Question

random vibrations

Problem 1 Two random variables x and y have the joint probability density function where c is a constant. Verify that x and y are statistically independent and find the value of c for plx, y) to be correctly normalized. Check that Elx) Elyl-0 and that Elx2] and Ely') are both infinite Problem 2. Each sample function x(t) of a random process x(t) is given by: where a, a2, wh, and w are constants but 61 and 62, although constant for each sample, vary randomly frorn one sample to the next. Ifo, and θ2 are statistically independent and each is uniformly distributed between 0 and 2rt, determine the auto-correlation function Rx(T). The corresponding sample functions of a separate random process are given by: y(t), bi sin [ant + t) + &J + b2 sin (apt + 3) where bih, and to are constants and θ3 varies randomly from one sample to the next. If es is statistically independent of th and 02 and is also uniformly distributed between 0 and 2π, find the cross-correlation functions Ry(T) and Ry(T). Problem 3. The single-sided spectral density function of the deflection x(t) of a machine component is: Go 0.01cm2/Hz over the frequency band 0 to 200Hz and is zero for frequencies above 200Hz. Determine the mean square deflection Ela2] and obtain an expression for the auto-correlation function Rzx(t) for the x(0) process, where the units of T are seconds. Problem 4 Consider the response yit) of an SDOF oscillator when it is excited by a stationary Gaussian process x(0) with spectral density So Find expressions for the following statistics of the oscillator's response: 1. The expected value of positive slope crossing of the level per unit time. 2 the probability density function for the distribution of peaks. 3. the expected value of positive slope zero-crossings and of the peaks per unit time. Explain why they are different and verify that, when there is very small damping, they become the same.

Answer 1

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