Non-stationary random vibration analysis of multi degree systems using auto-covariance orthogonal decomposition

An algorithm that integrates Karhunen–Loeve expansion (KLE) and finite element method (FEM) is proposed to carry out random vibration analysis of complex dynamic systems excited by stationary or non-stationary random processes. In KLE, the auto-covariance function of random process is discretized using orthogonal basis functions. During the response calculations, the eigenvectors of KLE are applied as forcing functions. Three methods are proposed to carry out the random vibration analysis termed as, Method 1A, Method 1B and Method 2. In Method 1A and Method 1B, the basis functions are chosen such that they include multiples of complete or half-cosine and sine functions over the selected time. In Method 2, the basis functions are chosen to be simple piecewise constants. The proposed algorithm is applied to a 2DOF system, a cantilever beam and a stiffened panel for both stationary and non-stationary excitations. Results show that three methods can describe the statistics of the dynamic response with sufficient accuracy. However, Method 1A results have a relatively larger error than that for Method 1B and Method 2 during initial transient time. The Method 2 results have an excellent agreement with analytical results. Moreover, the runtime of Method 2 algorithm is significantly less than both Method 1A and Method 1B algorithms even though its usage results in an increase in the number of KLE terms. Furthermore, Method 2, unlike Method 1A and 1B, neither yields negative and/or infinite eigenvalues for the auto-covariance function nor large inaccuracies.