Complex modal analysis of random vibrations

Introduction A modal analysis of random vibrations is developed in this Note. In contrast to real modal analysis, which is valid only for the classical damping case, complex analysis is valid for both classical and nonclassical damping cases. It is a time domain analysis most suitable for finding the covariances and correlation functions of the random responses. Analysis Consider a general n degree-of-freedom linear system, with the following governing differential equation: mx + ex + kx = w (t) (1) where m, c, and k are assumed to be nxn real symmetrical positive-definite matrices and w(t) a stationary white noise excitation, with the following properties: E [ w ( t ) ] = 0, E[w(t)w(t + T)] = 2irDd(T) where D is the autospectrum matrix of w(t).' The In eigenvalues /?/ and their corresponding eigenvectors Uj can be found by standard methods. When damping is below critical, the/?/ and u\ all appear in conjugate pairs. With them we can construct an n x In complex modal matrix, u= [ u l . . M 2 n ] and a In x 2n complex modal matrix, U=[Ul...U2a] = [Pu U] where P is an eigenvalue matrix, i.e., P = diag[pf-] By introducing state variables x and x, Eq. (1) can be written as