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Dynamic Systems
Vibration and Control
Fred Barez
Mechanical Engineering Department
College of.Engineering
San Jose State University
FALL 2015
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Dynamic Systems
Vibration and Control
Fred Barez
Mechanical Engineering Department
College of Engineering
San Jose State University
FALL 2015
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Preface
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This book is a compilation of materials related to mechanical vibration and control of
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dynamic systems.
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Currently, there is no single textbook available in the market in the market to cover these
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important subjects together. Vibrations and controls are two of the most essential areas of study
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for all mechanical engineering students. It is intended here, through the materials presented, than
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an overview, with some depth of the subject maters be provided to our readers.
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~ I would like to thank my students for making suggestions to improve the revision. Special
~ thanks are given to Mindy Kwan and Jenny Liu for preparing this manuscript. I also would like
~ to thank Kristen Steffe ck, Brissa Ortega, and Greg Calimpong for their contribution to this latest
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revision.
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Table of Contents
Table of Contents ........................................................................................... iii
Part 1 Mechanical Vibrations .......................................................................... 1-1
Introduction .............................................................................................. 1-2
Laws of Newtonian Mechanics ........................................................................... 1-4
Elements of a Vibratory System .......................................................................... 1-5
Vibration of Physical Systems ............................................................................ 1-6
Classification of Mechanical Vibrations ................................................................. 1-6
Free and Forced ....................................................................................................................... 1-6
Undamped and Damped ......................................................................................................... 1-7
Degrees ofFreedom ........................................................................................ 1-8
Linear and Non-Linear ..................................................................................... 1-8
Modeling and Analysis of Physical Vibratory Systems ................................................ 1-8
Derivation of the General Differential Equation ofMotion.................. . ................... 1-9
General Differential Equation of Motion: Force Method ............................................. 1-9
Undamped Free Vibration ............................................................................ 1-12
Undamped Rigid Body Vibration .................................................................... 1-15
Undamped Free Vibration of an Elastic Structure . . . . . . . . . . . . ........ .... .. ...... ...................... 1-17
Torsional Vibration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 1-17
Simple Pendulum................................................................................. .. 1-18
How to determine the Young's Modulus of an object in the form of a uniform beam? ............ 1-19
How to measure the mass and spring constant of a vibratory system? ..................................... 1-19
How to determine the mass moment of an irregularly shaped object? ..................................... 1-19
General Differential Equation of Motion: Energy Method .......................................... 1-20
Determination of the Natural Circular Frequency: Energy Method ............................... 1-21
General Differential Equation of Motion: D'Alembert's Method ................................... 1-22
Vibration of a Solid Body in Liquids ................................................................... 1-23
Damped Free Vibration .................................................................................. 1-24
Logarithmic Decrement................................................................................. 1-25
Forced Vibration of Particles ................................................................................... 1-28
Undamped Forced Vibration.................................................................. .. ..................... 1-28
a) Externally Applied Forces ................................................................................................. 1-28
b) Support Motion ................................................................................................................. 1-31
c) Unbalanced Rotating Forces (Self-Excitation) ................................................ 1-32
Beating ...................................................................................................... 1-33
Damped Forced Vibrations .............................................................................. 1-35
Externally Applied Forces ............................................................................. 1-35
Support Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... 1-39
Rotating Unbalanced Forces (Self-Excitation) ....................................... ... ...1-41
Transmitted forces .............................................................................................. 1-42
Undamped Forced Vibration.................................................................. . ....... 1-42
a) Externally Applied Forces...................................................... .. ............. 1-42
b) Support Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 1-42
iv
c) Rotating Unbalanced Forces ...................................................................... 1-43
Damped Forced Vibration ............................................................................... 1-43
Quality Factor .............................................................................................. 1-45
Raltive Motion and its Application to Vibration Measurements ..................................... 1-45
Seismic Instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-45
Vibration Measurement Instruments or Seismic Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 7
Seismometer: Instrument with Low Natural Frequency ............................................................ 1-47
Accelerometer: instrument with High Natural Frequency ......................................................... 1-48
Multidegree of Freedom Systems ....................................................................... 1-49
Eigenvalues and Eigenvectors ....................................................................... 1-52
Frequencies and Mode Shapes ....................................................................... 1-53
Equation of Motion ofMultidegree of Freedom System .......................................... 1-56
Vibration Desigtl and Control ........................................................................... 1-57
Shock and Vibration .................................................................................... 1-58
Vibration Isolation ...................................................................................... 1-59
Desigtl Considerations for Vibration Isolation ...................................................... 1-62
Vibration Absorbers... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1-64
Tuned Mass Dampers .................................................................................. 1-66
Vibration Frequency and Acceptable Diplomats ...................................................... 1-67
Distributed Parameter Systems ............................................................................................. 1-68
Longitudinal Vibration of a Uniform Rod ........................................................... 1-68
Torsional Vibration of a Uniform Rod ............................................................... 1-69
Transverse Vibration of a String or Cable.................................................... . ... 1-70
The Wave Equation ............................................................................................................. 1-71
Solution to Wave Equation .............................................................................. 1-72
Flow-Induced Vibrations .................................................................................. 1-73
Examples ................................................................................................... 1-76
Homework Problems .................................................................................... 1-107
Part 2 Control Systems ..•......•..•..•.....................................................•........•..•. 2-1
Introduction...................................................................................................................... 2-2
Physical Systems Mathematical Modeling ................................................................................. 2-2
Mechanical System..................................................................................... 2-2
Electrical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
Types of Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4
Open-Loop Control System ........................................................................... 2-4
Closed-Loop Control System . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4
Feedback Sensor or Signal Transformer . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . 2-5
Control System Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5
Manual and Automatic Feedback Control Systems.............................................. ... 2-6
Active and Passive Feedback Control Systems ......................................................... 2-6
Single-Input, Single-Output (SISO) and Multiple-Input, Multiple-Output (MIMO) Control
Systems ....................................................................................................... 2-6
Continuous-Data (Analog) and Simple-Data (Digital) Systems ..................................... 2-6
Examples of Control System .............................................................................. 2-6
Flow ofTraffic................... ... ........... ....... ....... ..... ..... ....... ....... ....... ..... ...... 2-7
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Vehicle Cruise Control ..................................................................................2 -7
House Heating System.................................................................... . ........... 2-7
Water Tank Level Control................................................................ . ........ 2-7
Missile Launcher Control...................................................................................... .. .......... 2-8
Control System Analysis and Design .. . . .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. . .. . .. ... 2-8
A Review of Laplace Transform Theory................................................... . . . . . . . 2-9
Model of a Simple Feedback Control System ........................................................ 2-11
Transfer Function ......................................................................................... 2-12
System Response .......................................................................................... 2-13
Block Diagram ............................................................................................. 2-14
Signal Flow Graph ........................................................................................ 2-16
Mason's Gain Formula ................................................................................... 2-17
Transfer Functidn and System Response ............................................................... 2-18
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Poles, Zeros, and System Response ..................................................................... 2-19
Poles of a yfansfer Function ........................................................................... 2-19
Zeros of a Transfer Function .......................................................................... 2-19
Response Characteristics .............................................................................. 2-19
System Stability ......................................................................................................................... 2-19
The Routh -Hurwitz Stability Criterion .................................................................................... 2-22
System Response and Stability Analysis ............................................................... 2-26
Transient Performance and the S-Plane ..................................................................................... 2-27
Response of High-Order Systems ....................................................................... 2-28
Nature of the Transient Response and Dominating Poles ......................................................... 2-29
Dominant Closed-Loop Poles .................................................................................................... 2-29
First-Order System: A Simple Lag System ............................................................................... 2-29
Second-Order System: A Quadratic Lag System ....................................................................... 2-31
Transient Response Specifications ............................................................................................ 2-33
Steady-State Response Performance Criteria ............................................................................ 2-34
System Types ............................................................................................................................. 2-35
Steady-State Errors .................................................................................................................... 2-36
Steady-State Error Due to a Step Input ................................................................................. 2-36
Steady-State Error Due to a Ramp Input ............................................................................... 2-37
Steady-State Error Due to a Parabolic Input ......................................................................... 2-38
Controller and Types ...................................................................................... 2-40
Proportional Controller ................................................................................ 2-41
Integral Controller ..................................................................................... 2-41
Derivative Controller ............................................................................................... 2-41
Proportional+ Integral ................................................................................ 2-42
Proportional + Derative ............................................................................... 2-42
Integral + Derivative ..................................................................................... 2-42
Proportional+ Integral+ Derivative ...............................................................................2 -42
State-Variable Modeling .................................................................................. 2-45
General Form of the State Variable Equation ......................................................... 2-46
Solution of State Equations ........................................................................................................ 2-47
Frequency Response Analysis........................................................................ 2-48
Frequency Response Function .......................................................................... 2-48
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Polar Plot of G Geo) .........................................•.•.........•.............•............................•• 2-50
Simple Lag System Example .................................................................................. 2-50
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-52
Relative Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-52
Nyquist Plot ............................................................................................................................... 2-54
Bode Plots .................................................................................................. 2-54
Frequency Response Performance Criteria ............................................................ 2-59
Open-Loop Frequency Response Performance Measures ......................................................... 2-60
Performance Specifications on the Bode Plot ......................................................... 2-62
Bandwidth .................................................................................................. 2-64
Root Locus ..................................................................................................... 2-69
How to Plot the Root Locus ....................................................................................................... 2-66
Examples ................................................................................................... 2-69
Homework Problems ..................................................................................... 2-121
Appendix A Fourier Series .......................•.•.•........•............................•.........•• A-1
Appendix B Matrix Algebra ..................•.......•............................•.....•.•..•.•.•..... B-1
Appendix C Mechanical Vibration Motion Summaries .•...............•.•...................... C-1
Appendix D Summary of Forced Vibration •...........•..•...•..••.•....•.••...•.......•........... D-1
Appendix E Vibration and Control Resources and Products ........•.•...•••.•..•..•..•....... E-1
Appendix G Vibration and Control Education Videos .•.•............................•...•..•..• F-1
Tables [1-52] ............................................................................................... T-1
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