How to implement a Monte Carlo Simulation for studying uncertainty in dynamic stiffness?

Question

I have a dynamic mechanical system of the form: $$\mathbf{M{\ddot q}+{\Omega_c}G{\dot q}+{Kq}=F(t)}$$

and I need to introduce some uncertainty for the $K$ parameter, which is a symetric matrix of $q$ degrees of freedom.

I did a bit of research of Monte Carlo simulation but I´m a bit confused of how different textbooks approach the problem.

For example , my actual stiffness value is of $1e8$, and I wish to generate values uniformly distributed between $K_l = 1e6$ and $K_u = 1e12$.

I found that I can do this by applying: $$\lambda_i = \mu_{\lambda_i} + \epsilon_i$$ where $\mu_{\lambda_i}$ is the mean value of $\lambda_i$ and $\epsilon$ is a random parameter, and also $$\sigma_{\lambda_i}^2 = E[{(\lambda_i-\mu_{\lambda_i})^2}]$$

I would be using Matlab for running the simulation , with the command rand I can generate a randomly distributed numbers between 0 and 1.

The problem I find with this is that I don't know know to input the boundaries of the $K$ stiffnes values I want.

Then I found another text book that explains how to generate random variables using: $$ X_i = A + (B-A)R_i$$ Where $R_i$ is the random number which can be generated by Matlab with rand command to generate a new random variable $X_i$, with lower and upper limits $A$ and $B$. So in matlab I would do a for loop like this:

for i=1:num
    Ki=Kl+(Ku-Kl)*rand;
end

My main question would actually be, what is the difference between these two methods and for the first one, would $\epsilon$ just be a random 0-1 number adding to the average designated value?

Thanks

Answer 1

My advice would be to forget about the mathematical theory and start thinking about what the engineering means. It's hard to think of any situation where the stiffness of a sensibly designed structure might vary "randomly" by a factor of a million, unless all the stiffness values are so high that the response is very insensitive to the stiffness value - and in that case, why bother doing a simulation at all?

Changing the stiffness of a dynamic system by a factor of 1 million will change the frequency of the fundamental modes by a factor of about 1 thousand. That isn't "a design" in any sense of the words that I can recognise in real life engineering.

Think about what causes your stiffness to change - geometrical tolerances, unknown material properties, poorly controlled nonlinear behaviour (e.g. buckling, plasticity, creep), etc. Then use those physical parameters to calculate the stiffness for each simulation point in your Monte Carlo study.

Note that this approach will change the different entries in the $q$-degrees of freedom stiffness matrix by different amounts, and that is probably much more realistic than multiplying the whole matrix by one scalar factor.