A High-Accurate Time Integration Method for Solving Structural Vibration Responses

A time integration method for the equations of motion is developed based on the Gauss implicit Runge-Kutta method to high-accurate solving the responses in structural vibration. The present method possesses the features of unconditional stability and self-starting and can achieve fourth-order accuracy in displacement, velocity, and acceleration simultaneously. The algorithm is a matrix form and no need to iterate in the calculation. The convergent accuracy is verified by a numerical example, and the effectiveness is also verified by solving the dynamic responses of a vibration isolation system and the vibration responses of a pylon structure with cyclic loads and earthquake loads.

[1]  J. Zhang.  A‐stable linear two‐step time integration methods with consistent starting and their equivalent single‐step methods in structural dynamics analysis , 2021, International Journal for Numerical Methods in Engineering.

[2]  Jie Zhang,et al.  A‐stable two‐step time integration methods with controllable numerical dissipation for structural dynamics , 2019, International Journal for Numerical Methods in Engineering.

[3]  K. Bathe,et al.  The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method , 2019, Computers & Structures.

[4]  Qingbin Li,et al.  A Robust and Efficient Composite Time Integration Algorithm for Nonlinear Structural Dynamic Analysis , 2015 .

[5]  Mohammad Rezaiee-Pajand,et al.  Nonlinear dynamic structural analysis using dynamic relaxation with zero damping , 2011 .

[6]  Severiano González-Pinto,et al.  A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems , 2010, ACM Trans. Math. Softw..

[7]  Basem S. Attili,et al.  The Hilber–Hughes–Taylor-α (HHT-α) method compared with an implicit Runge–Kutta for second-order systems , 2010, Int. J. Comput. Math..

[8]  K. Bathe,et al.  On a composite implicit time integration procedure for nonlinear dynamics , 2005 .

[9]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[10]  Kumar K. Tamma,et al.  The time dimension: A theory towards the evolution, classification, characterization and design of computational algorithms for transient/ dynamic applications , 2000 .

[11]  W. Zhong,et al.  A Precise Time Step Integration Method , 1994 .

[12]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[13]  C. W. Gear,et al.  Numerical Methods for Ordinary Differential Equations , 1989 .

[14]  T. J.R. Hughes,et al.  ANALYSIS OF TRANSIENT ALGORITHMS WITH PARTICULAR REFERENCE TO STABILITY BEHAVIOR. , 1983 .

[15]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[16]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[17]  John C. Houbolt,et al.  A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft , 1950 .