The objective of the current paper is to provide an accurate distributed parameter model for random vibration analysis of multi-floor buildings. The Hamilton's principle is employed to derive the equations governing the dynamic behavior of the system as well as the related kinematic and natural boundary conditions. The natural frequency and mode shapes of the developed model are then extracted analytically and validated using finite element simulations. It is also observed that the predictions of the proposed model for the natural frequencies of the system is far more accurate than those of that of the discrete model available in the literature. Using a single mode approximation in the Lagrange equation, the partial differential equations of the motion are reduced to a single ordinary differential equation. Assuming a band limited white noise for the acceleration of the support, the random response specifications (such as expected value, autocorrelation, spectral density and mean square) of the system is calculated by making use of the random vibration theory. The qualitative and quantitative nature of the response characteristic are also analyzed to reveal the effects of different design parameters on the system's response. The suggested modeling approach in this paper may be employed for prediction of the dynamic behavior of more complex structures to different types of deterministic or random excitations. Also the provided analytical method for the random response calculation of the system can be utilized to make more informed decisions in the design process.

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