Mechanical Vibration E-Book

Ivana Kovacic

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 236

Veröffentlichungsjahr: 2017

Beschreibung

Mechanical oscillators in Lagrange's formalism – a thorough problem-solved approach

This book takes a logically organized, clear and thorough problem-solved approach at instructing the reader in the application of Lagrange's formalism to derive mathematical models for mechanical oscillatory systems, while laying a foundation for vibration engineering analyses and design.

Each chapter contains brief introductory theory portions, followed by a large number of fully solved examples. These problems, inherent in the design and analysis of mechanical systems and engineering structures, are characterised by a complexity and originality that is rarely found in textbooks.

Numerous pedagogical features, explanations and unique techniques that stem from the authors’ extensive teaching and research experience are included in the text in order to aid the reader with comprehension and retention. The book is rich visually, including numerous original figures with high-standard sketches and illustrations of mechanisms.

Key features:

• Distinctive content including a large number of different and original oscillatory examples, ranging from simple to very complex ones.
• Contains many important and useful hints for treating mechanical oscillatory systems.
• Each chapter is enriched with an Outline and Objectives, Chapter Review and Helpful Hints.

Mechanical Vibration: Fundamentals with Solved Examples is essential reading for senior and graduate students studying vibration, university professors, and researchers in industry.

Leseprobe

Table of Contents

Cover

Title Page

About the Authors

Preface

1 Preliminaries

Chapter Outline

Chapter Objectives

1.1 From Statics

1.2 From Kinematics

1.3 From Kinetics

1.4 From Strength of Materials

2 Lagrange’s Equation for Mechanical Oscillatory Systems

Chapter Outline

Chapter Objectives

2.1 About Lagrange’s Equation of the Second Kind

2.2 Kinetic Energy in Mechanical Oscillatory Systems

2.3 Potential Energy in Mechanical Oscillatory Systems

2.4 Generalised Forces in Mechanical Oscillatory Systems

2.5 Dissipative Function in Mechanical Oscillatory Systems

References

3 Free Undamped Vibration of Single‐Degree‐of‐Freedom Systems

Chapter Outline

Chapter Objectives

Theoretical Introduction

4 Free Damped Vibration of Single‐Degree‐of‐Freedom Systems

Chapter Outline

Chapter Objectives

Theoretical Introduction

5 Forced Vibration of Single‐Degree‐of‐Freedom Systems

Chapter Outline

Chapter Objectives

Theoretical Introduction

6 Free Undamped Vibration of Two‐Degree‐of‐Freedom Systems

Chapter Outline

Chapter Objectives

Theoretical Introduction

7 Forced Vibration of Two‐Degree‐of‐Freedom Systems

Chapter Outline

Chapter Objectives

Theoretical Introduction

8 Vibration of Systems with Infinite Number of Degrees of Freedom

Chapter Outline

Chapter Objectives

8.1 Theoretical Introduction: Longitudinal Vibration of Bars

8.2 Theoretical Introduction: Torsional Vibration of Shafts

8.3 Theoretical Introduction: Transversal Vibration of Beams

9 Additional Topics

Chapter Outline

Chapter Objectives

9.1 Theoretical Introduction

9.2 Equivalent Two‐Element System for Concurrent Springs and Dampers

9.3 Nonlinear Springs in Series

9.4 On the Deflection and Potential Energy of Nonlinear Springs: Approximate Expressions

9.5 Corrections of Stiffness Properties of Certain Oscillatory Systems

Appendix Mathematical Topics

A.1 Geometry

A.2 Trigonometry

A.3 Algebra

A.4 Vectors

A.5 Derivatives (Notation:

)

A.6 Variation (Virtual Displacements)

A.7 Series

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1.1

Table 1.1.2

Table 1.3.1

Chapter 08

Table 8.3.1.1

Chapter 09

Table 9.4.1

Table 9.5.1

List of Illustrations

Chapter 01

Figure 1.2.1

Figure 1.2.2

Figure 1.2.3

Figure 1.2.4

Figure 1.2.5

Figure 1.2.6

Figure 1.2.7

Figure 1.2.8

Figure 1.2.9

Figure 1.3.1

Figure 1.3.2

Figure 1.3.3

Figure 1.3.4

Figure 1.3.5

Figure 1.4.1

Figure 1.4.2

Figure 1.4.3

Chapter 02

Figure 2.2.1

Figure 2.3.1

Figure 2.3.2

Figure 2.3.3

Figure 2.3.4

Figure 2.4.1

Chapter 03

Figure 3.1

Figure 3.1.1

Figure 3.1.2

Figure 3.2.1

Figure 3.2.2

Figure 3.3.1

Figure 3.3.2

Figure 3.3.3

Figure 3.4.1

Figure 3.4.2

Figure 3.5.1

Figure 3.5.2

Figure 3.5.3

Figure 3.6.1

Figure 3.6.2

Figure 3.7.1

Figure 3.7.2

Figure 3.8.1

Figure 3.8.2

Figure 3.8.3

Figure 3.9.1

Figure 3.9.2

Figure 3.10.1

Figure 3.10.2

Figure 3.11.1

Figure 3.11.2

Figure 3.11.3

Figure 3.11.4

Figure 3.12.1

Figure 3.12.2

Figure 3.12.3

Figure 3.13.1

Figure 3.13.2

Figure 3.14.1

Figure 3.14.2

Figure 3.14.3

Figure 3.14.4

Figure 3.15.1

Figure 3.15.2

Figure 3.15.3

Figure 3.15.4

Chapter 04

Figure 4.1.1

Figure 4.1.2

Figure 4.1.3

Figure 4.2.1

Figure 4.2.2

Figure 4.3.1

Figure 4.3.2

Figure 4.4.1

Figure 4.4.2

Figure 4.5.1

Figure 4.5.2

Figure 4.6.1

Figure 4.6.2

Figure 4.7.1

Figure 4.7.2

Figure 4.8.1

Figure 4.8.2

Figure 4.8.3

Figure 4.9.1

Figure 4.9.2

Figure 4.9.3

Figure 4.9.4

Figure 4.9.5

Figure 4.10.1

Figure 4.10.2

Figure 4.10.3

Figure 4.11.1

Figure 4.11.2

Figure 4.11.3

Figure 4.11.4

Chapter 05

Figure 5.1.1

Figure 5.1.2

Figure 5.1.3

Figure 5.2.1

Figure 5.3.1

Figure 5.3.2

Figure 5.3.3

Figure 5.3.4

Figure 5.4.1

Figure 5.4.2

Figure 5.4.3

Figure 5.4.4

Figure 5.4.5

Figure 5.4.6

Figure 5.5.1

Figure 5.5.2

Figure 5.5.3

Figure 5.5.4

Figure 5.5.5

Figure 5.5.6

Figure 5.6.1

Figure 5.6.2

Figure 5.6.3

Figure 5.6.4

Figure 5.7.1

Figure 5.7.2

Figure 5.7.3

Figure 5.7.4

Chapter 06

Figure 6.1.1

Figure 6.1.2

Figure 6.1.3

Figure 6.1.4

Figure 6.2.1

Figure 6.2.2

Figure 6.2.3

Figure 6.2.4

Figure 6.3.1

Figure 6.3.2

Figure 6.3.3

Figure 6.3.4

Figure 6.4.1

Figure 6.4.2

Figure 6.5.1

Figure 6.5.2

Figure 6.5.3

Figure 6.5.4

Figure 6.5.5

Figure 6.6.1

Figure 6.6.2

Figure 6.6.3

Figure 6.6.4

Figure 6.6.5

Figure 6.7.1

Figure 6.7.2

Figure 6.7.3

Figure 6.7.4

Figure 6.7.5

Figure 6.7.6

Chapter 07

Figure 7.1.1

Figure 7.1.2

Figure 7.1.3

Figure 7.1.4

Figure 7.1.5

Figure 7.2.1

Figure 7.2.2

Figure 7.2.3

Figure 7.2.4

Figure 7.2.5

Figure 7.3.1

Figure 7.3.2

Figure 7.3.3

Figure 7.3.4

Figure 7.3.5

Figure 7.3.6

Figure 7.4.1

Figure 7.4.2

Figure 7.4.3

Figure 7.4.4

Figure 7.4.5

Figure 7.5.1

Figure 7.5.2

Figure 7.5.3

Figure 7.5.4

Figure 7.6.1

Figure 7.6.2

Figure 7.6.3

Figure 7.6.4

Figure 7.6.5

Chapter 08

Figure 8.1.1

Figure 8.1.1.1

Figure 8.1.1.2

Figure 8.1.2.1

Figure 8.1.2.2

Figure 8.1.2.3

Figure 8.1.3.1

Figure 8.1.3.2

Figure 8.1.3.3

Figure 8.1.3.4

Figure 8.1.4.1

Figure 8.1.4.2

Figure 8.1.4.3

Figure 8.2.1

Figure 8.2.1.1

Figure 8.2.1.2

Figure 8.2.2.1

Figure 8.2.2.2

Figure 8.2.2.3

Figure 8.2.3.1

Figure 8.2.4.1

Figure 8.2.4.2

Figure 8.3.1

Figure 8.3.2

Figure 8.3.1.1

Figure 8.3.1.2

Figure 8.3.1.3

Figure 8.3.2.1

Figure 8.3.2.2

Figure 8.3.2.3

Figure 8.3.3.1

Figure 8.3.3.2

Figure 8.3.3.3

Figure 8.3.4.1

Figure 8.3.4.2

Figure 8.3.4.3

Figure 8.3.5.1

Figure 8.3.5.2

Figure 8.3.5.3

Chapter 09

Figure 9.1.1

Figure 9.2.1

Figure 9.2.2

Figure 9.2.3

Figure 9.2.4

Figure 9.2.5

Figure 9.2.6

Figure 9.2.7

Figure 9.2.8

Figure 9.5.1

Figure 9.5.2

Figure 9.5.3

Figure 9.5.4

Figure 9.5.5

Figure 9.5.6

Guide

Cover

Table of Contents

Begin Reading

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Mechanical Vibration

Fundamentals with Solved Examples

Ivana Kovačić

University of Novi SadSerbia

Dragi Radomirović

University of Novi SadSerbia

This edition first published 2018© 2018 John Wiley & Sons Ltd

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The right of Ivana Kovačić and Dragi Radomirović to be identified as the authors of this work has been asserted in accordance with law.

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Library of Congress Cataloging‐in‐Publication Data

Names: Kovačić, Ivana, author. | Radomirović, Dragi, author.Title: Mechanical vibration : fundamentals with solved examples / by Ivana Kovačić, University of Novi Sad, Serbia, Dragi Radomirović, University of Novi Sad, Serbia.Description: Singapore : Wiley, [2017] | Includes bibliographical references and index. |Identifiers: LCCN 2017026938 (print) | LCCN 2017027118 (ebook) | ISBN 9781118927571 (pdf) | ISBN 9781118927588 (epub) | ISBN 9781118675151 (cloth)Subjects: LCSH: Vibration.Classification: LCC TA355 (ebook) | LCC TA355 .K689 2017 (print) | DDC 620.3–dc23LC record available at https://lccn.loc.gov/2017026938

Cover design by WileyCover image: Courtesy of the authors

About the Authors

Ivana Kovačić graduated in Mechanical Engineering from the Faculty of Technical Sciences (FTN), University of Novi Sad, Serbia. She obtained her MSc and PhD in the Theory of Nonlinear Vibrations at the FTN. She is currently a Full Professor of Mechanics at the FTN and the head of the Centre of Excellence for Vibro‐Acoustic Systems and Signal Processing CEVAS at the same faculty. Kovačić is the Subject Editor of three academic journals: the Journal of Sound and Vibration, the European Journal of Mechanics A/Solids and Meccanica. Her research involves the use of quantitative and qualitative methods to study differential equations arising from nonlinear dynamics problems mainly in mechanical engineering, and recently also in biomechanics and tree vibrations.

Dragi Radomirović graduated in Mechanical Engineering from the Faculty of Technical Sciences (FTN), University of Novi Sad (UNS), Serbia. He obtained his MSc and PhD in Analytical Mechanics at the FTN. He is a Full Professor of Mechanics at the Faculty of Agriculture, UNS. His research interests are directed towards Mechanical Vibrations and Analytical Mechanics.

Preface

Mechanical Vibrations: Fundamentals with Solved Examples takes a logically organized, clear and thorough problem‐solved approach to instructing the reader in the application of certain mechanical principles to derive mathematical models for oscillatory systems, while laying a foundation for vibration engineering analysis and design.

The methodology is mainly based on Analytical Mechanics and Lagrange’s formalism, so that the approach, terminology and notation used are consistent, creating a coherent chain that links the chapters in the book. This enables the reader to learn gradually how to treat different oscillatory mechanical systems performing small oscillations.

Chapter 1 contains preliminaries, giving a brief overview of important facts and expressions which are needed as background knowledge from Statics, Kinematics, Kinetics and Strength of Materials. Chapter 2 provides theoretical basics related to Lagrangian Mechanics and the formation of equations of motion from Lagrange’s equations of the second kind. Chapters 3–7 contain a variety of examples in which Lagrange’s equations of the second kind are used to derive the corresponding equations of motion for small oscillations. Chapter 8 is concerned with oscillatory systems with infinite numbers of degrees of freedom, and Chapter 9 introduces some elements of equivalent stiffness and damping as well as Nonlinear Oscillations theory. There is also one Appendix with certain mathematical topics, based on which the book can be used on a stand‐alone basis. There are a Chapter Outline and Objectives at the beginning of each chapter as well as a Theoretical Introduction. This theoretical portion is followed by a large number of fully solved examples, the solutions of which are presented in detail. The examples are arranged in order of increasing difficulty, so that the most difficult exercises are given near the end of the chapters. Some of these problems, inherent in the design and analysis of mechanical systems and engineering structures, are characterised by a complexity and originality that is rarely found in textbooks. Numerous pedagogical features, extensive explanations and absolutely unique techniques that stem from the authors’ long‐term teaching and research experience are included in the text in order to aid the reader’s comprehension and retention. The book is visually rich and all the figures are original and produced by the authors themselves.

We, the authors, believe that the book can be used as a tutorial or supplementary material for self‐study for senior students, graduate students, university professors and researchers in mechanics and engineering. We hope that they will find it helpful while indulging themselves in the magnificent world of mechanical oscillatory problems.

Due to unfortunate circumstances, during 2015 both authors lost beloved members of their families: Milorad Radomirović, Nikola Kovačić and Smiljana Radomirović. This book is dedicated to them.

Novi Sad, Serbia, 2016

Ivana Kovačić and Dragi Radomirović

1Preliminaries

Chapter Outline

This chapter provides theoretical fundamentals for the considerations presented in the subsequent chapters. It is divided into four sections, containing some key concepts, facts and expressions from Statics, Kinematics, Kinetics and Strength of Materials.

Chapter Objectives

To present preliminaries from Statics, Kinematics, Kinetics and Strength of Materials

To focus only on the key concepts, facts and expressions of interest for the considerations presented in this book

To provide help to readers to enable them to use this book on a stand‐alone basis

1.1 From Statics

1.1.1 Mechanical Systems and Equilibrium Equations

Of interest here is the equilibrium of different systems of forces and torques lying in one plane. They are given separately in Table 1.1.1 together with the corresponding equilibrium equations and their descriptions. Note that before writing these equations, one must create a free‐body diagram for the object under consideration if it is not free. The way to do this is described in Table 1.1.2.

Table 1.1.1

Type

Mechanical model

Equilibrium equations

Concurrent forces

Sum of the projections of all forces on two orthogonal axes is equal to zero.

Parallel forces and torques in the same plane

Sum of the projections of forces on the axis parallel to the direction of the forces is equal to zero. Sum of the moments about any point A on or off the body is equal to zero.

Arbitrary forces and torques in one plane

Sum of the projections of all forces on two orthogonal axes is equal to zero.Sum of the moments about any point A on or off the body is equal to zero.The alternative equilibrium equations are:

three equilibrium equations with the zero sum of moments for any three points that are not on the same straight line;

one force equilibrium equation in an arbitrary

x

‐direction and two equilibrium equations with the zero sums of moments for any two points that must not lie on a line perpendicular to the

x

‐direction.

Table 1.1.2

Type

Schematics

Description

Smooth surface

Contact force is compressive and is normal to the tangent at the point of contact.

Weightless straight rod

Force exerted is the direction of the straight rod, but it can be tensile or compressive.

Rope/ Cable

Force exerted is always a tension away from the body in the direction of the rope.

Pin connection

Force exerted lies in the plane normal to the pin axis; this force is usually shown in terms of two orthogonal components (for example, horizontal and vertical).

Roller/ Rocker/ Ball support

Compressive force normal to the supporting surface/guide.

Fixed or built‐in support

Two orthogonal components of a force and a torque.

1.1.2 Constraints and Free‐Body Diagrams

When considering the equilibrium of an object or combinations of objects via equations of motion, it is essential to isolate them from all surrounding bodies. This isolation is accomplished by the free‐body diagram, which shows all active forces and active torques acting on the object or combinations of objects as well as forces and torques that exist due to mechanical contacts with surrounding bodies, which represent the so‐called mechanical constraints. There are several common types that can exist in a plane, and they are collected and described in Table 1.1.2. Note that these forces and torques are also called passive forces and passive torques.

1.1.3 Equilibrium Condition Via Virtual Work

Besides the approach based on the equilibrium equation, one can determine and investigate equilibria based on the principle of virtual work.

The virtual work of a force is the scalar product of the vector of the force and the virtual displacement of the point at which it acts. This can be further expressed in the rectangular/Cartesian coordinate system as follows:

(1.1.1)

The virtual work of a torque on the virtual rotation can be defined as

(1.1.2)

where the plus sign corresponds to the case when the torque helps to increase of the angle of rotation, while the minus sign holds in the opposite case.

For the case of a system of forces and torques, the overall virtual work is the sum of the virtual work of each of them:

(1.1.3)

If the position of the mechanical system is depicted by N generalised coordinates qi (i = 1,…, N), one can express Equation (1.1.3) in the form

(1.1.4)

where the coefficients represent the so‐called generalised forces (see Section 2.4). In the equilibrium position, the generalised forces are equal to zero:

(1.1.5)

Note that the number of homogeneous algebraic equations (1.1.5) is equal to the number of generalised coordinates, that is, the number of degrees of freedom. For example, if the system has one degree of freedom, there is one generalised force and one equation (1.1.5) to determine the equilibrium or any other parameter that yields it. For two‐degree‐of‐freedom systems, two equations (1.1.5) exist, and so on.

It should also be noted that in the case of ideal constraints (see Table 1.1.2), the virtual work of the corresponding forces and torques is equal to zero. This implies that when solving the examples related to static equilibria, one does not need to introduce reactions of ideal constraints as their virtual work is zero.

1.2 From Kinematics

Kinematics deals with the geometrical aspects of motion of particles and rigid bodies, as well as with the mathematical description of their motion and certain velocities and accelerations they have over time.

1.2.1 Kinematics of Particles

A particle moving in a plane is shown in Figure 1.2.1. To study its motion, different coordinates can be used. For example, a set of two mutually orthogonal axes x and y with the origin O can be arbitrary chosen, but the axes must be fixed (Figure 1.2.1a). The unit vectors are, respectively, i and j. The position vector is directed from the origin to the particle and is defined by:

(1.2.1)

Figure 1.2.1

where x and y, in general, change over time t, that is, and .

The particle’s velocity v and acceleration a are defined by:

(1.2.2)

(1.2.3)

where the overdot indicates differentiation with respect to time.

Besides this, one can use polar coordinates (Figure 1.2.1b) and with the moveable unit vectors r0 and c0. The velocity of the particle has two components:

(1.2.4)

1.2.2 Kinematics of Rigid Bodies

1.2.2.1 Rigid Body in Translatory Motion

During translatory motion every line in the body remains parallel to its original position at all times. One can distinguish rectilinear translation (Figure 1.2.2), when all points move along parallel straight lines, and curvilinear translation (Figure 1.2.3), when all points move along congruent curves. During translatory motion all points have the same velocity as noted in Figures 1.2.2 and 1.2.3. Thus, specifying the motion of one point enables one to describe completely the translation of the whole body.

Figure 1.2.2

Figure 1.2.3

1.2.2.2 Rigid Body in Fixed‐Axis Rotation

During rotation about a fixed axis (Figure 1.2.4) all points in a rigid body (other than those on the axis) move along concentric circular paths around the axis of rotation. Note that the axis of rotation in Figure 1.2.4 passes through the point O and is perpendicular to the plane of the figure. The position of the body is defined by the angle between the fixed line and a line attached to the body (this angle is labelled by ϕ in Figure 1.2.4) and the angular velocity of the body is . The velocity of each point is proportional to it, as well as to its distance with respect to the axis of rotation. Referring again to Figure 1.2.4, one can write

(1.2.5)

Figure 1.2.4

1.2.2.3 Rigid Body in General Plane Motion

General plane motion of a rigid body is a combination of translation and rotation around the axis perpendicular to the plane in which the translation takes place. Thus, the velocities of any two points of the body are mutually related by the expression (see Figure 1.2.5):

(1.2.6)

Figure 1.2.5

where .

Another way to consider general plane motion of a rigid body is to locate the point whose instantaneous velocity is equal to zero, which is the so‐called instantaneous centre of zero velocity. This point is labelled by P in Figure 1.2.6. If the directions of two non‐parallel velocities of any two points of the body are known, the instantaneous centre of zero velocity corresponds to the intersection point of the lines perpendicular to these velocities (see the lines perpendicular to vC and vD in Figure 1.2.6). Then, the angular velocity of the body is

(1.2.7)

Figure 1.2.6

All other points have velocities proportional to it. For example (see Figure 1.2.6):

(1.2.8)

It is also of interest to note the case when two bodies roll along each other, but there is no slipping between them, as shown in Figure 1.2.7a,b. Then, the equality of the arcs s1 and s2 depicted in Figure 1.2.7a must hold and the velocities of the contact points are equal (). Analogously, for the disc rolling without slipping along a horizontal fixed plane shown in Figure 1.2.7b, the equality of the arc length Rα and the distance s holds. The contact point is on the fixed plane, so its instantaneous velocity is zero and it represents the instantaneous centre of zero velocity ().

Figure 1.2.7

This case is shown in more detail in Figure 1.2.8. Relating the velocity of the centre of the disc to its angular velocity, one has

(1.2.9)

Figure 1.2.8

The velocities of several other points are also shown.

1.2.3 Kinematics of Particles in Compound Motion

Let us consider the motion of a point M (Figure 1.2.9) relative to a rigid body that moves relatively to a fixed coordinate system depicted by the axes x and y. The motion (trajectory, velocity, acceleration) of the point M with respect to the fixed coordinate system x–y is called absolute. The motion of the point M with respect the body and the coordinate system ξ–η attached to the body is called relative. The motion of the body with respect to the fixed coordinate systems is called transportation.

Figure 1.2.9

The theorem on composition of velocities for a compound (composition/resultant) motion states that the absolute velocity of the point is equal to the vector sum of the relative and transportation velocities of the point:

(1.2.10)