Advanced Dynamics of Mechanical Systems

Federico Cheli • Giorgio Diana

Advanced Dynamicsof Mechanical Systems

123

Federico CheliDepartment of Mechanical EngineeringPolytechnic University of MilanMilanItaly

Giorgio DianaDepartment of Mechanical EngineeringPolytechnic University of MilanMilanItaly

ISBN 978-3-319-18199-8 ISBN 978-3-319-18200-1 (eBook)DOI 10.1007/978-3-319-18200-1

Library of Congress Control Number: 2015939153

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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Preface

This book represents the natural evolution of the lecture notes of the course“Dynamics and Vibrations of Machines” held at the Politecnico di Milano in theacademic years 1981–1992 and of a book already published for the course“Simulation and modelling of mechanical systems” (academic years 1993–2014).

These collected works can be considered as a natural extension of the didacticwork carried out in this area, initially by Prof. O. Sesini and later by Profs.A. Capello, E. Massa and G. Bianchi. The contents of this book also sum updecades of experience gained by the research group which is part of the Section ofMechanical Systems at the Politecnico di Milano (former Institute of AppliedMechanics). It also draws upon the research topics developed by a research groupof the Department of Mechanics, to which the authors belong. Said research wasgenerally based on problems encountered in the industrial world, performed incollaboration with organizations and research centres including ABB, ENEL(Italian General Electricity Board), FS (Italian Railways), Bombardier, Alstom,Ansaldo, ENEL-CRIS, ISMES, Fiat, Ferrari and Pirelli, as well as countless others.In this context, the following research topics were considered of prime importance:

• analytical and experimental investigations on the vibration of power lines;• slender structures—wind interaction;• aeroelastic behaviour of suspension bridges;• dynamic behaviour of structures subjected to road and rail traffic;• rail vehicle dynamics, pantograph—catenary interaction, train—railway infra-structure interaction, etc.;

• ground vehicle dynamics; and• rotor dynamics.

These themes impacted significantly on the development of this book.The educational content of this volume is primarily addressed to students of

engineering taking courses in mechanics, aerospace, automation and energy, dis-ciplines introduced recently by the Italian Ministry of Education in compliance withthe New Italian University Order. However, given its organic structure and the

v

comprehensive overview of the subjects dealt with, the book could also serve as auseful tool to professionals in the industry.

In this book, an engineering approach for the schematization of a genericmechanical system, applicable both to rigid and deformable bodies, is introduced.Such an approach is necessary to identify the behaviour of a mechanical systemsubject to different excitation sources. In addition to the traditional aspects asso-ciated with the dynamics and vibrations of mechanical systems, the engineeringapproach illustrated herein allows us to reproduce the interaction of mechanicalsystems with different force fields acting on its various components (e.g. actionof fluids and contact forces), i.e. forces dependent on system motion, and, conse-quently, its state.

This concept, dealing with the interaction of force fields and mechanical systems,gives rise to a new system on which the dynamic behaviour is considered, focusing,in particular, on the analysis of motion stability.

Controlled systems, in which the action of the actuator, controlled in a closedloop, defines forces as a function of the state of the system, can also be assimilatedto systems interacting with force fields and, for this reason, dealt with in a similarway.

Traditionally, however, there are typical approaches in this area that cannot beignored and, for this reason, controlled systems are treated in a separate text.1

In this text, however, an effort has been made to reference the symbols and maintechniques used in control engineering, in order to create an easy interface formechanical engineers dealing with electronic control.

More specifically, in the first part of this book, we will analyse mechanicalsystems with 1 or more degrees of freedom (d.o.f.), generally in large motion and,subsequently, the small motion of systems in the neighbourhood of either thesteady-state motion or the static equilibrium position. In this phase, we will analyseboth discrete and continuous systems, for which certain discretization procedureswill be discussed (modal approach, finite elements).

Conversely, the second part of this text deals with the study of mechanicalsystems subjected to force fields, with many examples such as fluid–elastic inter-action, train and railway interaction, rotor dynamics, experimental techniquesrelated to parameter identification and random excitations.

The first part of the text can be a useful tool for undergraduate courses toapproach the dynamics and the vibration problems in the mechanical systems.

The second part is more suitable for graduate and Ph.D. students to analysemany real problems due to the interaction of mechanical systems with differentsurrounding fields of forces. The main problems related to the behaviour and sta-bility of these systems are fully described in the last part of the book and will bevery useful for the students.

1Diana and Resta [1].

vi Preface

We would like to extend our special thanks to all the lecturers and researchersof the Section of System Mechanics of the Department of Mechanics at thePolitecnico di Milano for all their help and input provided during the drafting of thisbook.

The authors would also like to express their gratitude especially to ProfessorBruno Pizzigoni for his hard and excellent work for the audit and the check of theEnglish text. It goes without saying that, as always, there are likely to be omissionsand errors for which we hope you will forgive us.

Federico CheliGiorgio Diana

Reference

1. Diana G, Resta F (2006) Controllo dei sistemi meccanici. Polipress, Milano

Preface vii

Contents

1 Nonlinear Systems with 1-n Degrees of Freedom . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Cartesian Coordinates, Degrees of Freedom,

Independent Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Writing Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Definition of the Various Forms of Energyas a Function of the Physical Variables . . . . . . . . . . . . 12

1.3.2 Definition of the Various Forms of Energyas a Function of the Independent Variables . . . . . . . . . 17

1.3.3 Application of Lagrange’s Equations. . . . . . . . . . . . . . 191.3.4 Lagrange Multiplier Method . . . . . . . . . . . . . . . . . . . 201.3.5 Method of Introducing Real Constraints

Using Force Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Nonlinear Systems with One Degree of Freedom . . . . . . . . . . . 23

1.4.1 Writing Equations of Motion “in the Large” . . . . . . . . 241.4.2 Writing Linearized Equations . . . . . . . . . . . . . . . . . . . 30

1.5 Nonlinear Systems with 2 Degrees of Freedom . . . . . . . . . . . . 371.6 Multi-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.6.1 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.6.2 Kinematic Analysis of the Rigid Body . . . . . . . . . . . . 481.6.3 Rotations and Angular Velocity of the Rigid Body . . . . 531.6.4 The Transformation Matrix of the Coordinates

in Terms of Cardan Angles . . . . . . . . . . . . . . . . . . . . 551.6.5 Relationship Between the Angular Velocities

and the Velocities in Terms of Cardan Angles . . . . . . . 571.7 The Dynamics of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . 57

1.7.1 Inertial Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591.7.2 External Excitation Forces . . . . . . . . . . . . . . . . . . . . . 621.7.3 Elastic and Gravitational Forces . . . . . . . . . . . . . . . . . 641.7.4 Dissipation Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 681.7.5 Definition of Kinetic Energy . . . . . . . . . . . . . . . . . . . 70

ix

1.7.6 Writing the Equations of Motion . . . . . . . . . . . . . . . . 751.7.7 The Cardinal Equations of Dynamics . . . . . . . . . . . . . 76

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2 The Dynamic Behaviour of Discrete Linear Systems . . . . . . . . . . . 832.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.2 Writing Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.2.1 Kinetic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.2.2 Dissipation Function . . . . . . . . . . . . . . . . . . . . . . . . . 882.2.3 Potential Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.2.4 Virtual Work of Active Forces . . . . . . . . . . . . . . . . . . 932.2.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.3 Some Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . 972.3.1 One-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . 972.3.2 Two-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . 992.3.3 An Additional Example of Two-Degree-of-Freedom

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.3.4 A Further Example of a Two-Degree-of-Freedom

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.3.5 n-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . . . 117

2.4 Solving the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 1252.4.1 One-Degree-of-Freedom System . . . . . . . . . . . . . . . . . 1252.4.2 Two-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . 1702.4.3 n-Degree-of-Freedom System. . . . . . . . . . . . . . . . . . . 198

2.5 Modal Approach for Linear n-Degree-of-Freedom Systems . . . . 2112.5.1 Modal Approach for Two-Degree-of-Freedom

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.5.2 Modal Approach for n-Degree-of-Freedom

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2152.5.3 Forced Motion in Principal Coordinates . . . . . . . . . . . 224

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

3 Vibrations in Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . 2413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.2 Transverse Vibrations of Cables . . . . . . . . . . . . . . . . . . . . . . . 241

3.2.1 Propagative Solution . . . . . . . . . . . . . . . . . . . . . . . . . 2453.2.2 Stationary Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 247

3.3 Transverse Vibrations in Beams . . . . . . . . . . . . . . . . . . . . . . . 2523.3.1 Transverse Vibrations in Beams Subjected

to an Axial Load (Tensioned Beam) . . . . . . . . . . . . . . 2653.4 Torsional Vibrations in Beams . . . . . . . . . . . . . . . . . . . . . . . . 2733.5 Analysis of the General Integral of the Equation

of Motion in Continuous Systems. . . . . . . . . . . . . . . . . . . . . . 275

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3.6 Analysis of the Particular Integral of Forced Motion . . . . . . . . . 2783.6.1 Hysteretic Damping in the Case

of a Taut Cable (Direct Approach) . . . . . . . . . . . . . . . 2823.7 Approach in Principal Coordinates . . . . . . . . . . . . . . . . . . . . . 284

3.7.1 Taut Cable Example . . . . . . . . . . . . . . . . . . . . . . . . . 286References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

4 Introduction to the Finite Element Method . . . . . . . . . . . . . . . . . . 3114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.2 The Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

4.2.1 Shape Function for the Taut Cable Element . . . . . . . . . 3154.2.2 The Shape Function for the Beam Element . . . . . . . . . 3184.2.3 Shape Function for Generic Finite Elements . . . . . . . . 320

4.3 The Equations of Motion of the System . . . . . . . . . . . . . . . . . 3224.4 Taut Cable Finite Element (an Application Example) . . . . . . . . 323

4.4.1 Discretization of the Structure . . . . . . . . . . . . . . . . . . 3234.4.2 Definition of the Stiffness [Kj] and Mass [Mj]

Matrix of the Taut Cable Finite Element in theLocal Reference System . . . . . . . . . . . . . . . . . . . . . . 326

4.4.3 Transformation of Coordinates: Local ReferenceSystem, Absolute Reference System . . . . . . . . . . . . . . 329

4.4.4 Definition of the Stiffness [Kj] and Mass [Mj]Matrix of the Taut Cable Elementin the Global Reference System . . . . . . . . . . . . . . . . . 333

4.4.5 Assembly of the Complete Structure . . . . . . . . . . . . . . 3334.4.6 Calculation of the Generalized Forces . . . . . . . . . . . . . 3394.4.7 Imposition of Constraints (Boundary Conditions) . . . . . 3464.4.8 Solving the Equations of Motion . . . . . . . . . . . . . . . . 3504.4.9 A Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . 353

4.5 An Application Example: Finite Beam Element . . . . . . . . . . . . 3564.5.1 Discretization of the Structure . . . . . . . . . . . . . . . . . . 3564.5.2 Definition of the Stiffness [Kl] and Mass [Ml]

Matrix of the Beam Element in the LocalReference System. . . . . . . . . . . . . . . . . . . . . . . . . . . 358

4.5.3 Definition of the Stiffness [Kj] and Mass [Mj]Matrix of the Beam Element in the GlobalReference System. . . . . . . . . . . . . . . . . . . . . . . . . . . 372

4.5.4 Writing of the Equations of Motionand Their Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 374

4.5.5 A Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . 374

Contents xi

4.6 Two-Dimensional and Three-Dimensional Finite Elements(Brief Outline) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3744.6.1 Definition of the Generic Shape Function . . . . . . . . . . 3784.6.2 General Definition of the Stiffness and Mass Matrices

of the Generic Three-Dimensional Finite Element. . . . . 3794.6.3 Two-Dimensional Elements (Membrane) . . . . . . . . . . . 3834.6.4 Three-Dimensional Elements (Brick Elements) . . . . . . . 3874.6.5 Plate Elements and Shell Elements . . . . . . . . . . . . . . . 3904.6.6 Isoparametric Elements . . . . . . . . . . . . . . . . . . . . . . . 391

4.7 Nonlinear Analysis in Structures Using the FiniteElement Method (Brief Outline) . . . . . . . . . . . . . . . . . . . . . . . 3944.7.1 Introduction to the Non-linear Problem . . . . . . . . . . . . 3954.7.2 Linearization of the Equations of Motion About

the Equilibrium Position . . . . . . . . . . . . . . . . . . . . . . 3974.8 Numerical Integration of the Equations

of Motion (Brief Outline) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4064.8.1 The Newmark Method in a Linear Field . . . . . . . . . . . 4094.8.2 The Newmark Method in a Nonlinear Field . . . . . . . . . 410

4.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

5 Dynamical Systems Subjected to Force Fields . . . . . . . . . . . . . . . . 4135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4135.2 Vibrating Systems with 1 DOF Perturbed Around

the Position of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 4175.2.1 System with 1 DOF Placed in an Aerodynamic

Force Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4225.3 Vibrating Systems with 2 d.o.f. Perturbed Around

the Position of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 4395.3.1 Two-Degree-of-Freedom System with Placed

in a Field of Purely Positional Forces . . . . . . . . . . . . . 4435.3.2 Two-Degrees-of-Freedom System Placed in a Field

of Position and Velocity Dependent Forces . . . . . . . . . 4615.4 Multi-Degree-of-Freedom Vibrating Systems Perturbed

Around the Position of Equilibrium . . . . . . . . . . . . . . . . . . . . 4805.4.1 The General Method for Analysing

a n-Degree-of-Freedom System Subjectto Non-conservative Forces . . . . . . . . . . . . . . . . . . . . 480

5.4.2 An Example: An Aerofoil Hit by a ConfinedFlow (n-Degree-of-Freedom System). . . . . . . . . . . . . . 484

5.5 Systems Perturbed Around the Steady-State Position. . . . . . . . . 4985.5.1 Systems with 1 d.o.f . . . . . . . . . . . . . . . . . . . . . . . . . 4985.5.2 Systems with 2 d.o.f . . . . . . . . . . . . . . . . . . . . . . . . . 502

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

xii Contents

6 Rotordynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5556.2 Description of the System Composed of the Rotor

and the Supporting Structure Interacting with It . . . . . . . . . . . . 5556.2.1 Schematising the Rotor . . . . . . . . . . . . . . . . . . . . . . . 5576.2.2 Schematising Bearings . . . . . . . . . . . . . . . . . . . . . . . 5616.2.3 Defining the Field of Forces in Seals or More

in General Between Rotor and Stator . . . . . . . . . . . . . 5746.2.4 Schematising the Casing and the Supporting

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5756.2.5 The Overall Model (an Example of Application) . . . . . 584

6.3 Analysing the Different Vibration ProblemsEncountered in Rotordynamics . . . . . . . . . . . . . . . . . . . . . . . . 592

6.4 Critical Speed, Response of the Rotor to Unbalance . . . . . . . . . 5956.4.1 Two Degree-of-Freedom Model Without Damping . . . . 5966.4.2 Two-Degree-of-Freedom Model with Damping . . . . . . 6006.4.3 Determining the Generalised Forces Acting

on a Rotor Due to Unbalance. . . . . . . . . . . . . . . . . . . 6026.5 Balancing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

6.5.1 Disk Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6106.5.2 Balancing a Real Rotor . . . . . . . . . . . . . . . . . . . . . . . 616

6.6 Two-Per-Rev Vibrations Excited by Different RotorStiffnesses, in Horizontal Shafts . . . . . . . . . . . . . . . . . . . . . . . 6446.6.1 Two-Degree-of-Freedom Model . . . . . . . . . . . . . . . . . 6476.6.2 Schematisation of the Problem on a Real Rotor . . . . . . 654

6.7 The Hysteretic Damping Effect . . . . . . . . . . . . . . . . . . . . . . . 6616.7.1 Two-Degree-of-Freedom Model . . . . . . . . . . . . . . . . . 661

6.8 The Gyroscopic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6706.9 Oil-Film Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

6.9.1 Estimating Instability Using the Eigenvalueand Eigenvector Solution. . . . . . . . . . . . . . . . . . . . . . 683

6.9.2 Estimating Instability with the Modal Method . . . . . . . 6836.9.3 Estimating Instability with the Forced Method . . . . . . . 6856.9.4 Effect of Load Variations on Supports

on the Conditions of Instability . . . . . . . . . . . . . . . . . 6876.10 Torsional Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

6.10.1 Methods for Reducing to an Equivalent System . . . . . . 6926.10.2 Schematisations with N-Degree-of-Freedom Systems . . 6946.10.3 Schematisation with Continuous Bodies . . . . . . . . . . . 6986.10.4 Finite Element Schematisation . . . . . . . . . . . . . . . . . . 6996.10.5 Elements that Can Be Adopted to Reduce

Torsional Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 700References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

Contents xiii

7 Random Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7097.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7097.2 Defining a Random Process . . . . . . . . . . . . . . . . . . . . . . . . . . 7127.3 Parameters Defining the Statistical Characteristics

of a Random Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7137.3.1 Calculating the Power Spectral Density Function

and Cross-Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . 7227.4 Defining the Random Stationary and Ergodic Process . . . . . . . . 7317.5 The Response of a Vibrating System to Random Excitation. . . . 733

7.5.1 Analysis with Several Correlated Processes . . . . . . . . . 7387.6 Some Examples of Application . . . . . . . . . . . . . . . . . . . . . . . 738

7.6.1 Response of a Structure to Turbulent Wind . . . . . . . . . 7387.6.2 Response of a Structure to Wave Motion . . . . . . . . . . 7497.6.3 Irregularities in the Road Profile. . . . . . . . . . . . . . . . . 760

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768

8 Techniques of Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

8.1.1 Identifying the Parameters of a MechanicalSystem in the Time and Frequency Domain. . . . . . . . . 774

8.1.2 The Least Squares Method . . . . . . . . . . . . . . . . . . . . 7768.2 Modal Identification Techniques. . . . . . . . . . . . . . . . . . . . . . . 778

8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7788.2.2 An Outline of the Basic Equations . . . . . . . . . . . . . . . 7788.2.3 Graphic Representations of the Transfer Function . . . . . 7838.2.4 Defining the Experimental Transfer Function . . . . . . . . 7858.2.5 Determining Modal Parameters . . . . . . . . . . . . . . . . . 7918.2.6 Applications and Examples . . . . . . . . . . . . . . . . . . . . 801

8.3 Identification in the Time Domain . . . . . . . . . . . . . . . . . . . . . 8038.3.1 The Ibrahim Method . . . . . . . . . . . . . . . . . . . . . . . . . 805

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

xiv Contents

Introduction

Dynamic analysis is a necessary step to design, verify, edit and then operate,diagnose and monitor a generic mechanical system. The term “dynamic analysis”refers to the study aimed at identifying the dynamic behaviour (displacements,velocities, accelerations, strains and stresses) of the different components of amechanical system subject to the different forces (excitation causes) occurringunder normal operating conditions. Mechanical systems are generally constitutedby mutually interconnected bodies subjected to forces that can be explicit functionsof time (i.e. independent from the dynamics of the system itself) or forces that,generically, depend on time and possibly on the motion of the system itself. In thelatter case, we speak generically of “force fields”. Figure 1 shows some examples ofmechanical systems in the presence of force fields.

Generically speaking, mechanical systems can perform large motions or smallmotions about a position of static equilibrium (rest): this allows for the immediateclassification of mechanical systems into two different categories:

• “structures”, i.e. systems that admit a static equilibrium position (rest position)about which “small motions” are studied;

• “machines”, i.e. systems in which this type of rest position is not always present,and which, therefore, either have generic large motions or are in “steady-state”motion.

For the structures, motion is permitted by the deformability of the variouscomponents: in this case, the dynamic analysis will concern the vibratory motionabout the equilibrium position.

Conversely, even without considering the deformability of their components,machines possess motion: typical examples are a shaft rotating around its own axis,a road or rail vehicle (Fig. 1), the slider–crank mechanism of an internal combustionengine or any machine element in motion (articulated systems, robots, etc.).

To evaluate the behaviour of a generic mechanical system at design level orduring tuning operation, it is necessary to realize a mathematical model thatattempts to simulate its dynamic behaviour.

xv

Fig. 1 Some examples of mechanical systems subject to force fields: high speed trains, gasturbines and new skyscrapers in Milan

xvi Introduction

The schematizations that can be adopted in the creation of mathematical modelsof mechanical systems can be more or less complex, depending on the type ofproblem that needs to be solved. Mathematical models may be more or lessaccurate:

• rigid body models [with one or more degree of freedom (d.o.f.)];• deformable bodies (i.e. with infinite degrees of freedom);• linear models or models linearized about an equilibrium condition (rest or steadystate) and;

• nonlinear models to simulate generically large motions.

In some cases, a scheme of interconnected rigid bodies can be sufficientlyaccurate; conversely, in others, it is necessary to consider the distributed defor-mability of the various elements. As an example, the analysis of the dynamics ofmotion of a rotor subjected to torques and resistant moments can be carried out byconsidering the rotor as rigid while for the study of bending and torsional motionsaround the steady-state speed of rotation, it is necessary to keep account of thedeformability of the rotor itself.

In addition to the mechanical system, it is also necessary to model the forces thatare applied to it. In this case too, different schematization “levels” of these forcesexist:

• models in which constraints are considered as ideal;• models where real constraints are considered, in which the contact forces due tothe same constraints can be attributed to force fields;

• models in which the interactions between any pair of bodies or between body andfluid are reproduced through the definition of force fields; and

• models in which a control action is taken into account and where both the controllogic and the actuators, used to impose the necessary forces, have to be described.

Therefore, in the analysis of a dynamic system, an in-depth knowledge of thedifferent force fields acting on the various elements is required (see Fig. 1, fluidaction, contact forces between bodies, action of electromagnetic fields, etc.), sincesuch fields can significantly affect the behaviour of the system itself.

An approach of this kind leads to an accurate, and, as such, complex modelling.Let us consider, as an example, the slider–crank mechanism of internal combustionengines: despite considering the crank, connecting rod and piston as rigid bodies,each of them should be allocated (though only considering plane motion for thesake of simplicity) three degrees of freedom, allowed by the deformabilityof the lubricant films present in the various pairs. Due to the presence of fluid in thevarious mechanism components (lubricant in the couplings, fluid contained in thecombustion chamber subject to thermodynamic transformation), force fields arise asa function of their positions and relative speeds. To write the equations of motionof the system, the definition of these force fields is essential. Depending on theparticular nature of the problem considered and the aim of the analysis, more simplemodels can be obtained. If, in the previous example, the law of motion of the piston

Introduction xvii

has to be calculated, it seems sufficiently accurate to neglect the relative motionsdue to the deformability of the lubricant fluid films in the mechanism pairs. In thiscase, the mechanism is reduced to a one degree-of-freedom system. If the objectiveof the analysis is, for example, to define the pressure and temperature distributionson the piston skirt, as previously mentioned, system modelling must be carried out.

Once the overall mathematical model of the mechanical system has been defined(i.e. mechanical system model and applied forces), the equations of motion must bewritten and solved to evaluate the dynamic response of same.

To write the equations of motions, various methods can be applied. Methods thatbest lend themselves to a systematic analysis of various problems are definitelythose of dynamic equilibrium and Lagrange’s equations:

• dynamic equilibrium can definitely be used for simple rigid bodyschematizations;

• in systems subject to ideal constraints, by using Lagrange’s method, it is notnecessary to introduce constraint reactions;

• Lagrange’s equations must be used if body deformability is considered, e.g. byusing a discretization method of the finite element type.

The differential equations that describe systems subject to large motions,obtained using the methods described above, appear to be nonlinear. Small motionsabout an equilibrium position are described by linear differential equations.

When analysing machines, it may often be useful to linearize the equations ofmotion about a static or dynamic equilibrium condition (i.e. about a rest or steady-state condition), like, for example, when calculating natural frequencies or ana-lysing incipient instability conditions.

At this point, the choice of the algorithm used to solve the equations of motionbecomes important: in the case of linear or linearized systems, the analysis oftenprovides solutions to various problems in a closed form while, for nonlinear sys-tems, a solution has to be found by using numerical techniques.

As regards the analysis of a generic mechanical system, a fundamental aspectthat will be taken into account in this book is the definition of a systematic approachto the schematization of the system itself. The study of the dynamics of systemswith rigid or deformable interconnected bodies has had recent developments, basedon the use of coordinate transformation matrices, thus giving rise to a method, nowknown as the “Multi-Body System Method”, particularly suited to computerimplementation in order to represent the kinematics of the various elements.2

The study of a dynamics problem consists in writing the equations of motion—generally nonlinear—in finding an existing equilibrium condition and in linearizingthe equations of motion around this condition, for stability analysis.

Dynamic analysis is also aimed at studying the transient motion performed topossibly attain a steady-state situation or limit cycle. Stability analysis is usually

2Cheli and Pennestrì [1], Shabana [2].

xviii Introduction

carried out by using a linear approach; conversely, the study of the transient motionor the determination of any limit cycles requires the solution of nonlinear problems.

For example, when wishing to tackle problems related to the dynamics of avehicle in motion on a straight road or on a curve, once the equations of motionhave been obtained, it is first necessary to find a steady-state solution (i.e. a vehiclemoving at a constant speed on a straight line or on a curve), subsequently going onto analyse the perturbed motion around the main trajectory. This analysis may eitherbe performed by means of equations linearized around steady-state conditions, orby integrating nonlinear equations with assigned initial conditions: in this way, inaddition to the analysis of the stability, any “large” motion of the vehicle can bedefined. Although it is not easy to make a clear distinction of subjects, it can bestated that the analysis of large motion is a subject of “Applied Mechanics”,3

although in Applied Mechanics courses simplified rigid body schematizations,typically either with one or a few degree of freedom, are usually presented.

Conversely, in this book, the foundations for an engineering approach to thegeneral problem are given, mainly by investigating the aspects associated with theanalysis of the stability and vibrations of mechanical systems. In the first part of thetext, we will consider the small motions of systems arising from perturbations ofstable static equilibrium conditions, which represent the classic study of vibratingsystems. By considering an energy approach (Lagrange’s equations), the forms ofenergy involved in the vibratory phenomenon are kinetic energy due to the massof the system, the elastic potential energy, possibly gravity and, finally, energydissipation either due to the imperfect elasticity of the materials or comparable toviscous effects. Mechanical systems falling into this category are termed dissipative.For this class of systems, we will consider motions arising from small perturbationsof static equilibrium conditions, perturbed free motions and the forced motions dueto external excitation forces, generally functions of time.

These systems will be analysed in order of complexity, starting gradually fromsystems with one degree of freedom, systems with 2-n degree of freedom (in thiscontext, multi-body methodologies for writing equations of motion will be men-tioned, see Chaps. 1 and 2), finally extending the discussion to systems with infinitedegree of freedom (continuous systems, Chap. 3).

As far as the latter are concerned, procedures pertaining to their discretizationwill be described, with particular reference to the modal approach (Chap. 3), thefinite element technique (Chap. 4) and, finally, the identification techniques ofmodal parameters from experimental tests (Chap. 8).

The second part of this book will be devoted to the study of mechanical systemssubjected to force fields (Chap. 5): In addition to the possible steady-state solution,the perturbed motion about same will be also studied. To achieve this, it is

3Bachschmid et al. [3].

Introduction xix

necessary to define the motion of the system in a more complete and complex formthan that used in the classical treatises of Applied Mechanics, i.e. by also consid-ering the deformability of the system, in the event of this being deemed important todefine its dynamic behaviour. The approach is of the general type: after writing theequations of motion, a steady-state solution is looked for. If such a solution isfound, the subsequent dynamic analysis can be performed by linearizing the systemabout this steady-state solution in order to check stability. The linearized equationsmay present constant coefficients or coefficients as functions of time. The meth-odology of analysis depends on the structure of these equations: more in particular,for the first category of equations, a systematic study is possible, while, withcoefficients as functions of time, analysis procedures are more complex. Thedynamic behaviour of the generic system can also be analysed by using a nonlinearapproach consisting of a numerical integration of the equations of motion or, forsome kind of nonlinear problems, of the use of approximate analytical methods. Towrite the equations that govern the mechanical system in its most general form, it isnecessary to consider not only the kinetic energy, potential energy and dissipativefunction, but also the presence of force fields surrounding the system: in addition tothe elastic one, force fields due to the action of a fluid, contact forces betweenbodies or electromagnetic fields may be present. The forces that the various ele-ments of a mechanical system exchange, not only with each other but also with thesurrounding environment, are generally functions of the independent variables thatdefine the motion of the system itself and their derivatives. In a broad sense, theseinteractions are considered to be due to “force fields”. Generally speaking, theseforce fields (Chap. 5) are nonlinear and characterize system behaviour significantly.

If the forces are solely functions of the configuration of the system, they will bereferred to as “positional” force field; on the contrary, if they depend on the velocityof the system, they will be termed “velocity” force field (force fields that have notbeen dealt with in other courses will be described briefly in the same chapter).

Not only does the presence of force fields condition steady state but also freemotion, influencing system stability: for example, owing to the fact that positionalnon-conservative force fields are able to introduce energy into the system, they cangenerate forms of instability. Velocity force fields are non-conservative by defini-tion, meaning that they are also capable of modifying the stability of a system. Thedevelopment of the study of mechanical systems subjected to force fields, inaddition to those regarding elastic and dissipative ones, is common to many dis-ciplines and falls within the analysis of mechanical systems in a more general sense.

Controlled systems can also be classified in this category, in the sense that acontrol system applies forces proportional to the values of the independent variablesor to the difference between a reference quantity value and the actual value of same(function of the independent variables). From this point of view, control systemscan also be considered as force fields. However, although this course is not aimed atthe systematic analysis of control problems, it definitely lays the foundations for

3 Bachschmid et al. [3].

xx Introduction

control engineering, with which it shares innumerable problems and solutionmethods, at least as far as the study of stability is concerned. Therefore, wherepossible, an effort will be made to approach the structure of equations and thesymbolism adopted in the field of control engineering, so as to deal with the studyof mechanical systems in a similar way to that of controlled systems. The equationsof motion of a mechanical system are generally written in the form:

½M�€xþ ½R� _xþ ½K�x ¼ Fðx; _x;€x; tÞ ð1Þ

where x is the vector of independent variables, F the applied external forces (due toforce fields or known functions of time or due to the action of a controller), ½M�, ½R�and ½K� the mass, damping and stiffness matrices. By linearizing these equationsabout an equilibrium or steady-state position, in addition to the structural matrices,terms arising from the linearization of the force fields also appear. These terms arecalled the Jacobians of the force fields: the equation of motion in this case can berewritten in the form:

½½M� þ ½MF ��€xþ ½½R� þ ½RF �� _xþ ½½K� þ ½KF ��x ¼ FðtÞ ð2Þ

where ½MF �, ½RF � and ½KF � are the matrices arising from the linearization of the forcefields while, on the right hand side, only the external forces, explicit functions oftime, remain. The analysis of the overall structure of the matrices (whether sym-metric, definite positive or not) allows us to check system stability. An approach ofthis type (see Eq. 2) is typical of mechanics; however, the same equations can berewritten in the equivalent form:

_z ¼ ½A�zþ uðtÞ ð3Þ

in which ½A� is formed by the matrices ½M�, ½R� and ½K�: in turn, this type of matrixcan be a function of time if the equations of motion arising from the linearization donot present constant coefficients. In Eq. (3), the vector of so-called state variables isindicated:

z ¼ _xx

� �ð4Þ

while uðtÞ is the vector of known terms, solely a function of time, owing to the factthat the state-dependent control forces are already included in a linear way in ½MF �,½RF � and ½KF �. The two approaches (1) and (3) only differ in terms of a symbolicaspect, even though matrix ½A� loses the physical meanings of the problem withrespect to matrices ½M�, ½R� and ½K�.

More in detail, the second part (Chap. 5) is structured as follows:

• a general discussion of the problem of systems subjected to force fields;

Introduction xxi

• an analysis of systems subjected to positional force fields, differing from gravi-tational and elastic ones, and a discussion of stability for systems with one andtwo degrees of freedom;

• a steady-state solution and linearization of the force field;• analysis of systems with one or two degrees of freedom subjected to positionaland velocity force fields; and

• the development of several examples including the analysis of a 2D airfoil flow,motion of a journal inside a bearing with hydrodynamic lubrication, motion of atrain axle and of a road vehicle, or extensions to continuous systems (finiteelement models).

In dedicated chapters, the course will also deal with problems related to:

• rotor dynamics (balancing, oil film instability, interaction with the foundation,etc., Chap. 6);

• the definition of different types of random excitation forces including those due toturbulent wind, waves and earthquakes (Chap. 7): in this chapter, the problem ofvortex shedding will be illustrated, referencing the more general problem ofvibrations induced by fluids, outlined in previous chapters; and

• the experimental identification of parameters of a real system (Chap. 8): thisaspect is fundamental in modelling.

The creation of mathematical models, targeted at defining the dynamic behav-iour of mechanical systems in the terms mentioned above, is a well-establisheddiscipline. Algorithms developed to simulate, as accurately as possible, thebehaviour of structures and machines subjected to different excitation causes havebecome essential tools for the design and operation of such systems.

References

1. Cheli F, Pennestrì E (2006) Cinematica e dinamica dei sistemi multibody. Casa EditriceAmbrosiana, Milano

2. Shabana AA (2005) Dynamics of multibody systems, 3rd edn. Cambridge University Press3. Bachschmid N, Bruni S, Collina A, Pizzigoni B, Resta F (2003) Fondamenti di meccanica

teorica ed applicata. McGraw-Hill

xxii Introduction