Linear Systems Under Gaussian White Noise Excitation: Exact Closed-Form Solutions

In Chap. 3 , the Wiener path integral formulation developed in Chap. 2 for first-order stochastic differential equations is extended to treat second-order stochastic differential equations modeling the dynamics of stochastically excited linear multi-degree-of-freedom structural systems. Specifically, considering Gaussian white noise excitation, it is shown that higher than second variations in the path integral functional expansion vanish. Thus, retaining only the first term (most probable path approximation) yields the exact response joint transition Gaussian probability density function. Two illustrative examples pertaining to a stochastically excited 2-degree-of-freedom linear oscillator and to a bending beam with Young’s modulus modeled as a stochastic field are considered to demonstrate the exact nature of the closed-form solutions.