Structural Dynamics: Random Response Flashcards
How can the mean of the output me computed by knowing the mean of the input?
mq = H(0)mf = K^-1mf, where F is the input and q is the output.
How can the variance of the output me computed using the impulse response?
If we want to compute in the time domain, the impulse response and the autocovariance of the input is required. The problem can be solved in frequency domain by calculating the power spectral density of the output and then transforming it to the time domain.
Can white noise be used to calculate the variance of the output?
Although the variance of the excitation is unbounded, the variance of the displacement is finite, provided there is some damping.
This is because even though the PSD is non zero in a large range, the only area where the two are intersecting is close to the resonance of the harmonic oscillator. So, we need to characterize the PSD of input in the range where the output is significant.
How can we derive Lyapunov’s equation?
Taking the space-state formulation, we calculate the autocovariance of x.
Taking the Δ of each variable (x,u,y), which is: Δ() = () - m() and then multiplying the first equation by Δx^T from the right hand side and computing the mean integral. The resulting equation, knowning that kxx(0)= σχχ^2 is:
σ^2x_dot,x = Aσ^2x_x + Bσ^2u_x
While for the scalar case it is possible to say that the cross-variance between a dof and its derivative is null, the same cannot be said for the vectorial case.
Now, multiplying Δx^T from the left hand side:
σ^2x_x_dot = σ^2x_xA^T + σ^2x_uB^T.
Knowing that σ^2x_dot = - σ^2x_x_dot and σ^2x_u = σ^2u_x^T and for a white noise input, the final form of the Lyapunov equation is:
Aσ^2xx + σ^2xxA^T + BWB^T, where W is the intensity of the white noise.
Ultimately, the Lyapunov equation is a matrix that allows to compute the cross-variance of the states of a vectorial state-space system subject to a white noise input. Knowing that Δy = CΔx + DΔu, the variance of the output can be computed σ^2yy.
What are shape filters? When are they used?
When the input is not a white noise, we can say that we are dealing with a shape filter system subjected to a white noise as input. Therefore the input is a white noise, which is filtered to give the colored noise (the original PSD Φuu), which is then inputed to the system.
Therefore, Φuu = |Hf(ω)|^2 w, where w s the white nosie input. Given an input vector u with an assigned PSD Φuu, we find the transfer matrix of the space-filter Hf that returns the same PSD when subject to a white noise w.
The shape filters dynamics are:
xf_dot = Afxf + Bfn
u = Cfxf
And then an optimization problem is solved in order to find the coefficients A,B,C so that the error goes to zero. We don’t need to represent exactly this PSD in all our frequencies, but we just need to have a representation in the range where the output is significant.
What is the Rice formula?
When a structure is subject to random vvariation for a largue number of cycles, failure may come from fatigue damages. In order to compute the damage, we have to count the number of times the quantity crosses a certain threshhold.
For example, we have a statistically variable bending moment at the root of the aircraft. We want to understand how many times in a certain number of flights or flight hours the bending moment will cross a certain threshold. this is related to the variance but not in a straightforward way.
From a probabilistic POV, this works, but it’s based on a very questionable hypothesis: every event at every instant in time is considered independent of what happened at the previous instant. This is not exactly true.
For the Rice formula, we have to fefine a counting statistical process N(b), where b is our desired threshold. To do so, we use the step (heaviside) function H, so that y is equal to one if x passes the threshold or y is equal to 0 if it is lower than the threshold. The derivative of the heaviside function will be diferent from zero each time H is discontinuous. Then, taking the derivative (or rate independently from the direction of crossing) N(b,t1,t2), we can compute the number of threshold crossing.
Then, we calculate the expected value for the number of threshold crossings per unit time and ultimatelty we reach a formula where we can compute the number of upward crossings. The variance of the output signal x and its derivative x_dot are needed. Ultimately, we can calculate the probability of not crossing as the limit of (1-Ndt), where dt is equal to limit T/n, where T is the total intervals of time considered.
