Modelling of Mechanical Systems: Discrete Systems

Francois Axisa

• Publisher: Butterworth-Heinemann
• Year: 2003
• ISBN:
  9780080511863
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

This first volume is concerned with discrete systems – the study of which constitutes the cornerstone of all mechanical systems, linear or non-linear. It covers the formulation of equations of motion and the systematic study of free and forced vibrations. The book goes into detail about subjects such as generalized coordinates and kinematical conditions; Hamilton's principle and Lagrange equations; linear algebra in N-dimensional linear spaces and the orthogonal basis of natural modes of vibration of conservative systems. Also included are the Laplace transform and forced responses of linear dynamical systems, the Fourier transform and spectral analysis of excitation and response deterministic signals.Forthcoming volumes in this series:
Vol II: Structural Elements; to be published in June 2005
Vol III: Fluid-structure Interactions; to be published in August 2006
Vol IV: Flow-induced Vibrations; to be published in August 2007* Presents the general methods that provide a unified framework to model mathematically mechanical systems of interest to the engineer, analyzing the response of these systems
* Focuses on linear problems, but includes some aspects of non-linear configuration
* Comprehensive coverage of mathematical techniques used to perform computer-based analytical studies and numerical simulations
* Discusses the mathematical techniques used to perform analytical studies and numerical simulations on the computer

Chapter 1

Mechanical systems and equilibrium of forces

This chapter is intended both as a review of the basic principles of the Newtonian mechanics and as an introduction to a few physical concepts and mathematical notations we shall use throughout the book. Here, equilibrium equations (dynamic or static) of discrete systems are still derived by using the vector mechanics, i.e. direct balancing of the forces, and/or moments acting in the system. This approach is certainly the most familiar one to the majority of students, who are assumed to be already well acquainted with the basic techniques of vector analysis of Newtonian mechanics. However, the notions emphasized here, which concern degrees of freedom, generalized coordinates and kinematical conditions, are the first necessary ingredients of the Lagrangian formalism which is the subject of the next three chapters. Although the physical content of analytical mechanics founded by Lagrange (1788) is the same as that found in the Principia of Newton (1687), the analytical approach is more appealing from the logical viewpoint than the vectorial one and, even if more abstract, quickly reveals itself as far easier to apply when dealing with most material systems encountered in theoretical physics and mechanical engineering.

1.1 Modelling of mechanical systems

Formulation of a mathematical model is the first step in the process of analysing the behaviour of any real system. However, to produce a useful model, one must first adopt a set of simplifying assumptions which have to be relevant in relation to the physical features of the system to be modelled and to the specific information one is interested in. Thus, the aim of modelling is to produce an idealized description of reality, which is both expressible in a tractable mathematical form and sufficiently close to reality as far as the physical mechanisms of interest are concerned.

Mechanical systems are made up of material bodies, i.e. finite portions of media endowed with mass. When excited by forces, or by prescribed motions, their position and shape change progressively with time, eventually reaching a new permanent equilibrium. When modelling such systems, it is necessary to specify five items listed below (and then discussed in the following order):

1. Geometry and mass distribution of the material system.

2. Space in which the motion is studied.

3. Coordinates used to define its position versus time.

4. Kinematical constraints connecting the relative motion of distinct parts of the system.

5. Laws of mechanical behaviour of the material, and/or of mechanical interaction between distinct parts of the system.

1.1.1 Geometry and distribution of masses

In the first instance, it is necessary to define the shape of the material system and the mass distribution within it. The simplest system that may be conceived is the mass-point, or particle. This concept refers to a material body whose geometrical dimensions are neglected when describing its motion. The total mass m of the real body is attributed to the point, m being a positive scalar quantity. More generally, a collection of particles P_j (finite or not) with masses m_j j = 1,2,… is called a discrete system.

At first sight, a body whose dimensions are not neglected must be modelled as a continuous system, which is described by using a continuous (hence uncountable) set of points. They are endowed with a mass per unit volume (mass density) ρ that may vary either continuously, or piecewise continuously, with position inside the body. However, provided the body can be ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• Foreword
• Preface
• Introduction
• Chapter 1: Mechanical systems and equilibrium of forces
• Chapter 2: Principle of virtual work and Lagrange’s equations
• Chapter 3: Hamilton’s principle and Lagrange’s equations of unconstrained systems
• Chapter 4: Constrained systems and Lagrange’s undetermined multipliers
• Chapter 5: Autonomous oscillators
• Chapter 6: Natural modes of vibration of multi degree of freedom systems
• Chapter 7: Forced vibrations: response to transient excitations
• Chapter 8: Spectral analysis of deterministic time signals
• Chapter 9: Spectral analysis of forced vibrations
• Appendices
• Bibliography
• Index
• Series synopsis: Modelling of Mechanical Systems