3 Modal Analysis Clarence W. de Silva

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The University of British Columbia

3.1 Introduction .......................................................................... 3-1

3.2 Degrees of Freedom and Independent Coordinates .......... 3-2

Nonholonomic Constraints

3.3 System Representation ......................................................... 3-4

Stiffness and Flexibility Matrices † Inertia Matrix † Direct

Approach for Equations of Motion

3.4 Modal Vibrations ................................................................ 3-10

3.5 Orthogonality of Natural Modes ...................................... 3-14

Modal Mass and Normalized Modal Vectors

3.6 Static Modes and Rigid-Body Modes ............................... 3-15

Static Modes † Linear Independence of Modal Vectors †

Modal Stiffness and Normalized Modal Vectors †

Rigid-Body Modes † Modal Matrix † Configuration Space and

State Space

3.7 Other Modal Formulations ............................................... 3-22

Nonsymmetric Modal Formulation † Transformed Symmetric

Modal Formulation

3.8 Forced Vibration ................................................................. 3-28

First Mode (Rigid-Body Mode) † Second Mode

(Oscillatory Mode)

3.9 Damped Systems ................................................................. 3-32

Proportional Damping

3.10 State-Space Approach ......................................................... 3-36

Modal Analysis † Mode Shapes of Nonoscillatory Systems †

Mode Shapes of Oscillatory Systems

Appendix 3A Linear Algebra ............................................ 3-41

Summary

This chapter presents the modal analysis of lumped-parameter mechanical vibrating systems. In the considered

systems, inertia, flexibility, and damping characteristics are lumped at a finite number of discrete points in the system.

Techniques for determining the natural frequencies and mode shapes of vibration are given. The orthogonality of

mode shapes is established. The existence of natural modes in damped systems is investigated. Proportional damping

is discussed. Both free vibration and forced vibration of multi-degree-of-freedom (multi-DoF) systems are analyzed.

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