5 Random Vibration Haym Benaroya Rutgers University

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5.1 Random Vibration ............................................................. 5-1

5.2 Single Degree of Freedom: The Response to

Random Loads .................................................................... 5-2

Formulation † Derivation of Equations † Response

Correlations † Response Spectral Density

5.3 Response to Two Random Loads ...................................... 5-7

5.4 Multi-Degree-of-Freedom Vibration ................................ 5-12

Deterministic Vibration † Solution by Frequency Response

Function † Modal Analysis

5.5 Multi-Degree-of-Freedom: The Response to

Random Loads .................................................................... 5-17

Response due to a Single Random Force † Response to

Multiple Random Forces † Impulse-Response Approach †

Modal Analysis Approach

5.6 Continuous System Random Vibration ........................... 5-29

Transverse Vibration of Beams † Random Transverse

Vibration

Summary

This chapter summarizes the key ideas of linear random vibration. This discipline focuses on determining the

response statistics of an oscillator or structure to input forces that are definable only in terms of their statistics.

Typical problems include the following: (1) given the power spectrum of the force, find the power spectrum of the

response; (2) given the mean value and variance of the force, find the mean value and variance of the response. The

methodology is built upon the linear theory of vibration for discrete single- and multi-degree-of-freedom (DoF)

systems, and continuous systems. The approaches are essentially the direct method and the modal analysis method.

The direct method may also be called a transfer matrix method (see Chapter 2). Modal analysis (see Chapters 3 and

4) has the same benefit in random vibration as is done in deterministic vibration studies: it can be computationally

more efficient. A number of examples are given, as are a list of representative references.

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