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Library of Congress Cataloging-in-Publication Data

Names: Inman, D. J., author.

Title: Vibration with control / Daniel John Inman.

Description: Second edition. | Chichester, West Sussex, UK ; Hoboken, NJ, USA:

John Wiley & Sons Inc., [2017] | Includes bibliographical references and index.

Identifiers: LCCN 2016048220 | ISBN 9781119108214 (cloth) |

ISBN 9781119108238 (Adobe PDF) | ISBN 9781119108221 (epub)

Subjects: LCSH: Damping (Mechanics). | Vibration.

Classification: LCC TA355 .I523 2017 | DDC 620.3--dc23 LC record available at

https://lccn.loc.gov/2016048220

Cover Images: (background) © jm1366/Getty Images, Inc.; (left to right) © Teun van den Dries / Shutterstock.com; © simonlong/Getty Images, Inc.; © Edi_Eco/Getty Images, Inc.; © karnaval2018 / Shutterstock.com; © Stefano Borsani / iStockphoto

Cover Design: Wiley

CONTENTS

Preface

About the Companion Website

1: Single Degree of Freedom Systems

1.1 Introduction
1.2 Spring-Mass System
1.3 Spring-Mass-Damper System
1.4 Forced Response
1.5 Transfer Functions and Frequency Methods
1.6 Complex Representation and Impedance
1.7 Measurement and Testing
1.8 Stability
1.9 Design and Control of Vibrations
1.10 Nonlinear Vibrations
1.11 Computing and Simulation in MATLAB^TM
Chapter Notes
References
Problems

2: Lumped Parameter Models

2.1 Introduction
2.2 Modeling
2.3 Classifications of Systems
2.4 Feedback Control Systems
2.5 Examples
2.6 Experimental Models
2.7 Nonlinear Models and Equilibrium
Chapter Notes
References
Problems

3: Matrices and the Free Response

3.1 Introduction
3.2 Eigenvalues and Eigenvectors
3.3 Natural Frequencies and Mode Shapes
3.4 Canonical Forms
3.5 Lambda Matrices
3.6 Eigenvalue Estimates
3.7 Computation Eigenvalue Problems in MATLAB
3.8 Numerical Simulation of the Time Response in MATLAB^tm
Chapter Notes
References
Problems

4: Stability

4.1 Introduction
4.2 Lyapunov Stability
4.3 Conservative Systems
4.4 Systems with Damping
4.5 Semidefinite Damping
4.6 Gyroscopic Systems
4.7 Damped Gyroscopic Systems
4.8 Circulatory Systems
4.9 Asymmetric Systems
4.10 Feedback Systems
4.11 Stability in the State Space
4.12 Stability of Nonlinear Systems
Chapter Notes
References
Problems

5: Forced Response of Lumped Parameter Systems

5.1 Introduction
5.2 Response via State Space Methods
5.3 Decoupling Conditions and Modal Analysis
5.4 Response of Systems with Damping
5.5 Stability of the Forced Response
5.6 Response Bounds
5.7 Frequency Response Methods
5.8 Stability of Feedback Control
5.9 Numerical Simulations in MATLAB
Chapter Notes
References
Problems

6: Vibration Suppression

6.1 Introduction
6.2 Isolators and Absorbers
6.3 Optimization Methods
6.4 Metastructures
6.5 Design Sensitivity and Redesign
6.6 Passive and Active Control
6.7 Controllability and Observability
6.8 Eigenstructure Assignment
6.9 Optimal Control
6.10 Observers (Estimators)
6.11 Realization
6.12 Reduced-Order Modeling
6.13 Modal Control in State Space
6.14 Modal Control in Physical Space
6.15 Robustness
6.16 Positive Position Feedback Control
6.17 MATLAB Commands for Control Calculations
Chapter Notes
References
Problems

7: Distributed Parameter Models

7.1 Introduction
7.2 Equations of Motion
7.3 Vibration of Strings
7.4 Rods and Bars
7.5 Vibration of Beams
7.6 Coupled Effects
7.7 Membranes and Plates
7.8 Layered Materials
7.9 Damping Models
7.10 Modeling Piezoelectric Wafers
Chapter Notes
References
Problems

8: Formal Methods of Solutions

8.1 Introduction
8.2 Boundary Value Problems and Eigenfunctions
8.3 Modal Analysis of the Free Response
8.4 Modal Analysis in Damped Systems
8.5 Transform Methods
8.6 Green's Functions
Chapter Notes
References
Problems

9: Operators and the Free Response

9.1 Introduction
9.2 Hilbert Spaces
9.3 Expansion Theorems
9.4 Linear Operators
9.5 Compact Operators
9.6 Theoretical Modal Analysis
9.7 Eigenvalue Estimates
9.8 Enclosure Theorems
Chapter Notes
References
Problems

10: Forced Response and Control

10.1 Introduction
10.2 Response by Modal Analysis
10.3 Modal Design Criteria
10.4 Combined Dynamical Systems
10.5 Passive Control and Design
10.6 Distributed Modal Control
10.7 Nonmodal Distributed Control
10.8 State Space Control Analysis
10.9 Vibration Suppression using Piezoelectric Materials
Chapter Notes
References
Problems

11: Approximations of Distributed Parameter Models

11.1 Introduction
11.2 Modal Truncation
11.3 Rayleigh-Ritz-Galerkin Approximations
11.4 Finite Element Method
11.5 Substructure Analysis
11.6 Truncation in the Presence of Control
11.7 Impedance Method of Truncation and Control
Chapter Notes
References
Problems

12: Vibration Measurement

12.1 Introduction
12.2 Measurement Hardware
12.3 Digital Signal Processing
12.4 Random Signal Analysis
12.5 Modal Data Extraction (Frequency Domain)
12.6 Modal Data Extraction (Time Domain)
12.7 Model Identification
12.8 Model Updating
12.9 Verification and Validation
Chapter Notes
References
Problems

A: Comments on Units

Reference

B: Supplementary Mathematics

B.1 Vector Space
B.2 Rank
B.3 Inverses
References

Index

EULA

List of Illustrations

Chapter 1

Figure 1.1 (a) A spring-mass schematic, (b) a free body diagram, and (c) a free body diagram of the static spring mass system.
Figure 1.2 Measurement of spring constant using static deflection caused by added mass.
Figure 1.3 Determination of the spring constant. The dashed box indicates the linear range of the spring.
Figure 1.4 The response of a simple spring-mass system to an initial displacement of x₀ = 0.5 mm and an initial velocity of . The period, defined as the time it takes to complete one cycle off oscillation, T = 2π/ω_n, becomes T = 2π/2 = πs.
Figure 1.5 (a) Schematic of spring-mass-damper system. (b) A free-body diagram of the system in part (a).
Figure 1.6 Response of an underdamped system illustrating oscillation with exponential decay.
Figure 1.7 Response of an overdamped system illustrating exponential decay without oscillation.
Figure 1.8 Response of critically damped system to an initial displacement and three different initial velocities indicating no oscillation.
Figure 1.9 (a) The schematic of the forced spring-mass-damper system, assuming no friction on the surface. (b) The free-body diagram of the system of part (a).
Figure 1.10 Magnification curves (dimensionless) for an SDOF system showing the normalized amplitude of vibration versus the ratio of driving frequency to natural frequency (r = ω/ω_n).
Figure 1.11 Step response of an SDOF system.
Figure 1.12 Complex s-plane of the poles (roots of the characteristics of Equation (1.39).
Figure 1.13 Block diagram representation of an SDOF system.
Figure 1.14 Plots of the real part and the imaginary part of the FRF.
Figure 1.15 Nyquist plot for Equation 1.44.
Figure 1.16 Bode phase plot for Equation (1.39) showing resonance at –90°.
Figure 1.17 Bode magnitude plot for Equation (1.39) showing resonance and values of mass and stiffness.
Figure 1.18 Graphic illustration of Euler's formula of the complex exponential.
Figure 1.19 Free decay measurement method.
Figure 1.20 Quadrature peak picking method.
Figure 1.21 Response of a stable system.
Figure 1.22 Response of an asymptotically stable system.
Figure 1.23 Response of a system with a divergent instability.
Figure 1.24 Response of a system with flutter instability.
Figure 1.25 Block diagram of closed-loop system.
Figure 1.26 Block diagram of an open-loop system.
Figure 1.27 Force (vertical axis) deflection (horizontal axis) curves for three different springs in dimensionless terms, indicating their linear range. The curve g(x) = kx is a linear spring (short dashed line), the curve f(x) = kx − βx³ is called a softening spring (solid line) and the curve h(x) = kx + βx³ is called a hardening spring (long dashed line), for the case k = 1000 and β = 10.
Figure 1.28 The response of an underdamped system (m = 100 kg, c = 25 kg/s and k = 1000 N/m) to the initial conditions x₀ = 0.01 m, v₀ = 0 plotted using MATLAB.
Figure 1.29 A plot of the numerical integration of the underdamped system of Example 1.10.1 resulting from the MATLAB code given in Example 1.10.2.
Figure 1.30 A plot of the numerical integration of the damped forced system resulting from the MATLAB code given in Example 11.1.3

Chapter 2

Figure 2.1 A two-degree of freedom spring-mass system on a frictionless horizontal surface.
Figure 2.2 Free body diagrams of the two masses in Figure 2.1.
Figure 2.3 Schematic of a simplified model of a spinning satellite.
Figure 2.4 Schematic of a simplified model of a gyroscope.
Figure 2.5 Pflüger's Rod: a simply supported bar subjected to uniformly distributed tangential forces.
Figure 2.6 Schematic of a two degree of freedom system with applied control force f₂. As in Figure 2.1, the masses are assumed to be moving on a horizontal friction-free surface.
Figure 2.7 Free-body diagram of the system in Figure 2.6.
Figure 2.8 Experimentally obtained magnitude versus frequency plot for a simple beam-like structure.
Figure 2.9 Vibration model of a simple machine part. The quantity M(t) denotes an applied moment. The disk rotates without translation.
Figure 2.10 An airplane in flight (a) presents a number of different vibration models, one of which is given in part (b). In (b) a vibration model of a wing in flight is sketched, which accounts for bending and torsional motion by modeling the wing as attached to ground (the aircraft body in this case) through a linear spring k₁ and a torsional spring k₂.

Chapter 3

Figure 3.1 The relationship between Gerschgorin circles and eigenvalues.
Figure 3.2 The response q₁(t) versus time (solid line) and q₂(t) versus time (dashed line), as computed in MATLAB using numerical integration.

Chapter 4

Figure 4.1 Schematic of a rotating shaft subject to an axial compression force.
Figure 4.2 Stability regions for the system of the conservative gyroscopic system of Figure 4.1, as a function of the stiffness.
Figure 4.3 (a) A pendulum made of a massless rod and tip mass-free to rotate 360° around the point O. (b) The corresponding free body diagram.

Chapter 5

Figure 5.1 The displacement response to the initial conditions and forcing function of Example 5.9.1.

Chapter 6

Figure 6.1 Transmissibility curve used in determining frequency values for vibration isolation.
Figure 6.2 Normalized amplitude of vibration of the primary mass as a function of the frequency ratio for several values of the damping in the absorber system for the case of negligible damping in the primary system (i.e. a plot of Equation (6.13) for μ = 0.25 and < ?TeX$β = 1 as < ?TeX$r varies).
Figure 6.5 An illustration of bar with absorbers embedded into it, suggesting the concept of a metastructure.
Figure 6.6 A lumped parameter model of a bar with embedded absorbers of mass < ?TeX$m_n and stiffness < ?TeX$k_n. The mass, damping and stiffness properties of the host structure are indicated as < ?TeX$m, < ?TeX$c and < ?TeX$k respectively and distributed throughout the model.
Figure 6.7 The dashed line is the response of the plain beam (no VAs) and the solid line is the response with 20 distributed and embedded absorbers (with VAs). The plot on the left is the frequency response and on the right is the impulse response (Reichl and Inman, 2016).
Figure 6.8 Example of (a) an uncontrollable mechanical system and (b) a controllable mechanical system.
Figure 6.9 A two degrees of freedom structure with two control forces acting on it.
Figure 6.10 Position, < ?TeX$x₁(t), versus time for the case < ?TeX$Q = R = I.
Figure 6.11 Velocity, < ?TeX$x₂(t), versus time for the case Q = R = I.
Figure 6.12 Control, < ?TeX$u^*(t), versus time for the case < ?TeX$Q = R = I.
Figure 6.13 Position, < ?TeX$x₁(t), versus time for the case < ?TeX$Q = I, R = 10I.
Figure 6.14 Velocity, < ?TeX$x₂(t), versus time for the case < ?TeX$Q = I, R = 10I.
Figure 6.15 Control, < ?TeX$u*(t), versus time for the case < ?TeX$Q = I, R = 10.I.
Figure 6.16 The comparison of the error vector < ?TeX$e(t) versus time and the components of the state vector < ?TeX$x(t) versus time for the initial condition .
Figure 6.17 The components of the error vector < ?TeX$e(t) versus time and the components of state vector < ?TeX$x(t) versus time for the initial .
Figure 6.18 The components of the state vector < ?TeX$x(t) versus time for the case of complete state feedback.
Figure 6.19 Response of the original state variables.
Figure 6.20 Response of the balanced state variables.
Figure 6.21 The response computed and plotted using the initial command.

Chapter 7

Figure 7.1 A string fixed at each end under tension τ subject to a distributed force f(x, t).
Figure 7.2 A free body diagram of an infinitesimal element of the string in Figure 7.1.
Figure 7.3 A schematic of a rod or bar indicating the direction of longitudinal vibration.
Figure 7.4 The first two mode shapes of a vibrating string fixed at both ends.
Figure 7.5 A shaft connected to a rotational spring on one end and a disc on the other end.
Figure 7.6 The graphical solution of the transcendental equation for tan λl = −1/λl.
Figure 7.7 A beam indicating the deflection defining transverse vibration (motion in the y-direction).
Figure 7.8 The free-body diagram of an infinitesimal element of a slender beam of Figure 7.7 undergoing small deflections.
Figure 7.9 (left) The angles used to represent the effects of shear deformation. (right) Free body diagram of a differential element showing the forces and moments.
Figure 7.10 Dimensions and parameters for a U beam. The figure on the right shows a cross-section.
Figure 7.11 Vibration model of a pipe carrying a fluid in transverse vibration w(x, t).
Figure 7.12 A beam with a piezoelectric patch layered between points x₁ and x₂.
Figure 7.13 A square membrane or plate illustrating vibration perpendicular to its surface.
Figure 7.14 The dimensions of a three-layer beam for transverse vibration analysis.
Figure 7.15 Dimensions of a piezoelectric “bimorph” beam, with adhesive of thickness t_a and piezoelectric wafer of thickness t_p.
Figure 7.16 A string fixed at one end and connected to a spring and mass at the other end.
Figure 7.17 A beam fixed at one end and connected to a spring at the other end.

Chapter 8

Figure 8.1 A statically deflected string fixed at both ends.

Chapter 9

Figure 9.1 A non-uniform beam (dashed outline) with unknown spectrum bracketed by two beams with analytically known spectrum.

Chapter 10

Figure 10.1 A beam with an attached lumped mass-spring system.
Figure 10.2 Passive vibration control by using a damping layer.

Chapter 11

Figure 11.1 A two-node, one-element model of a cantilevered beam.
Figure 11.2 A four-node, three-element model of a cantilevered beam.

Chapter 12

Figure 12.1 Components of a vibration measurement system for modal analysis.
Figure 12.2 Frequency spectrum of a typical hammer impact.
Figure 12.3 Some signals and their Fourier spectrum.
Figure 12.4 Sample time histories and their corresponding digital record.
Figure 12.5 An illustration of leakage.
Figure 12.6 A Hanning window and its effect on transform signals.
Figure 12.7 A sample random signal.
Figure 12.8 A plot of the coherence of a sample signal versus frequency.
Figure 12.9 An idealized frequency response function.
Figure 12.10 Measurement positions marked on a simple plate for performing a modal test.
Figure 12.11 A display of the mode shape of a simple test structure.

Preface

Advance level vibration topics are presented here, including lumped mass and distributed mass systems in the context of the appropriate mathematics along with topics from control that are useful in vibration analysis, testing and design. This text is intended for use in a second course in vibration, or a combined course in vibration and control. It is also intended as a reference for the field of structural control and could be used as a text in structural control. The control topics are introduced at the beginners' level with no prerequisite knowledge in controls needed to read the book.

The text is an attempt to place vibration and control on a firm mathematical basis and connect the disciplines of vibration, linear algebra, matrix computations, control and applied functional analysis. Each chapter ends with notes on further references and suggests where more detailed accounts can be found. In this way I hope to capture a “bigger picture” approach without producing an overly large book. The first chapter presents a quick introduction using single degree of freedom systems (second-order ordinary differential equations) to the following chapters, which extend these concepts to multiple degree of freedom systems (matrix theory and systems of ordinary differential equations) and distributed parameter systems (partial differential equations and boundary value problems). Numerical simulations and matrix computations are also presented through the use of MATLAB^TM.

New In This Edition – The book chapters have been reorganized (there are now 12 instead of 13 chapters) with the former chapter on design removed and combined with the former chapter on control to form a new chapter titled Vibration Suppression. Some older, no longer used material, has been deleted in an attempt to keep the book limited in size as new material has been added.

The new material consists of adding several modeling sections to the text, including corresponding problems and examples. Many figures have been redrawn throughout to add clarity with more descriptive captions. In addition, a number of new figures have been added. New problems and examples have been added and some old ones removed. In total, seven new sections have been added to introduce modeling, coupled systems, the use of piezoelectric materials, metastructures, and validation and verification.

Instructor Support – Power Point slides are available for presentation of the material, along with a complete solutions manual. These materials are available from the publisher for those who have adapted the book. The author is pleased to answer questions via the email listed below.

Student Support – The best place to get help is your instructor and others in your peer group through discussion of the material. There are also many excellent texts as referenced throughout the book and of course Internet searches can provide lots of help. In addition, feel free to email the author at the address below (but don't ask me to do your homework!).

Acknowledgements – I would like to thank two of my current PhD students, Katie Reichl and Brittany Essink, for checking some of the homework and providing some plots. I would like to thank all of my former and current PhD students for 36 years of wonderful research and discussions. Thanks are owed to the instructors and students of the previous edition who have sent suggestions and comments. Last, thanks to my lovely wife Catherine Ann Little for putting up with me.

Leland, MichiganDaniel J. Inman

daninman@umich.edu

Vibration with Control

Second Edition

Daniel John Inman

University of Michigan, USA

Preface

About the Companion Website