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A Unique Formulation of Piecewise Exact Method to Analyze a Nonlinear Spring System under Harmonic Excitation

Published online by Cambridge University Press:  01 December 2014

P.-S. Xie  and
P.-J. Shih
P.-S. Xie
Affiliation:
School of Civil and Transportation Engineering, Guangdong University of Technology, Guangzhou, China
P.-J. Shih*
Affiliation:
Department of Civil and Environmental Engineering, National University of Kaohsiung, Kaohsiung, Taiwan
*
* Corresponding author (pjshih@nuk.edu.tw
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Abstract

This paper introduces a unique, efficient, and exact formulation for solving a single-degree-of-freedom system with nonlinear stiffness under a harmonic loading. This formulation is one kind of the piecewise exact method, and its benefit lies in providing the closed-form exact solution in each displacement segment. Since the exact solution is given in each segment, the continuity between two segments can be confirmed. Consequently, no instability errors affect the analysis. To determine the exact solutions in these segments, this research develops a technique that shifts the equilibrium points of the piecewise linear segments, which are discretized from a nonlinear stiffness curve, to new equilibrium points in order to satisfy the typical linear exact solution. Thus, positive- and negative-stiffness linear segments can be solved with this technique. This formulation saves roughly 60% of the calculation time (error < 10−10) as compared to the numerical approximation.

Keywords

Piecewise exact methodNonlinear springStep-by-step procedureThe newmark explicit method
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 3 , June 2015 , pp. 337 - 344
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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References

1.Clough, R. W. and Penzien, J., Dynamics of Structures, McGraw-Hill Inc., New York, U.S., pp. 111–131 (1993).Google Scholar
2.Liu, J. L., “Exact Solution for Dynamic Response of Multi-Degree-of-Freedom Bilinear Hysteretic System,” Journal of Engineering & Mechanics, 129, pp. 1342–1350 (2003).Google Scholar
3.Aydinoglu, M. N. and Fahjan, Y. M., “A Unified Formulation of the Piecewise Exact Method for Inelastic Seismic Demand Analysis Including the P-Delta Effect,” Earthquake Engineering & Structural Dynamics, 32, pp. 871–890 (2003).Google Scholar
4.Shih, P.-J. and Shih, W.-P., “Impact Dynamics of Vibratory Microprobe for Microcoordinate Measurement,” Journal Applied Physics, 101, p. 113516 (2007).Google Scholar
5.Shih, P.-J., “Tip-Jump Response of an Amplitude-Modulated Atomic Force Microscope,” Sensors, 12, pp. 6666–6684 (2012).Google Scholar
6.Newmark, N. M., “A Method of Computation for Structural Dynamics,” Journal of the Engineering Mechanics Division, 85, pp. 67–94 (1959).CrossRefGoogle Scholar
7.Wilson, E. L., Farhoomand, I. and Bathe, K. J., “Nonlinear Dynamic Analysis of Complex Structures,” Earthquake Engineering & Structural Dynamics, 1, pp. 241–252 (1973).Google Scholar
8.Hughes, T. J. R., The Finite Element Method-Linear Static and Dynamic Finite Element Analysis, Prentice-Hall Inc., New Jersey, U.S. (1987).Google Scholar
9.Chang, S. Y., “A Technique for Overcoming Load Discontinuity in Using Newmark Method,” Journal of Sound and Vibration, 304, pp. 556–569 (2007).CrossRefGoogle Scholar
10.Chang, S. Y., “Accurate Integration of Nonlinear Systems Using Newmark Explicit Method,” Journal of Mechanical, 25, pp. 289–297 (2009).Google Scholar
11.Chang, S. Y., “Accurate Evaluation of Newmark Method Referring to Theoretical Solutions,” Journal of Earthquake Engineering, 12, pp. 1–16 (2008).CrossRefGoogle Scholar
12.Hilber, H. M., Hughes, Thomas J. R. and Taylor, R. L., “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,” Earthquake Engineering & Structural Dynamics, 5, pp. 283–292 (1977).CrossRefGoogle Scholar
13.Bathe, K. J. and Irfan Baig, M. M., “On a Composite Implicit Time Integration Procedure for Nonlinear Dynamics,” Computers & Structures, 83, pp. 2513–2524 (2005).Google Scholar
14.Chopra, A. K., Dynamics of Structures: Theory and Applications to Earthquake Engineering, 3rd Edition, Prentice Hall, Englewood Cliffs, New Jersey, U.S. (2012).Google Scholar