UNIVERSITY OF CINCINNATI
Date:
I,
February 25, 2003
John J. Kantura III
,
Hereby submit this as part of the requirements for the
degree of:
Master of Science
In:
Mechanical Engineering
It is entitled:
Test Rig Design and Evaluation:
Characterizing Nonlinearity of a Friction Joint
Approved by:
Dr. Edward Berger
Dr. Randall Allemang
Dr. David Brown
TEST RIG DESIGN AND EVALUATION:
CHARACTERIZING NONLINEARITY OF A FRICTION JOINT
A Thesis Submitted to:
Division of Research and Advanced Studies
Of The University of Cincinnati
In Partial Fulfillment of the
Requirements for the Degree of:
MASTER OF SCIENCE
In the Department of Mechanical, Industrial, and Nuclear Engineering
Of the College of Engineering
2003
By:
John J. Kantura III
B.S., University of Cincinnati, 2001
Committee Chair:
Committee:
Dr. Edward Berger
Dr. Randall Allemang
Dr. David Brown
ABSTRACT
Since many real systems are nonlinear, it is important to understand their
characteristics for creating safe, efficient designs. This is true for structural dynamics,
where the behavior of a system is influenced by joints, riveted connections, bolted
connections, etc. These connections can be a source of sizeable damping for a structure.
In addition, it is possible that these nonlinear connections will influence the system’s
structural dynamic response. A better understanding of these nonlinear behaviors will be
of great help for analytical models to account for such friction sources in the system
dynamics of a structure.
In order to complete this task, a sliding friction joint test rig was developed by
another student, Mr. Robert Watts. Mr. Watts designed a rig that constrains two beams in
contact. The test rig was designed in an attempt to isolate the beams and the friction
joint. One beam can be transversely excited at various levels. The applied excitation
levels and normal loads can be varied to investigate the joint from a full contact case to a
partial slip case. With the rig, a number of experiments can be performed to investigate
the contact of the beams.
This paper evaluates the test rig to ensure it performs as designed. A modal test
of the rig was performed and indeed verifies that the rig isolates the beams as a SDOF
system over a frequency range of 40 to 300 Hz. The test setup details, such as testing
software, hardware, loading, etc., are thoroughly documented to aid future research. Data
was collected and analyzed to show that the current rig can be effectively operated under
a full loading condition to a near slip condition. Characterization methods in both the
frequency and time domain are given, with expected nonlinear trends shown in the data.
Results such as FRF distortion and inverse FRF’s show a clear nonlinear damping. For
time domain results, plots of hysteresis loops show a static coefficient of friction value
near 0.60. Lastly, ways to improve specific areas such as beam alignment, guide rods,
and normal loading are discussed with a new design.
ACKNOWLEDGEMENTS
I would like to thank everyone that has made this research possible. Without the
support and guidance from the staff at the University of Cincinnati, friends, and family,
this work would have not occurred. First, I would like to thank my committee members,
Dr. Berger, Dr. Allemang, and Dr. Brown. You have helped me greatly not only with my
research, but also gave me the opportunity as a teaching assistant which allowed me the
ability to attend graduate school. I am deeply indebted to you; you have been excellent
professors and friends.
My thanks also go out to Jason Grundey, who was of great help with this project.
We spent many, many hours setting up the rig and acquiring data; with your help it was a
fun experience. Along the same lines, I wish the best of luck to all SDRL laboratory
members. The lab consists of a great group of people who are very knowledgeable and
make the day fun.
Giving me a solid foundation to start from, Robert Watts did a great job with the
original design and construction of the rig. I also greatly appreciate Jim Deel supplying
the MTS I-DEAS software which was used for all of the data collection.
Finally, I must thank my fiancée Kathy and my parents for all of their wonderful
support. You have been so supportive and kept me motivated in ways I cannot express.
Thank you all.
TABLE OF CONTENTS
Nomenclature ..................................................................................................................... x
Contact Mechanics.......................................................................................................... x
Modal ............................................................................................................................. xi
Chapter 1: Introduction ..................................................................................................... 1
1.1 Background ............................................................................................................... 1
1.2 Review of Relevant Literature .................................................................................. 1
1.2.1 Joint Models....................................................................................................... 2
1.2.2 Joint Experiments............................................................................................... 7
1.2.3 Characterization Methods ................................................................................ 10
1.3 Premise of Paper ..................................................................................................... 11
Chapter 2: Contact Mechanics Theory ........................................................................... 13
2.1 Hertz Model ............................................................................................................ 13
2.2 Sliding Contact........................................................................................................ 18
2.2.1 Partial Slip Case............................................................................................... 19
2.2.2 Gross Slip Case ................................................................................................ 22
2.3 Energy Dissipation.................................................................................................. 22
Chapter 3: Modal Theory ................................................................................................ 26
3.1 Linear Systems........................................................................................................ 26
3.1.1 SDOF System Overview.................................................................................. 26
3.1.2 Modal Testing of Linear Systems .................................................................... 30
3.2 Nonlinear Systems .................................................................................................. 31
3.2.1 Modal Testing of Nonlinear Systems .............................................................. 33
i
3.3 Sensor Mass Loading.............................................................................................. 35
3.4 Summary ................................................................................................................. 37
Chapter 4: Test Rig Design.............................................................................................. 38
4.1 Introduction............................................................................................................. 38
4.2 Original Design....................................................................................................... 40
4.3 Beam Contact Surfaces ........................................................................................... 43
4.4 Test Rig Modal Analysis ........................................................................................ 46
4.5 Test Rig Setup......................................................................................................... 50
Chapter 5: Testing Issues ................................................................................................ 55
5.1 Introduction............................................................................................................. 55
5.2 Hardware................................................................................................................. 55
5.3 Software .................................................................................................................. 55
5.4 Normal Load ........................................................................................................... 57
5.5 Final Test Setup ...................................................................................................... 58
Chapter 6: Data Analysis................................................................................................. 61
6.1 Characterization Methods ....................................................................................... 61
6.1.1 FRF Distortion ................................................................................................. 61
6.1.2 Complex Plots.................................................................................................. 62
6.1.3 Inverse FRF...................................................................................................... 64
6.1.4 Hysteresis Loops.............................................................................................. 65
6.1.5 Hilbert Transforms........................................................................................... 65
Chapter 7: Results............................................................................................................ 67
7.1 Testing Results Introduction ................................................................................... 67
ii
7.1.1 FRF, Phase, Autopower Plots .......................................................................... 68
7.1.2 Complex Plots.................................................................................................. 70
7.1.3 Inverse FRF...................................................................................................... 71
7.1.4 Hysteresis Loops.............................................................................................. 72
7.1.5 Hilbert Transforms........................................................................................... 77
7.2 Summary ................................................................................................................. 80
Chapter 8: Rig Redesign .................................................................................................. 81
8.1 Introduction............................................................................................................. 81
8.2 Rig Issues ................................................................................................................ 81
8.2.1 Guide Rods....................................................................................................... 81
8.2.2 Beam Alignment .............................................................................................. 82
8.2.3 Machining ........................................................................................................ 82
8.2.4 Normal Loading ............................................................................................... 82
8.2.5 Rig Size............................................................................................................ 83
8.2.6 Software Closed Loop Control ........................................................................ 84
8.3 Proposed Rig Changes ............................................................................................ 85
8.4 Rig Design Comparison.......................................................................................... 91
Chapter 9: Conclusions ................................................................................................... 95
9.1 Introduction............................................................................................................. 95
9.2 Testing Methods...................................................................................................... 95
9.3 Data Analysis .......................................................................................................... 96
9.4 Future Suggestions.................................................................................................. 96
References ........................................................................................................................ 98
iii
Appendix A: Additional Figures.................................................................................... 103
A1. Test Rig ................................................................................................................ 103
Appendix B: Additional Tables...................................................................................... 109
B1. Full Modal Test Calibration Table ....................................................................... 109
B2. Test Setup Calibration Table................................................................................ 110
Appendix C: MatLab Scripts ......................................................................................... 111
C1. Autopower Comparison Plots .............................................................................. 111
C2. FRF Distortion Plots............................................................................................. 111
C3. Complex Plots ...................................................................................................... 112
C4. Inverse FRF Plots ................................................................................................. 113
C5. Hysteresis Loop Calculations............................................................................... 114
C6. Hysteresis Loop Area Calculation........................................................................ 121
C7. Hilbert Transforms ............................................................................................... 123
iv
LIST OF FIGURES
Figure 1.1: Active Joint [3]................................................................................................. 3
Figure 1.2: Oblique Force Loading, at Angle α ................................................................. 5
Figure 1.3: Energy Dissipation, Oscillating Forces [8] ...................................................... 5
Figure 1.4: (A.) Monolithic and (B.) Jointed Beam Experiments [14]............................... 7
Figure 1.5: Test Setup, Friction Joint and Shaker [17] ....................................................... 9
Figure 1.6: Test Results, Acceleration, Velocity, and Hysteresis Loops [17] .................. 10
Figure 2.1: Contact of Bodies under Loading, Spheres and Cylinders [20] ..................... 13
Figure 2.2: Elastic Contact of Two Nonconforming Bodies ............................................ 15
Figure 2.3: Pressure Distribution, Elastic Half Space....................................................... 16
Figure 2.4: Contact of Sliding Surfaces............................................................................ 18
Figure 2.5: Stick and Slip Regions of Two Bodies in Contact ......................................... 20
Figure 2.6: Stick and Slip Regions: Curve A=No Slip, Curve B=Partial Slip ................. 21
Figure 2.7: Circular Contact from Oscillating Force ........................................................ 23
Figure 2.8: Hysteresis Loop.............................................................................................. 24
Figure 3.1: SDOF System ................................................................................................. 27
Figure 3.2: FRF and Phase Plot, Example SDOF System ................................................ 29
Figure 3.3: Complex Plot, Example SDOF System.......................................................... 30
Figure 3.4: Example of the Principle of Superposition, on a Hardening System [23] ..... 32
Figure 3.5: Analytical FRF Curve of a Nonlinear System................................................ 33
Figure 3.6: Nonlinear System FRF with Varying Input Levels & Random Excitation.... 34
Figure 3.7: Peak FRF values vs. Force Level ................................................................... 35
Figure 3.8: Sensor Mass Loading Effects ......................................................................... 36
v
Figure 4.1: Isometric View of Original Test Rig.............................................................. 39
Figure 4.2: Right Side View… ........................................................................................ 41
Figure 4.3: Front View………. ........................................................................................ 41
Figure 4.4: Close View of Test Columns.......................................................................... 42
Figure 4.5: Beam Contact Surfaces .................................................................................. 44
Figure 4.6: Bottom Aluminum Testing Beam, R=0.78114”............................................. 45
Figure 4.7: Top Aluminum Testing Beam, R=0.78565” .................................................. 45
Figure 4.8: Accelerometer Placement on Test Rig ........................................................... 48
Figure 4.9: Beams in Contact, Significant Motion of Beams in First Bending, 194 Hz .. 49
Figure 4.10: Isometric View of Original Test Rig Assembly........................................... 51
Figure 4.11: Isometric View of Test Rig Assembly ......................................................... 53
Figure 4.12: Right Side View of Test Rig Assembly ....................................................... 54
Figure 4.13: Top View of Test Rig Assembly.................................................................. 54
Figure 5.1: Autopower, Force Control and No Force Control.......................................... 57
Figure 5.2: Strain Gages Mounted on Beam..................................................................... 58
Figure 5.3: View of Testing Setup.................................................................................... 59
Figure 6.1: Complex Plot Distortions [23] ....................................................................... 63
Figure 7.1: Wideband FRF of Bottom Beam.................................................................... 68
Figure 7.2: FRF, Phase & Autopower .............................................................................. 69
Figure 7.3: Autopower & Complex Plots ......................................................................... 70
Figure 7.4: Inverse FRF’s ................................................................................................. 71
Figure 7.5: Hysteresis Loops, Forcing Level 54 Newtons................................................ 74
Figure 7.6: Hysteresis Loops, Forcing Levels All Below Gross Slip............................... 75
vi
Figure 7.7: Beam Acceleration Signals, forming a Lissajous Pattern .............................. 75
Figure 7.8: Energy Dissipation Follows a Cubic Trend ................................................... 77
Figure 7.9: Hilbert Transforms, Real Part ........................................................................ 78
Figure 7.10: Hilbert Transforms, Imaginary Part ............................................................. 79
Figure 8.1: Strain Gage Acquisition for VXI [37]............................................................ 83
Figure 8.2: Illustration of Linear Bearing [38] ................................................................. 85
Figure 8.3: New Beam Mounting ..................................................................................... 86
Figure 8.4: Isometric View of Rig Proposal ..................................................................... 88
Figure 8.5: Isometric View of New Rig Assembly........................................................... 89
Figure 8.6: Right Side View of New Rig Assembly......................................................... 90
Figure 8.7: Top View of New Rig Assembly ................................................................... 90
Figure 8.8: Front View of New Rig .................................................................................. 91
Figure 8.9: Isometric View of Both Rigs.......................................................................... 92
Figure 8.10: Right Side View of Both Rigs...................................................................... 93
Figure 8.11: Top View of Both Rigs ................................................................................ 93
Figure 8.12: Front / Side View of Both Rigs .................................................................... 94
Figure A.1: Alternate View of Original Rig Assembly .................................................. 103
Figure A.2: Alternate View of Test Rig Assembly......................................................... 104
Figure A.3: Rig During Testing ...................................................................................... 105
Figure A.4: Strain Indicator & Switch and Balance Boxes ............................................ 106
Figure A.5: Shaker During Testing................................................................................. 106
Figure A.6: VXI Setup… ............................................................................................... 107
Figure A.7: Beam Setup.. ............................................................................................... 107
vii
Figure A.8: PC, Amp, DSA During Testing................................................................... 107
Figure A.9: Alternate View of New Rig Assembly........................................................ 108
viii
LIST OF TABLES
Table 4.1: Test Rig Part List ............................................................................................. 43
Table 4.2: Modal Testing Equipment List ........................................................................ 47
Table 5.1: Testing Equipment List.................................................................................... 60
Table 7.1: Coefficient of Friction Values ......................................................................... 76
Table B.1: Load Cell / Accelerometer Data, Modal Test ............................................... 109
Table B.2: Load Cell / Accelerometer Data, Test Setup................................................. 110
ix
NOMENCLATURE
CONTACT MECHANICS
Q
Tangential Friction Force
Po
Normal Force
V
Sliding Velocity
a
½ Contact Area
c
½ Stick Contact Area
µ
Coefficient of Friction
W
Work
F*
Cyclic Oblique Force
G
Modulus of Rigidity
v
Poisson’s Ratio
E
Modulus of Elasticity
σx
Stress in the x-Direction
σy
Stress in the y-Direction
σz
Stress in the z-Direction
τ xy
Shear in the xy-Direction
δ
Material Deflection
R1
Radius of Body 1
R2
Radius of Body 2
z1
Point on Unloaded Body 1
z2
Corresponding Point on Unloaded Body 2
x
Separation between Points z1 and z2
h
∂
∂n
u z1
Partial Derivative with Respect to variable n
Displacement of Body at Point 1
MODAL
x
Displacement Vector
m
System Mass
k
System Stiffness
c
System Damping
F
Input Force
SDOF
Single Degree of Freedom
MDOF
Multiple Degrees of Freedom
SIMO
Single Input Multiple Output
MIMO
Multiple Input Multiple Output
FRF
Frequency Response Function
COH
Ordinary Coherence Function
H (ω )
Frequency Response Function
H (s)
Transfer Function
H 1 (ω )
FRF Estimator, Noise on the Response
H 2 (ω )
FRF Estimator, Noise on the Input
H v (ω )
FRF Estimator, Noise on Response and Input
[...]
Matrix Expression
xi
{...}
•
x
Vector Expression
First Derivative with Respect to Time
••
x
Second Derivative with Respect to Time
Re+ j Im
Complex Number, Real Part + Imaginary Part
H
Hilbert Transform
H-1
Inverse Hilbert Transform
∂
Partial Derivative with Respect to t
∂t
j
−1
N
Number of Modes of the System
No
Number of Inputs
Ni
Number of Outputs
t
Time Variable
∆f
Discrete Frequency Interval
∆t
Discrete Time Interval
λ
System Root, λ = σ + jω
ω
Frequency Variable
ωr
Damped Natural Frequency
Ωr
Undamped Natural Frequency
σ
Damping Variable
ζ
Damping Ratio
I (ω)
Inverse FRF, 1
H (ω)
xii
CHAPTER 1: INTRODUCTION
1.1 BACKGROUND
One of the fundamental assumptions used in much of modal analysis and
characterization is that of linearity. The assumption of linearity states that the ratio of a
system’s output over the input is constant. For many cases, this assumption is often valid
over certain operating ranges. Working with linear models is easier from both an
analytical and experimental standpoint.
However, nearly all systems are nonlinear over certain operating ranges. The key
question is, ‘To what extent?’ and ‘Over what range?’ For some systems, the
nonlinearity must be characterized in order to accurately describe the system and its
dynamics. Understanding these nonlinear phenomena will help with creation of
intelligent designs. If nonlinearities are overlooked, deviations from analytical models
versus experimental results will occur, and will lead to unexpected design results.
1.2 REVIEW OF RELEVANT LITERATURE
The long-term purpose of this study is to characterize the nonlinearities which
occur in a friction joint. There are many nonlinearities which may occur as a result of
friction. Some examples of friction nonlinearities include [1]:
ƒ
Velocity dependence
ƒ
Multi-valued function at zero velocity
ƒ
Memory effects
ƒ
Time dependent sticking friction
1
ƒ
Dynamic response dependence
ƒ
Discontinuities during velocity transitions
1.2.1 JOINT MODELS
The dynamic behavior of assembled structures can be greatly influenced by
friction effects at the connections. Gaul and Nitsche introduce the effects of friction in
mechanical joints with an extensive literature review [2]. This paper reviews many
articles on energy dissipation in bolted joints, pin connections, semi-active connections,
and more. Applications of friction models are given, such as using phenomenological
models for control tasks and active joints for vibration suppression. In addition, the work
also includes 134 references to other helpful papers.
Gaul and Nitsche [3] continue their research by examining active joints in an indepth discussion of a piezoelectric active joint. The piezoelectric material in the joint
allows the normal force on the joint to be varied, as illustrated in Figure 1.1. They
describe the joint using the LuGre friction model [4]. The LuGre model represents the
two contacting solid surfaces as bristles, which deflect like springs at the microscopic
level. The authors then propose a controller design to maximize the damping of the
system, a two beam model with pinned-pinned grounds, connected with the active joint.
The simulation results show an increase in system damping with the active joint when
compared to a simple bolted connection under a constant normal force.
2
Piezoelectric
Stack Disc
Voltage
Figure 1.1: Active Joint [3]
Continuing with the discussion of bolted connections, Lenz and Gaul use a two
mass resonator with Coulomb friction to investigate the macroslip of bolted connections
[5]. Macroslip is defined as relative motion between two surfaces. Microslip is defined
as no relative motion between two surfaces; however, localized slipping occurs at the
microscopic level. Many joints used in construction will fail if macroslip occurs.
However, joints can be effectively designed to allow microslip, a source of damping. To
investigate the microslip, two experiments are performed, in both longitudinal and
torsional directions. Hysteresis loops are plotted for the experimental cases and
compared to the Valanis friction model [6]. An advanced form of the Valanis model is
3
proposed which is able to describe the microslip behavior as well as the transition to
macroslip for a bolted lap joint.
Bindemann and Ferri investigate five friction models on a flexible beam structure
[7]. Three of the friction models allow the coefficient of friction to vary with velocity,
one of the models allows the coefficient to vary with the history of the slip velocity, and
the last model has a normal load dependence. The results of the study show that system
damping is not largely affected by the manner in which the coefficient of friction varies
with velocity or load; rather it is the average value of the friction coefficient that
determines the amount of damping.
The topic of energy dissipation is an important issue in friction studies. Johnson
[8] experimentally investigates the theory presented by Mindlin and Deresiewicz [9].
The experiment investigates the energy dissipation of two spheres in contact, excited by
oscillating forces. In previous work, Mindlin [10] showed that energy dissipation occurs
no matter how small the tangential oscillating excitation force. In the theory proposed by
Mindlin and Deresiewicz, if the excitation force is applied at an arbitrary angle α, energy
dissipation will not occur below a limiting value of tan α = µ. A schematic of this setup
is shown in Figure 1.2. Johnson’s experimental results, shown in Figure 1.3, agree fairly
well with the theory, except in the intermediate range of the oscillating forces. This
discrepancy can be attributed to elastic distortion of the surface asperities, as well as the
fact that repeated slip gives rise to a variation in the effective coefficient of friction.
4
Po
F
α
r
c
ao
a
c
ao
a
Figure 1.2: Oblique Force Loading, at Angle α
Figure 1.3: Energy Dissipation, Oscillating Forces [8]
5
Continuing along these lines, Goodman and Brown outline an experiment where a
sphere is clamped between two parallel plates [11]. The experiments verify work done
by Mindlin and Deresiewicz [9]. The sphere is excited tangentially at different levels, as
well as at different clamping forces. The final results for energy dissipation agree with
theory under a range of normalized displacement values, 0.45 to 1.00. For the lower
range of 0 to 0.45, material damping effects are cited for a possible cause in the
disagreement with the theory.
Iwan formulates a two-degree of freedom hysteretic system in his work [12]. The
system is comprised of two masses connected in series to ground, with springs and
Jenkins elements. Each Jenkins element consists of a linear spring in series with a
coulomb damper. Since this problem is difficult to solve analytically, the method of
slowly varying parameters was used. The frequency response solution is then found and
plotted. The results show if the system is excited above a critical level, the response is
unbounded at resonance. This is the case for even finite levels of excitation.
In a similar work by Iwan, a physical argument is used to derive a model to
investigate hysteresis loops, instead of the common mathematical motivation [13]. The
model consists of a plate connected to ground with a series of Jenkins elements. The
steady-state solution of the model is found, and hysteresis loops for different forcing
levels are given. The results are then compared to experimental data on a single story
structure with steel columns, and agree fairly well. The model was able to give within
about 10% the estimate of peak response amplitude along with the shape of the frequency
response curves. Finally, the response was found to be sensitive to the model parameters,
especially at low stress levels.
6
1.2.2 JOINT EXPERIMENTS
In two studies performed at Los Alamos National Laboratories, Moloney, Peairs,
and Roldan studied friction damping [14] on both monolithic and jointed beams. Figure
1.4 shows an example of the beams that were tested, (A.) is a monolithic beam and (B.) is
the lapped beam.
(A.)
(B.)
Figure 1.4: (A.) Monolithic and (B.) Jointed Beam Experiments [14]
For the experiment, the beams were excited with an impact, and the damping ratios were
computed for each beam. For the monolithic beam, the results showed a constant
damping ratio value of 0.002 when plotted versus velocity amplitude. For the jointed
beam, the damping ratio increased to an upper bound of 0.01 when plotted versus
velocity amplitude. These results confirm the monolithic beam is linear while the jointed
beam is nonlinear due to the joints.
7
Next, finite element models for both the monolithic and jointed beams are
discussed. The monolithic beam is a simple, linear model and the results agree well with
the experiments. The jointed beam uses a modified bristle friction model because it was
found that the standard model did not correctly predict the friction behavior. The
standard bristle model was modified by inserting an explicit dependence on velocity.
Once changed, the jointed beam FE model agreed with the experimental results.
Continuing the above work, Kess, Rosnow, and Sidle investigate the jointed beam
to determine the effects of bearing surfaces on the energy dissipation [15]. Three
experiments were conducted on the beam:
ƒ
The beams assembled with its full contact area
ƒ
The beams assembled with large washers reducing the contact area
ƒ
The beams assembled with small washers further reducing the contact area.
Using log decrement, the damping ratios were found to be strongly sensitive to velocity
for the large contact area, mildly nonlinear for the middle contact area, and linear for the
smallest contact area.
Continuing their work, a FE analysis is performed using two different friction
models, Dahl [16] and LuGre. When comparing the FE results to the experiments, the
LuGre model was found to more accurately match the experiment in the full contact case.
Another lap joint experiment is performed by Gregory, Smallwood, Coleman, and
Nusser [17]. A device was designed and built to isolate the frictional effects of a simple
bolted connection in shear. The damping associated with bolted interfaces has shown to
8
be very difficult to model. An experimental setup which can yield insight into the
damping phenomena will aid in the development of analytical models. The device allows
for the measurement of energy dissipation over a range of input levels, input types,
normal forces, material types, and surface conditions. The design consists of two lapped
pieces which are loaded with adjustable rollers, and are directly mounted to a shaker.
Figure 1.5 illustrates the design [17].
Figure 1.5: Test Setup, Friction Joint and Shaker [17]
The paper discusses important issues in the design such as alignment of the test
specimens, sensor calibration, and roller energy loss calculations. With these issues
accounted for, data was collected with a fixed sinusoidal excitation under varying normal
loads. The accelerations of each piece of the lap joints were measured and a sample set
of data is shown in Figure 1.6. The figure plots the acceleration across the joint in the
upper left, force versus relative acceleration in the upper right, force versus relative
velocity in the lower left, and force versus relative displacement in the lower right. An
interesting result is found in the upper left plot, as the two accelerations are found to
9
create a Lissajous pattern. This is discussed in the paper and attributed to the higher
harmonics which are indicative of nonlinear effects in the system.
Figure 1.6: Test Results, Acceleration, Velocity, and Hysteresis Loops [17]
Lastly, the energy per cycle is calculated for a number of loading conditions. The results
clearly show a cubic relationship, which agree with analytical work performed by
Goodman [22] in the low joint force case.
1.2.3 CHARACTERIZATION METHODS
The last two papers reviewed in this literature summary deal with characterization
methods for nonlinear systems. He and Ewins investigate a nonlinear SDOF system
excited by a sinusoid [18]. The commonly-practiced technique of force or response
10
control is discussed, and its difficulties are addressed. A method using inverse FRF’s and
force control at one low forcing level is proposed. The results show useful experimental
characterization methods for testing nonlinear systems.
Covering a broader range of characterization methods is a paper by Adams and
Allemang [19]. It provides an overview of nonlinearities in systems, discussing issues of
output, feedback, time invariance, and noise. Excitation methods are covered, along with
how they affect nonlinear systems. Detection and classification methods are provided for
both time and frequency domains, with applications to SDOF and MDOF systems.
Example characterization methods discussed are inverse FRF, time and frequency
domain Hilbert transforms, wavelets, and NARMAX models.
1.3 PREMISE OF PAPER
This paper explores experimental friction joint dynamics by implementing a test
rig. First, background information will be covered for the contact mechanics theory used
in this project. A brief discussion is made about Hertz’s theory and its assumptions.
Following this, the energy dissipation of two cylinders in contact is discussed because of
its relevance to the experimental system examined here. Continuing with background
information, modal theory for linear and nonlinear systems is discussed. Brief summaries
of some of the concepts used are covered. Once the theory is presented, some modal
testing considerations for the linear and nonlinear systems are discussed.
Once these background areas are introduced, the original rig design is presented.
A test rig was constructed to perform the testing on a friction joint. An exchange student,
Mr. Robert Watts, designed the majority of the test rig. This project began with the
11
completion of the test rig machining and assembly. Initial testing on the test rig was
performed, including a full modal test. As work progressed, modifications were made to
the rig in attempt to aid in useful outputs.
Next, various characterization methods are discussed. Similar to the test rig
discussion, the testing setup and significant testing issues are discussed. For this project,
I-DEAS testing software was obtained and implemented. This software package allows
for a sinusoidal input, along with the ability to control the input excitation level to a
constant as suggested in literature [18]. Once results were obtained, the characterization
methods presented earlier are used to process the initial data collection on the friction
joint. Frequency domain detection indicates the nonlinear damping of the system.
Hysteresis loops are created from time domain data to show the energy dissipation of the
joint.
After these topics, a modified rig design is proposed and discussed in an attempt
to resolve some of the testing issues that were outlined earlier. The proposed new rig
design is shown in a solid model highlighting all of the changes. Lastly, a
recommendation and discussion section is included. This area covers ideas for future
work that can be performed in this area of study.
12
CHAPTER 2: CONTACT MECHANICS THEORY
2.1 HERTZ MODEL
Contact mechanics as studied today is based upon the work and theories of
Heinrich Hertz, from the late 19th century. Two 3D surfaces contact at a point when
brought together. After the surfaces deform, the contact area becomes elliptical. For the
special case of two spheres, the contact area will be circular. For the 2D plane strain case
of two cylinders, initial contact will occur along a line. After surface deformation, the
contact area will become rectangular. These two special cases are illustrated in Figure
2.1. This figure is taken from Berger [20].
Side View
Top View
Side View
Top View
Contact Area
(A.) Two Spheres in Contact
Contact Area
(B.) Two Cylinders in Contact
Figure 2.1: Contact of Bodies under Loading, Spheres and Cylinders [20]
13
The overall assumptions made by Hertz for two bodies in contact are:
ƒ
Surfaces are continuous and nonconforming, such that the contact area is
much less then the radius
ƒ
Strains are small
ƒ
The bodies are elastic half spaces
ƒ
Surfaces are frictionless
Figure 2.2 below illustrates this contact. With an applied load N on the bodies, an
expression for the displacements is:
δ = δ1 + δ 2
u z1 + u z 2 = δ − Ax 2 − By 2
(2.1)
(2.2)
Where Ax2+By2 represents the body profile in the vicinity of the contact.
Hertz’s theory is based on two major assumptions. First, the material is assumed
linearly elastic over a small strain region. In order to meet this requirement, the
dimensions of the contact area must be small compared with the body dimensions and
with the radii of the contact surfaces. This assumption allows for the contact stress to be
treated separately. The second assumption is that the surfaces are frictionless. This
assumption allows only a normal force to be transmitted.
14
N
δ2
2
1
δ2
uz1
δ1
uz2
a
xy
Plane
a
δ1
Figure 2.2: Elastic Contact of Two Nonconforming Bodies
This project deals with a specific case investigated by Hertz, two cylinders in
contact. This case can also be illustrated in Figure 2.2 above, assuming plane strain
geometry into the page. The two cylinders contact along a line into the page, with a
contact area of 2a due to the applied load of N . The pressure distribution will take the
parabolic form shown in Figure 2.3. Again, since this study investigated two cylinders in
contact, the geometry will be symmetric, so the pressure distribution for this case will
also be symmetric.
15
P(x)
q(x)
a
a
xy
Plane
z
Figure 2.3: Pressure Distribution, Elastic Half Space
Now that the problem of two cylinders in contact has been setup, the equations for
the pressure distribution and normal load can be found. Johnson [21] gives a detailed
derivation of these equations, but for conciseness, only the results will be shown here.
The pressure distribution P( x) is found as:
P ( x) =
1
2N 2
2
2
−
(
a
x
)
π a2
(2.3)
16
In equation (2.3) the applied normal load on the bodies is denoted as N . It is defined as:
a=
4NR
π E*
(2.4)
The combined modulus of elasticity is needed to find the applied normal load. It is
defined in terms of Poisson’s ratio and the modulus of elasticity for the two materials.
This study involves two like materials in contact, so the combined modulus can be
defined as:
 1 −ν 2 
1
= 2

E*
 E 
(2.5)
The stress calculations can be made once the above equations are found.
Referring back to Figure 2.3, the elastic half space is shown with the applied pressure
distribution P( x) and traction q( x) . Recall Hertz’s theory is frictionless, so q( x) = 0 .
However, the derivation of the stresses is very in-depth and is shown in detail by Johnson
[21]. Both the normal applied force and the applied traction contribute to the stresses on
the bodies. Key points for the resulting stresses will instead be covered as follows:
ƒ
Stresses are largest at the axis of symmetry
ƒ
Far-field stresses asymptotically approach zero
ƒ
Stress on the surface σ z equals the applied pressure in the contact area
ƒ
Maximum principal shear stress occurs below the surface
17
2.2 SLIDING CONTACT
For this study an oscillating input force will be used to excite sliding in the
friction joint, which will dissipate energy. This section gives an overview of the sliding
contact of the two bodies according to Johnson [21]. A view of the two surfaces in
sliding contact can be seen in Figure 2.4. In this figure, the applied normal load is
denoted by N , with body 1 moving in the positive x direction. V is the sliding velocity
and 2a is the width of the contact strip.
N
2
Q
Q
V
1
a
x
a
z
Figure 2.4: Contact of Sliding Surfaces
If frictionless surfaces are assumed, then the contact area and pressure distribution
can be described by Hertz’s theory as outlined above. However, there is a frictional force
that will oppose the motion of the body. Accounting for the friction, let Q be the
tangential frictional force as shown in Figure 2.4.
18
It must now be determined if the tangential friction influences the size and shape
of the contact area, or if the distribution of the normal pressure is changed. It can be
shown [21] that the applied traction Q only slightly affects the pressure distribution
P( x) . Next, if the two materials are the same, as in this case, ( G1 = G2 and v1 = v2 ), then
the shape and size of the contact area are fixed by the normal load, and are independent
of the tangential force.
There are now three possibilities which may occur between the two bodies in
contact:
ƒ
Full stick between the two bodies
ƒ
Partial slip between the two bodies
ƒ
Gross slip between the two bodies
Since there is an applied tangential load for this study, only the last two cases will be
covered. The first case of full stick does not produce any relative motion between the
bodies and therefore does not dissipate any energy.
2.2.1 PARTIAL SLIP CASE
When an applied tangential force is less than the limiting frictional force, then no
gross slip between the two bodies will occur. However, a frictional traction does occur,
which tends to shear the body as illustrated in Figure 2.5 below. At the centerline there is
no displacement, but outside of this area are slip regions. This condition is referred to as
“stick” and “slip”.
19
N
δx2
δx2
ux1
2
Q
A
Q
B
ux2
1
a
a
δx1
A: Stick Region
B: Slip Region
δx1
Figure 2.5: Stick and Slip Regions of Two Bodies in Contact
The first case investigated will be that of partial slip for two cylinders in contact.
There is an applied tangential load that will not allow gross slip, 0 < Q < µ N . Much of
this initial work was performed by Mindlin and can also be referenced in his works, such
as [9, 10]. The tangential traction is described as the sum of the tractions over the regions
stick – slip regions:
q( x) = q '( x) + q ''( x)
(2.6)
20
The size of the stick region then can be calculated by integrating equation (2.6) over its
respective intervals of − a to a and −c to c , where:
c 
Q 
= 1 −

a  µN 
1
2
(2.7)
A
q
B
2
1
c
a
c
a
Figure 2.6: Stick and Slip Regions: Curve A=No Slip, Curve B=Partial Slip
We see from equation (2.7) that if the applied normal load N is held constant and
the tangential force Q is increased from zero, then slip begins at the edge of the contact,
a . As Q is increased, the slip region increases towards the centerline. Once Q ≥ µ N ,
then full slip will occur, and the bodies will have relative motion. This case is outlined
next.
21
2.2.2 GROSS SLIP CASE
Once the applied traction on the bodies is Q ≥ µ N , then the full slip case will
occur. The surface traction for full slip can be defined as:
q ( x) = ± µ N ( x)
(2.8)
Equation (2.8) is simple Coulomb friction. In terms of the applied normal load it is
defined as:
q ( x) = m
1
2µ N 2
2
2
−
a
x
(
)
2
πa
(2.9)
Note how the negative sign is associated with positive sliding velocity, V . The traction
is opposite in direction of the relative velocity.
2.3 ENERGY DISSIPATION
This section will show how an applied oscillating force causing partial slip will
dissipate energy in a system as discussed by Johnson [8]. The system analyzed will be
the contact of two spherical bodies, with an applied normal load N . There is an applied
oscillating tangential force that varies from +Q* to −Q* . The normal pressure can be
described by Hertz’s theory in a similar method as that illustrated above for a cylinder.
As in the previous section, the size of the stick region then can be calculated by
integrating the tractions over the stick and slip zones:
22
c 
Q 
= 1 −

a  µN 
1
3
(2.10)
Slip will occur in the annulus c ≤ r ≤ a as outlined by equation (2.10). The traction is
shown in Figure 2.7 below, curve A.
N
A
B
Q
r
C
c
c'
a
c
c'
a
A: Q = +Q*
B: Q = 0
C: Q = −Q*
Figure 2.7: Circular Contact from Oscillating Force
23
From the derived unloading and loading expressions for the displacement in the xdirection and the normal loading, a hysteresis loop can be created. Figure 2.8 below
illustrates this unloading and loading cycle for the applied tangential force.
Q
µN
Q*
δ
δ∗
-δ∗
-Q*
Figure 2.8: Hysteresis Loop
The work done by the system is the area in the hysteresis loop, which can be
calculated by the generalized expressions shown in [21]. For this case, the applied
tangential force is normal to the surface, so the expressions can be simplified. The work
done is shown as:
24
9µ 2 N 2  2 − ν 1 2 − ν 2
∆W =
+

G2
10a  G1
5
3
  
Q
5Q*

*

×
−
−
1
1


 −
6µ N
   µ N 
2
 
Q*  3 
1 −  1 −

µ N  
 

(2.11)
Finally, two more assumptions are made to reduce equation (2.11). If the
(
oscillation is small, then Q*
)
µ N << 1 and the materials are the same, then
∆W simplifies to:
∆W =
1  2 −ν 1 2 −ν 2
+

36a µ N  G1
G2
 3
 Q*

(2.12)
Goodman [22] was the first to notice this cubic dependence for the energy dissipation in
is research. He discovered that the energy dissipation per cycle is proportional to the
cube of the difference of Fmax − Fmin .
As shown from the above theory, slip and therefore energy dissipation can occur
at a tangential force level that is appreciably less than what is necessary for full slip of
two bodies. This dissipated energy is a source of damping in many built-up structures
and should be considered, leading to the long term focus of this study.
25
CHAPTER 3: MODAL THEORY
3.1 LINEAR SYSTEMS
The majority of modal theory is based on four major assumptions. These are that
the system is time invariant, observable, obeys Maxwell’s reciprocity law, and is linear.
For this study, it can be verified through standard testing procedures that the first three
conditions can be met. However, it is the last assumption, linearity, which will be
discussed in this work. A linear system can be described by a set of linear, second order
differential equations.
Mathematically, linearity can be defined by the principle of superposition, which
can be summarized as follows. The response of a system x1(t) to an applied force F1(t),
and the response of the same system x2(t) to a second force F2(t) will be additive. So if
an arbitrary force of αF1(t)+βF2(t) is applied, the system’s response will be αx1(t)+βx2(t),
where α and β are constants as described by Worden and Tomlinson [23].
There is a quick way to experimentally check the linearity of a system. For a
frequency response measurement, the output of a system is divided by the input. For a
linear system, doubling the input force into a system will yield double the response.
Since the FRF is this ratio, it will stay the same. Therefore, for any excitation level, the
FRF of a linear system will remain the same.
3.1.1 SDOF SYSTEM OVERVIEW
This section will cover a very brief overview of the calculations for a single
degree of freedom system. This will introduce the modal nomenclature and give the
reader a quick summary of the topics covered. For a much more comprehensive
26
discussion on this topic, the reader is referred to any good vibrations book, such as
Thomson and Dahleh [24] or Allemang [25].
For a given SDOF system:
F(t)
x(t)
c
m
k
Figure 3.1: SDOF System
••
•
m x (t ) + c x(t ) + kx(t ) = F (t )
(3.1)
If F (t ) = 0 , then
••
•
m x (t ) + c x(t ) + kx(t ) = 0
(3.2)
(ms 2 + cs + k ) Xe st = 0
(3.3)
Letting x = Xe st , then
27
λ1 = − c 2m +
(c 2m) − (k m)
2
λ2 = − c 2m −
(c 2m) − (k m)
2
(3.4)
Therefore,
x(t ) = Aeλ1t + Beλ2 t
λ2 = λ1* = σ 1 − jω1
λ1 = σ 1 + jω1
(3.5)
(3.6)
To determine the frequency response function of a system, the equation of motion is
written in the frequency domain,
(− mω
2
)
+ jcω + k X (ω ) = F (ω )
(3.7)
H (ω ) = X (ω )
(3.8)
The FRF is defined as,
F (ω )
With,
H (ω ) =
1
− mω + jcω + k
2
(3.9)
28
As an example, the FRF of a theoretical SDOF system is shown in Figure 3.2
below. The FRF is plotted as the top graph, and the phase is the lower graph. In Figure
3.3, the same SDOF system is plotted on a complex plane.
Log Magnitude of H(ω)
-3
Magnitude
10
-4
ωn=22.36
10
-5
10
-6
10
0
10
20
30
50
60
70
80
90
100
70
80
90
100
Phase Angle of H(ω)
0
Angle [Deg]
40
-50
-100
-150
-200
0
10
20
30
40
50
60
Frequency [Hz]
Figure 3.2: FRF and Phase Plot, Example SDOF System
29
-4
1
Complex Plot of H(ω)
x 10
Imaginary part of H(ω)
0
-1
-2
-3
-4
-2
-1
0
Real part of H(ω)
1
2
3
-4
x 10
Figure 3.3: Complex Plot, Example SDOF System
3.1.2 MODAL TESTING OF LINEAR SYSTEMS
Much of the modal testing and parameter estimation methods used today revolve
around the assumption of linearity. Typical inputs for modal testing are impact, random,
burst random, chirp, steady state sine, and swept sine. Each method has advantages and
disadvantages that must be weighed before implementation. Random signals are very
popular as they usually are a quick and efficient input to characterize a linear system.
There are also a number of popular frequency response function estimators that
can be applied to experimental data. The H1 estimator assumes that there is no noise on
the input but noise on the output. It is always a lower bound of the actual FRF. The H2
30
estimator is just the opposite, assuming no noise on the output but noise on the input.
This function is an upper bound to the actual FRF. Finally, the H3 estimator is an
“average” between H1 and H2, assuming error on both input and output. Some articles
such as by Sperling and Wahl [26] argue that the H3 estimator is the best choice for a
nonlinear system, when the goal is producing a linearized model of the nonlinear system.
3.2 NONLINEAR SYSTEMS
The principle of superposition states that a linear system can be characterized as
the addition of lesser inputs and outputs. For a nonlinear system, this is not the case. A
nonlinear system will deviate from this trend and cause unexpected results. However,
many systems can be linear in certain ranges. These points are illustrated in Figure 3.4
below.
In this figure, the linear and nonlinear systems overlay well at small values of
x(t ) . However, at higher excitation levels, the linear and nonlinear systems deviate.
This simple example illustrates how even a nonlinear system can be linear in certain
ranges.
31
F(t)
Nonlinear
F1+F2≠F3
Linear
F2
F1
x1
x2
x1+x2=x3
x(t)
Figure 3.4: Example of the Principle of Superposition, on a Hardening System [23]
The FRF of a nonlinear system will often have more then one solution at a
frequency, so its solution is not unique. An example of a FRF for a nonlinear system
illustrating this fact is shown in Figure 3.5. This “saw-tooth” effect of the FRF is often
missed during testing, as only one of the values on the curve can be acquired. Because of
this, it is a common practice to sweep up in frequency, and then sweep down. This will
yield the inner portion of the saw-tooth and tip of the saw-tooth respectively.
32
Magnitude
Freq [Hz]
Figure 3.5: Analytical FRF Curve of a Nonlinear System
3.2.1 MODAL TESTING OF NONLINEAR SYSTEMS
The typical excitation methods used in linear systems do not work as well for
nonlinear systems. As mentioned above, a random signal is a popular choice for linear
systems. The use of this type of signal will however average out a system’s nonlinearity
by its broadband, multi-amplitude nature. Therefore, other excitation signals are used in
nonlinear systems. Probably the most popular excitation method is that of a sinusoid. A
sinusoidal excitation concentrates all of the input energy at a specific frequency. Another
advantage of a sine input is that it has very good noise and harmonic control, by a result
of integration [27, 28]. However, a major disadvantage is the time necessary to complete
a test. Since the system is excited by a sine wave at a specific frequency, it takes
considerably more time to step through a frequency range compared to a random signal at
33
a wide frequency band. The system must reach steady-state, and with a lightly damped
system the time needed to reach steady-state is further increased.
Since much of modal testing today is built around linear systems, some of the
hardware and software requirements to perform a swept sine modal testing might not be
as readily available. A common technique used to analyze a nonlinear system with a
random excitation is shown in Figure 3.6. The plot represents actual FRF data collected
from the test rig described in Chapter 4. The variance in the FRF’s illustrates
nonlinearity in the system at different excitation levels. As mentioned in the SDOF
theory, the FRF is simply a ratio of output to input, so if the system was linear the FRF’s
would all be identical.
Log Magnitude
10
10
3
2
1
10
170
175
180
185
190
195
200
Frequency [Hz]
205
210
215
220
Figure 3.6: Nonlinear System FRF with Varying Input Levels & Random Excitation
34
In addition, a trend can be witnessed when the magnitudes of the FRF’s are
plotted versus excitation level. Figure 3.7 illustrates a fluctuating trend for the data
presented in Figure 3.6. These results found in the previous two figures foreshadow the
nonlinearity which will be investigated for this project.
1.2
1.1
1
Normalized Force
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1
2
3
4
5
6
Increasing Force Values
7
8
9
10
Figure 3.7: Peak FRF values vs. Force Level
3.3 SENSOR MASS LOADING
Throughout testing on the rig, a number of different sensors and normal loading
conditions were used for acquiring data. When processing the data, it was noticed that
the FRF’s of the system varied slightly, having different natural frequencies. The
35
accelerometers used varied in mass, and the number of accelerometers varied. It is
believed that the mass loading of the different accelerometers and the normal load on the
beams had an effect on the natural frequencies.
Figure 3.8 shows three testing setups on the beams. In the figure, the right FRF
peak occurs around 224 Hz. This setup had one small accelerometer on the tip of the
beam. The middle FRF peak occurs around 206 Hz. This setup had one large
accelerometer on the tip. Finally, the left FRF peak occurs around 194 Hz, and had two
large accelerometers on the tip, and one on the center of the beam, for a total of three
large accelerometers. This and the varying normal loading of the beams could have
influenced the stiffness of the beam, shifting the frequencies.
2.74
10
2.73
10
3 Large Accel's
1 Small Accel
Log Magnitude
2.72
10
1 Large Accel
2.71
10
2.7
10
2.69
10
2.68
10
2.67
10
180
190
200
210
Freq [Hz]
220
230
240
Figure 3.8: Sensor Mass Loading Effects
36
3.4 SUMMARY
The above information gives a brief overview of some of the characteristics of
nonlinear problems. It also illustrates how a nonlinear problem can be difficult to solve,
both analytically and experimentally. Care must be taken while acquiring data, or the
nonlinear characteristics of the system which of interest can be disguised or removed.
Testing parameters to note include but are not limited to:
ƒ
Excitation methods
ƒ
Excitation levels
ƒ
Frequency bands
ƒ
Acquisition software
Care must also be used when analyzing the data. Many of the readily available
tools and software make the assumption of linearity in their algorithms. As can be seen
in Figures 3.6 and 3.7, there is clearly some nonlinear trends with this project. Testing
the system without care would have quickly disguised the results.
37
CHAPTER 4: TEST RIG DESIGN
4.1 INTRODUCTION
In order to characterize the nonlinear friction joint in this project, a test rig was
designed and built. As mentioned earlier, Mr. Watts, an exchange student from the
University of Newcastle upon Tyne, initially designed the test rig [29]. He designed the
basic rig, creating 2D AutoCAD prints for some of the parts. Mr. Watts also specified
and ordered the necessary materials, and began machining of the rig.
The goal of the test rig was to allow the joint to be investigated in a wide variety
of ways, allowing for a wealth of future research. This thesis began with the completion
of the test rig that Mr. Watts was not able to finish. Once the machining and assembly of
the rig was completed, a solid model of the rig was created. The solid model of the rig
can be seen in Figure 4.1. The rig’s design accommodates such features as:
ƒ
Variable beam length
ƒ
Controllable normal load
ƒ
Consistent boundary conditions
ƒ
Flexible excitation location
ƒ
Simple structural dynamics
38
Figure 4.1: Isometric View of Original Test Rig
39
4.2 ORIGINAL DESIGN
The rig consists of two 12” square plates which are 1.5” thick. The bottom plate
is aluminum, while the middle plate is steel. The center plate is constructed of steel to
provide a larger normal force on the beams. A large force provides a sizeable contact
area between the beams and therefore a larger slip zone and more energy dissipation.
The plates both have a square area machined out of them to mount the beams. The beams
are held in place by adding a small shim that is the same dimensions of the test beams.
Finally, the beams are secured by tightening a bolt in each plate, which applies the
clamping force to the roots of the beams. A right side view and front view of the rig are
shown in Figures 4.2 and 4.3 respectively.
The steel center plate slides on four guide rods. Attached to the center plate are
pipe flanges, with steel pipe sections screwed into them. The steel pipe sections translate
on the four guide rods with the use of eight nylon bushings. This up and down motion
allows for a range of different beam lengths to be installed into the rig, while also
providing proper alignment of the contact area.
Next, the rig has a top aluminum plate, also 12.0” square. This plate mounts to
the top of the four guide rods. It is 0.5” thick, as it is only intended to torsionally stiffen
the structure and maintain alignment.
40
Figure 4.2: Right Side View
Figure 4.3: Front View
The final components of the rig are the two test beams. The beams are
constructed of aluminum, 9.0” long, 1.00” wide, and 0.50” thick. The lower beam has a
convex tip, while the upper beam has a concave tip, as shown in Figure 4.4. The
dimensions for these radii are 0.787” and 0.788” respectively, as outlined by Mr. Watts.
However, these given dimensions are difficult to manufacture and are discussed in detail
later in Section 4.3.
41
Figure 4.4: Close View of Test Columns
Lastly, the rig was intended to mount to the floor of the Structural Dynamics
Research Laboratory here at the University of Cincinnati. The four guide rods protrude
though the bottom aluminum plate and are threaded. These rods then can be screwed into
the threaded steel slabs that are inlaid in the laboratory floor.
For the original rig design, a list of the materials, dimensions, and quantity can be
seen in Table 4.1.
42
Description
White Nylon Bushing
Plain Steel Hex Nut
Square Head Bolt
Cap Screw
Aluminum Bar, Top Beam
Aluminum Bar, Bottom Beam
Threaded Black Welded Steel Pipe
Iron Threaded Pipe Flange
Carbon Steel Guide Rod
Aluminum Top Plate
Steel Middle Plate
Aluminum Base Plate
Dimensions
1" OD x 0.625" ID x 1"
0.625" – 11 UNC
0.375" – 16 UNC x 5.5"
0.5" – 20 UNF x 1.5"
0.5" x 1" x 9"
0.5" x 1" x 9"
1" ID x 14.5", 1.3125"
1.3125" – 1" NPT 11-1/2
0.625" – 11 UNC
0.5" x 12" x 12"
1.5" x 12" x 12"
1.5" x 12" x 12"
Quantity
8
12
2
16
1
1
4
4
4
1
1
1
Table 4.1: Test Rig Part List
4.3 BEAM CONTACT SURFACES
Under the original design, a critical aspect of the test rig is the design and
machining of the test beams. The beam contact surfaces are designed to give as much
contact as possible under normal loading. To achieve this, the conforming cylindrical
surfaces were designed to be 0.787” and 0.788”. Figure 4.5 illustrates the beam surfaces.
43
Figure 4.5: Beam Contact Surfaces
Initially, a first set of beams was machined at the university machine shop. After
inspection, it was not possible to meet the design dimensions. Therefore, it was decided
to machine another set of beams. The original set was used for initial testing and setup.
The second set of beams was machined out of house, and their dimensions were checked
with an optical comparator, to which the university does not have access. The results for
the second set of beams can be seen in Figures 4.6 and 4.7.
44
Figure 4.6: Bottom Aluminum Testing Beam, R=0.78114”
Figure 4.7: Top Aluminum Testing Beam, R=0.78565”
45
As can be seen from the above results, again the original design dimensions could
not be met. The outlined original dimensions are very difficult to achieve, since they are
only 0.001” different. From a contact mechanics view if the geometry of the two mating
parts is close, more material will be in contact and therefore more energy will be
dissipated. The final dimensions of 0.781” and 0.786” for these beams will still yield a
suitable contact area.
4.4 TEST RIG MODAL ANALYSIS
Upon completing the machining and assembly of the rig, some initial testing was
performed. A full modal test of the rig was performed to verify that the rig was actually
performing as intended. The goal of the rig was to isolate the beams in a first bending
mode, without any interaction from the rig itself.
To perform an initial shakedown of the test rig, impact testing was performed.
This method is quick to setup and provides a basic idea of the system parameters.
Testing was performed using the University of Cincinnati SDRL software MRIT [30].
This software was designed specifically for impact testing. The equipment used for this
testing can be seen in Table 4.2.
For testing, the rig was securely mounted to the laboratory floor. The beams were
brought into contact under a full normal load of 370 Newtons.
46
Description
Gateway PC
HP VXI Mainframe
HP VXI Communications Card
HP VXI Source / Data Acquisition Card
HP ICP / Voltage Box
PCB Patch Panel
Nuplaglas Impact Hammer
HP Dynamic Signal Analyzer
PCB Hand Held Calibrator
PCB Load Cell
PCB Accelerometers
Model Number
GP7-450
E8408A
E8491A
E1432A
070C29
35670A
394C06
See Appendix
Table B1
See Appendix
Table B1
Serial Number
0013379071
US39000125
US37000393
US37060141
US35370567
US35371399
3245A00229
1155
See Appendix
Table B1
See Appendix
Table B1
Table 4.2: Modal Testing Equipment List
Accelerometers were placed on the rig as shown in Figure 4.8. There were a total
of 38 locations on the rig. At these 38 locations, accelerometer blocks were mounted.
This allowed the accelerometers to be mounted in the X, Y, and Z-directions. For the
data collection, all of the accelerometers were oriented in one direction, and a set of data
was collected. The sensors were then reoriented, and again data was collected. This
process was performed for all directions and at various forcing levels. The impact
locations for all of the testing occurred on the lower beam, as this is the location of the
shaker in later testing.
47
Impact
Figure 4.8: Accelerometer Placement on Test Rig
Once all of the data was collected using MRIT, it was then analyzed in X-Modal
II [31]. X-Modal II is a modal analysis software program that runs on a MatLab engine,
48
created at the University of Cincinnati SDRL. The structure is digitized and a geometry
file is created to show the animated rig. A still frame of the animated rig can be seen in
Figure 4.9.
Beams in First Bending,
No Other Rig Response
Figure 4.9: Beams in Contact, Significant Motion of Beams in First Bending, 194 Hz
49
With the data processed and mode shapes animated, it was found that in the range
of 40 to 300 Hz, there was no significant rig response. In this range, there is only one
mode, at 194 Hz. This mode is the beams in first bending. This is also the mode shown
in Figure 4.9. This result is significant, as it shows that there are no other modes of the
rig influencing the results of the data.
4.5 TEST RIG SETUP
After the modal test of the rig was completed, setup of the rig was begun for data
collection. To start, it was decided to add a large base to the test rig. This is due to the
fact that the laboratory space was needed where the rig was originally intended to mount.
The added steel base is 26.0” x 15.0” in size, and is 1.0” thick. The base is threaded to
accept the four guide rods for secure mounting. An additional feature the base allows is
for shaker mounting. The base is threaded to rigidly mount a shaker, which is elevated to
the proper height on the beam with the use of steel c-channel. Rubber sheets are used in
between all of the equipment and the base in an attempt to isolate any vibration. The rig
setup can be seen in Figure 4.10 below.
Following mounting of the rig and shaker to the base plate, the rig was outfitted
for testing. The shaker was connected to the upper portion of the lower beam,
approximately 1.0” below the friction joint. The load cell was attached to the lower beam
with dental cement. Two accelerometers were mounted, one on each beam,
approximately 0.5” from the friction joint.
50
Base Plate
Figure 4.10: Isometric View of Original Test Rig Assembly
During this initial setup, it was discovered that the normal load on the beams was
too great. With the shaker used for excitation, it was not possible to slip the friction joint.
Originally, the rig was designed to provide a large normal force on the beams. This large
normal force in conjunction with close conforming beam radii of 0.787” and 0.788”, were
51
to provide a large contact area between the beams. To achieve this large normal load, the
large steel center plate was designed. For even greater normal loading, it was intended
that steel weights could be stacked on this center plate.
Since exciting the friction joint and achieving slip was the critical point of this
experiment, it was determined that the normal load needed to be reduced. There were a
few options to accomplish this. One option was remove weight from the large steel plate.
This could be accomplished by milling the plate thinner, or by drilling holes in the plate.
Another option was to “lift” some of the weight off of the beams. With some
available materials at hand, a threaded rod was attached to the center plate, using the
existing pipe flange bolt holes. This threaded rod protrudes through the top aluminum
plate. Lastly, a spring and nut are assembled on the top of the threaded rod. This setup
allows for the nut to be tightened down, lifting weight off the beams. It also allows for a
quick change of normal loads on the beams throughout the testing process. In the future,
if larger normal loads are need, weights can still be added to the center plate.
These changes can be seen in Figure 4.11. A right side view and top view of the
completed assembly are show in Figure 4.12 and 4.13 respectively. Figure 4.12 also
illustrates the alignment of the shaker on the bottom beam.
52
Mechanism to
Control Normal
Load
Figure 4.11: Isometric View of Test Rig Assembly
53
Figure 4.12: Right Side View of Test Rig Assembly
Figure 4.13: Top View of Test Rig Assembly
54
CHAPTER 5: TESTING ISSUES
5.1 INTRODUCTION
Once the rig was modified and setup, there were still some issues that need to be
resolved before data could be collected. Questions such as testing hardware, software,
testing methods, and monitoring normal load all need to be answered. This chapter will
cover these topics.
5.2 HARDWARE
To perform the testing on the rig, there were a few data acquisition hardware
choices. The first choice was to use a Hewlett Packard 3565 system. This system was a
good choice, as it is readily available at the university, and not under high demand by
other students. However, after using it for some initial testing, the disadvantages of this
older technology were apparent. The autoranging feature was very slow when compared
to more current data acquisition hardware. Accordingly, it was decided to use a Hewlett
Packard VXI system. This system is faster and more compact. The only disadvantage is
that it is often in demand by other students.
5.3 SOFTWARE
A big issue for this project was the acquisition of testing software. The in-house
University of Cincinnati SDRL testing software MIMO [32] only supports random
excitation signals, not a sinusoid. As discussed earlier, for a nonlinear system, a random
excitation will essentially average out the nonlinearity of the system. Therefore, it was
55
necessary to acquire a testing package that would interface with the university’s existing
testing hardware, and provide for the sine excitation.
It would be possible to modify the current MIMO software, but from a time
standpoint, other testing software was investigated. After some research, the Sine
Testing Package for I-DEAS was acquired from MTS Systems [33]. This package allows
for many of the features found in the university software, but also allows for sine and
swept sine excitation.
Another advantage of the I-DEAS testing package is that it offers force control.
Under typical testing conditions, the force input level drops at a resonance, as the system
is easily excited. The I-DEAS package has an option to activate a feedback controller on
a selected channel. An example of this phenomenon can be seen in Figure 5.1 below,
which is an autopower spectrum. Shown in the plot is the autopower curve on the force
channel. The drop in force can be seen at 220 Hz, which is the resonance of the system.
However, when the force controller is activated for the load cell, the input force is kept
constant, even through resonance. This phenomenon is also shown on the plot.
With this project, it was desired to keep the excitation force levels constant
throughout the resonance of the system. This is a common goal with frictional testing
[18]. In addition, with this option it would be possible to monitor the acceleration
channels, keeping an acceleration channel constant through resonance.
56
Autopower Spectrum, Input Channel
3
10
2
No Force Control
Excitation Force2 [N2]
10
1
10
0
Force Control
10
-1
10
200
205
210
215
220
Freq [Hz]
225
230
235
240
Figure 5.1: Autopower, Force Control and No Force Control
5.4 NORMAL LOAD
The last issue for the rig setup was a method of determining the normal load on
the beams. The most common method for this task would be with strain gages. With all
of the necessary equipment on hand, this was the selected method. The beams were
mounted with strain gages at the root of each beam. On each of the 1.00” faces of the
beams, a strain gage was mounted horizontally and vertically. The gages were then wired
to a switch box in a full bridge format. The switch box is then connected to a strain
indicator box, which digitally displays the normal load in micro strain. In this setup, the
strain gages will not read any bending influence of the beams, but will rather just read the
57
normal load as desired. A close up view of the strain gages mounted on a beam is shown
in Figure 5.2.
Figure 5.2: Strain Gages Mounted on Beam
5.5 FINAL TEST SETUP
As a final overview, this section will illustrate the entire testing setup. Figure 5.3
shows the entire setup used to acquire data. All of the equipment is listed in Table 5.1.
For completeness, additional figures of the setup can be seen in Appendix A. Figure A.3
illustrates the entire rig setup. Figure A.4 is a photo of the strain indicator setup. Figure
A.5 is a close-up view of the shaker mounting location on the lower beam. Figure A.6 is
58
the HP VXI equipment setup. A close-up view of the accelerometer mounting locations
is shown in Figure A.7. Finally, Figure A.8 shows the PC, amplifier, DSA, and function
generator equipment.
Figure 5.3: View of Testing Setup
59
Description
Gateway PC
HP VXI Mainframe
HP VXI Communications Card
HP VXI Source / Data Acquisition Card
MB Dynamics Amplifier
MB Dynamics Shaker
Strain Indicator Box
Switch & Balance Box
HP Dynamic Signal Analyzer
PCB Hand Held Calibrator
PCB Load Cell
PCB Accelerometers
Model Number
GP7-450
E8408A
E8491A
E1432A
SS250VCF
Modal 50
P-3500
SB-10
35670A
394C06
See Appendix
Table B2
See Appendix
Table B2
Serial Number
0013379071
US39000508
US37000393
US37060141
290418
000001
0121324
61104
3245A00229
1155
See Appendix
Table B2
See Appendix
Table B2
Table 5.1: Testing Equipment List
60
CHAPTER 6: DATA ANALYSIS
6.1 CHARACTERIZATION METHODS
There are numerous methods to investigate nonlinearity in a system. Results from
testing can be investigated in the frequency domain and the time domain, each having
results that may be more helpful then others. For data collected from this test rig, a
selected few methods are outlined and discussed in this chapter.
6.1.1 FRF DISTORTION
The first characterization method discussed will be FRF distortions. This method
was briefly discussed earlier in Section 3.2.1. The frequency response function is the
ratio of the output of the system divided by the input to the system. If a system is linear,
then this ratio will have the same value, regardless of input level, as the output will scale
accordingly. This equation is shown in equation (3.8), restated here:
H (ω ) = X (ω )
F (ω )
(6.1)
Since this friction joint is nonlinear, we expect that if we excite the system at a
forcing level and then a higher forcing level, we would not receive the same output
values. If we overlay the FRF’s at different forcing levels, there should be a noticeable
difference.
If the magnitudes of the FRF’s are different, then this occurrence can usually be
attributed to a change in damping of the system. If the natural frequency of the system
shifts, then this trend can usually be attributed to a change in stiffness of the system.
61
6.1.2 COMPLEX PLOTS
The second characterization method discussed is that of complex plots. Since the
FRF is a complex function, it contains both magnitude and phase information of the
system. Depending on the nonlinearity, one or both of these quantities can be affected.
This is where a complex plot may become useful. A complex plot is the imaginary part
of the FRF versus the real part of the FRF.
For a linear system, the complex plot will appear as a perfect circle. This was
illustrated earlier in Figure 3.3, on an analytical system. In a dissipative nonlinearity,
often the complex plots will be characterized by their oval shape and pear shapes. For
example, the complex plots for quadratic damping tend to decrease in size as the input
force is increased. In addition to the size reduction, they stretch along the real axis, into
an oval shape. In the case of Coulomb friction, the complex plots increase in size as the
input force is increased. Also, the shape changes from pear shape at low levels to circular
at higher levels. These phenomena can be seen in Figure 6.1 [23]. As a general rule, if
the complex plot deviates from a circle or near circle, then a nonlinearity in the system is
suspect.
62
Linear:
Re H(ω)
|H(ω)|
Freq
Im H(ω)
Quadratic Damping:
Re H(ω)
|H(ω)|
Freq
Im H(ω)
Coulomb Friction:
Re H(ω)
|H(ω)|
Freq
Im H(ω)
Figure 6.1: Complex Plot Distortions [23]
63
6.1.3 INVERSE FRF
Another useful tool used to investigate a nonlinear system is the use of inverse
FRF’s. This method inverts the frequency response function into what is referred to as
the dynamic stiffness function, or DSF [19, 23]:
In vers e FR F ( ω ) = I ( ω ) = −m ω 2 + jc ω + k
(6.2)
The inverse FRF is broken up into its real and imaginary parts:
Re {I ( ω )} = −m ω 2 + k
Im {I ( ω )} = jc ω
(6.3)
(6.4)
For linear systems, if Re {I ( ω )} is plotted versus ω 2 and Im {I ( ω )} is plotted
versus ω , the resulting plots should yield straight lines. The curve of the real part will
have an intercept of −k and slope of −m . The curve of the imaginary part will pass
through the origin and have a slope of c . For nonlinear systems, these plots will deviate
from straight lines, at the frequency of the nonlinearity. If the system has a nonlinear
stiffness, it will be viewed in the real plot, and conversely nonlinear damping will
influence the imaginary plot.
When implementing this characterization method, one must note the influence of
other nearby modes in MDOF systems. However, for this study it has been shown that
the influence of nearby modes is not significant.
64
6.1.4 HYSTERESIS LOOPS
In the time domain, hysteresis loops are useful for determining parameters of a
system. A hysteresis loop plots displacement versus load. One complete cycle of the
response creates one loop on the plot. Since the axes of the plot are force and
displacement, the area in the curve is the work performed. A sample figure of a
hysteresis loop was shown earlier in Chapter 2, Figure 2.8.
For this study, the force axis was normalized to yield the coefficient of friction of
the beams. Since the normal loading and excitation forces on the beams are known, it
was a simple matter of plotting µ = F
N
. However, the current rig’s design did not
allow the normal load to be as unvarying as desired. This issue could not be accounted
for with this data, and will be further discussed in Section 8.2.4.
6.1.5 HILBERT TRANSFORMS
The last characterization method covered is that of the Hilbert transform. There
are applications for the Hilbert transform in both the time and frequency domains [19,
34]. The frequency domain transformation is similar to that of the Fourier transform.
Mathematically, its kernel function is different, instead of e
j ωt
for the Fourier, the
Hilbert is −1 j π ( ω − ω0 ) :
Re {H } =
2 ω0
π
ω
∫
0
Re {FRF( ω )}
ω 2 − ω0 2
dω
(6.5)
65
Im {H } =
ω
2 ω Im {FRF( ω )}
dω
π ∫0
ω 2 − ω0 2
(6.6)
Through the use of complex algebra and Cauchy’s theorem, it can be shown that
the equalities hold for a linear system:
H {Im {FRF( ω )}} = − Re {FRF( ω )}
(6.7)
H {Re {FRF( ω )}} = Im {FRF( ω )}
(6.8)
If these equalities do not hold, then it can be an indicator of a nonlinearity in the
system. It must be noted however that there can be truncation errors due to the fact that
the analytical expression for the Hilbert transform is over the interval −∞ to ∞ . There
are specific correction methods developed to help resolve this, but again care must be
used. Truncation near resonance will yield poor results. In order to avoid these
problems, data was collected with a sizable frequency band before and after resonance, so
the truncation problems will not be an issue.
66
CHAPTER 7: RESULTS
7.1 TESTING RESULTS INTRODUCTION
Now that the theories for the characterization methods have been described in
Chapter 6, they will be applied to data taken from the rig. Much time was spent
collecting data, in both the time and frequency domains. As will be discussed in Chapter
8, there are a number of issues that make data collection challenging on the rig.
As a staring point in the data collection, FRF’s were collected on the beams over a
broad frequency range, 0 to 1400 Hz. This data was collected with a burst random input
to again verify that there was no other nearby modes that may influence the data
processing techniques. As can be seen in Figure 7.1, it appears the SDOF techniques
should be effective.
Once the wide frequency range data was collected, subsequent data was collected
at a smaller range. The smaller frequency range was selected in order to expedite testing,
as the step sine testing with feedback control is very time consuming.
In summary, the design of experiments for this study included varying the
following factors:
ƒ
Normal Load
ƒ
Shaker Input Force
ƒ
Frequency Band
ƒ
Sampling Frequency
67
Log Magnitude
FRF Plot, Bottom Beam
2
10
0
10
50
100
150
200
250
Freq [Hz]
300
350
400
100
150
200
250
Freq [Hz]
300
350
400
Phase Angle [Radians]
4
2
0
-2
-4
50
Figure 7.1: Wideband FRF of Bottom Beam
7.1.1 FRF, PHASE, AUTOPOWER PLOTS
The main data collection time was spent recording FRF data. Data was collected
from 180 to 240 Hz in steps of 0.1 Hz. For this data, the beams were fully loaded with
the weight of the center plate, 370 N. The lower beam was excited at five varying force
levels: 0.25 N, 0.50 N, 1.00 N, 1.50 N, and 2.00 N. These testing conditions of normal
load and excitation levels stick the joint, with microslip occurring at the edges of contact.
As can be seen in Figure 7.2, as the excitation force was increased, the FRF
becomes lower in magnitude and more rounded, indicating the increased damping that is
occurring. Shown in the bottom graph of the Figure 7.2 is the autopower of the input.
The controller was able to keep a fairly constant force throughout resonance, but it began
68
to deviate at 2.0 N. An input of 3.0 N was attempted, but the controller could not handle
this level, and the power spectrum drastically dropped at resonance.
FRF Plot
3
Log Magnitude
10
2
10
1
10
180
190
200
210
Freq [Hz]
220
230
240
Phase Plot
Phase Angle [Radians]
4
Force 1
Force 2
Force 3
Force 4
Force 5
3
2
1
0
-1
180
190
200
210
Freq [Hz]
220
230
240
220
230
240
AutoPower, Input Channel
6
Excitation Level2 [N2]
5
4
3
2
1
0
-1
180
190
200
210
Freq [Hz]
Figure 7.2: FRF, Phase & Autopower
69
7.1.2 COMPLEX PLOTS
Using the above data, at full normal load on the beams and five forcing levels,
complex plots were created. Shown in Figure 7.3, as the excitation level was increased,
the complex plots became smaller in diameter. This seems to agree with Figure 6.1,
which describes quadratic damping.
AutoPower, Input Channel
Excitation Level2 [N2]
6
4
2
0
180
190
200
210
Freq [Hz]
220
230
240
Nyquist Plots
1000
Force 1
Force 2
Force 3
Force 4
Force 5
Imag of FRF
800
600
400
200
0
-1500
-1000
-500
0
Real of FRF
500
1000
1500
Figure 7.3: Autopower & Complex Plots
70
7.1.3 INVERSE FRF
Figure 7.4 below shows the application of the inverse FRF method to the above
data. As described in Chapter 6, the inverse FRF method divides the FRF into its real
and imaginary components. Plotted in the first part of the figure is the real component of
the FRF. As can be seen, it is a fairly linear. The second plot is that of the imaginary
part. There is clearly an increasing dip at the resonance of the beams, as the excitation
force is increased. Since this deviation from linearity occurs in the imaginary
component, it points to a nonlinearity in damping. The second dip in this plot, around
238 Hz does not seem to vary with excitation levels, therefore it might be a system
characteristic, unnoticed with the other detection methods.
Inverse FRF Plots
Real of 1/FRF
0.5
0
-0.5
Force 1
Force 2
Force 3
Force 4
Force 5
3.5
4
4.5
5
5.5
Freq2 [Hz 2]
4
x 10
Imag of 1/FRF
0.01
0.005
0
-0.005
-0.01
180
190
200
210
Freq [Hz]
220
230
240
Figure 7.4: Inverse FRF’s
71
7.1.4 HYSTERESIS LOOPS
The next set of data presented deals with time domain data. A number of
hysteresis loops were created, at a fixed normal load. The normal load was selected to be
130 Newtons. This loading was selected in order to force the beams within a safe
operating range of the shaker, and still receive data in the stick, stick – slip, and gross slip
areas of the beams. Since the normal loading was held constant, the forcing levels were
varied to produce these results.
For this excitation force, a fixed sine wave excitation was used on the lower
beam. The sine wave had a frequency of 50 Hz; chosen because it is significantly lower
than the mode of the system. As mentioned earlier, the load axis was normalized to read
the value of the coefficient of friction. This axis will be useful to monitor the coefficient
values as the system is excited over its stick – slip range.
Data was first collected with a sampling step of 0.004 seconds, but this was found
to be too coarse to yield good results when plotting the hysteresis loops. Therefore, the
final set of data was collected with a sampling step of 0.0002 seconds.
Four increasing voltage levels were used to excite the joint. The voltages used
correspond to a tangential force of 18, 36, 54, and 72 Newtons. It was found at higher
forcing levels the data was too noisy due to the stick – slip nature of the joint to yield
good results. A sample result for these four excitation levels are presented in Figure 7.5.
The figure consists of five subplots. The top plot is the time signals for the load cell and
two accelerometers. The middle left plot is the beam acceleration signals plotted against
each other. The middle right plot is relative acceleration of the beams versus the
coefficient of friction. The bottom left is the relative velocity of the beams versus the
72
coefficient of friction. Lastly, the bottom right is the relative displacement versus the
coefficient of friction, also known as the hysteresis loop. This data is presented in a
similar matter as to how Gregory et al. presented it in Figure 1.6 [17]. The time signals
show the development of noise in the measurement as the excitation is increased, which
could be leading to some of the distortions in the hysteresis loops. At the higher forcing
levels the measurements are more difficult to acquire.
Figure 7.6 is an overlay of the hysteresis loops for the four cases. The loops show
the expected trend of increasing in area as the excitation force is increased. Figure 7.7 is
a close up view of the acceleration signals across the joint. As shown by Gregory et al. in
Figure 1.6 [17], the plot forms a Lissajous pattern. While the results may not be as clear
as those seen by Gregory, it is an indicator that the rig is performing in a similar manner
as their test setup.
73
Time Signals, Force and Acceleration
100
Magnitude [Volts]
50
Force
Top Beam
Bottom Beam
0
-50
-100
7.95
7.955
7.96
7.965
7.97
7.975
Time [Sec]
7.98
Acceleration vs. Acceleration
7.985
7.99
7.995
8
Relative Acceleration Loop
50
Coefficient of Friction
Bottom Beam Accel
0.6
0
0.4
0.2
0
-0.2
-0.4
-0.6
-50
-40
-20
0
20
Top Beam Accel
40
-20
0.6
0.6
0.4
0.4
0.2
0
-0.2
-0.4
-0.6
-0.02
20
Hysteresis Loop
Coefficient of Friction
Coefficient of Friction
Relative Velocity Loop
-10
0
10
Relative Acceleration
0.2
0
-0.2
-0.4
-0.6
-0.01
0
0.01
Relative Velocity
0.02
-5
0
Relative Displacement
5
-5
x 10
Figure 7.5: Hysteresis Loops, Forcing Level 54 Newtons
74
Hysteresis Loop
Forcing Level
Forcing Level
Forcing Level
Forcing Level
0.6
1
2
3
4
Coefficient of Friction
0.4
0.2
0
-0.2
-0.4
-0.6
-5
-4
-3
-2
-1
0
1
Relative Displacement
2
3
4
5
-5
x 10
Figure 7.6: Hysteresis Loops, Forcing Levels All Below Gross Slip
Acceleration vs. Acceleration
15
Bottom Beam Accel
10
5
0
-5
-10
-15
-15
-10
-5
0
Top Beam Accel
5
10
15
Figure 7.7: Beam Acceleration Signals, forming a Lissajous Pattern
75
Next, the coefficient of friction values are summarized in Table 7.1. The
coefficients of friction build as expected. At the next higher excitation level the joint
begins to stick and slip, yielding poor data. The static coefficient of friction for
aluminum on aluminum is 0.60 [35], which agrees closely to this last experimental value
of 0.55.
Forcing Level [N]
18
36
54
72
Coefficient of Friction
0.14
0.27
0.41
0.55
Table 7.1: Coefficient of Friction Values
Lastly, the energy dissipation was calculated. Since the hysteresis loops are a plot
of force versus displacement, simply calculating the area inside the loop yields the work.
The results for the four excitation levels are shown in Figure 7.8. The results seem to
agree with the experimental work by Johnson, as discussed earlier in Section 1.2.1
(Figure 1.3). When plotting the results on a log – log scale, the slope is found to be
approximately 2.53, which is close to the cubic relationship found by Goodman as
discussed in Section 2.3. It has been found in other research by Quinn, Segalman, and
others that the experimental value is actually closer to 2.7 [36]. While it is realized that
this trend is only observed with nine data points, these results are promising and can be
expanded upon by future research.
76
Energy Dissipation per Cycle
0.0090
-1.500
0.50
0.0080
Log of Energy Loss / Cycle
0.0070
Energy Loss / Cycle [J]
0.0060
0.0050
0.70
0.90
1.10
1.30
1.50
1.70
1.90
2.10
-2.000
-2.500
y = 2.5301x - 6.9455
-3.000
-3.500
0.0040
-4.000
Log of Force
0.0030
0.0020
0.0010
0.0000
0
10
20
30
40
50
60
70
Force [N]
Figure 7.8: Energy Dissipation Follows a Cubic Trend
7.1.5 HILBERT TRANSFORMS
The last analysis tool used is the Hilbert transform. The data processed with this
technique is the same as that used in the earlier characterization methods. The normal
load was 370 N, the step of the sine excitation was 0.1 Hz, and the excitation force levels
were 0.25 N, 0.50 N, 1.00 N, 1.50 N, and 2.00 N.
The Hilbert transform comparison is shown in Figures 7.9 and 7.10. If the system
was linear, the Hilbert transform and the FRF should perfectly overlay. However, as can
be seen in these plots, they do not. There are some clear differences near the resonance
of the system. This can be seen for both the real and imaginary parts of the system.
77
These results can indicate both a nonlinearity in the stiffness and the damping of the
system. Also note the truncation which occurs at the beginning and end of the data, a
result of the nature of the Hilbert transform.
Force 1
Real Hilbert
Real FRF
Magnitude
500
0
-500
180
190
200
210
Force 2
220
Magnitude
0
-500
180
190
200
210
Force 3
220
Magnitude
230
240
Real Hilbert
Real FRF
500
0
-500
180
190
200
210
Force 4
220
230
240
Real Hilbert
Real FRF
500
Magnitude
240
Real Hilbert
Real FRF
500
0
-500
180
190
200
210
Force 5
220
230
240
Real Hilbert
Real FRF
500
Magnitude
230
0
-500
180
190
200
210
Frequency [Hz]
220
230
240
Figure 7.9: Hilbert Transforms, Real Part
78
Force 1
Magnitude
800
Imag Hilbert
Imag FRF
600
400
200
0
-200
180
190
200
210
Force 2
220
Magnitude
800
230
240
Imag Hilbert
Imag FRF
600
400
200
0
-200
180
190
200
210
Force 3
220
Magnitude
800
230
240
Imag Hilbert
Imag FRF
600
400
200
0
-200
180
190
200
210
Force 4
220
Magnitude
800
230
240
Imag Hilbert
Imag FRF
600
400
200
0
-200
180
190
200
210
Force 5
220
Magnitude
800
230
240
Imag Hilbert
Imag FRF
600
400
200
0
-200
180
190
200
210
Frequency [Hz]
220
230
240
Figure 7.10: Hilbert Transforms, Imaginary Part
79
7.2 SUMMARY
The initial data collected from the test rig certainly shows promising trends. The
rig is able to isolate the friction area and with the proper testing techniques nonlinear
damping can be witnessed. The frequency domain data is easy to acquire at low
excitation levels, < 2.0 Newtons. The FRF’s and complex plots of the frequency data are
good indicators of the nonlinearity.
The time domain data is also showing the energy dissipation as the beams
oscillate. These results show the test rig is operating in a similar manner as the rig
designed by Gregory et al. [17]. The hysteresis loops also seem to accurately predict the
coefficient of friction for the beams, 0.60. The energy dissipation also shows the cubic
trend as outlined by Goodman [22].
Proposals are made in the next chapter to help yield improved data and ease in the
process of data collection. There are some changes and modifications which can be made
to the rig to make it more useful over a larger operation range, and allow more
investigation into the friction joint of interest.
80
CHAPTER 8: RIG REDESIGN
8.1 INTRODUCTION
Throughout the testing process, there were a number of reoccurring issues with
the test rig and testing methods. These issues made data collection difficult and could be
leading to errors. The main issues with the rig were discovered to be:
ƒ
Guide rods
ƒ
Beam alignment issues
ƒ
Tolerance for in-house machining
ƒ
Normal load
ƒ
Reduce size for testing convenience
ƒ
Force control
These issues will be discussed in this chapter and a proposal will be made to correct them
for future research.
8.2 RIG ISSUES
8.2.1 GUIDE RODS
The first area of improvement on the original rig was discovered to be with the
guide rods for the large steel center plate. These four guide rods allow the center plate to
move up and down for beams of different length. In addition, they control the alignment
for the beams and the normal load. The current design has four pipes with nylon
bushings inserted in them to ride on the guide rods. During testing, this design has
81
proven not to slide very well, often sticking. This sticking causes adjustment of the
normal load to be difficult and inaccurate during testing.
8.2.2 BEAM ALIGNMENT
The next issue was with beam alignment. With the current rig, the beams were
not perfectly aligned. When brought into contact, the beams are shifted slightly both in
the X and Y-directions.
8.2.3 MACHINING
The above two problems can be partially attributed to machining difficulties. The
rig was machined in house, and the tolerances are not held as closely as could be desired.
If funding would allow it, it is recommended that a professional construct the rig. The
original design would have better alignment if machined by a professional shop.
8.2.4 NORMAL LOADING
Another area to be investigated also deals with the center steel plate. As
mentioned earlier in the paper, the center plate was purposely designed to be large and
heavy, to give a large contact area on the beams. However, when the rig was setup, it
was discovered that this weight was too heavy to slip the joint.
Also concerning the normal loading is the measurement of the normal load during
testing. Currently, the beams are equipped with strain gages at their bases. The gages are
then connected to a digital strain indicator box. This box displays the microstrain on each
beam, but does not record it. There is a provision on the stain indicator boxes to read the
82
output voltage; this value could then be recorded to monitor the normal load. There is
also specific hardware available for the VXI system to measure this, for example Agilent
Technology’s card E1529A performs this task. Figure 8.1 is a schematic of the card’s
interface with a VXI system; the figure is taken from Agilent Technology’s web page
[37].
Figure 8.1: Strain Gage Acquisition for VXI [37]
8.2.5 RIG SIZE
The fact that the center plate is too heavy leads to the idea that the entire rig could
be reduced in size. A smaller designed rig would be more conducive to testing and
would allow it to be more portable for setup.
83
8.2.6 SOFTWARE CLOSED LOOP CONTROL
A final issue involves the testing software. A good feature with the I-DEAS
testing software is the closed loop control. After using this system for a while, however,
it became apparent that when monitoring the force control on the rig that it can only be
maintained constant through resonance at lower excitation levels. At higher excitation
levels, the controller is no longer able to accurately iterate between its ranges; therefore it
cannot find a correct level. The controller has two setting options if this event occurs, it
will either stop the measurement, or simply continue on after a set number of attempts.
In order to monitor the force control and watch these events, a digital signal
analyzer was attached to a PCB amplifier. The amplifier has a BNC connection on its
front panel which makes this a simple procedure.
After contacting MTS Systems, some information was gained on this topic. There
is file which controls some of the closed loop settings. In this file, the gain of the
controller and number of iterations can be adjusted. Modifying this file has yielded some
more successful results. However, at higher levels the problem still occurs.
A last attempt to remedy this problem is to manually adjust the amplifier gain.
While monitoring the DSA, the amplifier gain can be adjusted up or down to achieve the
desired voltage level to the keep the force constant.
84
8.3 PROPOSED RIG CHANGES
Taking the above topics into consideration, a proposal for some changes to the rig
is presented. The changes are discussed and a solid model is shown to illustrate the
changes.
The biggest suggested change for a new rig would be the use of linear bearings
for the center plate. A linear bearing will allow for a much smoother and consistent
adjustment of the center plate. An example of a linear bearing is illustrated in Figure 8.2,
taken from McMaster-Carr [38]. The selected linear bearings are held in an aluminum
pillow block, and can tolerate 3° of misalignment. This alignment tolerance will be
advantageous for the machining tolerances.
Figure 8.2: Illustration of Linear Bearing [38]
The next modification is on the beams themselves. In the current rig, the beams
are clamped into the two thick plates; inserted into two rectangular machined holes. If
these holes are not perfectly aligned in both plates, the beam’s friction interface will not
be aligned. This is the case with the current rig. Another issue with this clamping
85
method is its rigidity. If the clamping joint allows slight movement, it can cause a source
of damping. A new method for the beams is shown in Figure 8.3.
Figure 8.3: New Beam Mounting
Figure 8.3 shows how the new beams will be machined with a base on them.
While this is not the most efficient design from a machining standpoint, it will solve the
alignment and clamping methods. The beam base has a slot in it, so the beam can be
adjusted in two directions. In addition, the beam is secured by bolts into the plates. With
the installation of a lock washer, this should keep the beam secure.
86
Another change from the original design is the reduction of weight of the center
plate. With a lighter center plate, the joint can be slipped at lower excitation levels and
will provide for easier testing. The center plate in the new design is also simpler, as it is
just a plate of 0.5” aluminum. With the original design, it had to be thick (1.50”) to
clamp the upper beam, a feature no longer needed.
An isometric view of the new rig can be seen in Figure 8.4. The new linear
bearings are shown in blue. Because the linear bearings are mounted in aluminum pillow
block housings, new guide rails were designed. These are indexed into the center plate,
for alignment issues. They are also constructed from aluminum, for reduced weight. The
new, thin center plate is also shown.
The spring mechanism from the original rig was carried over to the new rig.
Through testing with the original rig, this setup proved efficient. A change to the new rig
however was the addition of another spring mechanism under the top plate of the rig.
This setup will allow for loading and unloading of the beams.
87
Figure 8.4: Isometric View of Rig Proposal
An assembly view of the new rig proposal on the steel base with the shaker is
shown in Figure 8.5. All of these components are carried over from the current rig.
Figure 8.6 shows a right side view of the rig, while Figure 8.7 shows a top view, and
Figure 8.8 a front view of the assembly. These figures illustrate how the shaker attaches
to the lower beam in the same fashion as the current rig.
88
Figure 8.5: Isometric View of New Rig Assembly
89
Figure 8.6: Right Side View of New Rig Assembly
Figure 8.7: Top View of New Rig Assembly
90
Figure 8.8: Front View of New Rig
8.4 RIG DESIGN COMPARISON
As a comparison, the solid models of the two rigs are shown in this section. The
new rig is smaller in all directions than the original design. Figure 8.9 is an isometric
view of the two rigs. The thinner bottom and center bases can be clearly seen in this
figure.
91
Figure 8.9: Isometric View of Both Rigs
Figure 8.10 is a right side view comparison of the two rigs. In this view the
height difference between the rigs is shown. Also note the shakers are in the same
locations for both designs.
92
Figure 8.10: Right Side View of Both Rigs
A top view of the comparison is shown in Figure 8.11. In this view, note how the
square plates were reduced in dimensions, but the location of the guide rods is
maintained.
Figure 8.11: Top View of Both Rigs
93
Lastly, Figure 8.12 is an angled view of the front and side of the original and
proposed rig designs.
Figure 8.12: Front / Side View of Both Rigs
94
CHAPTER 9: CONCLUSIONS
9.1 INTRODUCTION
The overall goal of this project was to evaluate the performance of the nonlinear
test rig. At the conclusion of this work, much experience was gained on the test rig
design and implementation. Methods for taking data were evaluated and documented
along with characterization methods which can be used in future research. Analyzing the
data showed the rig was yielding results which are very promising. This section will
summarize the most important lessons learned, in addition to making a proposal for
future work. Hopefully this paper will help document some of the advances on this rig so
more efficient testing and research can be performed in future work.
9.2 TESTING METHODS
With the current testing setup, there were a number of hurdles to overcome. The
use of I-DEAS Sine Measurements package allows for the use of swept sine testing,
which is one of the best excitation methods for nonlinearities. However, there are a few
issues that should be addressed. The ability to use a closed loop input only works at
lower excitation levels, where the beams will not slip. Once the joint begins to slip, the
system dynamics become very complicated with the stick – slip phenomenon. The
software controller is not able to successfully iterate under these conditions.
Another area to be addressed with the stick – slip motion is sensor selection. For
a selected sensor, it may overload when the beams slip, but when the beams stick, the
sensor is under-ranged. Finally, the stick – slip occurrence is not very consistent, as the
center plate of the rig does not freely float. It is recommended that data be collected at
95
lower excitation levels, where these events are less likely to occur. Of course, this
dramatically limits the amount of data that can be collected, as the stick – slip
phenomenon cannot be witnessed.
9.3 DATA ANALYSIS
The data analysis from this rig showed some promising trends. The FRF’s clearly
showed as the input level was increased, the magnitudes lowered and the curves become
wider. The complex plots also show the trend of quadratic damping. The hysteresis loop
data is showing some promising results as well. Under the recommend operating ranges,
the time data is producing the expected trends for the loops. As the excitation levels are
increased, the energy dissipation is increasing with a cubic relationship.
9.4 FUTURE SUGGESTIONS
The largest suggestion for future work on the nonlinear test rig is a redesign on
the implementation of its components. The rig’s intent is very good; it efficiently isolates
the two beams in contact. However, the weight of the center plate is too great, making
testing difficult the equipment available. In addition, the sliding of the center plate is
poor, again making acquisition of consistent data complicated.
For a first implementation, the current rig design was very efficient. There is still
data that can be taken and analyzed from the rig, if it is taken with care and within its
operating range, which is away from the stick – slip of the beams. The current rig will
allow for different beams to be tested, specifically different interfaces. A less
96
conforming interface between the beams may be helpful. A smaller conforming area will
allow the beams to slip easier, and possibly more smoothly and consistently.
97
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Berger, E. J. Friction Modeling for Dynamic System Simulation. Department of
Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, 2001.
[2]
Gaul, L. and Nitsche, R. The Role of Friction in Mechanical Joints. Applied
Mechanics Review, Vol. 54, No 2, pg 93-106, 2001.
[3]
Gaul, L. and Nitsche, R. Friction Control for Vibration Suppression. Academic
Press, 2000.
[4]
Canudas de Wit, C., Olsson, H., Astrom, J., and Lischinsky, P. A New Model for
Control of Systems with Friction. IEEE Transactions on Automatic Control, Vol.
40, No 3, pg 419-425, March 1995.
[5]
Lenz, J. and Gaul, L. The Influence of Microslip on the Dynamic Behavior of
Bolted Joints. Proceedings of the 13th International Modal Analysis Conference,
pg 248-254, 1995.
[6]
Valanis, K.C. A Theory of Viscoplasticity without a Yield Surface. Archive of
Mechanics., Vol. 23, No 4, pg 517-551, 1971.
[7]
Bindemann, A. C. and Ferri, A. A. The Influence of Alternative Friction Models
on the Passive Damping and Dynamic Response of a Flexible Structure.
Proceedings of the AIAA/ASME/ASCE/AHS 36th Structure, Structural
Dynamics, and Materials Conference, New Orleans, LA, April 10-12, 1996.
[8]
Johnson, K.L. Energy Dissipation at Spherical Surfaces in Contact Transmitting
Oscillating Forces. Journal Mechanical Engineer Science, Vol. 3, No 4, pg 362368, 1961.
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[9]
Mindlin, R.D., and Deresiewicz, H. Elastic Spheres in Contact Under Varying
Oblique Forces. Journal of Applied Mechanics, Vol. 20, pg 327-344, 1953.
[10]
Mindlin, R.D., Mason, W.P., Osmer, T.F., and Deresiewicz, H. Effects of
Oscillating Forces on the Contact Surfaces of Elastic Spheres. Proceedings:
First US National Congress of Applied Mechanics, Chicago, pg 203-208, 1953.
[11]
Goodman, L. E. and Brown, C. B. Energy Dissipation in Contact Friction:
Constant Normal and Cyclic Tangential Loading. Journal of Applied Mechanics,
pg 17-22, March 1962.
[12]
Iwan, W. D. The Steady State Response of a Two-Degree-of-Freedom Bilinear
Hysteretic System. Journal of Applied Mechanics, pg 151-156, March 1965.
[13]
Iwan, W. D. A Distributed-Element Model for Hysteresis and Its Steady-State
Dynamic Response. Journal of Applied Mechanics, pg 893-900, March 1966.
[14]
Moloney, Christopher W., Peairs, Daniel M., and Roldan, Enrique R.
Characterization of Damping in Bolted Lap Joints. Los Alamos National
Laboratory, 2000.
http://www.lanl.gov/projects/dss/Projects/2000/LapJoint/paper/paper.pdf
[15]
Kess, Harold R., Rosnow, Nathan J., and Sidle, Brian, C. Effects of Bearing
Surfaces on Lap Joint Energy Dissipation. Los Alamos National Laboratory,
2001.
http://www.lanl.gov/projects/dss/Projects/2001/LapJoint/paper/paper_damping.pdf
[16]
Dahl, Philip R. Solid Friction Damping of Spacecraft Oscillations. AIAA Paper
No 75-1104 Presented at the AIAA Guidance and Control Conference, Boston,
Mass, 1975.
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[17]
Gregory, Danny L., Smallwood, David O., Coleman, Ronald G., and Nusser
Michael A. Experimental Studies to Investigate Damping In Frictional Shear
Joints. Engineering Sciences Center, Sandia National Laboratories.
http://www.me.cmu.edu/faculty1/griffin/2001/presentations/smallwood_2.pdf
[18]
He, J. and Ewins, D. J. A Simple Method of Interpretation for the Modal Analysis
of Nonlinear Systems. Proceedings of the 5th International Modal Analysis
Conference, pg 626- 634, 1987.
[19]
Adams, Douglas, E. and Allemang, Randall J. Survey of Nonlinear Detection and
Identification Techniques for Experimental Vibrations. ISMA 23 International
Conference on Noise and Vibration Engineering, Vol. 1, pg 269-281, 1998.
[20]
Berger, E. J. Fundamentals of Tribology, 20-263-641. Department of
Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, 1999.
[21]
Johnson, K.L. Contact Mechanics. Cambridge University Press, Cambridge,
New York, 1985.
[22]
Goodman, L.E. A Review of Progress in Analysis of Interfacial Slip Damping.
Structural Damping, papers presented at a colloquium on structural damping held
at the ASME annual meeting in Atlantic City, N.J., in December 1959, pg 35-48.
[23]
Worden, K. and Tomlinson, G. R. Nonlinearity in Structural Dynamics:
Detection, Identification, and Modelling. Institute of Physics Publishing, Bristol,
2001.
[24]
Thomson, William T. and Dahleh, Marie Dillon. Theory of Vibration with
Applications. Prentice-Hall, Inc., New Jersey, 1993.
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[25]
Allemang, Randall J. Vibrations: Experimental Modal Analysis. UC-SDRL-CN20-263-663. University of Cincinnati Structural Dynamics Research Laboratory,
1999.
[26]
Sperling L. and Wahl, F. The Frequency Response for Weakly Nonlinear
Systems. Identification in Engineering Systems, Proceedings of the Conference
held at Swansea, Ed. Friswell, M.I. and Mottershead, J.E., pg 456- 465, March
1996.
[27]
Allemang, Randall J. Mechanical Vibrations II Class Notes. Course Number 20MECH-662. Vibrations: Analytical and Experimental Modal Analysis. Rev.
February 1999. http://www.sdrl.uc.edu/ucme662/ucme662.html
[28]
Allemang, Randall J. Mechanical Vibrations III Class Notes. Course Number
20-MECH-663/664. Vibrations: Experimental Modal Analysis. Rev. March
1999. http://www.sdrl.uc.edu/ucme663/ucme663.html
[29]
Watts, Robert. Investigating the Effects of Surface Contacts on Overall System
Dynamics. Exchange student to the University of Cincinnati, Summer 2001.
[30]
MRIT. University of Cincinnati, Structural Dynamics Research Laboratory.
http://www.sdrl.uc.edu/data_acquisition.html
[31]
X-Modal II. University of Cincinnati, Structural Dynamics Research Laboratory.
http://www.sdrl.uc.edu/X-ModalII.html
[32]
MIMO. University of Cincinnati, Structural Dynamics Research Laboratory.
http://www.sdrl.uc.edu/data_acquisition.html
[33]
MTS Systems Corporation. http://www.mts.com/nvd/Software/IDEAS.htm
101
[34]
Hahn, Stefen L. Hilbert Transforms in Signal Processing. Artech House, Inc.,
London, 1996.
[35]
NASA Explores. Laboratory Experiment, Information taken from:
http://www.nasaexplores.com/lessons/01-053/9-12_2.pdf
[36]
Quinn, Dane D. and Segalman, Daniel J. Using Series-Series Iwa-Type Models
for Understanding Joint Dynamics. Sandia National Laboratories, SAND20024120J.
[37]
Agilent Technology. www.aligent.com. Data acquisition card specifications:
http://cp.literature.agilent.com/litweb/pdf/5968-0432E.pdf
[38]
McMaster-Carr. http://www.mcmaster.com/ Catalog pg. 935.
102
APPENDIX A: ADDITIONAL FIGURES
A1. TEST RIG
Figure A.1: Alternate View of Original Rig Assembly
103
Figure A.2: Alternate View of Test Rig Assembly
104
Figure A.3: Rig During Testing
105
Figure A.4: Strain Indicator & Switch and Balance Boxes
Figure A.5: Shaker During Testing
106
Figure A.6: VXI Setup
Figure A.7: Beam Setup
Figure A.8: PC, Amp, DSA During Testing
107
Figure A.9: Alternate View of New Rig Assembly
108
APPENDIX B: ADDITIONAL TABLES
B1. FULL MODAL TEST CALIBRATION TABLE
Channel
Type
Direction
Model #
Serial #
Calibration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
33
34
35
36
37
38
39
Input
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
-Y
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
X, Y, Z
208A02
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
336C34
8339
12980
12945
12960
12956
12995
12959
12963
13055
12994
2982
13013
12977
12949
12961
2998
12962
8073
12997
13047
7389
7384
3519
9953
10015
7387
3491
12979
11080
9662
3672
6487
10002
12975
12981
10010
12990
12974
0.013 V/N
99.983 mV/g
98.873 mV/g
100.883 mV/g
101.922 mV/g
101.132 mV/g
100.123 mV/g
99.976 mV/g
96.927 mV/g
99.939 mV/g
102.180 mV/g
99.421 mV/g
100.347 mV/g
102.876 mV/g
100.990 mV/g
94.472 mV/g
102.190 mV/g
93.107 mV/g
100.106 mV/g
102.260 mV/g
95.653 mV/g
96.111 mV/g
96.904 mV/g
99.071 mV/g
98.098 mV/g
99.039 mV/g
95.970 mV/g
98.707 mV/g
95.491 mV/g
96.053 mV/g
99.007 mV/g
95.042 mV/g
98.446 mV/g
97.483 mV/g
99.731 mV/g
97.140 mV/g
100.680 mV/g
100.739 mV/g
Table B.1: Load Cell / Accelerometer Data, Modal Test
109
B2. TEST SETUP CALIBRATION TABLE
Channel
Type
Direction
Model #
Serial #
Calibration
1
2
3
4
5
6
7
Input
Output
Output
Output
Output
Output
Output
+X
+X
+X
+Y
-Y
+Y
+Y
208A02
336C34
336C34
336C34
336C34
336C34
336C34
8206
7392
12982
10011
9639
12994
12959
0.0122 V/N
96.02 mV/g
103.10 mV/g
102.39 mV/g
97.45 mV/g
101.90 mV/g
100.56 mV/g
Table B.2: Load Cell / Accelerometer Data, Test Setup
110
APPENDIX C: MATLAB SCRIPTS
C1. AUTOPOWER COMPARISON PLOTS
%
This Program Creates the Autopower Comparison Plots
clear
clc
close all
data_1=readadf('1N_Full.afu');
x_1=get(data_1,'Abscissa');
y_1=get(data_1,'Ordinate');
data_2=readadf('10N_Full.afu');
x_2=get(data_2,'Abscissa');
y_2=get(data_2,'Ordinate');
figure
semilogy(x_1(:,1),y_1(:,1),'r',x_2(:,1),y_2(:,1),'b')
xlabel('Freq [Hz]')
ylabel('Excitation Force^2 [N^2]')
title('Autopower Spectrum, Input Channel')
axis([200,240,1e-1,1e3])
text(202,150,'No Force Control');
text(202,1.5,'Force Control');
C2. FRF DISTORTION PLOTS
%
This Program Creates the FRF Distortion Plots
clear
clc
close all
choice=menu('Pick Number of Plots to View','1','2','3','4','5');
for ii=1:choice;
data=readadf
x=get(data,'Abscissa');
y=get(data,'Ordinate');
frf=menu('Choose Beam to
Investigate','1','2','3','4','5','6','7','8','9','15','34');
if frf==10;
frf=15;
end
if frf==11;
frf=34;
end
[n,m]=size(x);
figure(1)
subplot(311)
111
if ii==1;
color='r';
end
if ii==2;
color='b';
end
if ii==3;
color='g';
end
if ii==4;
color='m';
end
if ii==5;
color='k';
end
semilogy(x(1:n,frf),abs(y(1:n,frf)),color)
xlabel('Freq [Hz]')
ylabel('Log Magnitude')
title('FRF Plot')
hold on
subplot(312)
plot(x(1:n,frf),unwrap(angle(y(1:n,frf))),color)
xlabel('Freq [Hz]')
ylabel('Phase Angle [Radians]')
title('Phase Plot')
hold on
subplot(313)
plot(x(1:n,1),y(1:n,1),color)
xlabel('Freq [Hz]')
ylabel('Excitation Level^2 [N^2]')
title('AutoPower, Input Channel')
axis([min(x(:,1)) max(x(:,1)) -1 10])
hold on
end
legend('Force 1','Force 2','Force 3','Force 4','Force 5',1)
legend boxoff
C3. COMPLEX PLOTS
%
This Program Creates the Complex Plots
clear
clc
close all
start=10;
choice=menu('Pick Number of Plots to View','1','2','3','4','5');
for ii=1:choice;
data=readadf
x=get(data,'Abscissa');
y=get(data,'Ordinate');
112
frf=menu('Choose Beam to
Investigate','1','2','3','4','5','6','7','8','9','10');
[n,m]=size(x);
figure(1)
subplot(211)
if ii==1;
color='r';
end
if ii==2;
color='b';
end
if ii==3;
color='g';
end
if ii==4;
color='m';
end
if ii==5;
color='k';
end
plot(x(start:n,1),y(start:n,1),color)
xlabel('Freq [Hz]')
ylabel('Excitation Level^2 [N^2]')
title('AutoPower, Input Channel')
hold on
subplot(212)
plot(real(y(start:n,frf)),imag(y(start:n,frf)),color)
grid on
axis equal
hold on
end
xlabel('Real of FRF')
ylabel('Imag of FRF')
title('Complex Plots')
subplot(212)
legend('Force 1','Force 2','Force 3','Force 4','Force 5',1)
legend boxoff
C4. INVERSE FRF PLOTS
%
This Program Creates the Inverse FRF Plots
clear
clc
close all
start=1;
choice=menu('Pick Number of Plots to View','1','2','3','4','5');
for ii=1:choice;
data=readadf
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x=get(data,'Abscissa');
y=get(data,'Ordinate');
frf=menu('Choose Channel to
Investigate','1','2','3','4','5','6','7','8','9','10');
[n,m]=size(x);
figure(1)
subplot(211)
if ii==1;
color='r';
end
if ii==2;
color='b';
end
if ii==3;
color='g';
end
if ii==4;
color='m';
end
if ii==5;
color='k';
end
plot((x(start:n,frf)).^2,real(1./y(start:n,frf)),color)
xlabel('Freq^2')
ylabel('Real of 1/FRF')
hold on
subplot(212)
plot(x(start:n,frf),imag(1./y(start:n,frf)),color)
xlabel('Freq')
ylabel('Imag of 1/FRF')
hold on
end
subplot(211)
legend('Force 1','Force 2','Force 3','Force 4','Force 5',4)
legend boxoff
title('Inverse FRF Plots')
C5. HYSTERESIS LOOP CALCULATIONS
%
This Program Computes the Hysteresis Loop from the Time Signals
clear
clc
close all
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
load data
data=readadf('0_5V_15uS_50Hz.ati');
%
get information from data
xx=get(data,'Abscissa');
yy=get(data,'Ordinate');
114
N=103;
[a b]=size(yy);
for ii=1:2;
y=yy((a-(ii*N))+1:a-((ii-1)*N),:);
x=xx((a-(ii*N))+1:a-((ii-1)*N),:);
%
calculate shaker force
load_cell=.0122;
strain=15;
A=.5;
E=10.4e6;
v=.32;
normal_load_lb=(A*E*(strain*1e-6))/(2+2*v);
normal_load=normal_load_lb*4.44822;
Forces=(1/1).*y(:,1);
Normalized=Forces./(normal_load);
%
integrate
vel_1=cumtrapz(y(:,2)).*(x(10,1)-x(9,1));
vel_2=cumtrapz(y(:,3)).*(x(10,1)-x(9,1));
vel_1=vel_1-mean(vel_1);
vel_2=vel_2-mean(vel_2);
disp_1=cumtrapz(vel_1).*(x(10,1)-x(9,1));
disp_2=cumtrapz(vel_2).*(x(10,1)-x(9,1));
disp_1=disp_1-mean(disp_1);
disp_2=disp_2-mean(disp_2);
figure(1)
subplot(311)
plot(x(:,1),y(:,1),x(:,2),y(:,2),x(:,3),y(:,3))
legend('Force','Top Beam','Bottom Beam',2)
legend boxoff
title('Time Signals, Force and Acceleration')
xlabel('Time [Sec]')
ylabel('Magnitude [Volts]')
axis([7.95,8,-100,100])
hold on
subplot(323)
plot(y(:,2),y(:,3))
title('Acceleration vs. Acceleration')
xlabel('Top Beam Accel')
ylabel('Bottom Beam Accel')
axis([-40,40,-50,50])
hold on
subplot(324)
plot(y(:,2)-y(:,3),Normalized)
title('Relative Acceleration Loop')
xlabel('Relative Acceleration')
ylabel('Coefficient of Friction')
axis([-20,20,-.75,.75])
hold on
subplot(325)
plot((vel_1-vel_2),Normalized)
title('Relative Velocity Loop')
xlabel('Relative Velocity')
ylabel('Coefficient of Friction')
axis([-.02,.02,-.75,.75])
115
hold on
subplot(326)
plot((disp_2-disp_1),Normalized)
title('Hysteresis Loop')
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
axis([-5e-5,5e-5,-.75,.75])
hold on
end
figure(5)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2-disp_1),Normalized,'b')
hold on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
load data
data=readadf('1_0V_15uS_50Hz.ati');
%
get information from data
xx=get(data,'Abscissa');
yy=get(data,'Ordinate');
N=103;
[a b]=size(yy);
for ii=1:2;
y=yy((a-(ii*N))+1:a-((ii-1)*N),:);
x=xx((a-(ii*N))+1:a-((ii-1)*N),:);
%
calculate shaker force
load_cell=.0122;
strain=15;
A=.5;
E=10.4e6;
v=.32;
normal_load_lb=(A*E*(strain*1e-6))/(2+2*v);
normal_load=normal_load_lb*4.44822;
Forces=(1/1).*y(:,1);
Normalized=Forces./(normal_load);
%
integrate
vel_1=cumtrapz(y(:,2)).*(x(10,1)-x(9,1));
vel_2=cumtrapz(y(:,3)).*(x(10,1)-x(9,1));
vel_1=vel_1-mean(vel_1);
vel_2=vel_2-mean(vel_2);
disp_1=cumtrapz(vel_1).*(x(10,1)-x(9,1));
disp_2=cumtrapz(vel_2).*(x(10,1)-x(9,1));
disp_1=disp_1-mean(disp_1);
disp_2=disp_2-mean(disp_2);
figure(2)
subplot(311)
plot(x(:,1),y(:,1),x(:,2),y(:,2),x(:,3),y(:,3))
legend('Force','Top Beam','Bottom Beam',2)
116
legend boxoff
title('Time Signals, Force and Acceleration')
xlabel('Time [Sec]')
ylabel('Magnitude [Volts]')
axis([7.95,8,-100,100])
hold on
subplot(323)
plot(y(:,2),y(:,3))
title('Acceleration vs. Acceleration')
xlabel('Top Beam Accel')
ylabel('Bottom Beam Accel')
axis([-40,40,-50,50])
hold on
subplot(324)
plot(y(:,2)-y(:,3),Normalized)
title('Relative Acceleration Loop')
xlabel('Relative Acceleration')
ylabel('Coefficient of Friction')
axis([-20,20,-.75,.75])
hold on
subplot(325)
plot((vel_1-vel_2),Normalized)
title('Relative Velocity Loop')
xlabel('Relative Velocity')
ylabel('Coefficient of Friction')
axis([-.02,.02,-.75,.75])
hold on
subplot(326)
plot((disp_2-disp_1),Normalized)
title('Hysteresis Loop')
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
axis([-5e-5,5e-5,-.75,.75])
hold on
end
figure(5)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2-disp_1),Normalized,'r')
hold on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
load data
data=readadf('1_5V_15uS_50Hz_a.ati');
%
get information from data
xx=get(data,'Abscissa');
yy=get(data,'Ordinate');
N=103;
[a b]=size(yy);
for ii=1:2;
y=yy((a-(ii*N))+1:a-((ii-1)*N),:);
x=xx((a-(ii*N))+1:a-((ii-1)*N),:);
%
calculate shaker force
load_cell=.0122;
strain=15;
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A=.5;
E=10.4e6;
v=.32;
normal_load_lb=(A*E*(strain*1e-6))/(2+2*v);
normal_load=normal_load_lb*4.44822;
Forces=(1/1).*y(:,1);
Normalized=Forces./(normal_load);
%
integrate
vel_1=cumtrapz(y(:,2)).*(x(10,1)-x(9,1));
vel_2=cumtrapz(y(:,3)).*(x(10,1)-x(9,1));
vel_1=vel_1-mean(vel_1);
vel_2=vel_2-mean(vel_2);
disp_1=cumtrapz(vel_1).*(x(10,1)-x(9,1));
disp_2=cumtrapz(vel_2).*(x(10,1)-x(9,1));
disp_1=disp_1-mean(disp_1);
disp_2=disp_2-mean(disp_2);
figure(3)
subplot(311)
plot(x(:,1),y(:,1),x(:,2),y(:,2),x(:,3),y(:,3))
legend('Force','Top Beam','Bottom Beam',2)
legend boxoff
title('Time Signals, Force and Acceleration')
xlabel('Time [Sec]')
ylabel('Magnitude [Volts]')
axis([7.95,8,-100,100])
hold on
subplot(323)
plot(y(:,2),y(:,3))
title('Acceleration vs. Acceleration')
xlabel('Top Beam Accel')
ylabel('Bottom Beam Accel')
axis([-40,40,-50,50])
hold on
subplot(324)
plot(y(:,2)-y(:,3),Normalized)
title('Relative Acceleration Loop')
xlabel('Relative Acceleration')
ylabel('Coefficient of Friction')
axis([-20,20,-.75,.75])
hold on
subplot(325)
plot((vel_1-vel_2),Normalized)
title('Relative Velocity Loop')
xlabel('Relative Velocity')
ylabel('Coefficient of Friction')
axis([-.02,.02,-.75,.75])
hold on
subplot(326)
plot((disp_2-disp_1),Normalized)
title('Hysteresis Loop')
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
axis([-5e-5,5e-5,-.75,.75])
hold on
end
figure(5)
118
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2-disp_1),Normalized,'k')
hold on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
load data
data=readadf('2_0V_15uS_50Hz_a.ati');
%
get information from data
xx=get(data,'Abscissa');
yy=get(data,'Ordinate');
N=103;
[a b]=size(yy);
for ii=1:2;
y=yy((a-(ii*N))+1:a-((ii-1)*N),:);
x=xx((a-(ii*N))+1:a-((ii-1)*N),:);
%
calculate shaker force
load_cell=.0122;
strain=15;
A=.5;
E=10.4e6;
v=.32;
normal_load_lb=(A*E*(strain*1e-6))/(2+2*v);
normal_load=normal_load_lb*4.44822;
Forces=(1/1).*y(:,1);
Normalized=Forces./(normal_load);
%
integrate
vel_1=cumtrapz(y(:,2)).*(x(10,1)-x(9,1));
vel_2=cumtrapz(y(:,3)).*(x(10,1)-x(9,1));
vel_1=vel_1-mean(vel_1);
vel_2=vel_2-mean(vel_2);
disp_1=cumtrapz(vel_1).*(x(10,1)-x(9,1));
disp_2=cumtrapz(vel_2).*(x(10,1)-x(9,1));
disp_1=disp_1-mean(disp_1);
disp_2=disp_2-mean(disp_2);
figure(4)
subplot(311)
plot(x(:,1),y(:,1),x(:,2),y(:,2),x(:,3),y(:,3))
legend('Force','Top Beam','Bottom Beam',2)
legend boxoff
title('Time Signals, Force and Acceleration')
xlabel('Time [Sec]')
ylabel('Magnitude [Volts]')
axis([7.95,8,-100,100])
hold on
subplot(323)
plot(y(:,2),y(:,3))
title('Acceleration vs. Acceleration')
xlabel('Top Beam Accel')
ylabel('Bottom Beam Accel')
119
axis([-40,40,-50,50])
hold on
subplot(324)
plot(y(:,2)-y(:,3),Normalized)
title('Relative Acceleration Loop')
xlabel('Relative Acceleration')
ylabel('Coefficient of Friction')
axis([-20,20,-.75,.75])
hold on
subplot(325)
plot((vel_1-vel_2),Normalized)
title('Relative Velocity Loop')
xlabel('Relative Velocity')
ylabel('Coefficient of Friction')
axis([-.02,.02,-.75,.75])
hold on
subplot(326)
plot((disp_2-disp_1),Normalized)
title('Hysteresis Loop')
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
axis([-5e-5,5e-5,-.75,.75])
hold on
end
figure(5)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2-disp_1),Normalized,'g')
hold on
axis([-5e-5,5e-5,-.75,.75])
legend('Forcing Level 1','Forcing Level 2','Forcing Level 3','Forcing Level
4',2)
legend boxoff
axis([-5e-5,5e-5,-.75,.75])
legend boxoff
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
load data
data=readadf('1_0V_15uS_50Hz.ati');
%
get information from data
xx=get(data,'Abscissa');
yy=get(data,'Ordinate');
N=103;
[a b]=size(yy);
for ii=1:4;
y=yy((a-(ii*N))+1:a-((ii-1)*N),:);
x=xx((a-(ii*N))+1:a-((ii-1)*N),:);
%
calculate shaker force
load_cell=.0122;
strain=15;
A=.5;
E=10.4e6;
v=.32;
120
normal_load_lb=(A*E*(strain*1e-6))/(2+2*v);
normal_load=normal_load_lb*4.44822;
Forces=(1/1).*y(:,1);
Normalized=Forces./(normal_load);
%
integrate
vel_1=cumtrapz(y(:,2)).*(x(10,1)-x(9,1));
vel_2=cumtrapz(y(:,3)).*(x(10,1)-x(9,1));
vel_1=vel_1-mean(vel_1);
vel_2=vel_2-mean(vel_2);
disp_1=cumtrapz(vel_1).*(x(10,1)-x(9,1));
disp_2=cumtrapz(vel_2).*(x(10,1)-x(9,1));
disp_1=disp_1-mean(disp_1);
disp_2=disp_2-mean(disp_2);
figure(6)
plot(y(:,2),y(:,3))
title('Acceleration vs. Acceleration')
xlabel('Top Beam Accel')
ylabel('Bottom Beam Accel')
axis([-17,17,-17,17])
hold on
end
C6. HYSTERESIS LOOP AREA CALCULATION
%
This Program Computes the Areas of the Hysteresis Loops
clear
clc
close all
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
load data
%data=readadf('0_5V_15uS_50Hz.ati');
%rr=40;
% %
load data
%data=readadf('1_0V_15uS_50Hz.ati');
%rr=36;
% %
load data
data=readadf('1_5V_15uS_50Hz_a.ati');
rr=53;
% %
load data
%data=readadf('2_0V_15uS_50Hz_a.ati');
%rr=55;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
get information from data
xx=get(data,'Abscissa');
yy=get(data,'Ordinate');
121
N=103;
[a b]=size(yy);
for ii=1:2;
y=yy((a-(ii*N))+1:a-((ii-1)*N),:);
x=xx((a-(ii*N))+1:a-((ii-1)*N),:);
%
calculate shaker force
load_cell=.0122;
strain=15;
A=.5;
E=10.4e6;
v=.32;
normal_load_lb=(A*E*(strain*1e-6))/(2+2*v);
normal_load=normal_load_lb*4.44822;
Forces=(1./load_cell).*y(:,1);
Normalized=Forces./(normal_load);
%
integrate
vel_1=cumtrapz(y(:,2)).*(x(10,1)-x(9,1));
vel_2=cumtrapz(y(:,3)).*(x(10,1)-x(9,1));
vel_1=vel_1-mean(vel_1);
vel_2=vel_2-mean(vel_2);
disp_1=cumtrapz(vel_1).*(x(10,1)-x(9,1));
disp_2=cumtrapz(vel_2).*(x(10,1)-x(9,1));
end
disp_1=disp_1-mean(disp_1);
disp_2=disp_2-mean(disp_2);
figure(1)
subplot(311)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2(1:rr)-disp_1(1:rr)),Normalized(1:rr),'b')
%axis([-5e-5,5e-5,-.41,.41])
min1=min(Normalized(1:rr));
figure(3)
subplot(211)
plot((disp_2(1:rr)-disp_1(1:rr)),Normalized(1:rr)-min1,'b')
area1=trapz((disp_2(1:rr)-disp_1(1:rr)),Normalized(1:rr)-min1);
figure(1)
subplot(312)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2(rr+1:N)-disp_1(rr+1:N)),Normalized(rr+1:N),'b')
%axis([-5e-5,5e-5,-.41,.41])
min2=min(Normalized(rr+1:N));
figure(3)
subplot(212)
plot((disp_2(rr+1:N)-disp_1(rr+1:N)),Normalized(rr+1:N)-min2,'b')
122
area2=trapz((disp_2(rr+1:N)-disp_1(rr+1:N)),Normalized(rr+1:N)-min2);
figure(1)
subplot(313)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2-disp_1),Normalized,'b')
%axis([-5e-5,5e-5,-.41,.41])
total_area=-area1+area2
figure(2)
xlabel('Relative Displacement')
ylabel('Coefficient of Friction')
title('Hysteresis Loop')
plot((disp_2-disp_1),Normalized,'b')
%axis([-5e-5,5e-5,-.41,.41])
grid on
C7. HILBERT TRANSFORMS
%
This Program Computes the Frequency Hilbert Transform
clear
clc
close all
choice=menu('Pick Number of Plots to View','1','2','3','4','5');
for ii=1:choice;
data=readadf
x=get(data,'Abscissa');
y=get(data,'Ordinate');
[n,m]=size(x);
frf=menu('Choose Beam to
Investigate','1','2','3','4','5','6','7','8','9','10');
%
hilbert transforms
h11=hilbert(imag(y(:,frf)));
h12=real(y(:,frf));
h21=hilbert(real(y(:,frf)));
h22=imag(y(:,frf));
%
plots
figure(1)
if ii==1;
color='r';
end
if ii==2;
color='b';
end
if ii==3;
color='g';
end
if ii==4;
color='m';
end
123
if ii==5;
color='k';
end
if choice==1;
subplot(211)
plot(x(:,frf),imag(h11),'b',x(:,frf),h12,'r')
legend('Real Hilbert','Real FRF')
legend boxoff
xlabel('Frequency [Hz]')
ylabel('Magnitude')
title('Hilbert Transforms, Frequency Domain')
hold on
subplot(212)
plot(x(:,frf),-imag(h21),'b',x(:,frf),h22,'r')
legend('Imag Hilbert','Imag FRF')
legend boxoff
xlabel('Frequency [Hz]')
ylabel('Magnitude')
hold on
end
if choice~=1;
subplot(211)
plot(x(:,frf),imag(h11),color,x(:,frf),h12,color)
legend('Real Hilbert','Real FRF')
legend boxoff
xlabel('Frequency [Hz]')
ylabel('Magnitude')
title('Hilbert Transforms, Frequency Domain')
hold on
subplot(212)
plot(x(:,frf),-imag(h21),color,x(:,frf),h22,color)
legend('Imag Hilbert','Imag FRF')
legend boxoff
xlabel('Frequency [Hz]')
ylabel('Magnitude')
hold on
end
end
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