Vibration in Continuous Media

Jean-Louis Guyader

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781905209279
• eBook ISBN: 9781118613689
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

Three aspects are developed in this book: modeling, a description of the phenomena and computation methods. A particular effort has been made to provide a clear understanding of the limits associated with each modeling approach. Examples of applications are used throughout the book to provide a better understanding of the material presented.

Chapter 1

Vibrations of Continuous Elastic Solid Media

1.1. Objective of the chapter

This work is addressed to students with a certain grasp of continuous media mechanics, in particular, of the theory of elasticity. Nevertheless, it seems useful to recall in this chapter the essential points of these domains and to emphasize in particular the most interesting aspects in relation to the discussion that follows.

After a brief description of the movements of the continuous media, the laws of conservation of mass, momentum and energy are given in integral and differential form. We are thus led to the basic relations describing the movements of continuous media.

The case of small movements of continuous elastic solid media around a point of static stable equilibrium is then considered; we will obtain, by linearization, the equations of vibrations of elastic solids which will be of interest to us in the continuation of this work.

At the end of the chapter, a brief exposition of the equations of linear vibrations of viscoelastic solids is outlined. The equations in the temporal domain are given as well as those in the frequency domain, which are obtained by Fourier transformation. We then note a formal analogy of elastic solids equations with those of the viscoelastic solids, known as principle of correspondence.

Generally, the presentation of these reminders will be brief; the reader will find more detailed presentations in the references provided at the end of the book.

1.2. Equations of motion and boundary conditions of continuous media

1.2.1. Description of the movement of continuous media

To observe the movement of the continuous medium, we introduce a Galilean reference mark, defined by an origin O and an orthonormal base

. In this reference frame, a point M, at a fixed moment T, has the co-ordinates (x₁ , x₂ , x₃).

The Euler description of movement is carried out on the basis of the four variables (x₁ , x₂ , x₃ , t); the Euler unknowns are the three components of the speed

of the particle which is at the point M at the moment t.

[1.1]

Derivation with respect to time of quantities expressed with Euler variables is particular; it must take into account the variation with time of the co-ordinates x_i of the point M.

Figure 1.1. Location of the continuous medium

For example, for each acceleration component γ_i of the particle located at the point M, we obtain by using the chain rule of derivation:

and noting that:

we obtain the expression of the acceleration as the total derivative of the velocity:

or in index notation:

[1.2]

In the continuation of this work we shall make constant use of the index notation, which provides the results in a compact form. We shall briefly point out the equivalences in the traditional notation:

– partial derivation is noted by a comma:

– an index repeated in a monomial indicates a summation:

The Lagrangian description is an alternative to the Euler description of the movement of continuous media. It consists of introducing Lagrange variables (a₁ , a₂ , a...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Chapter 1: Vibrations of Continuous Elastic Solid Media
• Chapter 2: Variational Formulation for Vibrations of Elastic Continuous Media
• Chapter 3: Equation of Motion for Beams
• Chapter 4: Equation of Vibration for Plates
• Chapter 5: Vibratory Phenomena Described by the Wave Equation
• Chapter 6: Free Bending Vibration of Beams
• Chapter 7: Bending Vibration of Plates
• Chapter 8: Introduction to Damping: Example of the Wave Equation
• Chapter 9: Calculation of Forced Vibrations by Modal Expansion
• Chapter 10: Calculation of Forced Vibrations by Forced Wave Decomposition
• Chapter 11: The Rayleigh-Ritz Method based on Reissner’s Functional
• Chapter 12: The Rayleigh-Ritz Method based on Hamilton’s Functional
• Bibliography and Further Reading
• Index