Acta Mechanica Solida Sinica

, Volume 19, Issue 4, pp 289–296

# A predict-correct numerical integration scheme for solving nonlinear dynamic equations

• Jianping Fan
• Tao Huang
• Chak-yin Tang
• Cheng Wang
Article

## Abstract

A new numerical integration scheme incorporating a predict-correct algorithm for solving the nonlinear dynamic systems was proposed in this paper. A nonlinear dynamic system governed by the equation = F (v, t) was transformed into the form as = Hv + f (v, t). The nonlinear part f (v, t) was then expanded by Taylor series and only the first-order term retained in the polynomial. Utilizing the theory of linear differential equation and the precise time-integration method, an exact solution for linearizing equation was obtained. In order to find the solution of the original system, a third-order interpolation polynomial of v was used and an equivalent nonlinear ordinary differential equation was regenerated. With a predicted solution as an initial value and an iteration scheme, a corrected result was achieved. Since the error caused by linearization could be eliminated in the correction process, the accuracy of calculation was improved greatly. Three engineering scenarios were used to assess the accuracy and reliability of the proposed method and the results were satisfactory.

## Key words

nonlinear dynamics direct integration method internal resonance saturation phenomenon jumping phenomenon bifurcation

## References

1. [1]
Lu H.X., Yu H.J. and Qiu C.H., Direct integration methods with integral model for dynamic systems, Applied Mathematics and Mechanics, Vol.22, No.2, 2001, 151–156.
2. [2]
Zhong W.X. and Yang Z.S., On the computation of the main eigen-pairs of the continuous-time linear quadratic control problem, Applied Mathematics and Mechanics, Vol.12, No.1, 1991, 45–50.
3. [3]
Zhong W.X., On precise time-integration method for structural dynamics, Journal of Dalian University of Technology, Vol.34, No.2, 1994, 131–136.
4. [4]
Zheng Z.C., Shen S. and Su Z.X., Accurate solution of nonlinear dynamic systems by numerical integration method, Acta Mechanica Sinica, Vol.35, No.3, 2003, 284–295.Google Scholar
5. [5]
Zhang X.N. and Jiang J.S., The precise integration algorithm for nonlinear dynamic equations of structures, Chinese Journal of Applied Mechanics, Vol.17, No.4, 2000, 164–168.Google Scholar
6. [6]
Qiu C.H., Lu H.X. and Zhong W.X., On segmented-direct-integration method for nonlinear dynamics equations, Acta Mechanica Sinica, Vol.34, No.3, 2002, 369–378.Google Scholar
7. [7]
Liu Y.Z. and Chen L.Q., Nonlinear Vibrations, Beijing: Higher Education Press, 2001.Google Scholar
8. [8]
Nayfeh A.H. and Moot D.T., Nonlinear Oscillations, New York: Wiley-Interscience, 1979.Google Scholar

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2006

## Authors and Affiliations

• Jianping Fan
• 1
• Tao Huang
• 1
• Chak-yin Tang
• 2
• Cheng Wang
• 1
1. 1.Institute of Engineering Calculation & SimulationHuazhong University of Science & TechnologyWuhanChina
2. 2.Department of Industrial and Systems EngineeringThe Hong Kong Polytechnic UniversityHong KongChina