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- Sprache: Englisch
VIRTUAL EXPERIMENTS in MECHANICAL VIBRATIONS
The first book of its kind to explain fundamental concepts in both vibrations and signal processing using MATLAB virtual experiments
Students and young engineers with a strong grounding in engineering theory often lack the practical skills and knowledge required to carry out experimental work in the laboratory. Fundamental and time-consuming errors can be avoided with the appropriate training and a solid understanding of basic concepts in vibrations and/or signal processing, which are critical to testing new designs.
Virtual Experiments in Mechanical Vibrations: Structural Dynamics and Signal Processing is designed for readers with limited knowledge of vibrations and signal processing. The intention is to help them relate vibration theory to measurements carried out in the laboratory. With a hands-on approach that emphasizes physics rather than mathematics, this practical resource explains fundamental concepts in vibrations and signal processing. It uses the concept of a virtual experiment together with MATLAB to show how the dynamic properties of vibration isolators can be determined, how vibration absorbers can be designed, and how they perform on distributed parameter structures.
Readers will find that this text:
- Allows the concepts of experimental work to be discussed and simulated in the classroom using a physics-based approach
- Presents computational virtual experiments using MATLAB examples to determine the dynamic behaviour of several common dynamic systems
- Explains the rationale of virtual experimentation and describes typical vibration testing setups
- Introduces the signal processing tools needed to determine the frequency response of a system from input and output data
- Includes access to a companion website containing MATLAB code
Virtual Experiments in Mechanical Vibrations: Structural Dynamics and Signal Processing is a must-have resource for researchers, mechanical engineers, and advanced undergraduate and graduate students who are new to the subjects of vibrations, signal processing, and vibration testing. It is also an invaluable tool for universities where the possibilities of doing experimental work are limited.
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Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
List of Abbreviations
List of Symbols
About the Companion Website
1 Introduction
1.1 Introduction
1.2 Typical Laboratory‐Based Vibration Tests
1.3 Relationship Between the Input and Output for a SISO System
1.4 A Virtual Vibration Test
1.5 Some Notes on the Book
References
2 Fundamentals of Vibration
2.1 Introduction
2.2 Basic Concepts – Mass, Stiffness, and Damping
2.3 Single Degree‐of‐Freedom System
2.4 Free Vibration
2.5 Impulse Response Function (IRF)
2.6 Determination of Damping from Free Vibration
2.7 Harmonic Excitation
2.8 Frequency Response Function (FRF)
2.9 Other Features of the Receptance FRF
2.10 Determination of Damping from an FRF
2.11 Reciprocal FRF
2.12 Summary
References
3 Fourier Analysis
3.1 Introduction
3.2 The Fourier Transform (FT)
3.3 The Discrete Time Fourier Transform (DTFT)
3.4 The Discrete Fourier Transform (DFT)
3.5 Inverse Fourier Transforms
3.6 Summary
References
4 Numerical Computation of the FRFs and IRFs of an SDOF System
4.1 Introduction
4.2 Effect of Sampling on the FRFs
4.3 Effect of Data Truncation
4.4 Effects of Sampling on the IRFs Calculated Using the IDFT
4.5 Summary
References
5 Vibration Excitation
5.1 Introduction
5.2 Vibration Excitation Devices
5.3 Vibration Excitation Signals
5.4 Summary
References
6 Determination of the Vibration Response of a System
6.1 Introduction
6.2 Determination of the Vibration Response
6.3 Calculation of the Vibration Response of an SDOF System
6.4 Summary
References
7 Frequency Response Function (FRF) Estimation
7.1 Introduction
7.2 Transient Excitation
7.3 Random Excitation
7.4 Comparison of Excitation Methods and Effects of Shaker–Structure Interaction
7.5 Virtual Experiment – Vibration Isolation
7.6 Summary
References
8 Multi‐Degree‐of‐Freedom (MDOF) Systems: Dynamic Behaviour
8.1 Introduction
8.2 Lumped Parameter MDOF System
8.3 Continuous Systems
8.4 Summary
References
9 Multi‐Degree‐of‐Freedom (MDOF) Systems: Virtual Experiments
9.1 Introduction
9.2 Two Degree‐of‐Freedom System: FRF Estimation
9.3 Beam: FRF Estimation
9.4 The Vibration Absorber as a Vibration Control Device
9.5 Summary
References
Appendix A: Numerical Differentiation and Integration
A.1 Differentiation in the Time Domain
A.2 Integration in the Time Domain
A.3 Differentiation and Integration in the Frequency Domain
Reference
Appendix B: The Hilbert Transform
References
Appendix C: The Decibel: A Brief Description
Reference
Appendix D: Numerical Integration of Equations of Motion
D.1 Euler's Method
D.2 The Runge–Kutta Method
References
Appendix E: The Delta Function
E.1 Properties of the Delta Function
E.2 Fourier Series Representation of a Train of Delta Functions
Reference
Appendix F: Aliasing
References
Appendix G: Convolution
G.1 Relationship Between Convolution and Multiplication
G.2 Circular Convolution
References
Appendix H: Some Influential Scientists in Topics Related to This Book
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Relationships between IRFs and FRFs for an SDOF system.
Table 2.2 Definitions of the FRFs and their reciprocals.
Chapter 3
Table 3.1 Fourier transforms and inverse Fourier transforms for a displacem...
Chapter 4
Table 4.1 Effects on the FRFs of an SDOF system using the IDFT of their res...
Chapter 5
Table 5.1 Some characteristics of excitation signals.
Chapter 6
Table 6.1 Some ways to calculate the output of a linear SDOF force‐excited ...
Chapter 7
Table 7.1 Random errors for the various estimators.
Table 7.2 Summary of methods to obtain an FRF.
Chapter 8
Table 8.1 Natural frequencies and mode shapes for some rod configurations....
Table 8.2 Natural frequencies and mode shapes for some beam configurations....
Table 8.3 Summary of the expression for the FRF and IRF of a lumped paramet...
Table 8.4 Summary of the expressions for the FRF and IRF of a rod.
Table 8.5 Summary of the expressions for the FRF and IRF of a beam.
Appendix A
Table A.1 Differentiation and integration in the time and frequency domains...
List of Illustrations
Chapter 1
Figure 1.1 The subject of vibration engineering.
Figure 1.2 Schematic diagram showing the scope of the book.
Figure 1.3 Typical experimental set‐ups to measure a frequency response func...
Figure 1.4 Block diagram representing of a simple single‐input, single‐outpu...
Figure 1.5 The process and rationale for a virtual vibration experiment.
Chapter 2
Figure 2.1 Fundamental lumped parameter elements. (ai) Linear spring. (aii) ...
Figure 2.2 SDOF mass‐spring‐damper system. (a) Conventional diagram. (b) Alt...
Figure 2.3 Free vibration of an SDOF system.
Figure 2.4 An impulsive force and its idealised representation as a delta fu...
Figure 2.5 Delayed IRFs for an SDOF system when a delta function force impul...
Figure 2.6 Response of an SDOF mass‐spring‐damper system to harmonic excitat...
Figure 2.7 Schematic diagram showing the amplitudes of the stiffness, dampin...
Figure 2.8 Frequency domain representation of an SDOF mass‐spring‐damper sys...
Figure 2.9 Modulus and phase of the receptance of an SDOF mass‐spring‐damper...
Figure 2.10 Modulus of the receptance of an SDOF mass‐spring‐damper system s...
Figure 2.11 Complex representation of the receptance for an SDOF mass‐spring...
Figure 2.12 Amplitude and phase of the displacement FRF at frequencies close...
Figure 2.13 Example of the displacement FRF used for damping estimation, sho...
Figure 2.14 Dynamic stiffness of an SDOF system. The real part is plotted as...
Figure 2.15 Displacement IRF and FRF for an SDOF mass‐spring‐damper system....
Chapter 3
Figure 3.1 A periodic displacement time history and its Fourier components....
Figure 3.2 Basis functions for the complex Fourier series: two contra rotati...
Figure 3.3 Modulus and phase spectrum of the complex Fourier series.
Figure 3.4 A periodic signal with period
T
p
that is extended to infinity.
Figure 3.5 Block diagram of an SDOF system.
Figure 3.6 Double‐sided receptance FRF.
Figure 3.7 A continuous time history
x
(
t
) and the same time history sampled ...
Figure 3.8 An impulse train of delta functions.
Figure 3.9 Impulse train representation of a sampled time history.
Figure 3.10 DTFT of
x
(
n
Δ
t
) compared to the FT of
x
(
t
).
Figure 3.11 DFT of
h
(
n
Δ
t
) compared to the FT of
h
(
t
) of the displacement IRF...
Figure 3.12 Illustration of the difference between a DFT of a signal sampled...
Figure 3.13 IDFT of the receptance of an SDOF system.
Chapter 4
Figure 4.1 Theoretical IRFs and FRFs of an SDOF system.
Figure 4.2 Receptance FRF of an SDOF system, showing the effects of aliasing...
Figure 4.3 Velocity IRF of an SDOF system. The inset shows the details of sa...
Figure 4.4 Mobility FRF of an SDOF system, showing the effects of sampling a...
Figure 4.5 Acceleration IRF of an SDOF system. The inset shows the details o...
Figure 4.6 Accelerance FRF of an SDOF system, showing the effects of samplin...
Figure 4.7 Illustration of the effects of data truncation.
Figure 4.8 Illustration of the effects of data truncation on the displacemen...
Figure 4.9 Illustration of the effects of windowing data in the frequency do...
Figure 4.10 Illustration of the effects of windowing data in the frequency d...
Figure 4.11 Illustration of the effects of windowing data in the frequency d...
Chapter 5
Figure 5.1 Vibration excitation of a structure using an electrodynamic shake...
Figure 5.2 Vibration excitation of a structure using an instrumented impact ...
Figure 5.3 Simple model of a force applied to a structure by an instrumented...
Figure 5.4 Signals supplied to an electrodynamic shaker or generated by an i...
Figure 5.5 A windowed section of a signal used for analysis.
Figure 5.6 Illustration of the assumption of periodicity in the time domain ...
Figure 5.7 Power spectral density of a truncated sine wave with a window of ...
Figure 5.8 Power spectral density of a truncated sine wave with a window of ...
Figure 5.9 Power spectral densities of sampled sine waves (from 0 to
f
s
/2)....
Figure 5.10 A typical random signal split into time segments for averaging p...
Figure 5.11 Effect of averaging on the variance of the estimate of the PSD....
Figure 5.12 Rectangular and Hanning windows.
Figure 5.13 Hanning window applied to a random signal for segment averaging....
Figure 5.14 Time history of a random signal and its single‐sided PSD estimat...
Figure 5.15 A linear chirp and its single‐sided ESD estimate.
Figure 5.16 Time history of a half‐sine pulse, and its FT and DFT (from 0 to...
Chapter 6
Figure 6.1 Block diagram of a simple vibrating system.
Figure 6.2 Determination of the response of a vibrating system by convolutio...
Figure 6.3 Calculation of the displacement response of an SDOF system excite...
Figure 6.4 Calculation of the displacement response of an SDOF system excite...
Chapter 7
Figure 7.1 Block diagram of a simple vibrating system.
Figure 7.2 Block diagram of a force‐excited system with added noise in the m...
Figure 7.3 Individual measurements to determine the FRF of an SDOF system fr...
Figure 7.4 Process of FRF estimation of an SDOF system due to transient forc...
Figure 7.5 Relationship between the coherence and SNR at each frequency for ...
Figure 7.6 Estimates of the FRF of an SDOF system excited with a half‐sine i...
Figure 7.7 Estimates of the FRF of an SDOF system excited with a half‐sine i...
Figure 7.8 Process of FRF estimation of an SDOF system due to random force e...
Figure 7.9 Frequency domain data resulting from processing the random data s...
Figure 7.10 Plots showing the effect of the window size compared to on the l...
Figure 7.11 Structure excited by an electrodynamic shaker and a simple model...
Figure 7.12 Frequency domain representation of the division of the generated...
Figure 7.13 Block diagram showing the relationship between the generated for...
Figure 7.14 Two basic vibration isolation situations.
Figure 7.15 Force or displacement transmissibility of an SDOF system.
Figure 7.16 Schematic diagram of an experiment to determine the stiffness an...
Figure 7.17 Illustration of the vibration of a suspended mass on an isolator...
Chapter 8
Figure 8.1 Simple lumped‐parameter model of an MDOF system.
Figure 8.2 Point and transfer receptances of a 3DOF lumped parameter system....
Figure 8.3 Mode shapes of the 3DOF system where the masses are all equal to ...
Figure 8.4 Relationship between resonance and anti‐resonance frequencies and...
Figure 8.5 Point and transfer receptances of a symmetric 3DOF lumped paramet...
Figure 8.6 Lumped parameter MDOF system in the physical and the modal domain...
Figure 8.7 Modal decomposition of the different FRFs for the MDOF system in ...
Figure 8.8 FRF and IRF of an MDOF system together its modal components.
Figure 8.9 Two continuous systems – a rod undergoing axial vibration and a b...
Figure 8.10 FRF and IRF of a fixed–free rod.
Figure 8.11 First three mode shapes of a pinned–pinned beam, a fixed–fixed b...
Figure 8.12 FRF and IRF of a fixed–free (cantilever) beam.
Chapter 9
Figure 9.1 Procedure to carry out a virtual experiment.
Figure 9.2 Measurement of the accelerance of a 2DOF system.
Figure 9.3 Measurement of receptance FRFs and modal properties of a cantilev...
Figure 9.4 Connecting a vibration absorber to a general vibrating system.
Figure 9.5 A vibration absorber attached to an SDOF host structure or a sing...
Figure 9.6 Normalised displacement FRFs of the absorber mass and the host st...
Figure 9.7 Experiment to measure the accelerance of the host structure, and ...
Figure 9.8 Experiment to measure the apparent mass of the vibration absorber...
Figure 9.9 Experiment to measure the effectiveness of the vibration absorber...
Figure 9.10 Experiment to measure the effective of a vibration absorber atta...
Figure 9.11 Experiments required to design the vibration absorber for the be...
Figure 9.12 Schematic diagram showing the dynamics of a host structure and a...
Appendix A
Figure A.1 Part of a velocity signal as a function of time.
Appendix B
Figure B.1 Amplitude‐ and phase‐modulated signal together with its envelope ...
Appendix C
Figure C.1 An illustration of the differences between the linear, log
10
, and...
Appendix D
Figure D.1 An illustration of Euler's method in the numerical integration of...
Figure D.2 An illustration of the Runge–Kutta method.
Appendix E
Figure E.1 The unit impulse and the delta function. (a) A unit impulse. (b) ...
Figure E.2 Sinc function approximation of a delta function.
Figure E.3 Relationship between a delta function in one domain and its equiv...
Figure E.4 A train of delta functions.
Appendix F
Figure F.1 Illustration of the sampling (and aliasing) of a rotating vector....
Figure F.2 Illustration of the effect of aliasing on the receptance FRF of a...
Figure F.3 Figure showing how the theoretical receptance FRF and the infinit...
Appendix G
Figure G.1 A schematic diagram of an LTI vibrating system.
Figure G.2 Illustration of a force input described as a series of impulses, ...
Figure G.3 An example of convolution for an SDOF system.
Figure G.4 Illustration of convolution in the frequency domain for an SDOF s...
Figure G.5 Determination of the response of a vibrating system by convolutio...
Figure G.6 Illustration of the difference between convolution and circular c...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface
List of Abbreviations
List of Symbols
About the Companion Website
Begin Reading
Appendix A Numerical Differentiation and Integration
Appendix B The Hilbert Transform
Appendix C The Decibel: A Brief Description
Appendix D Numerical Integration of Equations of Motion
Appendix E The Delta Function
Appendix F Aliasing
Appendix G Convolution
Appendix H Some Influential Scientists in Topics Related to This Book
Index
End User License Agreement
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Virtual Experiments in Mechanical Vibrations
Structural Dynamics and Signal Processing
Michael J. BrennanDepartment of Mechanical EngineeringSão Paulo State University (UNESP)Ilha SolteiraBrazil
Bin TangSchool of Energy and Power EngineeringDalian University of TechnologyChina
This edition first published 2023© 2023 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Michael J. Brennan and Bin Tang to be identified as the authors of this work has been asserted in accordance with law.
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Limit of Liability/Disclaimer of WarrantyMATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging‐in‐Publication Data Applied for:
Hardback ISBN: 9781118307977
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To Our Wives Laura and Xiudan, and Our Children Emma, Jingde, and Jinghui
“All models are wrong, but some are useful”
George Box
Preface
The idea to write this book came about from many years of interacting with students, both undergraduate and postgraduate. There seemed to be a disconnect between the theoretical treatment of mechanical vibrations and the signal processing procedures needed to measure vibration in the laboratory. They are often treated as separate subjects, sometimes taught in different departments by different lecturers. When the first author of the book came to UNESP Ilha Solteira in Brazil at the end of 2010, he decided to teach a course that combined the two approaches. The notes developed for that course form the basis of this book.
At the beginning of 2010 Bin Tang came as an academic visitor, supported from the China Scholarship Council (Grant No. 2009821053), to the Institute of Sound and Vibration Research (ISVR) in Southampton, UK, where Mike Brennan had a position as professor of engineering dynamics. They worked together for about one year on research related to nonlinear vibrations. Bin Tang then returned to his position as an assistant professor at Dalian University of Technology (DUT), and Mike departed for Brazil. The following year Mike visited Bin Tang in DUT, and about two years later, Bin Tang came to Brazil as an academic visitor, supported by the Brazilian National Council for Scientific and Technological Development (CNPq). He stayed for two years, and during this time they had many discussions about the topics in this book, honing the ideas and concepts. A decision was made to write the book, but this never really began in earnest until the COVID 19 pandemic struck in 2020. This curtailed the much‐enjoyed academic activity of travelling and meeting colleagues around the world, and freed up some time to work on the book.
The authors are extremely grateful for the many discussions with both colleagues and students over the years that have helped to form the perspective from which the book is written. The authors would like to acknowledge the financial support of the Brazilian National Council for Scientific and Technological Development (CNPq), (Grant No. 401360/2012-1) and the National Natural Science Foundation of China (Grant No. 11672058). It is hoped that students who are new to the topic, or those who are more experienced in some areas of either vibration or signal processing will find the book useful.
Michael J. Brennan
São Paulo State University (UNESP)
Ilha Solteira
Brazil
Bin Tang
Dalian University of Technology
China
January 2022
List of Abbreviations
CPSD
cross power spectral density
DFT
discrete Fourier transform
DOF
degrees‐of‐freedom
DTFT
discrete time Fourier transform
Env
envelope
ESD
energy spectral density
FEM
finite element method
FFT
fast Fourier transform
FRF
frequency response function
FS
Fourier series
FT
Fourier transform
IDFT
inverse discrete Fourier transform
IDTFT
inverse discrete time Fourier transform
IFT
inverse Fourier transform
Im
imaginary part
IRF
impulse response function
ln
natural logarithm
LTI
linear time‐invariant
MDOF
multi‐degree‐of‐freedom
ODE
ordinary differential equation
PDE
partial differential equation
PSD
power spectral density
Re
real part
rms
root mean square
SDOF
single‐degree‐of‐freedom
SIMO
single input multiple outputs
SISO
single input single output
SNR
signal-to-noise ratio
List of Symbols
Symbol
Description
Units
a
(
t
)
Analytic signal (displacement)
[m]
(
A
n
,
l
)
p
Modal constant for the
p
‐th mode
[1/kg]
c
Viscous damping coefficient
[Ns/m]
c
B
Phase speed in a beam
[m/s]
c
R
Phase speed in a rod
[m/s]
C
Damping matrix
[Ns/m]
Modal damping matrix
[Ns/m]
E
Young's modulus
[N/m
2
]
E
Expectation operator
f
Frequency
[Hz]
f
c
(
t
)
Damping force
[N]
f
e
(
t
)
Excitation force
[N]
Force impulse
[Ns]
f
i
(
t
)
Force applied at point
i
[N]
f
k
(
t
)
Stiffness force
[N]
f
m
(
t
)
Inertia force
[N]
f
n
Natural frequency
[Hz]
f
s
Sampling frequency
[Hz]
f
(
t
)
Vector of forces
[N]
Vector of complex force amplitudes
[N]
ℱ
Fourier transform operator
ℱ
−1
Inverse Fourier transform operator
Complex force amplitude
[N]
Force amplitude
[N]
Complex damping force
[N]
Complex stiffness force
[N]
Complex inertia force
[N]
F
(j
ω
)
FT of
f
e
(
t
)
[N/Hz]
g
Modal force vector
[N]
g
p
(
t
)
Modal force for the
p
‐th mode
[N]
Estimate of the single‐sided PSD of
x
(
t
)
[m
2
/Hz]
h
(
t
)
Displacement impulse response function
[m/Ns]
Velocity impulse response function
[m/Ns
2
]
Acceleration impulse response function
[m/Ns
3
]
H
(j
ω
),
H
(
f
)
Receptance FRF
[m/N]
H
(j
ω
)
Receptance matrix
[m/N]
H
vel
(j
ω
)
Mobility FRF
[m/Ns]
H
acc
(j
ω
)
Accelerance FRF
[m/Ns
2
]
H
1
(j
ω
)
H
1
estimator
[m/N]
H
2
(j
ω
)
H
2
estimator
[m/N]
i
(
t
)
Train of delta functions
[1/s]
i
s
(
t
)
Current supplied to the shaker
[A]
I
Second moment of area for the cross-section of a beam
[m
4
]
j
k
Stiffness
[N/m]
Modal stiffness of the
p
‐th mode
[N/m]
K
(j
ω
)
Dynamic stiffness
[N/m]
K
Stiffness matrix
[N/m]
Modal stiffness matrix
[N/m]
m
Mass
[kg]
Modal mass of the
p
‐th mode
[kg]
M
(j
ω
)
Apparent mass
[kg]
Complex moment amplitude
[Nm]
M
Mass matrix
[kg]
Modal mass matrix
[kg]
q
Vector of modal displacements
[m]
q
p
(
t
)
Modal participation factor of the
p
-th mode
[m]
R
ij
Residual for the modal model
[m/N]
S
Cross‐sectional area of a rod or a beam
[m
2
]
S
ff
(
ω
)
PSD of
f
e
(
t
)
[N
2
/Hz]
Estimate of the PSD of
f
e
(
t
)
[N
2
/Hz]
S
fx
(j
ω
)
CPSD between
f
e
(
t
) and
x
(
t
)
[Nm/Hz]
Estimate of the CPSD between
f
e
(
t
) and
x
(
t
)
[Nm/Hz]
S
xx
(
f
)
PSD of
x
(
t
)
[m
2
/Hz]
Estimate of the PSD of
x
(
t
)
[m
2
/Hz]
S
xf
(j
ω
)
CPSD between
x
(
t
) and
f
e
(
t
)
[Nm/Hz]
Estimate of the CPSD between
x
(
t
) and
f
e
(
t
)
[Nm/Hz]
t
Time
[s]
T
Time duration
[s]
T
d
Damped natural period
[s]
T
n
Undamped natural period
[s]
T
p
Fundamental period of a periodic signal
[s]
u
(
t
)
Heaviside function
u
(
x
,
t
)
Axial displacement of a rod
[m]
Complex axial displacement amplitudes for a rod
[m]
w
(
x
,
t
)
Lateral displacement of a beam
w
(
t
),
W
(
f
)
windows in the time domain and its FT
[m]
Complex displacement amplitude of a beam
[m]
Complex lateral displacement amplitude for a beam
[m]
x
(
t
)
Displacement
[m]
x
(
t
)
Vector of displacements
[m]
Velocity
[m/s]
Vector of velocities
[m/s]
Acceleration
[m/s
2
]
Vector of accelerations
[m/s
2
]
Vector of complex displacement amplitudes
[m]
Complex displacement amplitude
[m]
Displacement amplitude
[m]
Amplitude of the
n
‐th harmonic of the Fourier series
[m]
Complex amplitude of the
n
‐th harmonic of the complex
[m/Hz]
Fourier series
[m]
X
(j
ω
),
X
(
f
)
FT of
x
(
t
)
X
s
(
f
)
DTFT of
x
(
t
)
[m]
X
(
k
Δ
f
)
DFT of
x
(
n
Δ
t
)
[m]
Z
(j
ω
)
Impedance
[Ns/m]
Greek Symbols
α
Time delay
[s]
β
R
Wavenumber for a rod
[1/m]
β
B
Wavenumber for a beam
[1/m]
δ
(
t
)
Delta function
[1/s]
δ
(
x
)
Delta function
[1/m]
Δ
f
Frequency resolution
[Hz]
Δ
t
Time resolution
[s]
ε
Time duration
[s]
φ
,
θ
,
ψ
Phase angle
[rad]
φ
p
(
x
)
Mode shape for the
p
‐th mode of a rod or a beam
φ
p
Mode shape vector for the
p
‐th mode of a lumped parameter system
Φ
Matrix of mode‐shape vectors for a lumped parameter system
Coherence function between
f
and
x
γ
Ratio of absorber nat. freq. to host structure nat. freq.
η
Structural loss factor
μ
Mass ratio
ρ
Density
[kg/m
3
]
σ
Standard deviation
ζ
Damping ratio
ζ
p
Damping ratio of the
p
‐th mode
ω
Circular excitation frequency
[rad/s]
ω
a
Undamped natural frequency of a vibration absorber
[rad/s]
ω
d
Damped natural frequency
[rad/s]
ω
n
Undamped natural frequency or the
n
‐th harmonic
[rad/s]
ω
p
Undamped natural frequency of the
p
‐th mode
[rad/s]
Ω
Non‐dimensional frequency
About the Companion Website
This book is accompanied by a companion website which has MATLAB files.
www.wiley.com/go/brennan/virtualexperimentsinmechanicalvibrations
1Introduction
1.1 Introduction
Knowledge of the dynamic behaviour of systems and structures becomes increasingly important as organisations and companies strive to produce devices and products that outperform the competition. This means that engineers from a wide range of disciplines covering, for example, automotive, acoustical, aeronautical, aerospace, civil, mechanical, and marine engineering, are required to have knowledge of vibration engineering. Of course, some will need to be experts in this discipline, but others will simply need to be aware of some basic issues. This means that university engineering programmes for all the disciplines mentioned above generally have a course in mechanical vibrations. These courses tackle the subject in different ways, depending on the particular discipline. For example, civil engineers start from the study of the static behaviour of structures. Once this has been mastered, they move to the dynamic behaviour of structures, i.e. they start at a frequency of 0 Hz, and then investigate the behaviour as frequency increases. This sequence of study is similar for many disciplines, with the exception, perhaps, of physicists and acoustical engineers, who may tackle the subject using a wave description of the structural dynamics. Acoustical engineers generally restrict their frequency range of interest to that of human hearing, which is from about 20 Hz to 20 kHz. Thus, the way in which mechanical vibration is taught may vary enormously from course to course. To illustrate the diversity of the topic, Michael Brennan, the first author of this book, started his career in vibration engineering by investigating high‐frequency (>500 Hz) structure‐borne noise through a helicopter gear box support strut, whereas Bin Tang, the second author of this book, started his career by investigating the relatively low‐frequency torsional vibration (<30 Hz) of a ship's propellor shaft.
The terms ‘mechanical vibration’ and ‘engineering vibration’ are used interchangeably in this book. To master this topic from a theoretical and a practical point of view, the student is required to have some knowledge of physics, mathematics, and engineering. This is illustrated schematically in the Venn diagram shown in Figure 1.1. It is acknowledged that not all vibration engineers have the same profile. For example, some have a much more mathematical bias, focusing on theoretical aspects of the subject, perhaps working as researchers in universities, and others follow a much more practical career, working on the implementation of vibration control strategies in consulting or engineering companies. Notwithstanding this, it is the firm belief of the authors, that engineers/researchers will only gain mastery of the topic, if their knowledge base is in the area of the overlapping circles shown in Figure 1.1.
Figure 1.1 The subject of vibration engineering.
It is not the aim of this book to provide basic knowledge in mechanical vibrations, although it is expected that the reader will gain some insight into the dynamical behaviour of a simple vibrating system. There are several textbooks devoted to basic vibration theory, for example, Tse et al. (1978), Thompson (2002), Clough and Penzien (2003), de Silva (2006), Inman (2007), and Rao (2016). There is also the classic book (Den Hartog, 1956) that offers some excellent physical descriptions of vibrating systems. The aim of this book is to provide a text that will help to bridge the gap between vibration theory and laboratory‐based experimental work. Many students study vibration from a purely theoretical point of view. In some institutions, the lecturers may not even be experts in vibration engineering, and so they teach by closely following a textbook. Inevitably, this is often a mathematical exposition, with the underlying physics being frequently masked by mathematical complexity. Accordingly, many students do not gain the necessary physical insight, which would be helpful in their future careers. One way to overcome this problem is to formulate vibration problems in a more physical way in terms of variables that are measurable in a laboratory setting. In many situations, these are forces applied to the system or structure and the resulting accelerations/velocities/displacements. This means that before the theory is taught, some thought should be given to an accompanying experiment, to ensure that the output from the theoretical model involves measurable variables. It is, of course, desirable that any course has a practical component to complement and support the theory.
Much of the physical insight gained in vibration engineering, whether it be theoretical or experimental, occurs by viewing data in the frequency domain. However, all vibration signals are measured in the time domain, so these signals must be transformed to the frequency domain using signal processing techniques. This, of course, means that the vibration engineer should have some knowledge of the way in which this is done, and the mathematical basis behind the techniques. The way data are processed in practice is to first sample the data and then to work on them in digitised form using a computer. Processing sampled data brings further complications, which are discussed in Chapter 4. Many students of vibration engineering may have studied some signal processing techniques, such as Fourier analysis, but often this is done in a mathematics department, and therefore is often not related directly to the vibration theory taught in the engineering departments. There can thus be a chasm between the taught vibration theory and the way in which corresponding experimental data are captured and processed to enable comparisons between predictions and reality. It is the intention of this book to bring together these two disciplines and to give the reader some experience in applying the required signal processing techniques on simulated vibration data. There is one book on signal processing, which is specifically tailored for sound and vibration engineers (Shin and Hammond 2008), and there are other more general textbooks on the subject, which may help the reader with some of the more theoretical aspects, for example Papoulis (1962, 1977), Oppenheim and Schafer (1975), Oppenheim et al. (1997), and Bendat and Piersol (1980, 2000).
The scope of the book is encapsulated in the Venn diagram shown in the top part of Figure 1.2. It can be seen that it contains three elements, vibration theory, vibration experiments, and signal processing. At the end of the book the reader will have been exposed to elements of these three topics and will have carried out some ‘virtual’ experiments using simulated data. Through the theoretical development and exercises in the book, some proficiency should be gained, which hopefully will result in improved physical insight into both vibration theory and the rationale between the choices to be made in the signal processing procedures. At the end of the book, the reader should be in a position to carry out an experiment in the laboratory and process the measured signals, provided that the experimenter has been given some additional tuition on the practical aspects of how to set up an experiment and how to handle the transducers correctly.
Figure 1.2 Schematic diagram showing the scope of the book.
1.2 Typical Laboratory‐Based Vibration Tests
Two typical vibration tests are shown in Figure 1.3. In the top part of the figure, an electrodynamic shaker is used to excite the structure under test, and in the lower part of the figure an instrumented impact hammer is used to excite the structure. In both cases, the resulting vibration response is measured using an accelerometer, as shown in the figure. Details of some typical signals, which are used to drive the shaker and the type of force signal generated by the impact hammer, are discussed in Chapter 5. For the shaker excitation, a signal is provided by a signal generator, which is then passed through a power amplifier, before supplying the shaker. The signal then has enough power to drive the shaker. In many cases the signal generator forms part of a software package in a computer. The force is measured using a force gauge attached to the structure, and this signal together with the signal from the accelerometer are passed through conditioning amplifiers before entering the signal analyser, and being viewed in analogue form using the oscilloscope. For hammer excitation, the force gauge is in the tip of the hammer and measures the force applied to the structure during the impact. The signals from the force gauge and the accelerometer are processed in a similar way for both shaker and force excitation. Further details on how to set up a vibration experiment similar to that shown in Figure 1.3 are given in Waters (2013). General textbooks on vibration testing have been written by Ewins (2000), McConnell and Varoto (2008), Brandt (2011), and Avitabile (2017).
Figure 1.3 Typical experimental set‐ups to measure a frequency response function (FRF).
Source: Modified from Waters (2013) / Taylor & Francis.
The test set‐ups shown in Figure 1.3. are designed to measure a single input and single output (SISO). More accelerometers can be added at different points on the structure to form a single input multi‐output (SIMO) system, and an example of this type of measurement is described in Chapter 9. As mentioned above, more insight is gained by examining the data in the frequency domain – specifically the output for a given input at each frequency of excitation. This is achieved by studying this relationship which is called the frequency response function (FRF). The FRF is the backbone of this book, both theoretically and experimentally. It is derived analytically for a simple vibrating system in Chapter 2, and the way in which it is estimated from measurements or simulations using time domain force and acceleration data is described in Chapter 8.
1.3 Relationship Between the Input and Output for a SISO System
The relationships between the signals from a vibration measurement are shown in Figure 1.4. However, note that in this figure, displacement rather than acceleration is the response variable. This has been chosen for convenience, but also note that acceleration signals can easily be converted to velocity or displacement, by time‐domain integration as discussed in Appendix A. The engineering units are shown for all the variables in Figure 1.4, as this is considered to be important in the context of this book and is rarely provided in books on signal processing. The input to the system is a force fe(t) which has the SI unit of N, and the displacement response x(t) which has the unit of m. The vibrating system connecting the input to the output has a time domain description h(t), which is the impulse response function (IRF) and has units of m/Ns. The displacement output can be determined by convolving fe(t) with h(t), which is discussed further in Chapter 2, and is used extensively throughout the book.
Figure 1.4 Block diagram representing of a simple single‐input, single‐output vibration test. Note that the response in this case is displacement for convenience, which can be obtained by integrating acceleration twice as described in Appendix A.
As mentioned above, it is necessary to transform the data to the frequency domain. This is achieved by using the Fourier transform. The Fourier transform of the force time history is given by ℱ{fe(t)} and results in F(jω), where and ω is angular frequency, which has units of rad/s; F(jω) has units of N/Hz. Note that in this book time domain quantities are denoted by lower‐case italic symbols and frequency domain quantities are denoted by upper‐case italic symbols. The Fourier transform of the displacement time history is given by X(jω) = ℱ{x(t)}, which has units of m/Hz. Chapter 3 is devoted to the Fourier transform applied to continuous and sampled time histories. Note that frequency domain data can be transformed to the time domain, and this is achieved using the inverse Fourier transform, which is also discussed in Chapter 3. The frequency domain description of the system is the FRF, denoted by H(jω). This is related to the IRF by the Fourier transform, i.e. H(jω) = ℱ{h(t)} and has units of m/N. The output in the frequency domain X(jω) can be determined by multiplying F(jω) with H(jω), and this is discussed in Chapter 2.
You will become aware as you read this book that most of the analysis is conducted using FRFs. The theoretical FRFs shown are analytical because the systems discussed are relatively simple. However, if modelling is carried out using numerical tools such as finite element analysis (FEA), Petyt (2010), which is used extensively in industry, it is also important that structures are modelled so that FRFs can be easily extracted for analysis and comparison with measurements.
1.4 A Virtual Vibration Test
As mentioned previously, the aim of this book is to bridge the gap between vibration theory and vibration experiments. The book can also be used by students who do not have access to a laboratory to conduct experiments. They can carry out ‘virtual’ experiments. In a real experiment both force input and displacement output are measured, but in a virtual experiment the output data are generated using a model of the system. The concept is shown in Figure 1.5. The virtual experiment has a major advantage as a learning tool, in that the processed data in terms of an IRF or FRF, can be compared with the original model, which was used to generate the output time series. In this way, any artefacts in the data due to the processing can be clearly identified, which is not always possible in a real experiment.
Several methods can be used to determine the displacement output data, three of which are used in this book, and are described in Chapter 6. They are:
If the differential equation(s) of the vibrating system are known, then the response can be calculated by numerical integration of the equation(s) of motion. Generally, this is a straightforward procedure using a computer and is described in Appendix D.
If the IRF of the vibrating system is known, the response can be determined using convolution. Again, this is a relatively straightforward procedure and is described in Appendix G.
If the FRF of the vibrating system is known, the input force time history can be transformed to the frequency domain using the Fourier transform. The frequency domain response can then be calculated by multiplying this by the FRF, which can then be transformed to the time domain using the inverse Fourier transform to give the time history of the response. Alternatively, the FRF can be transformed to give the IRF and then the method in 2 can be used.
Figure 1.5 The process and rationale for a virtual vibration experiment.
1.5 Some Notes on the Book
The first thing that should be highlighted, is that this book is written in a heuristic way. Some mathematical proofs and details are omitted for ease of understanding. The more mathematically minded reader can readily find these details in the literature cited in this chapter. Secondly, the book is designed to be followed chapter by chapter to cover all the basic topics needed to conduct a virtual experiment. However, if the reader has detailed knowledge of certain topics, for example vibration theory, then Chapter 2 can be skipped, or if the reader is proficient in Fourier analysis, then Chapter 3 can be skipped.
The book is written with a novice in mind, so that very little previous knowledge is assumed. Accordingly, the book could be used as an undergraduate or a postgraduate text. The treatment of most topics, however, is brief, even though it is self‐contained, so many readers may need to consult other basic texts, for example Rao (2016) for vibration or Shin and Hammond (2008) for signal processing, for more detailed information.
Each chapter contains some computer programs written using MATLAB, which are provided to illustrate some of the concepts and to give the reader some practice in applying the techniques presented to consolidate their understanding. The programs can be found on the accompanying website. Although MATLAB is used for convenience to illustrate the computational procedures, the code can be readily modified to run in other software packages such as GNU Octave1 or python2, which are open source.
References
Avitabile, P. (2017).
Modal Testing: A Practitioner's Guide
. Wiley.
Bendat, J.S. and Piersol, A.G. (1980).
Engineering Applications of Correlation and Spectral Analysis
. Wiley.
Bendat, J.S. and Piersol, A.G. (2000).
Random Data: Analysis and Measurement Procedures
, 3
rd
Edition. Wiley‐Interscience.
Brandt, A. (2011).
Noise and Vibration Analysis: Signal Analysis and Experimental Procedures
. Wiley.
Clough, R.W. and Penzien, J. (2003).
Dynamics of Structures
, 3
rd
Edition. Computers & Structures, Inc.
Ewins, D.J. (2000).
Modal Testing: Theory, Practice and Application
, 2
nd
Edition. Research Studies Press.
Den Hartog, J.P. (1956).
Mechanical Vibrations
, 4
th
Edition. McGraw‐Hill.
Inman, D.J. (2007).
Engineering Vibration
, 3
rd
Edition. Pearson.
McConnell, K.G. and Varoto, P.S. (2008).
Vibration Testing: Theory and Practice
, 2
nd
Edition. Wiley.
Oppenheim, A.V. and Schafer, R.W. (1975).
Digital Signal Processing
. Prentice Hall International.
Oppenheim, A.V., Willsky, A.S., and Hamid Nawab, S. (1997).
Signals and Systems
, 2
nd
Edition. Prentice Hall International.
Papoulis, A. (1962).
The Fourier Integral and Its Applications
, McGraw‐Hill.
Papoulis, A. (1977).
Signal Analysis
, McGraw‐Hill.
Petyt, M. (2010).
Introduction to Finite Element Vibration Analysis
, 2
nd
Edition. Cambridge University Press.
Rao, S.S. (2016).
Mechanical Vibrations
, 6
th
Edition. Pearson.
Shin, K. and Hammond, J.K. (2008).
Fundamentals of Signal Processing for Sound and Vibration Engineers
. Wiley.
de Silva, C.W. (2006).
Vibration: Fundamentals and Practice
, 2
nd
Edition. CRC Press.
Thompson, W.T. (2002).
Theory of Vibration with Applications
, 3
rd
Edition. CBS Publishers & Distributors.
Tse, F.S., Morse, I.E., and Hinkle, R.T. (1978).
Mechanical Vibrations – Theory and Applications
, 2
nd
Edition. Ally and Bacon, Inc.
Waters, T.P. (2013).
Vibration Testing
,
Chapter 9 in Fundamentals of Sound and Vibration
,
(eds. F.J. Fahy and D.J. Thompson). CRC Press.
Notes
1
https://www.gnu.org/software/octave/index
(accessed 27 December 2021)
2
https://www.python.org/
(accessed 27 December 2021)
2Fundamentals of Vibration
2.1 Introduction
A vibrating system can be characterised in both the time and frequency domain. The quantities used to characterise the system can be obtained theoretically or experimentally, and are used extensively in this book. This chapter is devoted to deriving these quantities for a simple mechanical system. Thorough knowledge of such a system is essential for the deeper understanding of mechanical vibrations in general. Further, an understanding of the dynamics of a vibrating system in terms of its physical properties is extremely helpful in the interpretation of experimental data. No previous knowledge of vibrations is assumed in this chapter, as all the results are derived from first principles, requiring only a basic understanding of mechanics.
2.2 Basic Concepts – Mass, Stiffness, and Damping
There are three fundamental physical properties of a vibrating system. They are mass, stiffness, and damping. Although they tend to exist in a distributed form in the real world, for an initial study of vibration it is convenient to represent them in lumped parameter form using idealised elements as shown in Figure 2.1. Note that only translational linear elements are considered for simplicity, rather than rotational and/or nonlinear elements, which also exist in the real world. The interested reader is referred to more‐in‐depth texts on linear and nonlinear vibration, such as Tse et al. (1978), Inman (2007), Worden and Tomlinson (2001), Thomsen (2003), Kovacic and Brennan (2011), and Rao (2016).
The stiffness element is represented by a linear, massless spring with stiffness k, which has units of N/m. It is shown in Figure 2.1ai. The equations relating the forces at each end of the spring to the corresponding displacements are given by
and
Summing Eqs. (2.1a) and (2.1b) results in f2(t) = − f1(t), which shows that a force passes unattenuated through the stiffness element. If the right‐hand end of the spring is blocked, i.e. it is connected to a rigid foundation as shown in Figure 2.1aii, then it is described simply by fk(t) = kx(t), where fk(t) = f1(t) and x(t) = x1(t).
Figure 2.1 Fundamental lumped parameter elements. (ai) Linear spring. (aii) Linear spring with one end attached to ground. (bi) Linear viscous damper. (bii) Linear viscous damper with one end attached to ground. (ci) Lumped mass. (cii) Lumped mass with one free end.
For convenience, the damping element is represented by a linear viscous damper, with damping coefficient c, which has units of Ns/m. The damper has infinitely stiff (rigid), massless links that connect to the damping element, which is shown in Figure 2.1bi. The equations relating the forces at each end of the damper to the corresponding velocities are given by
and
where the overdot denotes differentiation with respect to time. Summing Eqs. (2.2a) and (2.2b) results in f2(t) = − f1(t
