بهبود روش انتگرال‌گیری دقیق مرتبه اول برای تحلیل دینامیکی سازه‌ها با معکوس‌سازی ماتریس حالت

For solving the dynamic equilibrium equation of structures, several second-order numerical methods have so far been proposed. In these algorithms, conditional stability, period elongation, amplitude error, appearance of spurious frequencies and dependency of the algorithms to the time steps are the crucial problems. Among the numerical methods, Newmark average acceleration algorithm, regardless of existence of spurious frequencies, is very popular in the structural dynamics due to its unconditionally stability status of the method. Recently, several first-order methods have been introduced for resolving the accuracy and stability issues. However, in these methods stability, accuracy and error in inversion of the state matrix are known as major issues. When the state matrix became singular or ill conditioned, numerical errors will occure in the computational process. Many of the available first-order methods were to improve the stability and accuracy and also to remove the error of inversion. Even though the introduced methods are conditionally stable, no investigation on errors, occuring during dynamic loading, has been reported for them. The main purpose of this paper is to utilize a specific decomposition method based on Singular Value Decomposition (SVD) for modifying PIM algorithm. Using the SVD inversion technique, the singularity problem of the state matrix has been resolved. In this paper, the modified method is called PIMS. As well, by applying the developed method for dynamic loading, the error of responses has been investigated. The results show that PIMS algorithm is stable and, comparing with secoend order Newmark and other available first order methods, has more accuracy.