Nonstationary random vibration of one-degree and multidegree of freedom systems

Abstract

In ref. [1] for nonstationary random processes such as $$\xi {\text{(}}t{\text{) = }}\smallint _{{\text{R}}_{\text{1}} } f(t,\omega )\exp [j\omega t]Z(d\omega ),{\text{ }}j = \sqrt { - 1} $$ we defined the spectral density $$S_\xi (t,{\text{ }}\omega ) = |f(t,{\text{ }}\omega )|^2 S(\omega )$$ where Z(A)is an orthogonal random measure, S(ω)is the spectral density of the stationary random process $$\xi _0 (t) = \smallint _{R_1 } \exp [j\omega t]Z(d\omega )$$ f(t, ω)is a complex function with two real variables, it is called modulation function and it is satisfied by $$\smallint _{R_1 } |f(t,\omega )|^2 S(\omega )d\omega< \infty $$ In ref. [1] we obtained the main results: for the linear dynamic systems which is described by such equations $$a_0 (t)\frac{{d^n y}}{{dt^n }} + a_1 (t)\frac{{d^{n - 1} y}}{{dt^{n - 1} }} + ... + a_z (t)y = \xi (t)$$ we obtained under certain conditions that $$\begin{gathered} y(t) = \smallint _{R_1 } K(t,\omega )\exp [j\omega t]Zd(\omega ) \hfill \\ Ey(t)\overline {y(s)} \smallint _{R_1 } K(t,\omega )\overline {K(s,\omega )} \exp [j\omega (t - s)]S(\omega )d\omega \hfill \\ E|y(t)|^2 = \smallint _{R_1 } K(t,\omega )|^2 S(\omega )d\omega \hfill \\ S_y (t,\omega ) = |K(t,\omega )|^2 S(\omega ) \hfill \\ where \hfill \\ K(t,\omega ) = \smallint _{ - \infty } ^t W(t,\tau )f(\tau ,\omega )\exp [ - j\omega (t - \tau )]d\tau \hfill \\ \end{gathered} $$ here W(t, τ) is the response function of the systems to the unit pulse. This paper is continued on ref. [1]. We obtain some results for the one-degree and multidegree of freedom systems.