الفهرس 
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Abstract
Abstract The second order ordinary differential equation of vibration is considered. First we deal with stochastic equation of undamped system under random initial conditions, then we study the variation of initial conditions on the two hinged frame. Then this study is followed by the solution stochastic equation under random excitation and random initial conditions. Finally, we study the stochastic equation of a damped system under random initial conditions and then under random excitation. As a random excitation, we use an earthquake model. The time domain technique is used to obtain the general solution of the above cases using the superposition method to evaluate the impulse response function. We use stochastic analysis to evaluate the statistical moments of the solution.