Random vibrations/engineering probability & statistics

[L.Richter]

Homework Statement

Stress :
S(t) = a₀ + a₁X(t) + a₂X²(t)
where X(t) is the random displacement, a Gaussian random process, and stationary.

1) determine the PDF of S(t)
2) determine the joint PDF of stress and stress velocity, S(t) and S'(t).
3) how would you determine the PDF of S'(t)?

Homework Equations

for 1) P_X(x) = 1/sqrt(2∏σ²) exp [-(x-μ)²/2σ²]
where μ= ∫xP_X(x)dx
σ²= ∫(x-μ)²P_X(x)dx

for 2) ??

for 3) extract the PDF of S'(t) from the joint PDF setting one of the bounds of S(t) to ∞.

The Attempt at a Solution

For 2) I was going to get the joint PDF by using a linear combination of the PDF of each through computing the mean, the variance and the standard deviation. If I assume independence then I could use P_S(t)S'(t) = P_S(t)P_S'(t) and that would lead me into a circle with 3). If anyone can make a suggestion how to approach the problem I would be greatly appreciative!

 

[mighty2000]

Thank you!

1) To determine the PDF of S(t), we can use the properties of Gaussian random processes. Since X(t) is a Gaussian random process, it is fully characterized by its mean and covariance functions. Therefore, we can write S(t) as a linear combination of Gaussian random variables, and the resulting PDF will also be a Gaussian distribution. The mean of S(t) is given by a0 + a1μx + a2μx2, where μx is the mean of X(t). Similarly, the variance of S(t) is given by a1σx + a2σx2, where σx is the standard deviation of X(t). Therefore, the PDF of S(t) is:

PS(t) = 1/√(2π(a1σx + a2σx2))exp[-(S(t) - (a0 + a1μx + a2μx2))^2/(2(a1σx + a2σx2))]

2) To determine the joint PDF of stress and stress velocity, we can use the properties of multivariate Gaussian distributions. Since S(t) and S'(t) are both linear combinations of Gaussian random variables, their joint distribution will also be a multivariate Gaussian distribution. The mean of the joint distribution is given by [μs μs'], where μs is the mean of S(t) and μs' is the mean of S'(t). The covariance matrix of the joint distribution is given by:

Σ = [[σs^2 σss'], [σss' σs'^2]]

where σs^2 and σs'^2 are the variances of S(t) and S'(t) respectively, and σss' is the covariance between S(t) and S'(t).

Therefore, the joint PDF of S(t) and S'(t) is:

PS(t)S'(t) = 1/(2π√(σs^2 σs'^2 - σss'^2))exp[-(S(t) - μs)^2/2σs^2 - (S'(t) - μs')^2/2σs'^2 + 2σss'(S(t) - μs)(S'(t) - μs')/(2σsσs')]

3) To determine the PDF of S'(t), we can extract it from the joint