Skip to main content

A Practical Approach To Dynamical Systems For Engineers

In Order to Read Online or Download A Practical Approach To Dynamical Systems For Engineers Full eBooks in PDF, EPUB, Tuebl and Mobi you need to create a Free account. Get any books you like and read everywhere you want. Fast Download Speed ~ Commercial & Ad Free. We cannot guarantee that every book is in the library!

A Practical Approach to Dynamical Systems for Engineers

A Practical Approach to Dynamical Systems for Engineers Book
Author : Patricia Mellodge
Publisher : Woodhead Publishing
Release : 2015-11-19
ISBN : 0081002246
Language : En, Es, Fr & De

GET BOOK

Book Description :

A Practical Approach to Dynamical Systems for Engineers takes the abstract mathematical concepts behind dynamical systems and applies them to real-world systems, such as a car traveling down the road, the ripples caused by throwing a pebble into a pond, and a clock pendulum swinging back and forth. Many relevant topics are covered, including modeling systems using differential equations, transfer functions, state-space representation, Hamiltonian systems, stability and equilibrium, and nonlinear system characteristics with examples including chaos, bifurcation, and limit cycles. In addition, MATLAB is used extensively to show how the analysis methods are applied to the examples. It is assumed readers will have an understanding of calculus, differential equations, linear algebra, and an interest in mechanical and electrical dynamical systems. Presents applications in engineering to show the adoption of dynamical system analytical methods Provides examples on the dynamics of automobiles, aircraft, and human balance, among others, with an emphasis on physical engineering systems MATLAB and Simulink are used throughout to apply the analysis methods and illustrate the ideas Offers in-depth discussions of every abstract concept, described in an intuitive manner, and illustrated using practical examples, bridging the gap between theory and practice Ideal resource for practicing engineers who need to understand background theory and how to apply it

Dynamics for Engineers

Dynamics for Engineers Book
Author : Soumitro Banerjee
Publisher : John Wiley & Sons
Release : 2005-12-13
ISBN : 0470868457
Language : En, Es, Fr & De

GET BOOK

Book Description :

Modelling and analysis of dynamical systems is a widespread practice as it is important for engineers to know how a given physical or engineering system will behave under specific circumstances. This text provides a comprehensive and systematic introduction to the methods and techniques used for translating physical problems into mathematical language, focusing on both linear and nonlinear systems. Highly practical in its approach, with solved examples, summaries, and sets of problems for each chapter, Dynamics for Engineers covers all aspects of the modelling and analysis of dynamical systems. Key features: Introduces the Newtonian, Lagrangian, Hamiltonian, and Bond Graph methodologies, and illustrates how these can be effectively used for obtaining differential equations for a wide variety of mechanical, electrical, and electromechanical systems. Develops a geometric understanding of the dynamics of physical systems by introducing the state space, and the character of the vector field around equilibrium points. Sets out features of the dynamics of nonlinear systems, such as like limit cycles, high-period orbits, and chaotic orbits. Establishes methodologies for formulating discrete-time models, and for developing dynamics in discrete state space. Senior undergraduate and graduate students in electrical, mechanical, civil, aeronautical and allied branches of engineering will find this book a valuable resource, as will lecturers in system modelling, analysis, control and design. This text will also be useful for students and engineers in the field of mechatronics.

Nonlinear Ordinary Differential Equations

Nonlinear Ordinary Differential Equations Book
Author : Dominic William Jordan,Department of Mathematics D W Jordan,Peter Smith,Both in the Mathematics Department Peter Smith,Senior Lecturer D W Jordan
Publisher : Oxford University Press, USA
Release : 1999
ISBN : 9780198565628
Language : En, Es, Fr & De

GET BOOK

Book Description :

The text of this edition has been revised to bring it into line with current teaching, including an expansion of the material on bifurcations and chaos. It is directed towards practical applications of the theory with examples and problems.

Modelling Simulation and Control of Non linear Dynamical Systems

Modelling  Simulation and Control of Non linear Dynamical Systems Book
Author : Patricia Melin,Oscar Castillo
Publisher : CRC Press
Release : 2001-10-25
ISBN : 1000611965
Language : En, Es, Fr & De

GET BOOK

Book Description :

These authors use soft computing techniques and fractal theory in this new approach to mathematical modeling, simulation and control of complexion-linear dynamical systems. First, a new fuzzy-fractal approach to automated mathematical modeling of non-linear dynamical systems is presented. It is illustrated with examples on the PROLOG programming la

Control Engineering Solutions

Control Engineering Solutions Book
Author : P. Albertos,P. Albertos Pérez,Roland Strietzel,Neil Mort
Publisher : IET
Release : 1997
ISBN : 9780852968291
Language : En, Es, Fr & De

GET BOOK

Book Description :

This book collects together in one volume a number of suggested control engineering solutions which are intended to be representative of solutions applicable to a broad class of control problems. It is neither a control theory book nor a handbook of laboratory experiments, but it does include both the basic theory of control and associated practical laboratory set-ups to illustrate the solutions proposed.

A Relaxation Based Approach to Optimal Control of Hybrid and Switched Systems

A Relaxation Based Approach to Optimal Control of Hybrid and Switched Systems Book
Author : Vadim Azhmyakov
Publisher : Butterworth-Heinemann
Release : 2019-02-14
ISBN : 012814789X
Language : En, Es, Fr & De

GET BOOK

Book Description :

A Relaxation Based Approach to Optimal Control of Hybrid and Switched Systems proposes a unified approach to effective and numerically tractable relaxation schemes for optimal control problems of hybrid and switched systems. The book gives an overview of the existing (conventional and newly developed) relaxation techniques associated with the conventional systems described by ordinary differential equations. Next, it constructs a self-contained relaxation theory for optimal control processes governed by various types (sub-classes) of general hybrid and switched systems. It contains all mathematical tools necessary for an adequate understanding and using of the sophisticated relaxation techniques. In addition, readers will find many practically oriented optimal control problems related to the new class of dynamic systems. All in all, the book follows engineering and numerical concepts. However, it can also be considered as a mathematical compendium that contains the necessary formal results and important algorithms related to the modern relaxation theory. Illustrates the use of the relaxation approaches in engineering optimization Presents application of the relaxation methods in computational schemes for a numerical treatment of the sophisticated hybrid/switched optimal control problems Offers a rigorous and self-contained mathematical tool for an adequate understanding and practical use of the relaxation techniques Presents an extension of the relaxation methodology to the new class of applied dynamic systems, namely, to hybrid and switched control systems

System Dynamics

System Dynamics Book
Author : Karl A. Seeler
Publisher : Springer
Release : 2014-08-26
ISBN : 1461491525
Language : En, Es, Fr & De

GET BOOK

Book Description :

This unique textbook takes the student from the initial steps in modeling a dynamic system through development of the mathematical models needed for feedback control. The generously-illustrated, student-friendly text focuses on fundamental theoretical development rather than the application of commercial software. Practical details of machine design are included to motivate the non-mathematically inclined student.

Modeling of Dynamic Systems with Engineering Applications

Modeling of Dynamic Systems with Engineering Applications Book
Author : Clarence W. de Silva
Publisher : CRC Press
Release : 2017-10-16
ISBN : 1498798640
Language : En, Es, Fr & De

GET BOOK

Book Description :

MODELING OF DYNAMIC SYSTEMS takes a unique, up-to-date approach to systems dynamics and related controls coverage for undergraduate students and practicing engineers. It focuses on the model development of engineering problems rather than response analysis and simulation once a model is available, though these are also covered. Linear graphing and bond graph approaches are both discussed, and computational tools are integrated thoughout. Electrical, mechanical, fluid, and thermal domains are covered, as are problems of multiple domains (mixed systems); the unified and integrated approaches taken are rapidly becoming the standard in the modeling of mechatronic engineering systems.

Inelasticity of Materials

Inelasticity of Materials Book
Author : Arun R Srinivasa,Sivakumar M Srinivasan
Publisher : World Scientific Publishing Company
Release : 2009-07-09
ISBN : 9813107391
Language : En, Es, Fr & De

GET BOOK

Book Description :

With the advent of a host of new materials ranging from shape memory alloys to biomaterials to multiphase alloys, acquiring the capacity to model inelastic behavior and to choose the right model in a commercial analysis software has become a pressing need for practicing engineers. Even with the traditional materials, there is a continued emphasis on optimizing and extending their full range of capability in the applications. This textbook builds upon the existing knowledge of elasticity and thermodynamics, and allows the reader to gain confidence in extending one's skills in understanding and analyzing problems in inelasticity. By reading this textbook and working through the assigned exercises, the reader will gain a level of comfort and competence in developing and using inelasticity models. Thus, the book serves as a valuable book for practicing engineers and senior-level undergraduate/graduate-level students in the mechanical, civil, aeronautical, metallurgical and other disciplines. The book is written in three parts. Part 1 is primarily focused on lumped parameter models and simple structural elements such as trusses and beams. This is suitable for an advanced undergraduate class with just a strength of materials background. Part II is focused on small deformation multi-dimensional inelasticity and is suitable for a beginning graduate class. Sufficient material is included on how to numerically implement an inelastic model and solve either using a simple stress function type of approach or using commercial software. Case studies are included as examples. There is also an extensive discussion of thermodynamics in the context of small deformations. Part III focuses on more advanced situations such as finite deformation inelasticity, thermodynamical ideas and crystal plasticity. More advanced case studies are included in this part. • This textbook takes a new, task- or scenario-based approach to teaching and learning inelasticity. The book is written in an active learning style that appeals to engineers and students who wish to design or analyze structures and components that are subject to inelasticity. • The book incorporates thermodynamical considerations into the modeling right from an early stage. Extensive discussions are provided throughout the book on the thermodynamical underpinnings of the models. • This textbook is the first to make extensive use of MATLAB to implement many inelasticity models. It includes the use of concepts such as Airy stress functions to solve plane problems for inelastic materials. The MATLAB codes are listed in the appendix for one to modify with their own models and requirements. • Step-by-step procedures for formulations and calculations are provided for the reader to readily adapt to the inelastic problems that he or she attempts to solve. • A large number of problems, exercises and projects for one to teach or learn from are included. These can be assigned as homework, in-class exercises or projects. • The book is written in a modular fashion, which provides adequate flexibility for adaptation in classes that cater to different audiences such as senior-level students, graduate students, research scholars, and practicing engineers.

Soft Computing for Control of Non Linear Dynamical Systems

Soft Computing for Control of Non Linear Dynamical Systems Book
Author : Oscar Castillo,Patricia Melin
Publisher : Physica
Release : 2012-12-06
ISBN : 3790818321
Language : En, Es, Fr & De

GET BOOK

Book Description :

This book presents a unified view of modelling, simulation, and control of non linear dynamical systems using soft computing techniques and fractal theory. Our particular point of view is that modelling, simulation, and control are problems that cannot be considered apart, because they are intrinsically related in real world applications. Control of non-linear dynamical systems cannot be achieved if we don't have the appropriate model for the system. On the other hand, we know that complex non-linear dynamical systems can exhibit a wide range of dynamic behaviors ( ranging from simple periodic orbits to chaotic strange attractors), so the problem of simulation and behavior identification is a very important one. Also, we want to automate each of these tasks because in this way it is more easy to solve a particular problem. A real world problem may require that we use modelling, simulation, and control, to achieve the desired level of performance needed for the particular application.

Nonlinear Approaches in Engineering Applications

Nonlinear Approaches in Engineering Applications Book
Author : Liming Dai,Reza N. Jazar
Publisher : Springer Science & Business Media
Release : 2011-12-22
ISBN : 1461414695
Language : En, Es, Fr & De

GET BOOK

Book Description :

Nonlinear Approaches in Engineering Applications focuses on nonlinear phenomena that are common in the engineering field. The nonlinear approaches described in this book provide a sound theoretical base and practical tools to design and analyze engineering systems with high efficiency and accuracy and with less energy and downtime. Presented here are nonlinear approaches in areas such as dynamic systems, optimal control and approaches in nonlinear dynamics and acoustics. Coverage encompasses a wide range of applications and fields including mathematical modeling and nonlinear behavior as applied to microresonators, nanotechnologies, nonlinear behavior in soil erosion,nonlinear population dynamics, and optimization in reducing vibration and noise as well as vibration in triple-walled carbon nanotubes.

Modelling and Parameter Estimation of Dynamic Systems

Modelling and Parameter Estimation of Dynamic Systems Book
Author : J.R. Raol,G. Girija,J. Singh
Publisher : IET
Release : 2004-01-01
ISBN : 0863413633
Language : En, Es, Fr & De

GET BOOK

Book Description :

This book presents a detailed examination of the estimation techniques and problems in dynamic systems. Containing several illustrations and computer programs, the book promotes a better understanding of system modelling and parameter estimation. Parameter estimation involves observation of a dynamic system to develop mathematical models that represent the system dynamics. With the increasing use of high speed digital computers, elegant and innovative techniques like filter error method, H° and artificial neural networks are finding more and more use in parameter estimation problems. The material is presented in an accessible manner and enables the user to implement and execute the programs and, therefore, gain first-hand experience of the estimation progress.

Engineering System Dynamics

Engineering System Dynamics Book
Author : Forbes T. Brown
Publisher : CRC Press
Release : 2006-08-15
ISBN : 9780849396489
Language : En, Es, Fr & De

GET BOOK

Book Description :

For today's students, learning to model the dynamics of complex systems is increasingly important across nearly all engineering disciplines. First published in 2001, Forbes T. Brown's Engineering System Dynamics: A Unified Graph-Centered Approach introduced students to a unique and highly successful approach to modeling system dynamics using bond graphs. Updated with nearly one-third new material, this second edition expands this approach to an even broader range of topics. What's New in the Second Edition? In addition to new material, this edition was restructured to build students' competence in traditional linear mathematical methods before they have gone too far into the modeling that still plays a pivotal role. New topics include magnetic circuits and motors including simulation with magnetic hysteresis; extensive new material on the modeling, analysis, and simulation of distributed-parameter systems; kinetic energy in thermodynamic systems; and Lagrangian and Hamiltonian methods. MATLAB® figures prominently in this edition as well, with code available for download from the Internet. This code includes simulations for problems that appear in the later chapters as well as code for selected thermodynamic substances. Using a step-by-step pedagogy accompanied by abundant examples, graphs, illustrations, case studies, guided exercises, and homework problems, Engineering System Dynamics: A Unified Graph-Centered Approach, Second Edition is a text that students will embrace and continue to use well into their careers. While the first half of the book is ideal for junior-level undergraduates, the entire contents are suited for more advanced students.

Microengineering MEMS and Interfacing

Microengineering  MEMS  and Interfacing Book
Author : Danny Banks
Publisher : CRC Press
Release : 2006-03-23
ISBN : 1420015419
Language : En, Es, Fr & De

GET BOOK

Book Description :

MEMS devices are finding increasingly widespread use in a variety of settings, from chemical and biological analysis to sensors and actuators in automotive applications. Along with this massive growth, the field is still experiencing growing pains as fabrication processes are refined and new applications are attempted. Anyone serious about entering the field must have a realistic knowledge of just what is possible with MEMS technologies as well as the myriad issues involved in fabrication and device integration. Microengineering, MEMS, and Interfacing: A Practical Guide provides a straightforward, down-to-earth overview of the current state of MEMS technology. The first section systematically reviews the various bulk and surface micromachining methods, photolithography masks, and nonsilicon processes, examining their capabilities, limitations, and suggested uses. Next, the author details the characteristics of individual devices and systems, their advantages and shortcomings, and how they can be combined to achieve desired functionality. He includes condensed introductions to relevant chemistry and biochemistry and then demonstrates applications of MEMS in these areas. Beginning with a short introduction to electronics, the final section explores the issues involved in interfacing MEMS components with other systems. With judicious use of illustrations to clarify the discussion, Microengineering, MEMS, and Interfacing: A Practical Guide offers hands-on tools for solving specific problems along with the insight necessary to use them most effectively.

Modeling Analysis And Control Of Dynamical Systems With Friction And Impacts

Modeling  Analysis And Control Of Dynamical Systems With Friction And Impacts Book
Author : Olejnik Pawel,Feckan Michal,Awrejcewicz Jan
Publisher : #N/A
Release : 2017-07-07
ISBN : 9813225300
Language : En, Es, Fr & De

GET BOOK

Book Description :

This book is aimed primarily towards physicists and mechanical engineers specializing in modeling, analysis, and control of discontinuous systems with friction and impacts. It fills a gap in the existing literature by offering an original contribution to the field of discontinuous mechanical systems based on mathematical and numerical modeling as well as the control of such systems. Each chapter provides the reader with both the theoretical background and results of verified and useful computations, including solutions of the problems of modeling and application of friction laws in numerical computations, results from finding and analyzing impact solutions, the analysis and control of dynamical systems with discontinuities, etc. The contents offer a smooth correspondence between science and engineering and will allow the reader to discover new ideas. Also emphasized is the unity of diverse branches of physics and mathematics towards understanding complex piecewise-smooth dynamical systems. Mathematical models presented will be important in numerical experiments, experimental measurements, and optimization problems found in applied mechanics.

Differential Dynamical Systems

Differential Dynamical Systems Book
Author : James D. Meiss
Publisher : SIAM
Release : 2007-01-01
ISBN : 0898716357
Language : En, Es, Fr & De

GET BOOK

Book Description :

Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics.Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems conceptsflow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics. Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple, Mathematica, and MATLAB software to give students practice with computation applied to dynamical systems problems. Audience This textbook is intended for senior undergraduates and first-year graduate students in pure and applied mathematics, engineering, and the physical sciences. Readers should be comfortable with elementary differential equations and linear algebra and should have had exposure to advanced calculus. Contents List of Figures; Preface; Acknowledgments; Chapter 1: Introduction; Chapter 2: Linear Systems; Chapter 3: Existence and Uniqueness; Chapter 4: Dynamical Systems; Chapter 5: Invariant Manifolds; Chapter 6: The Phase Plane; Chapter 7: Chaotic Dynamics; Chapter 8: Bifurcation Theory; Chapter 9: Hamiltonian Dynamics; Appendix: Mathematical Software; Bibliography; Index

Modeling and Analysis of Dynamic Systems Second Edition

Modeling and Analysis of Dynamic Systems  Second Edition Book
Author : Ramin S. Esfandiari,Bei Lu
Publisher : CRC Press
Release : 2014-04-24
ISBN : 146657495X
Language : En, Es, Fr & De

GET BOOK

Book Description :

Modeling and Analysis of Dynamic Systems, Second Edition introduces MATLAB®, Simulink®, and SimscapeTM and then uses them throughout the text to perform symbolic, graphical, numerical, and simulation tasks. Written for junior or senior level courses, the textbook meticulously covers techniques for modeling dynamic systems, methods of response analysis, and provides an introduction to vibration and control systems. These features combine to provide students with a thorough knowledge of the mathematical modeling and analysis of dynamic systems. See What’s New in the Second Edition: Coverage of modeling and analysis of dynamic systems ranging from mechanical to thermal using Simscape Utilization of Simulink for linearization as well as simulation of nonlinear dynamic systems Integration of Simscape into Simulink for control system analysis and design Each topic covered includes at least one example, giving students better comprehension of the subject matter. More complex topics are accompanied by multiple, painstakingly worked-out examples. Each section of each chapter is followed by several exercises so that students can immediately apply the ideas just learned. End-of-chapter review exercises help in learning how a combination of different ideas can be used to analyze a problem. This second edition of a bestselling textbook fully integrates the MATLAB Simscape Toolbox and covers the usage of Simulink for new purposes. It gives students better insight into the involvement of actual physical components rather than their mathematical representations.

System Safety Engineering and Risk Assessment

System Safety Engineering and Risk Assessment Book
Author : Nicholas J. Bahr
Publisher : CRC Press
Release : 2018-10-08
ISBN : 1466551615
Language : En, Es, Fr & De

GET BOOK

Book Description :

We all know that safety should be an integral part of the systems that we build and operate. The public demands that they are protected from accidents, yet industry and government do not always know how to reach this common goal. This book gives engineers and managers working in companies and governments around the world a pragmatic and reasonable approach to system safety and risk assessment techniques. It explains in easy-to-understand language how to design workable safety management systems and implement tested solutions immediately. The book is intended for working engineers who know that they need to build safe systems, but aren’t sure where to start. To make it easy to get started quickly, it includes numerous real-life engineering examples. The book’s many practical tips and best practices explain not only how to prevent accidents, but also how to build safety into systems at a sensible price. The book also includes numerous case studies from real disasters that describe what went wrong and the lessons learned. See What’s New in the Second Edition: New chapter on developing government safety oversight programs and regulations, including designing and setting up a new safety regulatory body, developing safety regulatory oversight functions and governance, developing safety regulations, and how to avoid common mistakes in government oversight Significantly expanded chapter on safety management systems, with many practical applications from around the world and information about designing and building robust safety management systems, auditing them, gaining internal support, and creating a safety culture New and expanded case studies and "Notes from Nick’s Files" (examples of practical applications from the author’s extensive experience) Increased international focus on world-leading practices from multiple industries with practical examples, common mistakes to avoid, and new thinking about how to build sustainable safety management systems New material on safety culture, developing leading safety performance indicators, safety maturity model, auditing safety management systems, and setting up a safety knowledge management system

The Optimal Homotopy Asymptotic Method

The Optimal Homotopy Asymptotic Method Book
Author : Vasile Marinca,Nicolae Herisanu
Publisher : Springer
Release : 2015-04-02
ISBN : 3319153749
Language : En, Es, Fr & De

GET BOOK

Book Description :

This book emphasizes in detail the applicability of the Optimal Homotopy Asymptotic Method to various engineering problems. It is a continuation of the book “Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches”, published at Springer in 2011 and it contains a great amount of practical models from various fields of engineering such as classical and fluid mechanics, thermodynamics, nonlinear oscillations, electrical machines and so on. The main structure of the book consists of 5 chapters. The first chapter is introductory while the second chapter is devoted to a short history of the development of homotopy methods, including the basic ideas of the Optimal Homotopy Asymptotic Method. The last three chapters, from Chapter 3 to Chapter 5, are introducing three distinct alternatives of the Optimal Homotopy Asymptotic Method with illustrative applications to nonlinear dynamical systems. The third chapter deals with the first alternative of our approach with two iterations. Five applications are presented from fluid mechanics and nonlinear oscillations. The Chapter 4 presents the Optimal Homotopy Asymptotic Method with a single iteration and solving the linear equation on the first approximation. Here are treated 32 models from different fields of engineering such as fluid mechanics, thermodynamics, nonlinear damped and undamped oscillations, electrical machines and even from physics and biology. The last chapter is devoted to the Optimal Homotopy Asymptotic Method with a single iteration but without solving the equation in the first approximation.

Rotordynamics

Rotordynamics Book
Author : Agnieszka Muszynska
Publisher : CRC Press
Release : 2005-05-20
ISBN : 1420027794
Language : En, Es, Fr & De

GET BOOK

Book Description :

As the most important parts of rotating machinery, rotors are also the most prone to mechanical vibrations, which may lead to machine failure. Correction is only possible when proper and accurate diagnosis is obtained through understanding of rotor operation and all of the potential malfunctions that may occur. Mathematical modeling, in particular modal modeling, is key to understanding observed phenomena through measured data and for predicting and preventing failure. Rotordynamics advances simple yet adequate models of rotordynamic problems and phenomena related to rotor operation in its environment. Based on Dr. Muszy(n ́)ska's extensive work at Bently Rotor Dynamics Research Corporation, world renowned for innovative and groundbreaking experiments in the field, this book provides realistic models, step-by-step experimental methods, and the principles of vibration monitoring and practical malfunction diagnostics of rotating machinery. It covers extended rotor models, rotor/fluid-related phenomena, rotor-to-stationary part rubbing, and other related problems such as nonsynchronous perturbation testing. The author also illustrates practical diagnoses of several possible malfunctions and emphasizes correct interpretation of computer-generated numerical results. Rotordynamics is the preeminent guide to rotordynamic theory and practice. It is the most valuable tool available for anyone working on modeling rotating machinery at the machine design stage or performing further analytical and experimental research on rotating machine dynamics.