3:45 PM - 4:00 PM
[SSS10-08] Proposal of a time integration method for computing the equations of motion for multi-degree-of-freedom systems with damping terms based on the theoretical solutions
Keywords:multi-degree-of-freedom, equation of motion, impulse response function, Duhamel's integral
One of the methods of seismic analysis is to discretize the analysis domain, treat it as a multi-degree-of-freedom (MDOF) system, and compute the dynamic response to seismic motion. For applying this method to a system approximated by a semi-infinite domain, such as the ground, the degrees of freedom become large, so it takes a long time to compute the time integration. Therefore, time integration methods solving the equations of motion for MDOF systems more rapidly and more accurately are required.
In recent years, methods have been proposed to compute the time integration of differential equations based on theoretical solutions (e.g., Al-Mohy et al., 2015; Besten, 2022). These methods give highly accurate results and compute without convergence calculations of simultaneous equations. However, these methods target equations of motion for MDOF systems without damping terms, which are readily to solve the theoretical solutions. Although the theoretical solutions of the equations with damping terms are difficult to obtain, we have found that the theoretical solutions of the homogeneous equations, which are the equations with damping terms but no external force terms, are readily to obtain if the damping mechanism is Rayleigh damping. This study proposes a method for computing the response to arbitrary external force based on the theoretical solutions for inhomogeneous equations with damping terms by Duhamel's integral with the impulse response functions, which is computed using the theoretical solutions of homogeneous equations if the system is linear. In addition, this study adopts the method of Tomobe et al. (2024), which combines the time integration based on the theoretical solutions and digital filtering techniques to remove the spurious high frequencies component, for computing the impulse response functions.
The proposed method is used to analyze a one-dimensional elastic wave propagation problem. To evaluate the performance of the proposed method, calculation results of the proposed method are compared to those of Newmark-beta method (Newmark, 1959). The material constants were set to the parameters used in Sharma et al. (2022). The benchmark demonstrates the results of the proposed method agree with those of Newmark-beta method, even though the proposed method is not required to solve simultaneous equations. Therefore, this method may be useful for a faster calculation method of the time integration for the linear and Rayleigh damping systems. However, when the digital filtering do not apply to the proposed method, the results of the proposed method do not agree with those of Newmark-beta method. Although Rayleigh damping strongly attenuates the high-frequency component, the results highlight that the proposed method must be combined with digital filtering.
This study proposed a new method for computing the time integration of equations of motion for MDOF systems with damping terms based on the theoretical solutions. The benchmark demonstrated that the results of the proposed method agree with those of Newmark-beta method for the linear and Rayleigh damping systems. In future research, introducing the numerically absorbing boundary to the proposed method needs further investigation. In addition, obtaining a theoretical solution of inhomogeneous equations is also necessitated to apply the non-linear problems.
In recent years, methods have been proposed to compute the time integration of differential equations based on theoretical solutions (e.g., Al-Mohy et al., 2015; Besten, 2022). These methods give highly accurate results and compute without convergence calculations of simultaneous equations. However, these methods target equations of motion for MDOF systems without damping terms, which are readily to solve the theoretical solutions. Although the theoretical solutions of the equations with damping terms are difficult to obtain, we have found that the theoretical solutions of the homogeneous equations, which are the equations with damping terms but no external force terms, are readily to obtain if the damping mechanism is Rayleigh damping. This study proposes a method for computing the response to arbitrary external force based on the theoretical solutions for inhomogeneous equations with damping terms by Duhamel's integral with the impulse response functions, which is computed using the theoretical solutions of homogeneous equations if the system is linear. In addition, this study adopts the method of Tomobe et al. (2024), which combines the time integration based on the theoretical solutions and digital filtering techniques to remove the spurious high frequencies component, for computing the impulse response functions.
The proposed method is used to analyze a one-dimensional elastic wave propagation problem. To evaluate the performance of the proposed method, calculation results of the proposed method are compared to those of Newmark-beta method (Newmark, 1959). The material constants were set to the parameters used in Sharma et al. (2022). The benchmark demonstrates the results of the proposed method agree with those of Newmark-beta method, even though the proposed method is not required to solve simultaneous equations. Therefore, this method may be useful for a faster calculation method of the time integration for the linear and Rayleigh damping systems. However, when the digital filtering do not apply to the proposed method, the results of the proposed method do not agree with those of Newmark-beta method. Although Rayleigh damping strongly attenuates the high-frequency component, the results highlight that the proposed method must be combined with digital filtering.
This study proposed a new method for computing the time integration of equations of motion for MDOF systems with damping terms based on the theoretical solutions. The benchmark demonstrated that the results of the proposed method agree with those of Newmark-beta method for the linear and Rayleigh damping systems. In future research, introducing the numerically absorbing boundary to the proposed method needs further investigation. In addition, obtaining a theoretical solution of inhomogeneous equations is also necessitated to apply the non-linear problems.