Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
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Solving Dynamic Equation Using Combination of Both Trigonometric and Hyperbolic Cosine Functions for Approximating Acceleration

  • Quoc Do Kien (Department of Civil Engineering, Ho Chi Minh City University of Technology) ;
  • Phuoc Nguyen Trong (Department of Civil Engineering, Ho Chi Minh City University of Technology)
  • Published : 2005.01.01
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Abstract

This paper introduces a numerical method for integration of the linear and nonlinear differential dynamic equation of motion. The variation of acceleration in two time steps is approximated as a combination of both trigonometric cosine and hyperbolic cosine functions with weighted coefficient. From which all necessary formulae are elaborated for the direct integration of the governing equation. A number of linear and nonlinear dynamic problems with various degrees of freedom are analysed using both the suggested method and Newmark method for the comparison. The numerical results show high advantages and effectiveness of the new method.

Keywords

References

  1. Clough, R. W. and Penzien, J., 1993, Dynamics of Structures, McGraw-Hill
  2. Daniel, W. J., 1997, Trevor, The Subcycled Newmark Algorithm, Computational Mech., Springer-Verlag, No. 20, pp. 272-281 https://doi.org/10.1007/s004660050248
  3. Do Kien Quoc and Nguyen Trong Phuoc, 2000, 'A New Numerical Method for Resolving Nonlinear Dynamic Problems Approximating Acceleration by the Hyperbolic Cosine Function,' Proc. of the Inter. Coll. in Mech. of Solids, Fluids, Structures and Interactions, Vietnam
  4. Do Kien Quoc and Nguyen Trong Phuoc, 2002, 'Evaluation of Numerical Methods applied in Dynamic Problems using Energy Criteria,' Proc. of the 7th National Conference on Mech., Hanoi Hanoi
  5. Do Kien Quoc and Pham Thi Hai, 1999, 'A New Numerical Method for Resolving Nonlinear Vibrations Problem Approximating Acceleration by Cosine Function,' Proc. of the 6th National Conference on Mech. of Solids, Ha Noi
  6. Fung Tat Ching, 1998, 'Complex Time Step Newmark Methods with Controllable Numerical Dissipation,' Inter. J. for Numerical Methods in Eng., No. 41, pp. 65-93 https://doi.org/10.1002/(SICI)1097-0207(19980115)41:1<65::AID-NME270>3.0.CO;2-F
  7. Hahn, G. D., 1991, 'A Modified Euler Method for Dynamic Analyses,' Inter. J. for Num. Methods in Eng., No. 32, pp. 943-955 https://doi.org/10.1002/nme.1620320502
  8. Newmark Nathan M., 1959, A Method of Computation for Structural Dynamics, J. of the Eng. Mech. Division, Proc. of the ASCE, pp. 63-95
  9. Nguyen Trong Phuoc and Do Kien Quoc, 2003, 'A Study of the Stability of the Trigonometric and Hyperbolic Cosine Acceleration Methods in Structural Dynamic,' Proc. of the Inter. Symposium on Dynamics and Control, Ha Noi