configuration one, a single
lower shaft element, and the resistive impeller load model. The speed reducer
models used in the comparison were that of Figure 2-31 and Figure 2-32. A
comparison of the modeling options can be seen in Figure 2-42.
Speed Reducer Model Study Utilizing System Model Version 2
LB-
3 phase ACmotor
40hp, 324T
1800HPM
Lfc. cjInpul fiia"
^"
| 1 oWu,| Td, | ^0pr* t>
| i[*
tow.,_SM
"
.74 (Moj
__. L-doutJa laduclioa.
__| J^J /\-f\, __
T Imp"
*CcS^'^
Speed Reducer Model Configuration Two:
Single-Reduction Model
o-tf
Speed Reducer Configuration Three:
Double-Reduction Model
o-tf
Figure 2-42 Speed Reducer Modeling Study: Simulink Models
75
2-4.6-4 Shaft Discretization Study
The purpose of the shaft discretization study is to examine the differences in
system response when the shaft is treated as a single lump or multiple lumps and
determine if a single lump model is sufficient. For the multiple-lump models, the
shaft length per section was divided into equal lengths. The system configuration
used was the same as that of the speed reducer study, System Model Version
Two. The lower shaft options investigated were one-lump, two-lump, and four-
lump models as represented in Figure 2-43.
Shaft Model Study Utilizing System Model Version 2
load
torque
3 phase AC motor
40hp, 324T
1B00RPM
torque cell
Lebow1105H-2K
inpul
OutputT* IWH
.
!
17 4 Ratio
doujjla.rertlction
ip(_)*
\
"^
Shaft Model 1 :
1 -Lump Shaft
Shaft Model 3:
4-Lump Shaft
Ts
Otf
Otf
Transfer Fcn1
-KD
<D
UfSum6
Transfer Fcn6
Transfer Fcnl
Shaft Model 2:
2-Lump Shaft
('FVfb
u
win r^
Sum7Transfer Fcn7
Transfer Fcn1
u
Transfer Fcrt9
rKDJshaft s ^^
w out
<D
Sum5Transfer FcnS
rKDJshaft s V-7
w out
<D
Figure 2-43 Shaft Discretization Modeling Study: Simulink Models
76
2.4.6.5 Impeller Load Modeling Study
The purpose of the impeller load modeling study is to examine several different
impeller load modeling configurations and compare the results of each. The load
configurations investigated were the2nd
order resistance model and the constant
effort (load) model. Also investigated was the influence fluctuating impeller load
has on the system for a2nd
order resistance with alternating load, and a constant
effort model with alternating load. The system configuration for the remainder of
the subsystems were based on System Model Version Two. The different
impeller load modeling options explored are represented in Figure 2-44. For the
constant effort loading, the impeller load was modeled as a step input equal to the
full load motor torque scaled by the total gear ratio of the reducer. The fluctuating
torque component for both the2nd
order resistance and constant effort model
were two sinusoidal efforts. In keeping with the loading described for Impeller
Model 3 (see Section 2.4.5), one sinusoid had an amplitude equal to 25% of the
nominal impeller torque and a frequency of oscillation equal to the nominal
impeller rotational speed. The second sinusoid had an amplitude equal to 10% of
the nominal impeller torque and a frequency of oscillation equal to the blade
passing frequency (nominal impeller speed multiplied by the number of blades, 3).
The total torque fluctuation is a combination of both amplitudes which results in
the nominal torque +/-35%.
77
Impeller Model Study Utilizing System Model Version 2
motor
full
load
torque
com r
3 phase AC motor
40hp, 324T
1800RPM
torque cell
Lebow1105H-2K
input
Output
17.4 Ratio
double reduction
T impl
lower_shaft
'4| lmp_1 |
Impeller Model 1:
2nd Order Resistance Model
Impeller Model 3:
Alternating Effort Model
o-
cosh
? "3
Step
H-lSine Wave
N
Sum
T imp
Sine Wave 1
Impeller Model 2:
Constant Effort Model
o -?ha
T imp
01Step
Impeller Model 4:
2nd Order Resistance and
Alternating Effort Model
ft! sh
oTimp
Sine Wavel
Figure 2-44 Impeller Load Modeling Study: Simulink Models
78
2.5 Model Verification
2.5.1 Torsional Frequency Analysis Using Finite Element Techniques
As a verification step for the Simulink Model, a finite element beam, mass, and
spring model was constructed in Ansys. Three separate FE analyses were
performed based on mixer shaft discretization of Hump, 2-lumps, and 4-lumps. A
diagram of the FE model can be seen in Figure 2-45.
Motor
Inertia
Cpgs
inertia
Reducer
HS shaft
Impeller
Inertia
3^<X
Reducer
Inertia
Reducer
HS Shaft K
HS Gear
Reduction
Figure 2-45 Finite Element Model Diagram
In the model, the gear reductions are achieved through the use of lever arms and
link elements. Link elements (or spars) cannot transmit rotation, only translation.
So for small deflections, such as in a modal analysis, two lever arms connected by
79
a link act as a contact point (the link element in Ansys is similar based on the
simple beam element and releasing the rotational DOFs). Therefore, any rotation
imparted on the high speed gear reduction through the high speed shaft will be
transformed (reduced) by the ratio of the lever arms. The rotational speed will
then be reduced again by the low speed gear reduction to the proper output
rotation. A similar scenario holds for the effort except that the effort is amplified
by the gear reductions instead of reduced. The four beam elements at both the
impeller location and motor location are massless and are used for mode shape
display purposes only. The lever arm and link components are modeled with
extremely large stiffness so as not to significantly effect the system frequencies
(other than acting as transformers).
One aspect of the FE model that was not incorporated in the Simulink model
was the independent stiffness terms of the second flex coupling and the reducer
high speed shaft. For model simplicity, and to avoid a derivative causal
relationship, the coupling stiffness and reducer high speed shaft stiffness were
combined into a single stiffness term based on rules for springs in series.
=
Kcpg2KHS
Kcpg2+KHS
The Simulink subsystem models that match the FE model are as follows:
Motor: Figure 2-22
Torque Cell: Figure 2-29
Speed Reducer: Figure 2-30
Lower Shaft: Figure 2-33
Impeller: N/A
80
The impeller resistance is excluded from the modal simulation since only the
system natural frequencies are of interest. Theses frequencies were obtained by
analyzing the frequency content of the impulse response of the system.
2.5.2 Full Scale Testing
Full scale test results of an instrumented mixer operating in process were
obtained for comparison purposes to the simulation method. The quantities which
were measured for the test were the torque cell torque and the lower shaft speed
at the speed reducer output. The data obtained from the testing are in the form of
time-domain and frequency-domain (power spectrum) plots of torque cell
transducer voltage and tachometer voltage calibrated and scaled to give the units
of in-lbs and rpm respectively. Two results sets were supplied for the same
configuration: one set captured response data from rest to full nominal speed and
the other captured the steady-state operation only. Since the mixer test setup was
intended for characterization of a proprietary process demonstration, only non
proprietary information relevant to the investigation at hand is discussed herein.
Throughout the full test effort (of which only a portion of the data was
obtained), several different operating speeds were investigated as well as single
and dual impeller configurations. The configuration that the test data was
obtained from had two impellers on the mixer shaft however only the lower
impeller was submerged. From a simulation modeling standpoint the inertia of the
81
upper impeller was included in the model but the impeller resistance term was not.
Some of the important test parameters are included below in Table 2-1 .
Full Scale Mixer Test Parameters
Configuration 1 2
Number of imps 1 1
Total Gear Ratio 17.421 17.421
Motor Input Freq, Hz 80 80
Motor synch RPM 2400 2400
Shaft nominal RPM 138 138
Shaft Diameter, inches 3.5 3.5
Shaft Length, inches 129 129
Impeller Spacing, inches 33 33
Operating Range start steady
Operating Range Descriptions:
start = From rest to steady state at nom speed
steady= Steady state at nominal speed
Table 2-1 Full Scale Mixer Test Parameters
The impeller sizing was such that the tests were run at 138rpm nominal shaft
speed to load the motor. This required that the motor operate at 2400 rpm. To
accomplish this a variable frequency drive and inverter duty motor were used with
input frequency to the motor being 80Hz. In addition to the parameters indicated
in Table 2-1 the equipment used in the testing was as defined below (specs for
individual subsystems can be found in the Appendix A):
Motor: 40hp, 1800rpm, NEMA Design B, 3-phase AC induction motor
Torque Cell: Eaton-Lebow, 1 105H-2K
Flexible Couplings: Falk 1060T flexible grid type
Speed Reducer: 17.421 ratio double reduction right-angle drive
82
Impeller: 3-bladed high solidity impeller
For the Simulink simulation of this system, the motor model with approximated
speed torque was used as was the full torque cell model and double reduction
speed reducer. Since an extra impeller was located on the shaft but not operating
in the fluid, the inertia was included but not the resistance. For shaft
discretization, 4 shaft lumps were used (2 for each impeller inertia). The impeller
model used was that of the resistive model incorporating the fluctuating torque as
discussed in Section 2.4.5 and exhibited in Figure 2-37. To simulate the 10 sec
ramp due to the experimental variable frequency drive settings, the reference
simulation torque in the motor subsystem model was ramped from 0 to 300 ft-lbs
over 10 seconds instead of just the constant 300 ft-lb value used in the simulation
model studies noted in Section 2.4.6.
83
3. RESULTS
The following section contains the results for the model verification tests and
simulation model studies as described in Sections 2.4 and 2.5. The results are
presented as time domain and frequency domain plots of torque and rotational
velocity. In many cases the same results are presented twice with different time
(or frequency) scales to better view the data plots. To insure that the frequencies
being observed are not aliases of higher frequencies, several different frequency
domain plots were constructed varying the sample times and fft resolution. All
discussion pertaining to the results contained in this section are presented in
Chapter 4.
3. 1 Simulation Model Studies
3.1.1 Motor Modeling
As discussed in Section 2.4. 1,the first model simulation study to perform was
the motor no-load analysis to verify that the motor speed-torque characteristics
were modeled properly. The results of the motor simulation model no-load
validation are presented in Figures 3-1 and 3-2. In Figure 3-1 the motor torque
and motor output speed are plotted vs. time on the same scale with the torque
curve units being in-lbs, and the speed curve units being RPM. The torque starts
at a value of 3,600 in-lbs and has a steady state value of 0 in-lbs. The motor
output speed starts at 0 RPM and has a steady state value of 1800 RPM. The
speed and torque curves were normalized based on synchronous motor speed
and motor full load torque respectively. The normalizedtorque vs normalized
84
speed were plotted in Figure 3-2. The curve crosses the 100% motor full load
torque line at a value of 98.8% of full load speed which corresponds to 1778 rpm
for an 1 800 rpm motor.
4000
3500
3000h
-Q
i
~
2500cu
cr
Motor Speed-Torque Response
2000
ir 1500
CD
|-1000
500
0
I 1
\ Torque
Spe?ed = 1 800 rpm
i i
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time, sec
Figure 3-1 Motor No-Load Model Validation
85
300
250
g 200
3
LL
2 150o
S 100
Motor Torque vs Speed
50
0
0 10 20 30 40 50 60 70 80 90 100
Speed, % Synchronous
Figure 3-2 Motor No-Load Torgue vs Speed
The results of the motor modeling study are presented in Figures 3-3, through
3-7. The results of the dual-step model and the speed-torque model are nearly
superimposed in all but Figure 3-7. Figure 3-3 is a plot of output torque vs time
over a time span of 1 sec with the results for each modeling option indicated by
the arrows. The results of all four modeling options approach a steady-state
torque of 2056 ft-lbs. The torque response (torque magnitude of output torque
relative to input) vs frequency is presented in Figure 3-4 for all four modeling
options. The first peak is at a frequency of 19.5Hz and the second at a frequency
of 266.3 Hz (as indicated in Table 3-1). The frequencies for all modeling options
were the same with the difference being the value of the magnitudesfor each.
Figure 3-5 is the impeller speed vs time for the same time span as presented for
the torque data. For all three options the steady-state speed is approaching 102
86
rpm. Figure 3-6 is the speed reducer output speed at the lower shaft vs time and
Figure 3-7 is a zoomed view of Figure 3-6 from 0.8 to 0.9 sec to better observe the
speed response. For both plots the steady-state speed is approaching 102 rpm.
The rise time for each system model (obtained graphically from Figure 3-3) was
found to be 0.084, 0.48, 0.084, 0.135 sec respectively for the speed-torque,
constant effort, dual-step, and ramp-step models.
SystemTorsional Freq, Hz
Model Option Mode 1 Mode 2 Mode 3
Speed-Torque Model 19.5 267
Constant Effort Model 19.5 267
Dual-Step Model 19.5 267
Ramp-Step Model 19.5 267
Table 3-1 Torsional Modal Frequencies: Motor Study
2500 r
Output Torque:Effect of Speed-Torque Curve
Dual-Step Model
0.1 0.2 0.3 0.4 0.5 0.6
Time, sec
0.7 0.8 0.9
Figure 3-3 Output Torque vs Time: Motor Study
87
Effect of Speed-Torque Curve
CD
-oZJ
18000
16000
14000
12000
10000
8000
6000
4000
2000
0 *gg^^M^m*^i^t)^t!Qi^^
50 100 150 200 250 300 350 400 450 500
Frequency, Hz
0
Figure 3-4 Output Torque Frequency Response: Motor Study
Impeller Speed:Effect of Speed-Torque Curve
Figure 3-5 Impeller Speed vs Time: Motor Study
88
Lower Shaft Speed: Effect of Speed-Torque Curve
Dual-Step Model
Speed-Torque Model
Constant Effort Model
Ramp-Step Model
1 I I L_
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0..
Time, sec
0.9 1
Figure 3-6 Reducer Output Speed vs Time: Motor Study
Lower Shaft Speed: Effect of Speed-Torque Curve
CL
CL
"S 101.85CD
"101.8
101.75
101.7
101.65
101.6
Speed-Torque Model
Ramp-Step Model j\ j\ j\j\f
, n a a A A /
Constant Effort Model
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
Time, sec
Figure 3-7 Reducer Output Speed vs Time: Motor Study (Zoomed)
89
3.1.2 Torque Cell Modeling
The results of the torque cell modeling study are presented in Figures 3-8
through 3-1 5. Figure 3-8 is a plot of output torque vs time from startup to 0.4
seconds comparing the system model with the torque cell to the system model
with the flexible coupling. Both options have a steady state torque of 2056 ft-lbs.
Figure 3-9 is the frequency response of the output torque for both modeling
options. The system model with the torque cell has four system torsional
frequencies and the system model with the flexible coupling has two. The
frequencies for each mode are presented in Table 3-2. Figure 3-10 is a plot of the
torque vs time as measured in the torque cell for both system model options.
Obviously the flexible coupling option has no torque cell so the torque for that
option is based on the flexible coupling output torque. Both options have steady-
state torque values approaching 118 ft-lbs. Figure 3-1 1 is the torque response in
the torque cell and has the same frequency content as the output torque response
shown in Figure 3-9 but the magnitudes are different for each mode.
Figure 3-12 represents the impeller speed vs time for both modeling options
which have steady-state values of 102 rpm. Figure 3-13 illustrates the difference
in the output speed as measured at the impeller and at the speed reducer output.
The speed reducer output (which is the start of the lower shaft) is the typical
location to place a tachometer for speed measurement purposes. Figures 3-14
and 3-1 5 are torque vs time plots for output torque vs torque cell torque for the
model incorporating the torque cell. This comparison is made to observe the
difference between the actual impeller load as compared to the measured load at
90
the torque cell. The cell torque has been multiplied by the total ratio of the speed
reducer to plot on an equal scale. Figure 3-15 presents the same information as
Figure 3-14 on a contracted time scale for illustrative purposes.
Model Option
Torque-Cell Model
Flexible Cpg Model
System Torsional Frequency, Hz
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
18
19.5
132
267
323 535
Table 3-2 Torsional Modal Frequencies: Torque Cell vs Flexible Coupling
2500
2000
co 1500.Q
CD
1000
500
Output Torque: Effect of Torque Cell
0.05 0.1 0.15 0.2 0.25
Time, sec
0.3 0.35 0.4
Figure 3-8 Output Torque vs Time: Torque Cell Study
91
CD
3
CO
2
7
O
15<104 Output Torque:Effect of Torque Cell
Torque Cell Model
Mode 2
10 -
Flexible Coupling Model
Mode 2
Torque Ce Model
5
Both Models
Mode 1
I
Mc le 3
Torque Cell Model
nk J L , JMode 4
I , A , , A0 100 200 300 400 500 600
Frequency, Hz
Figure 3-9 Output Torque Frequency Response: Torque Cell Study
Cell Torque: Effect of Torque Cell
200
180
160
140
120
CD= 100
80
60
40
20 |
0
ii
:,
I
Torque Cell Model
'
\ ''
pliff-jfe
Flexible Coupling Model
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time, sec
Figure 3-10 Cell Torque vs Time: Torque Cell Study
92
x 10Cell Torque:Effect of Torque Cell
2.5
CD
Z!
CD
CO1.5
CD
cr
0.5
Torque C
Moc
Both Models
Mode 1
ill Model
32
Flexirjfle Coupling Mo
Mode 2
a.
del
'
"prque
M
Cell Model
de 3
_Z^- J V
Torque Ceill Model
Mods 4
100 200 300 400
Frequency, Hz
500 600
Figure 3-11 Cell Torque Frequency Response: Torque Cell Study
Impeller Speed:Effect of Torque Cell
Torque Cell Model
Flexible Coupling Model
0.15 0.2 0.25 0.3 0.35 0.4
Time, sec
Figure 3-12 Impeller Speed vs Time: Torque Cell Study
93
110
100
90
80
70
60
CD au
Q.
40
30
20
10
Tachometer Speed vs Impeller Speed
0
Speed Reducer
Output (Tachometer)Speed
Impeller Speed
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time, sec
Figure 3-13 Comparison of Tachometer Speed and Impeller Speed
3000
2500
2000
Output Torque vs Measured Torque
Cell Torque
Output Torque
0.15 0.2 0.25
Time, sec
0.3 0.35 0.4
Figure 3-14 Comparison of Cell Torque and Output Torque
94
Output Torque vs Torque Cell
i 1 1 1 r
Output Torque
0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35
Time, sec
Figure 3-15 Comparison of Cell Torque and Output Torque (Zoomed)
95
3.1.3 Speed Reducer Modeling
The results of the speed reducer model study are presented in Figures 3-16
through 3-21 . Figure 3-1 6 contains gearbox output torque vs time for both the
simple single-reduction system, and the refined double-reduction model. The
steady-state output torque value for both modeling options approaches 2056ft-
lbs. Figure 3-17 is the frequency response of the output torque with the
frequencies corresponding to the modes indicated in Figure 3-17 presented in
Table 3-3. Figure 3-18 is the torque in the torque cell vs. time and Figure 3-19 is a
zoomed view of Figure 3-18 for the time interval of approximately 0.14 to 0.25 sec.
The steady-state torque value at the torque cell for both modeling options
approaches 1 1 8 ft-lbs. Figure 3-20 is a comparison of the frequency response of
the cell torque for both options. Figure 3-21 is a plot of impeller speed vs time for
both options with 102 rpm as a steady-state value for both.
Model Option
Single-Red. Model
Double-Red. Model
System Torsional Frequency, Hz
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
18
18
132
113
323
328
535
563 818
Table 3-3 Torsional Modal Frequencies: Reducer Study
96
2500
Output Torque:Discretized Speed Reducer
0 100 200 300 400 500 600 700 800 900 1000
Frequency, Hz
Figure 3-17 Output Torque Frequency Response: Reducer Study
97
Cell Torque:Discretized Speed Reducer
\
Response NearlyIdentical for Both
Modeling Options
0.05 0.1 0.15 0.2 0.25
Time, sec
0.3 0.35 0.4
Figure 3-18 Cell Torque vs Time: Reducer Study
Cell Torque:Discretized Speed Reducer
Single-Reduction
ModelDouble-Reduction
Model
0.14 0.16 0.18 0.2 0.22 0.24
Time, sec
Figure 3-19 Cell Torque vs Time: Reducer Study (Zoomed)
98
Cell Torque:Discretized Speed Reducer
Both Model
Options
Double- Reduction
Model
Mode 5
100 200 300 400 500 600 700 800 900 1000
Frequency, Hz
Figure 3-20 Cell Torque Frequency Response: Torque Cell Study
Impeller Speed:Discretized Speed Reducer
110 -
100 -
90 -
80 -
70 -
1 60 -
1 50
W
401
30
20
iili
i
10 _ /
0/ i
Response NearlyIdentical for Both
Modeling Options
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time, sec
Figure 3-21 Impeller Speed vs Time: Reducer Study
99
3.1.4 Shaft Discretization Study
The results of the shaft discretization study are presented in Figures 3-22
through 3-27. Figure 3-22 is a plot of the output torque vs time for the one-lump,
two-lump and three-lump shaft modeling options, respectively. All three simulation
results are nearly superimposed on one another in the plot and have steady-state
torque values that approach 2056 ft-lbs. There is a difference in the higher
frequency content of all three signals as indicated in Figure 3-23 which is zoomed
view of Figure 3-22. Figure 3-24 is the frequency response of the output torque
for all three modeling options. Where only a mode number is presented without a
label describing the model, this indicates that the frequency of the mode is the
same for all three models. The frequencies corresponding to the modes indicated
in Figure 3-24 are presented in Table 3-4. Figure 3-25 is the torque in the torque
cell vs. time. The steady-state torque for all three shaft modeling options
approaches 118 ft-lbs. Figure 3-26 is a comparison of the frequency response of
the cell torque for all three options and has the same frequency content as Figure
3-24. Figure 3-27 is a plot of impeller speed vs time for all three options with 1 02
rpm as the common steady-state value.
Model Option
1-Lump Shaft Model
2-Lump Shaft Model
4-Lump Shaft Model
System Torsional Frequency, Hz
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
18
18
18
132
132
132
323
272
292
535
325
328
535
535
Table 3-4 Torsional Modal Frequencies: Shaft Study
100
2500
2000
1500
1000
500
Output Torque: Shaft Discretization
A
\s
\ I
\ I
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time, sec
Figure 3-22 Output Torque vs Time: Shaft Study
Output Torque: Shaft Discretization
0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.295 0.3
Time, sec
Figure 3-23 Output Torque vs Time: Shaft Study (Zoomed)
101
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
x 10 Output Torque: Shaft Discretization
All Three
Model
Options
Mode 1
All Three
Model
Options
Moce2
2-lump & 4-lumpMode 3
L JL L
1-lumpMode 3
And
2-lump & 4-lumpMode 4
L
All
M.
free
lei
0| >p9ns
M
JL0 100 200 300 400 500 600
Frequency, Hz
Figure 3-24 Output Torque Frequency Response: Shaft Study
Cell Torque: Shaft Discretization
200
180
160
140
120
| 100
80
60
40 F
20
0
ill I
! III.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Time, sec
Figure 3-25 Cell Torque vs Time: Shaft Study
0.4
102
x 10 Cell Torque: Shaft Discretization
CD
1.8
1.6
1.4
1.2
CD
CD
5"
0.8
0.6
0.4
0.2
0
All
MDdel
Of
All Three
Model
Options
Mode 1
r nree
tons
Mo< e 2
2-lump & 4-lumpMode 3
1-lumpMode 3
And
2-lump & 4-lumpMode 4
All Tl iree
Moc If il
Opt c ns
Moc k 5
M0 100 200 300 400 500 600
Frequency, Hz
Figure 3-26 Cell Torque Frequency Response: Shaft Study
Impeller Speed: Shaft Discretization
o.
rx
"O<D
CDCl
CO
110
100
90
80
70
60
50
40
30
20
10
0
0.05 0.1 0.15 0.2 0.25
Time, sec
0.3 0.35 0.4
Figure 3-27 Impeller Speed vs Time: Shaft Study
103
3.1.5 Impeller Load Modeling
For the impeller load modeling study, results for the comparison of the2nd
order resistance model (Impeller Model 1) to the effort source model (Impeller
Model 2) are presented in Figures 3-28 through 3-32. The comparison of the
effort source with alternating effort model (Impeller Model 3) to the2nd
order
resistance with alternating effort model (Impeller Model 4) is presented in Figures
3-33 through 3-36. The comparison of the2nd
order resistance model, with and
without alternating effort (Impeller Model 1 vs Impeller Model 4), is presented in
Figures 3-37 though 3-39.
Figure 3-28 is a plot of output torque vs time from startup to 2 seconds for the
2nd
order resistance model and the effort source model. Both options have a
steady state torque approaching 2056 ft-lbs with the effort source model exhibiting
less damping (larger fluctuation). Figure 3-29 is the frequency response of the
output torque for both modeling options. Both models have the same frequency
content and are the same as that presented in Table 3-4 (Section 3.1.4)
for the
one-lump shaft option. Figure 3-30 is a plot of the torque vs time as measured in
the torque cell for both options and both options have steady-state torque values
approaching 118 ft-lbs. Figure 3-31 is the torque response in the torque cell and
has the same frequency content as the output torque response but different
magnitudes. Figure 3-32 represents the impeller speed vs time for both modeling
options which have steady-state values of 102 rpm.
Figure 3-33 is the output torque vs time for load models that have the resistive
load with alternating effort and the step load with alternating effort. Both options
104
have the same forced response of alternating torque centered about a mean value
of 2056 ft-lbs. Similarly, the cell torque plots of Figure 3-34 have an alternating
torque centered about a mean value of 1 18 ft-lbs. Figure 3-35 is the frequency
response of the cell torque from 0 to 20Hz. In the figure the shaft nominal
operating frequency ( 1 .7 Hz) and blade passing frequency (5.1 Hz) are evident.
Output Torque: Resistive vs Effort Model
2 Order Resistance
Model
Figure 3-28 Output Torque vs Time: Resistive vs Effort Model
105
x 10Output Torque:Resistive vs Effort Model
CD
a=1
CD
CO
CD
CT
|2 2
0
Mode 1
Mode 2
Mode 3
1Mode 4
0 100 200 300 400 500 600
Frequency, Hz
Figure 3-29 Output Torque Frequency Response: Resistive vs Effort Model
250 n
200
150
Cell Torque: Resistive vs Effort Model
2a
Order Resistance
Model
CD
tr
\/\/V\/Vv"v^
100
Figure 3-30 Cell Torque vs Time: Resistive vs Effort Model
106
CDT3
a
CD
TO
5
4.5
4
3.5
3
2.5
x 10
CD
2"
2.o
1.5
1
0.5
0
Cell Torque: Resistive vs Effort Model
IV ode 3
J.L
Mods
J V
100 200 300 400 500 600
Figure 3-31 Cell Torque Frequency Response: Resistive vs Effort Model
150r
100
0_
cr
"oCDCDQ.
CO
Impeller Speed: Resistive vs Effort Model
2
Order Resistance
Figure 3-32 Impeller Speed vs Time: Resistive vs Effort Model
107
Output Torque: Alternating Load: Resistive vs Effort Model
2n
Order Resistance and
Alternating Effort Source Model
0 0.5 1 1.5 2 2.5
Time, sec
Figure 3-33 Output Torque vs Time: Alternating Impeller Load
Cell Torque: Alternating Load: Resistive vs Effort Model
500 r
450
400
350
300
2 Order Resistance and
Alternating Effort Source Model
Step Load and
Iternating Effort Source Model
1.5 2 2.5
Time, sec
Figure 3-34 Cell Torque vs Time: Alternating Impeller Load
108
x104 Cell Torque: Alternating Load: Resistive vs Effort Model
3.5
=3
CCD
co o
CD3
o 1.5
0.5
Blade Passing
Mode 1
Shaft Speed
t
i~ |-~
r _i i_
6 8 10 12 14 16 18 20
Figure 3-35 Cell Torque Frequency Response: Alternating Impeller Load
150r
100
0.
rx
-dCDCDQ.
CO
50
Impeller Speed: Alternating Load: Resistive vs Effort Model
2n
Order Resistance and
Alternating Effort Source Model
Step Load and
Alternating Effort Source Model
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 3-36 Impeller Speed vs Time: Alternating Impeller Load
109
3000 r
2500
2000
d 1500
cr
O
1000
500
Output Torque: Resistive vs Alternating and Resistive
2"u
Order Resistance Model
2n
Order Resistance and
Alternating Effort Source Model
0.5 1.5 2 2.5
Time, sec
3.5
Figure 3-37 Output Torque vs Time: Resistive Model with Alternating Effort
Cell Torque: Resistive vs Alternating and Resistive
2 Order Resistance and
Alternating Effort Source Model
0.5 1.5 2 2.5
Time, sec
3.5
Figure 3-38 Cell Torque vs Time: Resistive Model with Alternating Effort
110
120
Impeller Speed: Resistive vs Alternating and Resistive
2 Order Resistance and
Alternating Effort Source Model
0 0.5 1 1.5 2.5 3.5
Figure 3-39 Impeller Speed vs Time: Resistive Model with Alternating Effort
111
3.2 Model Verification:
3.2.1 Finite Element Analysis Results
The results of the finite element modal analysis and equivalent Simulink model
simulation are presented in Table 3-5 and Figure 3-40. The frequency information
for the first five torsional modes of both methods for Hump, 2-lump, and 4-lump
shaft models is presented in Table 3-5. The Simulink model frequency response
to a 0.001 second duration, 118 ft-lb amplitude pulse is illustrated in Figure 3-40.
For modes 1,2 and 5, all three shaft options have the same frequency response.
However, the 1-lump model has one less frequency in the range and is indicated
by the empty value in Table 3-5.
Figures 3-41 a) and b) contain the mode shapes for the torsional vibration
modes. The analysis from which the results were derived was the 2-lump FEA
model. The plots were constructed in Excel based on taking the nodal rotational
displacements at each station and unit normalizing the results. In the figures the
stations represent nodal locations in the finite element model. Station 1 is at the
motor inertia, station 2 is the first flexible coupling, station 3 is at the second
flexible coupling, station 4 is at the lumped reducer inertia, station 5 is at the
middle of the lower shaft and station 6 is at the impeller.
112
System Torsional Frequency, H2r
Model Option Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
FEA: 1-lump 16.0 38.2 247.2 526.0
FEA: 2-lump 16.2 117.6 292.3 341.5 523.5
FEA: 4-lump 16.2 117.6 289.4 321.2 523.5
Simulation: 1-lump 17 132 301 528
Simulation: 2-lump 17 132 272 304 528
Simulation: 4-lump 17 132 288 311 528
Table 3-5 Torsional Modal Frequencies: FEA vs Simulation Model
Cell Torque: FEA Equiv Model
100 200 300 400
Frequency, Hz
500 600
Figure 3-40 Cell Torque Frequency Response: Simulation Model
113
Finite Element Analysis Results: Mode Shapes for 2-Lump Shaft OptionMode Shapes for Unit Normalized Nodal Rotation
MODE: 1
FREQ = 16.2 Hz
STATION ROT
======= =====
Motor 0.818
Cpgl 0.762
Cpg2 0.729
Reducer 0.029
Shaft -0.487
Impeller -1.000
1.000
0.500
0.000
-0.500
-1.000
Motor Cpg 1 Cpg 2 Reducer
Station
Shaft Impeller
MODE: 2
FREQ = 117.6 Hz
STATION ROT
======= =====
Motor -0.226
Cpgl 0.619
Cpg 2 1.000
Reducer 0.077
Shaft 0.046
Impeller -0.002
1.000
0.500
0.000
-0.500
-1 .000
Motor Cpg 1 Cpg 2 Reducer
Station
Shaft Impeller
Figure 3-41 a Torsional Mode Shapes: Modes 1, 2
114
Finite Element Analysis Results: Mode Shapes for 2-Lump Shaft Option
Mode Shapes for Unit Normalized Nodal Rotation
MODE: 3
FREQ = 292.3 Hz
STATION ROT
======= =====
Motor -0.048
Cpgl 1.000
Cpg 2 0.805
Reducer -0.067
Shaft -0.306
Impeller 0.003
1.000
0.500
0.000
-0.500
-1.000
Motor Cpg 1 Cpg 2 Reducer
Station
Shaft Impeller
MODE: 4
FREQ = 341.5 Hz
STATION ROT
======= =====
Motor -0.034
Cpgl 0.892
Cpg 2 0.558
Reducer -0.084
Shaft 1.000
Impeller -0.008
1.000
0.500
0.000
-1.000
Motor Cpg 1 Cpg 2 Reducer
Station
Shaft Impeller
MODE: 5
FREQ = 523.5 Hz
STATION ROT
======= =====
Motor 0.014
Cpgl -1.000
Cpg 2 0.959
Reducer -0.014
Shaft 0.008
Impeller 0.000
0.500
0.000
-0.500
-1.000
Motor Cpg 1 Cpg 2 Reducer
Station
Shaft Impeller
Figure 3-41 b Torsional Mode Shapes: Modes 3, 4, 5
115
3.2.2 Full Scale Test Results
The results of a full-scale mixer test and the simulated output of that test are
presented in Figures 3-42 to 3-50 and in Table 3-6. The results of the full scale
test with a 1 0 sec ramp-in and an overall operation range of 66 sec is presented in
Figure 3-42a though 3-42i. All of the plots are of the same data set examining the
time response of torque measured at the torque cell and speed at different time
intervals. The system was turned on at the 9 sec mark and run until 75 sec
(hence the 66 sec operating range stated above). The torque has some
overshoot to approximately 440 in-lbs then"settles"
to an approximate average
steady state value of 325 in-lbs. The tachometer speed doesn't exhibit any
overshoot and has an approximate average steady state value of 137 rpm. Figure
3-43a through 3-43d are plots of a second data set of the same configuration
capturing only steady state response well beyond the startup period. The steady
state torque and speed approach the same values as that of the data with the
ramp-in. The units for torque in all of the plots are in-lbs, and the units for impeller
speed are RPM. Figure 3-44a and 3-44b are plots of the power spectrum
magnitude of the torque cell voltage for the test unit. The units are Hz for the
frequency axis and decibels for the magnitude. Values for the frequency peaks
plotted in Figure 3-44 were tabulated in Table 3-6 along with identification of
frequency as either a system or forced frequency.
Results of the full scale test indicated that the torque fluctuation was
approximately 45% full load torque peak-to-peak. Hence the simulation model
was changed from the original +/-35% fluctuation value to +/-22.5% with all results
116
for the simulation that are presented in this section incorporating the 22.5%
fluctuation. The results of the simulation are presented in Figures 3-45 through 3-
50. Figure 3-45 is a plot of the tachometer speed vs time from start-up through 40
seconds. The average steady state speed is approximately 137.5 rpm with a
peak-to-peak oscillation of about 0.2 rpm. Figure 3-46 is a plot of the same
information as Figure 3-45 from 22-23 seconds. Torque cell torque vs time is
presented in Figure 3-47. The average measured torque is approximately 325in-
lbs with a peak-to-peak oscillation of 150 in-lbs. Figure 3-48 is also torque cell
torque vs time but over the time scale of 22 to 23 seconds. Figures 3-49 and 3-50
contain plots for the simulation power spectrum of the torque cell torque. Both
plots are based on the same data with Figure 3-50 examining a narrower
frequency scale. As seen in Table 3.7, the forcing frequencies identified in the
plots are the lower shaft speed (or impeller rotational frequency) at 2.27 Hz, blade
passing at 6.81 Hz, twice the shaft frequency at 4.5 Hz, twice the blade passing at
13.6 Hz, and three times the shaft speed at 9.05 Hz. The first four system
torsional modes identified were 15.4, 59.4, 113, and 322 Hz respectively.
All of the data obtained from the full scale test unit was sampled at 1000 Hz.
The power spectrum plots utilized a Hanning window and the source data was
averaged 4 times for the test results. The simulation power spectrum plots were
generated using the Matlab psd function with 4096 pts and Hanning window.
117
450.0-
400.0-
350.0-
300.0-
250.0-
200.0-
150.0-
100.0-
50.0-
0.0-
9 0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60 0 65.0 70.0 75 0
a) Speed and Torque vs Time: 9 to 75 sec
140.0
130.0
120.0
110.0
100.0
30.0
80.0
70.0-
rr
60.0
13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 170 17.5 13.0 18.5 13.0
b) Speed vs Time: 13 to 19 sec
450.0-
400.0-
350.0-
300.0-
250.0-
200.0-
130.0-
13.0 13.5 H.O 14.5 15.0 15.5 16.0 16.5 17.0 17.5 180 18.5 19.0
c) Torque vs Time: 13 to 19 sec
Figure 3-42 System Response at Torque Cell with 10 sec Ramp Up (Trial 1)
118
200.0
15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 13.5 20 0 20.5 21.0 2l'.5 220 22,5 23.0 23.5 24.0 24.5 25.0
d) Speed and Torque vs Time: 15 to 25 sec
140.0-
130.0-
120.0-
110.0
100.0
90.0'
15.0 15.5 1G.0 16.5 17.0 17 5 13.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0
e) Speed vs Time: 15 to 25 sec
30.0 302 30.4 30 6 30.8 31.0 312 314 31.6 31.8 32.0 32 2 32.4 32.6 32.8 33.0 33.2 33 4 33.6 33.8 34.0 34.2 34 4 34.6 34.3 35.0
f) Speed and Torque vs Time: 30 to 35 sec
Figure 3-42 System Response at Torque Cell with 10 sec Ramp Up (Trial 1)
119
375.0-
350.0-
325.0
300.0
275.0
250.0-
225.0
200.0
175.0-
150.0
125.0-,30.0
./.i, Aa MJ W "^l "'! I I II I
W V
30.1 30.2 30.3 30.4 30.5 30 6 30.7 30.8 30 3 31.0
g) Speed and Torque vs Time: 30 to 31 sec
138 2-
138 0-
137.5-
137.0-
136.5-
136.2-
M.
*kt/ 'V \
-0.44 sec
)\\ i
J]
Vj .J i
\w
W
k \
!,^wJVl^M
v*y
30.0 30.1 30.2 30.3 30.4 30.5
\ i #\
30 6 30 7 30. S 30.9 31.0
h) Speed vs Time: 30 to 31 sec
375.0
350.0
325 0-
300.0
275.0-
250.0
225.0
il vi \ ft \ d
s. i ftji
u
1 < /|
imiK!iAf
Mi
I1w
30 0 30.1 30.2 30.3 30.4 30 5 30 6 30 7 30.S 30 3 310
i) Torque vs Time: 30 to 31 sec
Figure 3-42 System Response at Torque Cell withIO sec Ramp Up (Trial 1)
120
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60,0 65.0 70.0 75.0
a) Speed and Torque vs Time: 0 to 75 sec
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0
b) Speed vs Time: 0 to 75 sec
Figure 3-43 Steady State System Response at Torque Cell (Trial 2)
121
139.0-
138.5-
138.0-
137.5-
137.0
136.5
136.0-
>f-Vi'
jJi
-0.43 sec
\kf.M^ftrt,
/V /PHP
S^/
30.4 30.6 30.730.0 30.1 30.2 30.3
c) Speed vs Time: 30 to 31 sec
30.8 30.9 310
400 0
380.0
360.0
340.0-
320.0-
300.0
280.0
260.0-
30.0
I i
h
'\U
y
r I
uII ''III.
ill
Iff I
kifil
30.1 30.2 30.3 30.4 30. S 30 6 30 7 30. E 30.3 31.0
d) Torque vs Time: 30 to 31 sec
Figure 3-43 Steady State System Response at Torque Cell (Trial 2)
Measured Frequency Peaks
Number Description f, Hz f, rpm PS Mag, db
1 Lower Shaft Speed 2.29 137 -49.2
2 Impeller Blade Passing 6.87 412 -43.0
3 Intermediate Shaft Speed 9.31 559 -33.7
4 2*Blade Passing 13.67 820 -47.0
5 1st Torsional Mode 16.00 960 -46.0
6 Input Shaft Speed 39.85 2391 -39.9
7 VFD Operating Frequency 79.68 4781 -35.7
8 3*lnput Shaft Speed 121.19 7271 -36.3
9 4*lnput Shaft Speed 159.37 9562 -50.9
10 5*lnput Shaft Speed 205.82 12349 -48.3
11 6*lnput Shaft Speed 242.41 14545 -48.2
12 8*lnput Shaft Speed 318.67 19120 -58.8
Table 3-6 Torque Cell Measured Frequencies
122
dBVrms
-10.8-
-20.0-
-30.0-
-40.0-
-50.0-1
-60.0-
-70.0-
-80.0-
-90.0-
-100.0-
-110.0-
-118.1-
nia.
iikilUUi m
Hz
Uiiik.iJ4lkiJkmL..iJ...Li.Wiii i
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0
a) 0 to 500 Hz
0.0 5.0 10.0 15.0 20.0 25.0 30.0
b) 0 to 50 Hz
Figure 3-44 Torque Cell Power Spectrum
123
150
Tachometer Speed: Simulation of Full Scale Test
100
Q_
DC
TD0)
<D
Q.
CO
50
0
Compare with
Figure 3-42e
0 5 10 15 20 25 30 35 40
Time, sec
Figure 3-45 Tachometer Speed vs Time: Simulation Model
Tachometer Speed: Simulation of Full Scale Test
137.6
137.4
Compare with
Figure 3-42h and
Figure 3-43c
21.8 22 22.2 22.4 22.6 22.8 23
Figure 3-46 Tachometer Speed vs Time: 22 to 23 sec
124
400
Cell Torque: Simulation of Full Scale Test
Compare with
Figure 3-42a
10 15 20 25 30 35 40
Time, sec
Figure 3-47 Cell Torque vs Time: Simulation Model
Cell Torque: Simulation of Full Scale Test
400
350
CD=!
cr
,o
300
250
22 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 23
Time, sec
Figure 3-48 Cell Torque vs Time: 22 to 23 sec
125
Simulated System Frequency Peaks
Number Description f, Hz f, rpm
1 Lower Shaft Speed 2.27 136
2 2*Lower Shaft Speed 4.50 270
3 Blade Passing 6.81 409
4 3*Lower Shaft Speed 9.05 543
5 2*Blade Passing 13.60 816
6 1st Torsional Mode 15.40 924
7 2nd Torsional Mode 59.40 3564
8 3rd Torsional Mode 113.00 6780
9 4th Torsional Mode 322.00 19320
Table 3-7 Simulated Torque Cell Measured Frequencies
Simulation Model Power Spectrum
CD
CD
3
CD
CO
o
Q.
0 50 100 150 200 250 300 350 400 450 500
Frequency, Hz
Figure 3-49 Simulation Model Power Spectrum: 0 to 500 Hz
126
m
33.
CD"O=1
cCD
CD
E
2
oCDCl
CO
CD
So
100
80
60
40
20
0
-20
-40
-60
-80
-100
Simulation Model Power Spectrum
!
4096 pt. FFTw
l |"
( ;
th 12 .27 Fianning Window
t);
4.; >
9""
5.4 ! I
f Ts-e : {
I i I i i i i
10 15 20 25 30 35 40 45 50
Frequency, Hz
Figure 3-50 Simulation Model Power Spectrum: 0 to 50 Hz
127
4. DISCUSSION OF RESULTS
Chapter 4 contains the discussion of the results presented in the previous
chapter (Chapter 3). The discussion will examine the validity of the modeling
options used and explain the effect each subsystem has on the overall system
dynamics. The results of the modeling studies will be discussed in Section 4. 1
and the results of the FE analysis and full scale model will be discussed in Section
4.2.
4.1 Model Studies
4.1.1 Motor Modeling
The motor was analyzed to first determine whether the desired speed-torque
characteristics were behaving as predicted. This was done by examining the no-
load response of the motor as it accelerates from rest to full speed as discussed in
Section 2.4. 1 . The no-load speed and torque responses as presented in Figure 3-
1 were as expected. The motor starts from rest and reaches a final speed of 1800
rpm (which is the motor synchronous speed) while the torque starts at the
breakdown value and vanishes as the motor reaches synchronous speed. Also,
the torque vs speed, which was plotted in Figure 3-2, matches the approximated
curve developed in Section 2.3.1 and plotted in Figure 2-10. The nominal full load
speed of the approximated speed torque curve was 1778 rpm, which represents a
0.2% difference form the motor rated full load speed of 1775 rpm. These points
indicate that the motor speed-torque relationship has been properly incorporated
into the modeling for the purposes it is intended. One negative impact of the
128
approximated speed torque curve is the effect it has on the initial rise time of the
system from rest to nominal operating speed. Since the approximated curve
initiates at the breakdown torque instead of the start-up torque, the simulated
system accelerates to full speed faster than the real system (or a model
incorporating the actual speed torque relationship) would.
The torque response and speed response plots of Figures 3-3 through 3-6
indicate that even though the four motor torque options investigated have the
same frequency content, the response time of the model incorporating the speed
torque curve was much faster than the constant effort input. The dual-step model
demonstrates that a reasonable approximation can be made without incorporating
the speed-torque relationship. This is due to the relatively steep relationship
between motor speed and torque from the breakdown torque to motor full load
torque. The ramp-step model is also a closer approximation than the constant
effort model however it is difficult to match both the rise time and settling of the
speed-torque model and the dual-step model. If only steady-state response is of
interest then all four options are adequate. However, if startup and/or transient
response is of importance then the speed-torque model and dual-step input model
more adequately capture the system behavior. One aspect of the speed-torque
model, which is evident in Figures 3-4 and 3-7, is that the constant effort, ramp-
step, and dual step approximations posses less damping than the speed-torque
model data.
An advantage of the speed-torque model over the dual-step model is that the
rise time is handled by the modeling and will automatically adjust to different
129
system inertia values. An approximate of the rise time can be calculated
assuming a constant torque and knowing all of the system inertia values. The
total inertia has to be referenced to the input shaft so the rules for reflecting inertia
values through a gearbox need to be applied. The relationship is based on the
time required to accelerate an inertial value from a reference speed to another
speed, and is as follows:
JxNt = sec
114.5x7
Where t is the time to accelerate the system in seconds, J is the overall system
inertia as seen by the motor with units of lb-in2, N is the speed change (rpm), T is
the applied torque (ft-lbs), and 1 14.5 is a collection of unit conversion terms.
Based on this relationship, the expected time to accelerate this system from rest
to 1775 rpm (at the motor) with a 300ft-lb torque would be 0.1 13 sec and with a
118 ft-lb torque would be 0.288 sec. The rise time determined from the results of
the constant effort source was found to be 0.48 sec. The higher torque values of
the other three modeling options resulted in faster rise times (0.084 sec for dual-
step and speed-torque model, and 0.135 sec for ramp-step model). As is evident
from the results (see Figures 3-3 and 3-5), the calculated rise times aren't overly
accurate but are close enough for a rough cut.
130
4.1.2 Torque Cell & Flexible Coupling Modeling
The primary purpose of the torque cell model study was to compare the results
of a system with a torque cell to that with only a single flexible coupling connecting
the motor to the speed reducer. The configuration of the torque cell introduces
two more stiffness terms and two more inertia terms into the system model. It is
obvious from Figures 3-8 through 3-12 that the extra stiffness terms of the torque
cell affect the system dynamics. From a frequency standpoint, the extra terms
result in two additional degrees of freedom in the system which directly results in
there being two additional system frequencies as seen in Figures 3-9 and 3-1 1 .
Also, Table 3-2 and Figures 3-9 and 3-1 1 indicate that the second system modal
frequency with the torque cell is much less (approximately half) than that of the
system with the flexible coupling only. For the single impeller configuration
examined, the frequency of the second mode is high enough (132 Hz or 7920
rpm) such that it is not near any potential system forcing functions. However, if a
additional impellers are introduced into the system, or a more flexible system is
modeled, then some of the higher system frequencies may be within the range of
the motor shaft forcing frequency creating the potential for a resonant condition.
The secondary purpose of the torquecell model study was to observe the
difference between torque and speed values from the point of characterization to
the point of measurement. This was accomplished by comparing the speed
measured at the impeller to that measured at the lower shaft (speed reducer
output) and comparing the torquemeasured at the impeller to that measured at
the torque cell. The speed reducer output is a usual location for the placement of
131
a tachometer to measure shaft speed. From Figures 3-14 and 3-15 it is very
evident that the torque response at the torque cell is different from that at the
impeller. The frequency content is the same, however the amplitude of the
oscillating torque is significantly different at the two measurement locations. This
is also true for the measured speed at the tachometer compared to the speed at
the impeller as seen in Figure 3-13 (the amplitude of the oscillating speed is
higher at the tachometer). Both plots indicate that the higher frequency
oscillations are amplified at the torque cell (or flexible coupling) when compared
with the lower shaft.
The high frequency amplification becomes important when considering that the
rules and relationships developed at the R&D level for impeller types is most likely
going to exhibit additional response characteristics based on torque cell
measurement. A pitfall which needs to be avoided is characterizing some of the
response signal as that of the forcing signal (at the impeller and at the motor). In
the current study, two signals from different locations in the model were compared
to determine which part of the measured response is due to system dynamics.
However, in a real system, the measurement cannot be performed at the impeller,
which leads to the potential for some aspects of the system response being
characterized as impeller loading. This could be accounted for at the testing
stage if the dynamics of the system are understood and the system frequencies
are determined ahead of time through dynamic simulation, modal analysis, or
both.
132
In general, the intended of use of the model described herein will determine
the significance of the response at the different measurement locations. As stated
previously, most systems for commercial use do not employ the use of a torque
cell. Therefore, if the model is to be used as a design tool then the specific
response at each location can be considered in the design and specification of
each subsystem and component. If the model is to be used as an impeller R&D
or process characterization tool, then the relative effects of the impeller torque and
measurement torque should be considered in the analysis.
4.1.3 Speed Reducer Modeling
The speed reducer study examined the influence that the level of discretization
of the speed reducer had on simulated system dynamics. The two submodels
studied were a single-reduction model (with a single stiffness term, single
transformer, and single inertia) and a double-reduction model with terms for the
high speed, intermediate speed and low speed subsystem components. It is
apparent from the results of the speed reducer model study that thedouble-
reduction model for this configuration yields minor differences from the single-
reduction model. The first, third, and fourth system torsional modal frequencies
(as presented in Table 3-3) are nearly identical. However, there is approximately
15% difference (lower) between the second mode for each option. Also, an
additional mode at 818 Hz is present for the double-reduction model that wasn't
present in the single-reduction model. This frequency is very high (49,000 rpm)
and is far removed from most system forcing frequencies. It could possibly be
133
near a gear tooth frequency (shaft rpm times number of gear teeth) on the high
speed gear in which case it would be important with regards to speed reducer
noise or potential for gear fretting. If a different speed reducer is modeled it is
possible that the frequency may be substantially lower bringing it within the range
of select system operating frequencies.
4.1.4 Shaft Modeling
The shaft discretization study was conducted to determine the sensitivity of the
simulation results to the number of shaft elements used for the lower shaft.
Figures 3-22 through 3-27 and Table 3-4 indicate that, for this system, all three
model options investigated give similar results. While the 1-lump shaft model
does a sufficient job at identifying several of the modal frequencies in the
frequency range investigated, it did fail to identify one of the system frequencies
(the3rd
modal frequency of the multi-lump models). Modal frequencies 1,2,
and 5
of the multi-lump models were identical to modes 1 ,2 and 4 of the 1 -lump model
(18, 132, and 535 Hz with respect to mode number). Mode 4 of the multi-lump
models were within 5hz (1.5%)
of mode 3 of the Hump model (323, 325, and 328
Hz with respect to 1,2 and 4-lump models). The 1-lump shaft model failed to
predict the3rd
modal frequency of 272 Hz for the 2-lump model and 292 Hz for the
4-lump model. There is roughly a 7% difference in the frequencies of the3rd
mode for the 2-lump and 4-lump models. Since convergence does not improve
much (or at all for modes 1,2,
and 5) with finer shaft discretization, a 2-lump model
should be sufficient for shaft modeling.
134
4.1.5 Impeller Load Modeling
As presented in Section 2.4.5 several different modeling options were
considered to characterize the impeller load. This included the2nd
order
resistance model and the constant effort sink with and without additional
fluctuating torque components for both model types. Comparison of the
simulation results of the different methods were presented in Section 3.1 .5 in
Figures 3-28 through 3-39. As expected, the resistive model exhibits considerably
more damping than the effort model. This is evident in the time domain and
frequency domain plots for torque and speed in Figures 3-28 through 3-32. For
frequency response, both modeling configurations have the same frequency
content but the magnitude of the first torsional mode is significantly different at the
impeller and at the torque cell due to the damping. This leads to the conclusion
that the effort sink model is a poor approximation if transient response is of
importance to the analysis.
It has been stated previously in this report that a mixing impeller is subjected to
varying loads as it operates. To try and account for this, sinusoidal effort sources
were incorporated into the impeller model as described in Section 2.4.5 with
frequencies equal to the shaft rotational speed and the impeller blade passing
frequency. The influence of this modeling option has on the system response
simulation can be seen in Figures 3-33 through 3-39. Similar to the model options
without fluctuating loads, there is an obvious difference between the effort sink
and the resistive load models. Both model types converge to the forced response
135
but, due to the increased damping, the resistive model reaches steady state more
quickly. One source of inaccuracy using this method for characterization of the
alternating loads is that, as modeled, the amplitude has been fixed based on the
nominal steady state impeller load. A more accurate representation would be
fluctuating torque values that are a percentage of the time varying torque instead
of a fixed value. This would affect the transient response through the startup
range but would have little effect on the steady state response.
136
4.2 Model Verification
4.2.1 FE Model
The finite element analysis was performed to verify that the simulation model
was behaving properly from a system torsional frequency standpoint. Nearly
equal system discretization was compared. The system models could not be
exactly duplicated in each analysis package due to the inability to simply scale
variables via a transformer in the finite element modal method as is possible in
Simulink (however, the method by which this can be overcome with the finite
element model was discussed in Section 2.5.1). Also, the lower shaft elements in
the FE model are beam elements with values for stiffness and inertia that are
more distributed (at endpoint nodes) than the lumped stiffness and inertia values
of the simulation model.
The frequency results presented in Table 3-5 indicate that the 1-lump models
in both analyses are not discretized enough to capture the 4th mode. The reason
is obvious when observing the mode shape for this mode with a 2-lump model as
indicated in Figure 3-41 b. The mode is dominated by the flexible couplings and
the upper shaft section (which doesn't exist on the Hump models). The results
also prove why one-element beams should never beused in FE analysis. While a
finer shaft discretization effects results in both model types, the Hump simulation
model predicted a second modal frequency much closer to the 2-lump and 4-lump
models than did a Hump FE model.
As was discussed in the shaft discretization section, the 2-lump and 4-lump
models give nearly identical results with theexception of the 4th mode. The FE
137
results and simulation results were within 5% for mode 1,11% for mode 2, 7% for
mode 3, 12% for mode 4 and 1% for mode 5 with a 2-lump shaft model for each.
The correlation for modes 3 and 4 improve to 1% and 3% with 4-lump models.
For both models, modes 1,2 and 5 do not converge any further with a 4-lump
shaft than with a 2-lump shaft.
Improvement of the correlation of the modal frequencies of the two modeling
methods could be improved by pursuing a different modeling scheme for the finite
element model. Instead of explicitly modeling the two gear reductions, the
effective inertia and stiffness values could be reflected to either the low speed end
or the high speed end (similar to the single-reduction simulation model). This
would enable the determination of the impact that the finite element modeling
scheme has on the frequency results. Even though this could improve agreement
between the two methods, it doesn't necessarily mean that it would improve the
accuracy of either method with respect to a real system.
138
4.2.2 Full Scale Testing
The objective of any modeling scheme is to accurately predict the behavior of
the real world system it is trying to simulate. At first glance it appears from the
results in Figures 3-42 through 3-44 that the fluid mixer studied here has a
significant amount of system noise and random response that the simulation did
not capture. However, comparison of the graphical data for the test unit and the
simulation indicate that the simulation model did accurately characterize some of
the system behavior, particularly the speed response. The closeness in form of
the speed response of the simulation and test unit are much more discemable
than that of the torque cell response. The oscillations at the shaft frequency and
blade passing frequency are obvious in Figure 3-42h and 3-43c. Based on the
time between major peaks of 0.44 sec and 0.43 sec which are graphically
depicted on the plots, the speed response for each has a dominant frequency of
2.27 Hz (136.4 rpm) and 2.33 Hz (139.5 rpm) respectively. Also apparent in the
plots are the three additional peaks between the major peaks due to the number
of impeller blades.
Comparison of the time domain response of the simulation cell torque to that
of the test unit indicates that simulation fails to identify the higher frequency
oscillation. On the power spectrum plots, the blade passing is easily identifiable
however the shaft rotational frequency is not. The dominant frequency (highest
magnitude) in the power spectrum is 9.31 Hz representing the intermediate shaft
speed. The speed reducer used in the testing (and modeled in the simulation)
had a primary reduction of 4.273, therefore a motor shaft speed of 2,395 rpm
139
(39.9 Hz) results in an intermediate shaft speed of 560 rpm (9.3 Hz). Or put
another way, a lower shaft speed of 137.5 rpm results in a 560 rpm intermediate
shaft speed. A similar frequency is present in the simulation power spectrum
(9.05 Hz) but it appears to be a harmonic of the shaft frequency and not
representative of the intermediate shaft. The closeness in frequency values is
purely coincidental (not to be confused with modal coincidence). Two other
significant frequencies (according to magnitude) that the simulation failed to
identify were the input shaft frequency of 39.9 Hz and the VFD frequency of 79.68
Hz. It was not expected that the simulation would identify the VFD frequency
since it was in no way incorporated into the motor model. It is possible that some
or all of the power magnitude due to the VFD is electrical interference (although it
was discussed previously that VFDs can cause torque spikes in the motor). A
potentially confusing aspect of the test system results is that the VFD frequency is
almost identically equal to twice the shaft frequency. The coincidence of these
frequencies makes it difficult to differentiate whether or not some of the even-
multiple harmonics of the shaft frequency are shaft dependent, VFD dependent,
or a combination of the two.
Other system modal frequencies (aside from the1st
mode) are not easily
identifiable in the test unit power spectrum due to the high frequency noise and
harmonics of the input shaft and VFD. There is a slight increase in magnitude
near 70 Hz which may represent the 60 Hz system frequency found in the model
simulation however the magnitude is much less than some adjacent peaks. The
113 Hz system frequency identified in the simulation is close to the 121.19 Hz
140
frequency for the test unit. It is difficult to discern whether or not this frequency for
the test data is a system mode, a harmonic, or a combination of the two.
Similarly, the 322 Hz modal frequency is very near the 318.7 Hz harmonic making
it difficult to identify it as a system frequency for the test unit.
An interesting observation of the test unit power spectrum is that all of the
peaks with significant magnitudes were either forcing frequencies or harmonics of
forcing frequencies. This indicates that the system natural frequencies were not
being excited to any appreciable extent.
As is evident in the start-up response of Figure 3-42a even with a gentle ramp-
in of 10 seconds there is some overshoot in the measured torque. It has been
demonstrated throughout that the mixer torsional response is typically very fast
leading to a simulated model that has no overshoot as seen in Figure 3-48. This
is not due to any of the characteristics of the torsional system. Even though the
torsional system has a fast response, the mixing medium may not. The resistance
value used in the simulation is based on a steady state characterization of the
torque draw. At start-up, the velocity gradients and fluid shear gradients are
significantly different than at steady state,such that the impeller is initially
experiencing a higher resistance than the modelingmethod would indicate.
Since the test data was obtained from a unit operating in water, additional
complexities such as chemical reactions and/or non-uniform fluid properties were
not factors. One other factor which can influence the resistance is varying
temperature. The addition of a large amount of energy in a confined volume can
141
increase the temperature of the fluid significantly (aside from any chemical
reactions which may take place) thus affecting its physical properties.
For the simulation model there are several modeling refinements which could
improve the correlation of the simulation to the real system. To better
characterize the input energy, a more appropriate approach would be to develop
the relationship between torque and VFD frequency and develop a reference
signal that would simulate this relationship over the desired ramp-in period and at
steady state. Also, the impeller torque oscillations need to be more closely tied to
the time varying speed and impeller torque instead of using a fixed frequency
content.
142
5. CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
The purpose of this project was to develop a methodology and modeling
techniques to simulate the torsional system dynamics of a mixing system. Several
different modeling scenarios were investigated for each of the subsystems as well
as input and load considerations. It was demonstrated that the models developed
to this point do a fair job simulating system response and identifying some of the
system frequencies, however refinements need to be made to some of the
subsystem models. Since this investigation concentrated on a single
configuration, the conclusions arrived at pertaining to the adequacy of certain
subsystem configurations and level of discretization may not have the same
relative effect or sensitivity that a different configuration may have. In the end it
will be up to the analyst to decide which system characteristics are important
based on the intent of the analysis to be performed.
It is important to accurately characterize all of the subsystems and components
comprising a model to determine the overall system response. The two most
important subsystems are the motor and the impeller since this is where energy is
entering and leaving the system. The model studies indicated that the variations
for the motor options used and impeller load modeling used had the greatest
effect on system dynamics leading to the conclusion that the impeller and motor
models should be as accurate as possible. The speed-torque motor model and
the resistive load with alternating load impeller model are better representations of
a real system than the other modeling options considered (see Sections 2.4.6.1
143
and 4.1 .1). Further enhancements to each of these models would result in better
system characterization.
For the geometry considered in the investigations of this thesis, a single
element (1-lump) shaft model may not be an accurate enough representation of
the system (see Sections 2.4.6.4 and 4.1 .4). The single lump shaft system
investigated failed to identify one of the intermediate frequency shaft torsional
modes even though it was fairly accurate at predicting several others. The shaft
study performed also indicates that there is not much difference in the results
between a 2-lump shaft and 4-lump shaft model. Both of these points indicate
that a 2-lump shaft model will be sufficient for most analyses of single impeller
systems. If multiple impellers are present then two shaft lumps should be used for
each impeller.
The speed reducer model study (Sections 2.4.6.3 and 4.1.3)
indicated that
there was little impact on the system response due to modeling the reducer using
either a single or double reduction configuration leading to the conclusion that
speed reducer discretization isn't that important. This opinion changed when
examining the results of the full scale test unit. One of the dominant frequencies
of the torque cell response for the test unit was the intermediate shaft (bevel
pinion shaft) speed which does not exist in the single reduction simulated system.
The reason for such a large relative magnitude as compared with other
frequencies in the power spectrum of this unit has not been identified. Possible
explanations could be mechanical conditions such as shaft alignment or a gear
tooth or shaft bearing anomaly.
144
Along these lines, a torsional system model could be used in conjunction with
frequency analysis of the real system to identify potentially harmful frequencies for
the speed reducer or flexible coupling connections. Complications that can arise
from high frequency oscillation are fretting of the high speed shaft at the flexible
coupling hubs and keyways or fretting of the gear sets. In addition to damaging
conditions such as fretting, inconveniently located system modal frequencies may
amplify system noise introduced by forced system frequencies such as high speed
bearing ball passing frequency or gear meshing frequencies. A torsional
frequency analysis could be used to identify potential problems before they occur.
With regards to measurement, it was evident that the torque and rotational
speed at the impeller are significantly different than at the measurement locations
(particularly the torque). The practicality of measuring the torque at the high
speed end cannot be disputed when considering the equipment size, cost and
availability as opposed to measuring at the low speed end. However, it was
observed that system transients more readily manifest themselves at the torque
cell location and at greater amplitudes. Any testing performed to characterize
impeller power and torque properties should include a torsional analysis of the
system to avoid characterizing some of the system response as impeller load.
145
5.2 Usage Recommendations
The model developed for this investigation is extremely flexible in regards to
varying configurations of the system and subsystems. Different motor, coupling,
torque cell, speed reducer, shaft, and impeller parameters can be easily
incorporated into the modeling approach. This would include equipment of
different size and/or material. The only caveat in regards to different
configurations which must be followed is the proper flow of power variables and
keeping track of which variables each signal line represents. As demonstrated,
the level of discretization of some of the subsystems can be changed to meet the
needs of the analysis. Finer shaft modeling (increased number of shaft lumps)
and speed reducer modeling are fairly easy to accomplish by following the
guidelines set forth in this demonstration. Also, with the impeller modeling options
presented, higher order or more complex impeller load simulation models can be
developed and analyzed.
From a system design standpoint, the modeling and methods could be used
for frequency tuning of a mixer. Parametric studies could be performed varying
flexible coupling stiffness (or any other system parameter for that matter) to
minimize the effects of torsional fluctuations as well as design around potentially
harmful resonant conditions.
The "drag anddrop"
nature of the modeling and simulation method make it a
useful tool to characterize other torsional systems. The subsystem models
developed could be used to simulate any torsional mechanical system with or
146
without a torsional damper (or multiple dampers). This could include (but is not
limited to) pumps, compressors, turbines, and marine propeller drive systems.
Obviously any parameters or behavior unique to those systems would have to be
considered in the modeling.
147
5.3 Recommendations for Further Study
5.3.1 Refined Impeller Modeling
The first refinement to investigate should be an impeller model which
incorporates the individual impeller blades instead of the lumped model used.
This would require further reduction of the impeller torsional relationships
presented to characterize each blade and loading in the lateral reference frame.
The complexities this will add to the modeling effort is the calculation of the
impeller blade lateral stiffness and determining the effective radius of the impeller
blade load. Also, the phase of each blade with respect to the other blades must
be maintained in the rotary reference frame.
The primary goal of this modeling configuration would be to determine whether
or not having each blade of the impeller modeled will allow prediction of the
effects of uneven blade loading. Currently, the calculated impeller lateral loading
which creates the shaft bending moment that typically limits/determines unit sizing
is based on empirical data which has no direct relation (in equation development)
to the lateral system. If a torsional impeller model can be developed that properly
predicts lateral load based on the uneven blade loading then a system modeling
approach can be developed for use in testing and diagnostics. The relationship
developed between torque and lateral load for a given system could be used to
establish a lateral load measuring method using only a torque cell. This would
greatly simplify the procedure for measuringlateral load (shaft bending) which
currently requires either load cellsor a strain-gauged rotating spool.
148
Once a comprehensive impeller model is complete, tests should be run on a
mixer which is instrumented to measure individual blade loading as well as torque
and lateral bending. The test results should then be used to either validate the
model or lead to further model refinements to best simulate the system.
Another impeller related investigation would be exploring the effects of relative
phasing of multiple impeller systems (i.e. effect of having blades of adjacent
impellers in-line vs staggered). The individual blades of a single 3-bladed impeller
would be staggered at 120deg. If a second impeller was present it could be offset
from the first impeller by 7i/nbld deg (60deg for 3 bladed impeller) out of phase. Of
primary interest for such an investigation would be the net lateral loads of each
impeller and the phase angle between the two loads. Of secondary importance
would be the phase between the torsional fluctuations of each impeller with
respect to one another.
5.3.2 Refined Motor Subsystem Incorporating Electrical Subsystem
A fairly detailed explanation of AC induction motors as well as the effects of
variable frequency were discussed in the modeling section. All of the modeling
presented was based on exclusion of the electromagnetic components and
developing models that included only the speed, torque and inertia characteristics.
As indicated, some approximations had to be made in the modeling of the motor
speed-torque relationship. To more accurately model the motor and fully capture
its behavior in the simulations, it may be necessary to include the electrical and
magnetic characteristics into the model. This would provide the means to fully
149
model the speed-torque curve, examine the voltage and current characteristics, as
well as aid in the study of the effects that different types of motor control options
have on the system.
5.3.3 Lateral Subsystem Modeling and Analysis
Another important modeling effort which should be undertaken is the modeling
and analysis of a lateral mixer system. This modeling effort would encompass
only the mixer shaft and impeller(s) since the other subsystems do not have any
significant lateral effects (unless of course a complete system is modeled that
incorporates the mixing vessel and mixer mounting supports). As demonstrated
with the torsional system model, a finite element modal analysis could be used to
verify the model from a system frequency standpoint. One topic that would
require a significant amount of investigation (and possibly testing) is determination
of the lateral damping of the impeller due to the mixing medium.
5.3.4 Torsional Subsystem Interaction with Lateral Subsystem
Once models have been completed for both the torsional and lateral
subsystems and a relationship is developed between torsional loading and lateral
loading, a comprehensive system model can be developed. The importance of
this effort would be to observe the effects loading scenarios in one reference
frame have on the dynamics in the other. The benefits of this, in combination with
some other modeling refinements, is presented in the next section regarding load
monitoring.
150
5.3.5 Load Monitoring
If all of the topics for further study in Sections 5.3. 1 through 5.3.4 are
investigated and unfold properly then all of the pieces would be in place to explore
the potential for using motor voltage and current measurements to measure mixer
torsional and lateral loading. Of course this would depend on the degree of signal
filtering which would take place in the various subsystems and components and
would have to be verified through testing. The potential benefit of this would be
the ability to use electrical power monitoring to measure torque as well as lateral
loads and bending. This is one step further than the lateral load measurement by
a torque cell as discussed in Section 5.3. 1 .
Once a system model is verified with measurements from the actual system it
could be used to predict loading and frequency information and expected effect on
input power. It would then be possible to monitor the electrical power into the
system and use the information to predict and monitor loading on the shaft and
impellers. It is much easier to instrument a motor for the voltage and current draw
than it is to instrument a mixer for torque and bending measurement. This could
greatly assist field assessment of operatingconditions and also be used as a
means of protecting equipment fromsevere overload conditions. Overload
protection could be achieved by interlocking the load measurement signal with the
motor control algorithms.
151
5.3.6 Mixing Application Effect on Loading and System Dynamics
One obvious result of the comparison of the simulated system to the full scale
test mixer is that a more intricate modeling of the impeller's interaction with the
mixing medium needs to be developed. As is evident in the results of the full
scale test, there appears to be a significant amount of noise or as yet
uncharacterized system behavior associated with mixing the fluid. If it is random
in nature, some manner of relating the magnitude of the system noise to some
mixing process related parameters needs to be developed.
5.3.7 Non-linear Coupling Stiffness
As presented in Section 2.3.2, flexible couplings of the type employed in mixer
applications are typically non-linear and the stiffness varies with the applied load.
A possible model refinement would be to incorporate the non-linear aspects of the
coupling and run a parametric study to determine the significance of non-linear vs
linear behavior.
5.3.8 Refined Speed Reducer Modeling
There are further refinements to the speed reducer subsystem model which
could be investigated. A fairly simple refinement would be the incorporation of the
reducer losses due to churning and friction as discussed in Section 2.3.3. Other
refinements could be the inclusion of gear backlash, lateral subsystem modeling
of reducer shafting, or even the stiffnessterms of individual gear teeth as they
152
come into contact with (and then out of contact with) teeth on the adjacent gear.
Many of the possible refinements would depend on the intended purpose of the
model and which characteristics of the speed reducer one wants to examine.
5.3.9 Modal Damping & Modal Resonance Study
Occasionally a system will be configured such that the shaft is extremely long
and slender resulting in first torsional modes that are at or near the blade passing
frequency. The purpose of this study would be to examine the effects of operating
a system such that the blade passing frequency is coincident with the shaft first
torsional frequency and attempt to force a resonant condition. Validation of the
modeling methods could lead to its use as a tool to determine system damping
coefficients for use with modal analysis studies. An often difficult endeavor when
performing vibration modal analysis of advanced systems is the determination of
the system damping coefficient for harmonic and spectral analysis. Many times
the damping is mode dependent and test data is not available. As such, a means
of determining system damping coefficients would greatly reduce the uncertainty
in harmonic and spectral analyses.
153
6. APPENDIX A
6. 1 Test Equipment Specifications
Motor: 40hp, 1800rpm, NEMA Design B, 3-phase AC induction motor, Reliance
Electric/Rockwell Automation
Torque Cell: Eaton-Lebow, 1105H-2K
Flexible Couplings: Falk 1060T flexible grid type
Speed Reducer: 17.4 ratio double reduction right-angle drive
Lower Shaft: 3.5 inch diameter by 209 inches long 1020 steel
Impeller: 3-bladed high solidity impeller
154
6.2 Manufacturer Data Sheets
6.2.1 Motor Performance Sheets
EEL,
S.O,FRAME HP TYPE
PHASE/
HERTZRPM MOLTS
324T 40 P 3/60 17 75 460
AMPS DUTYAMBC/
INSUL.s,F,
NEMA
DESIGN
CODE
LETTEREHCL.
47,7 CONT 4 0/F 1,15 B G FCXE
E/s ROTORTEST
s,o.
TEST
DATE
STATOR RES,e25 C
OHMS (BETWEEN LINES)
489103 418140 -31NE --- ...
,172
PERFORMANCE
LOAD HP AMPERES RPMPOWER FACTOR EFFICIENCY
NO LOAD 0 16 ,3 1800 4,14 0
1/4 10,0 20,0 1794 50.9 92,4
2/4 20,0 27 ,3 178 8 72,5 94,6
3/4 30,0 36 ,9 178 2 80,4 94,7
4/4 40,0 47,7 177 5 83,3 94,2
5/4 50,0 59.5 176 7 84, 3 93,4
SPEED TORQUE
RPM
TORQUE
% FULL LOAD
TORQUE
LB, -FT,
AMPERES
LOCKED ROTOR 0 186 220 287
PULL UP 720 152 18 0 268
BREAKDOWN 1675 245 290 163
FULL LOAD 1775 100 118 47,7
AMPERES SHDWW foe 46 0, \iOLT connect ion . IF OTHER VOLTAGE CONNECTIONS ABE available, the
AMPERES WILL VARY INVERSELY WITH THE RATED VOLTAGE
REMARKS: TYPICAL DATA
XE MOTOR-NEMA NOM. EFF , 94.1 %
GUARANTEED MXN, EFF, 93.6%
RockwellAutomation
l'i I
CI. I
APT
djot
Y P.M. BTCRDA-C MOTOR
woolfl9.A.A003PERFORMANCE
DATA ISS'JE DATE 06/10/93
Y W. 1. SHIT*
B* W. I. .SMTH
OS/10/03
155
Motor Performance Sheets (continued)
tTL*-1775 *>. 1,15 * 418140-31NE
niAMn 324T volts 4go htma ltsighb -asis.o. TYPICAL DATA
** 4 0 *"?$ 47^7 CODE IITITJtg TEST LATE
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pcxE statoi ses .a as ft, 17 2I-MUI/HEMT 3/60 MB5S/DBUL 4 ()/f I/ 489108 0LS (mWLTH LIKES)
1 1 1 1 1 1 1 1 1
54
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sumyns .oohh rojt 4^0 VOLT COHHECTIOH, IT CTHEJt VOLTAGE COHKCCTIOHS ARE AVAILABLE, THE
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156
6.3 Equation Development: Damped Free Response (adapted from Ref 9)
Mx + Cx + Kx- F(t) (governing differential equation)
x-e'
{Ms2+Cs +K\es' = 0
'1,2
c
2M
+
( C^2
K2Mj
K_M
x =A-es'r +B-es*'
A and B are constants solved for by setting initial conditions x{o)= 0 and i(o) = 0
x = e Ae
_C_X_K_
2M J M
+Be
C Y K
2M ) M
V
( r Y k> : Overdamped
II.
\1Mj
\2M;
M
is
< : Underdamped
M
M
: Critically damped
Cr = 2MJ = 2Mcon= 2JKM
cVM
C
C
2M
= C
'C
^
y2Mj
C-
'1,2 =(-^A/rriK
i
Jc + 2^-^i+^-x = F(0
157
7. APPENDIX B
7.1 Bond Graph Theory
The bond graph approach [Refs 3,7] is a system modeling approach that
allows one to model and analyze complex dynamical systems by dividing it into its
subsystems and base components. It works especially well with lumped
parameter systems such as the one being studied. One of the key benefits is its
ability to accommodate systems with parameters from different physical domains
as well as develop analogous relationships across physical domains. It can be
used to model a single system that has mechanical translation, mechanical
rotation, electromagnetic, and hydraulic components. This is achieved by dividing
a system into components according to how energy behaves within that
component. As discussed previously, energy can be stored potentially, stored
kinetically, dissipated, or transformed depending on the physical nature of the
component.
7.1 .1 Power and Energy Variables
The kinetic energy is stored in the mass in the form of momentum and
potential energy in the displacement of the spring. Since momentum, p(t) and
displacement, q(t) describe the two basic forms of energy storagein a system they
are considered to be the energy variables. Two other variables which are
important in the understanding of systems are effort, e(t) and flow, /ft). The
momentum and displacement are, respectively, the time integrals of the effort and
flow as presented in the following equations:
158
' t
p(t) = \e{t)dt = p0+ je(t)dtto
' t
q(t)=\f(t)dt = q0+jf(t)dt
Eqn 7.1
Eqn 7.2
rO
The effort, flow, momentum, and displacement can represent different
quantities in different types of systems. The relationship of these general vari
ables to those in real physical systems is presented in Table 7-1 :
Physical System Effort Flow Momentum Displacement
Mechanical
Translation
Force Velocity Momentum Displacement
Mechanical
Rotation
Torque Angular Velocity Angular
Momentum
Angular
Displacement
Electrical Voltage Current Flux Linkage Charge
Hydraulic Pressure Volumetric
Flowrate
Pressure
Momentum
Volume
Table 7-1 Energy and Power Variables
For Example: Effort can represent force, torque, voltage or pressure and flow can
represent linear velocity, angular velocity, current, or volumetric flowrate.
In the simple system discussed previously, effort could have represented the
initial force required to displace the mass (spring) and flow could have
represented the velocity of the mass. Additional effort variables can be
considered between the mass and the spring, between the spring and ground,
between the mass and the dashpot, and between the dashpot and ground.
Similarly, flow variables exist between the spring and mass, dashpot and mass,
spring and ground, anddashpot and ground. At the locations where each
subsystem is connected to another, energy is transferred and power flows
between the two. The product of effort and flow between two subsystems is the
159
instantaneous power transferred thus leading to the effort and flow variables being
termed the power variables and yielding the following relationship:
P(f) = e(t)f(t) Eqn 7.3
The time integral of the power transferred between two subsystems is the
energy per the following:
t t
E(0 = JP(f)A = \e(t)f(t)dt Eqn 7.4
7.1.2 Elements
The connection points where the components or subsystems are bonded are
referred to as ports. Subsystems that exchange energy at a single port are
referred to as 1 -ports and those that exchange energy at more than one location
are referred to as multiports [Refs x,x]. In the bond graph modeling technique,
each connection (port) is considered to be a bond with effort acting from one end
of the bond to the other and flow acting in the opposite direction of effort. The
objects at either end of the bond are referred to as elements of which there are
three basic types: components, sources, and junctions. The product of the effort
and flow within a bond at any given time is the instantaneous power transferred
between the elements.
1-Port Component Elements
As discussed, each bond is considered a port. For components, there are
three distinct 1-port element types: resistive, capacitive, and inertial, for which the
bond graph representation would be as indicated in Figure 7-2.
160
t' p e-
*-R fc-c ^1
I I Ia) Resistive Element b) Capacitive Element c) Inertial Element
Figure 7-1 One-Port Component Elements
The general mathematical relationship between effort and flow for the resistive
element can be described as follows:
* = <>(/) Eqn 7.5
/=<V'(e) Eqn 7.6
The general function0 (and its inverse) defines the constitutive relationship
between the variables. Resistance here is a generic term which can represent
any type of passive energy dissipation (electrical resistance, mechanical friction,
dampers, dashpots, etc). For the simple linear case, Eqns 7.5 and 7.6 reduce to
the following:
e = Rf Eqn 7.7
f = e/REqn 7.8
However, it is not limited to just the linear case. If the effort was related to the
square of the flow then the relationship would be per the following equations:
e =af2 Eqn 7.9
f =a-U2eu2 Eqn 7.10
where a is a scalar factor.
161
A similar approach is used to define the constitutive relation for a capacitive
element. Since a capacitor is an energy storage device, effort and flow do not
map directly into one another. For a capacitor, or any general potential energy
storage device, the effort is related to the displacement, not the flow. This results
in the following generalized relationship with Oc being the general function relating
the two.
q= c(e) Eqn. 7.11
e = Oc~\q) Eqn. 7.12
For a mechanical system compliance is the terminology used instead of
capacitance as in electrical systems. In such systems, Oc can represent the
inverse of the spring constant, k.
The approach used for inertial elements (and all general kinetic energy storage
devices) is similar to that used for capacitive elements except that the variables
being related by constitutive laws are the flow and the momentum. This results in
the following generalized relationship with O/ being the general function relating
the two variables.
p= *,{/) Eqn. 7.13
/ = 0,-'(p) Eqn. 7.14
162
1-Port Source Elements
There are two more 1-port element types, in addition to the three already
discussed, and they are the effort source, Se, and the flow source, 5/. The bond
graph representation for each is as follows:
a) Effort Source b) Flow Source
Figure 7-2 One-Port Source Elements
Both types are idealized sources with load having no impact on the source. The
sources can be step input, ramp input, or virtually any time dependent or constant
value.
2-Port Component Elements
There are two types of 2-port elements: transformers and gyrators. Both are
ideal elements and follow power conservation laws. The transformer element
maps an effort variable to another effort variable which is scaled by the
transformer modulus. Similarly, a flow variable is mapped to a flow variable
scaled by the inverse of the transformer modulus. The variables need not be of
the same physical domain. For example: a lever arm can transform a force
(effort) to a torque (effort) which is scaled by the moment arm length (transformer
modulus). The linear velocity (flow) of the end of the lever is transformed to a
angular velocity (flow) proportional to the linear velocity by the inverse of the
moment arm. The bond graph representation of a transformer is found in the
following figure.
163
^TF - **
Figure 7-3 Transformer Element
The constitutive laws for the effort and flow variables are as follows:
^=MTe2 Eqn 7.15
mt/i=/2 Eqn 7.16
where MT is the transformer modulus.
The gyrator element is very similar to the transformer element except that it
maps an effort variable to a flow variable and a flow variable to an effort variable.
The bond graph diagram for a gyrator element is as follows:
ei ^ ^/ e2^GY-
Figure 7-4 Gyrator Element
The constitutive laws for the effort and flow variables are as follows:
e{=MGf2 Eqn 7.17
MGf] =e2 Eqn 7.18
where MG is the gyrator modulus. One common example of a gyrator is an
electrical motor. The motor converts voltage (effort) to a rotational speed (flow) of
the output shaft and torque (effort) to current draw (flow).
3-Port Junction Elements
The 3-port junction elements are locations in the system where there is
interaction between more than two subsystems or components. The name 3-port
164
is used to describe the simplest version of these elements however there is no
real limit on the number of bonds (ports) that can be used at a junction. At each
junction power is conserved leading to the development of algebraic relationships
between the subsystems or components connected by the bonds. The two types
of junctions are common effort junctions (O-junctions) and common flow junctions
(1 -junctions). A graphical representation of the two junction types is as follows:
e
0
l<a) Common Effort Junction b) Common Flow Junction
Figure 7-5 Three-Port Junction Elements
As the name indicates, a common effort junction dictates that the effort on each
bond be equal. Since power is conserved and the efforts are equal, the sum of
the flows vanishes as indicated in the following equations.
ej]+e2f2+e3f,=0 Eqn 7.19
ei=
e2=
e3Eqn 7.20
/,+/2+/3=0 Eqn 7.21
The common flow junction is similar in nature to the common effort junction except
that the flow on each bond is equal and the efforts sum to zero. See the following
equations:
eJx+eJt+eJ^Q Eqn 7.22
/,=/2=/3E^n7"23
1+e2+e3=0Eqn 7.24
165
7.1.3 Causality
To this point, the direction of the power bonds has been established but not
the direction of the power and energy variables. In the bond graph method, this is
addressed by the concept of causality. In a simplified definition, causality is a
direction assigned to the cause and effect relationships between elements
(subsystems or components) within a system. Each element can dictate only one
variable per bond, it can either impose an effort or impose a flow condition on the
element at the other end of the bond. If element one imposes effort on element
two then by definition element two is imposing flow on element one. A visual
representation of this can be seen in the following figure.
Et | E2 Ej | E2
a) Effort Causality b) Flow Causality
Figure 7-6 Causal Strokes
In Figure 7-6a element one (Et) is imposing effort on element two {E2) and in
Figure 7-6b element one (Ei) is imposing flow on element two (E2). The vertical
line is referred to as a causal stroke.
The causal relationships for the one-port, two-port, and three-port elements as
described herein are graphically representedin Table 7-2. The causality
assignment along with direction of the power flow has important effects on the
form of the constitutive relationships of all elements (particularly for the one-port
166
energy storage devices). For capacitive and inductive energy storage elements,
the combination of power flow and causal relationships determines whether the
constitutive laws are integral or derivative (i.e. is the output signal from a storage
device an integral or derivative of the input signal). From a system standpoint, the
number of independent energy storage devices determines the order of the
system (i.e. 6 integral causal energy storage devices results in an6th
order
system). An attempt should be made in the bond graph construction and system
characterization to generate a bond graph that has only integral causal storage
devices. Derivative causality can lead to coupled components which indicate
interdependent system components, and can in some cases create numerical
problems for solvers if not algebraically resolved in the model equation
development. Therefore an integral causal bond on an energy storage device
implies that a state variable exists whereas derivative causality implies that
variable coupling exists.
An example of a coupled arrangement which would result in a derivative
causal relationship is two springs connected in series. Considered together, both
springs have an effective stiffness. However, the energy stored in each spring
depends on the value of the other spring leading to the coupled relationship. In
more complex cases, the coupled components may not be as conveniently
located as the adjacent spring example leading to difficulty in isolating and
characterizing the independentdynamic behavior of subsystem or component.
The two-port elements have no integral or derivative considerations since they
do not store energy. Transformer elements transfer causality between the two
167
bonds. Therefore if flow causality is being imposed on the transformer by element
one, then the transformer will impose flow causality on element two. Conversely,
the gyrator element reverses causality such that an effort causality imposed upon
it will be switched to a flow causality imposed on the following element.
The junction structures (3-ports) behave such that one bond dictates the
causality for the entire junction. For a common effort junction (0-junction) only one
bond can dictate (impose) effort forcing the other bonds at the junction to impose
flow conditions. At a common flow junction (1 -junction) only one bond can dictate
flow and the remaining bonds all impose effort.
168
Element Type Causal Form
Effort Source
Flow Source
Resistor
Capacitor
Inductor
Transformer
Gyrator
O-Junction
1 -Junction
s,
R
R H*
C <*-
^
i V*
I -*-
TF
TF
GY-
GY
0
i
1
Causal Equation
e = E(t)
f = F(t)
jfdt
\edt
v y
e =
J?
e, =MTe2
f2=MTf]
e2=ejMT
fi=filMT
e2=Mcft
f^=e2lMG
fi=ejMG
/,=-(/2+/3)
fi=f2=fi
e,=
-(e2+e3)
source
Table 7-2 Causal Forms
There are two elements that have causalityassigned automatically. An effort
always imposes effort on a systemand a flow source always imposes flow
169
on a system. The remainder of the causal strokes are propagated from the
junctions that sources are acting on until automatic causal strokes cannot be
assigned. The next place to start assigning causal strokes is at key energy
storage devices. If possible, causality should be assigned based on creating an
integral relationship between power variables due to reasons already discussed.
Causality is then propagated until automatic assignment stalls and the process is
repeated until all bonds have a causality assignment.
Returning to the simple spring, mass, dashpot system, a bond graph model
can be developed rather easily. At the mass where the spring and dashpot are
connected, all three components would experience the same velocity. Following
the guidelines just presented, this would be an ideal location to place a common
flow (1 -junction). Ground is also a location of common velocity (in this case zero
velocity). The forces in the spring and dashpot follow a parallel path so each
component would be graphed as represented in Figure 7-7a. If it is assumed that
the ground location will not move, the bond graph can be simplified as indicated in
Figure 7-7b. Since power is conserved, the bond graph can be further reduced to
the configuration of Figure 7-7c.
170
V,: 1 Se
V,: 1 Se
1 :V
a) b)
Se Se
c) d)Figure 7-7 Example System Bond Graph Development
What remains to complete the bond graph is the assignment of causal strokes.
The effort source automatically imposes effort causality on the 1 -junction. The
causality rules for a 1 -junction dictate that only one flow condition can be causally
imposed. Since the causality applied by the effort source cannot be automatically
propagated any further, a decision needs to be made as to the next causal
assignment. The next logical choice is to choose an energy storage device and
assign integral causality. In this case, assigning integral causality to the
inductance term (mass) sets flow causality on the 1 -junction. This requires that all
other bonds on the graph (the capacitor and the resistor) be set to effort causal.
This results in the capacitance term (spring) also becoming integral-causal with
171
the resulting bond graph of Figure 7-7d. The resulting system has two integral
causal elements in it and therefore has two state variables making it a2nd
order
system as was exhibited in Section 2.1 . 1 .
7.2 Block Diagram Modeling
To convert the system representation to a more computationally usable form, it
is necessary to convert the bond graph model to a block diagram model. The
main difference between the bond graph and block diagram models is that block
diagram connections represent the power and energy variables (effort, flow,
momentum, and displacement) instead of the power. A block diagram is a more
convenient modeling form than a bond graph due to the availability of analysis
software specifically designed for block diagrams. Each of the element types
discussed in the previous section have equivalent block diagram representations.
One advantage of the block diagram model is that the constitutive relations are
more explicit in the diagrams than bond graphs.
Once a causal bond graph has been constructed, the appropriate energy flow
direction is established and the relationship between elements is defined. For 1-
port elements, the block diagram representations are constructed based on the
constitutive relations developed in Eqn 7-5 through Eqn 7-14 and are also
presented in Table 7-2. The block diagram equivalents of the 1-port bond graph
elements are presented in Figure 7-8. Similar development results in block
diagram models for 2-port elements and 3-port junctions as presented in Figures
7-9 and 7-10 respectively.
172
1-1
TT^C
e
R
I
e
Ki
"
1
e
K,
I(
cCldt
i
e
J'
;i
e
<ilcU
*Il
Figure 7-8 One-Port
Element Block
Diagrams
TF
TF-
GYr
tH
M^cy-^-H =
1
MT!
MT
/,
1/MT
1/MT*, I:
e
MceI
XMc
(, h
e
1/MCe2
X1/MC
/, {-
Figure 7-9 Two-Port
Element Block
Diagrams
^
M o-*-
3*,
c
i
^>\-
e
V',
e
3
i
^
Figure 7-10 Three-Port Junction Block Diagrams
173
Using the relationships presented in the previous figures, the bond graph for
the spring-mass-dashpot system constructed in Figure 7-7 can be converted to
the following block diagram model.
*;
e4 ^,
se
e/r^
e4
^R
t,e
3 "t,t,
o1
J.
Figure 7-1 1 Example System Block Diagram
The effort source (forcing function) Se supplies ei which enters the summing
junction. The output of the summing junction, e2, is integrated to yield momentum
p and scaled by the inverse of O/ with/2 (velocity) being the output. The flow,/2,
then proceeds to the common flow junction where it branches into /},/?, and/4.
Note that in this diagram fi,f2,J3, and/4 are all equal. Flow/, is integrated to
yield displacement q and scaled by the inverse of 4>c yielding an effort value that
is fed back to the effort summing junction. Similarly, flow/* is scaled by the
resistor element and fed back to the effort summing junction. Although not as
notationally compact as a bond graph, the blockdiagram representation is easier
to follow with respect to energy flow and system variables.
174
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2. Fox, Robert W., Alan T. McDonald, Introduction to FluidMechanics, 3rd Ed.,
Wiley and Sons, Inc. New York, 1985
3. Karnopp, Dean C, Donald L. Margolis and Ronald C. Rosenberg, System
Dynamics: A UnifiedApproach, 2nd Ed., Wiley and Sons, Inc. New York, 1990
4. Oldshue, James Y., FluidMixing Technology, Chemical Engineering, McGraw-
Hill, Inc., New York, 1983
5. Oldshue, J. Y., N. R. Herbst and T. A. Post, A Guide to Fluid Mixing, Third
Printing, Lightnin, a Unit of General Signal Corporation, Rochester, New York,1995
6. Phipps, Clarence, Variable Speed Drive Fundamentals, 2nd Ed., The Fairmont
Press, Inc., 1997
7. Rosenberg, Ronald O, Dean C. Karnopp, Introduction to Physical System
Dynamics, McGraw-Hill, Inc., New York, 1983
8. Shepherd, D. G., Principles of Turbomachinery, Macmillan, Inc., New York,
1956
9. Thomson, William T., Theory of Vibration with Applications, 4th Ed., Prentice
Hall, Englewood Cliffs, New Jersey, 1993
lO.Hanselman, Duane and Bruce Littlefield, The Student Edition ofMatlab, User's
Guide Version 5, Prentice Hall, Upper Saddle River, New Jersey, 1998
1 1 . Dabney, James B. and Thomas L. Harman, The Student Edition of Simulink,
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175