4 Distributed-Parameter Systems Clarence W. de Silva

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The University of British Columbia

4.1 Introduction ....................................................................... 4-1

4.2 Transverse Vibration of Cables ......................................... 4-2

Wave Equation † General (Modal) Solution † Cable with

Fixed Ends † Orthogonality of Natural Modes † Application

of Initial Conditions

4.3 Longitudinal Vibrations of Rods ..................................... 4-13

Equation of Motion † Boundary Conditions

4.4 Torsional Vibration of Shafts ........................................... 4-19

Shaft with Circular Cross Section † Torsional Vibration of

Noncircular Shafts

4.5 Flexural Vibration of Beams ............................................. 4-26

Governing Equation for Thin Beams † Modal Analysis †

Boundary Conditions † Free Vibration of a Simply Supported

Beam † Orthogonality of Mode Shapes † Forced Bending

Vibration † Bending Vibration of Beams with Axial

Loads † Bending Vibration of Thick Beams † Use of the

Energy Approach † Orthogonality with Inertial Boundary

Conditions

4.6 Damped Continuous Systems .......................................... 4-50

Modal Analysis of Damped Beams

4.7 Vibration of Membranes and Plates ................................ 4-52

Transverse Vibration of Membranes † Rectangular Membrane

with Fixed Edges † Transverse Vibration of Thin Plates †

Rectangular Plate with Simply Supported Edges

Summary

This chapter presents the analysis of continuous (or distributed-parameter) mechanical vibrating systems. In these

systems, inertial, elastic, and dissipative effects are found continuously distributed in one, two, or three dimensions.

Examples such as strings, rods, shafts, beams, membranes, and plates are studied. Modal analysis is carried out

using the separation of time and space. The orthogonality property of mode shapes is established. Boundary

conditions are derived. Free vibration and forced vibration are analyzed.

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