Response of Aging Linear Systems to Ergodic Random Input

Response of an aging linear system exposed from a certain age $t_0$ to ergodic random input is analyzed. It is shown that the response, while non‐stationary with respect to time and age, is stationary and ergodic with regard to the birth time τ (the time when the system was built). Consequently, the instantaneous statistical characteristics of all possible response realizations at a chosen age, t, may be determined as the characteristics of the response at age t (and at a fixed exposure age, $t_0$) as the birth time τ is varied, i.e., as the input history is shifted in time against the instant when the system was built. Based on this new idea, the spectral method is generalized for aging systems, using a frequency response function and a spectral density of response that depend on both the current age t and the age $t_0$ when the exposure begins. The relation between the spectral densities of input and response is algebraic, similar to the case of stationary response of nonaging systems. For the special case of non‐stationary response of nonaging systems, the proposed new method is simpler than the existing methods. The new method can be applied, e.g., to shrinkage stresses in an aging linearly viscoelastic structure (a concrete structure) exposed to relative humidity fluctuations of weather. A simple illustrative example is solved in a closed form. Another possible application is earthquake motion of a structure undergoing progressive damage, provided the problem is approximated as linear.