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Single Degree-of-Freedom Linear Oscillator Read chapter Read first chapter

2013 | OriginalPaper | Chapter

8. Random Vibration

Author : André Preumont

Published in: Twelve Lectures on Structural Dynamics

Publisher: Springer Netherlands

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Abstract

This is the first of two chapters devoted to the response of structures excited by random loads. The first part of this chapter summarizes the main theoretical concepts that characterize the single input-single output relationship of a linear system excited by a stationary Gaussian random process: correlation function, power spectral density, cumulative mean square response, spectral moments, etc... Rice’s formulae and the envelope of a narrow band process are also discussed. The second part of this chapter is devoted to the response of multi input-multi output systems and the specification of the random field of excitation, with examples taken from boundary layer noise, wind response of tall buildings and vehicle moving on rough ground. The chapter concludes with the seismic response of a n-storey building and a set of problems.

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previous chapter Seismic Excitation
next chapter Peak Factor and Random Fatigue
Footnotes
1
\(E[.]\) stands for the mathematical expectation,
$$\begin{aligned} E[x]=\int _{-\infty }^{\infty }xp(x)\,dx \end{aligned}$$
where \(p(x)\) is the probability density function of the random variable \(x\).
 
2
Similarly, for the cross PSD,
$$\begin{aligned} \varPhi _{xy}(\omega )=\frac{1}{2\pi }\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-j\omega \tau }d \tau =\lim _{T\rightarrow \infty }\frac{1}{2\pi T}\, E\left[ X(\omega ,T)Y^*(\omega ,T)\right] \end{aligned}$$
 
3
The wind results from a multitude of pressure gradients distributed randomly at the surface of the earth, or a seismic acceleration may be considered as resulting from a large number of micro slips along the fault line, each one being the source of a random wave.
 
4
Observe, once again, the duality of the Fourier transform: infinitely long in the frequency domain means infinitely short in the time domain.
 
5
The half power bandwidth is defined as the width of the spectral density diagram when \(\varPhi _{yy}=\frac{1}{2}\varPhi _{max}\) (Fig. 8.4b).
 
6
\(m_4\) is often difficult to compute in practice, but \(\nu _1\) and \(\nu _0\) can easily be estimated from a sample record, by counting the maxima and the zero up-crossings.
 
7
the notation \(\varPsi \) is used instead of the more traditional \(\varPhi \), because \(\varPhi \) is used for the power spectral densities, also according to tradition.
 
8
The finite element mesh is usually governed by a good representation of the stiffness of the structure.
 
9
Neglecting the term in \(\varepsilon _j^2\) is justified in wind engineering and the drag forces are also Gaussian. Such an approximation is no longer acceptable in the analysis of off-shore structures under wave excitation.
 
10
e.g. see (7.​71).
 
11
particularly in “back-of-an-envelope” calculations, because the global response is treated as the sum of independent responses of linear oscillators.
 
12
According to Eq.(8.30), the contribution of mode \(i\) to the RMS value varies according to \(\xi _i^{-1/2}\).
 
Metadata
Title
Random Vibration
Author
André Preumont
Copyright Year
2013
Publisher
Springer Netherlands
Book
Twelve Lectures on Structural Dynamics

Print ISBN: 978-94-007-6382-1

Electronic ISBN: 978-94-007-6383-8

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-94-007-6383-8_8

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