6 Numerical Techniques Marie D. Dahleh Harvard University

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6.1 Introduction ....................................................................... 6-1

6.2 Single-Degree-of-Freedom System ................................... 6-2

Forced Single-Degree-of-Freedom System † Summary of

Single-Degree-of-Freedom System

6.3 Systems with Two or More Degrees of Freedom ............ 6-8

Example † Summary of Two-Degree-of-Freedom System

6.4 Finite Difference Method for a Continuous System ...... 6-11

Bar † Beam † Summary of Finite Difference Methods for a

Continuous System

6.5 Matrix Methods ................................................................. 6-14

Example: Three-Degree-of-Freedom System † Bisection

Method † Directly Calculating the Eigenvalues and

Eigenvectors from the Matrix Equation † Summary of

Matrix Methods

6.6 Approximation Methods for the Fundamental

Frequency ........................................................................... 6-18

Rayleigh Method † Dunkerley’s Formula † Summary of

Approximations for the Fundamental Frequency

6.7 Finite Element Method ..................................................... 6-20

Bar Element † Beam † Summary of Finite Element Method

Appendix 6A Introduction to MATLABw ...................... 6-24

Summary

This chapter gives an overview of numerical techniques for vibration analysis. The centered difference

approximation for the first, second, and fourth derivative are given. These form the basis for the finite difference

approximation of both spring – mass systems and the continuous problem. The fourth-order Runge – Kutta method

is presented. Both of these approaches are used to solve the single-degree-of-freedom (single-DoF) system. In order

to demonstrate these techniques for the multiple-degree-of-freedom (multi-DoF) system a two-degree-of-freedom

(two-DoF) system is explored. Finite element and finite difference methods are presented as solution techniques for

the continuous problem (also see Chapter 9). The bar and beam are used for examples. The Rayleigh method and

Dunkerley’s formula are presented. These are methods for computing the fundamental frequency.

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