Determination of the exact mode frequencies of multi-storey structures by state-space method and a comparison with mode superposition method

In this paper, a comparative research obtaining the time-consuming exact solution and approximate solution of linearly damped linear frame buildings has been carried out. State-space method and mode superposition method are respectively utilized to analyze the exact and approximate vibration solution of frame buildings. In mode su-perposition method, the proportional damping matrix of frame buildings has been constructed in two different schemes, such as modal combination of mass with stiff-ness matrices (Rayleigh) and disregarding the off-diagonal elements of non-classical damping (DOEA) matrix. Meanwhile, the damping matrix in state-space method is non-proportional and constructs in exact circumstance. By imposing different initial conditions on the system, it has been proved that the transient vibration is mostly greater than the steady-state vibration. These observations have individually been investigated to render the approximate response as close as possible to the exact re-sponse. The findings of current research indicate that the most suitable parameter which brings closer the approximate response to the exact response is DOEA ap-proach. In addition, a MATLAB code, which solves the equation of exact and approx-imate method, is generated in this research.