Linear Stability Analysis
The Lyapunov exponent for the stability of the second moments of a linearized reference solution can be determined by the Ito analysis. The nonlinear stiffness matrix in Equation (46) can be expanded into an asymptotic series with respect to a static loading condition. Under the assumption that the fluctuating part is small enough this series can be truncated after the linear term:
My + Cy + (K (xstat) + f(t)Ki) y = 0. (55)
This equation of motion is projected into a subspace of dimension m and then transformed into its state space description analogous to Equation (51):
z = [A + Bf(t)] z, (56)
where the coefficient matrices A and B are constant. The fluctuating part of the loading function is assumed to be Gaussian white noise. Then Equation (56) represents a first order stochastic differential equation. For this system the Lyapunov exponent X2 for the second moments can be easily derived by applying the Ito calculus (e. g. Soong and Grigoriu, 1992; Lin and Cai, 1995).
The Lyapunov exponents for almost sure stability can be approximated for linear SDOF-systems analytically (Lin and Cai, 1995):
where &>0 is the eigenfrequency, Z0 is the modal damping ratio and Sff is the power spectral density of the white noise excitation. The Lyapunov exponent X2 for the second moments can be calculated with
By exploiting this, the Lyapunov exponent for the samples can be appproximated from the second moment exponent according to
^ _ Л2 Zom
~ ~4 2~
This equation can also be applied on MDOF-systems, it should be mentioned that the term ~Z0m0 corresponds then to the Lyapunov exponent of the system without random parametric excitation.