Abstract
In this paper, a procedure for the optimal design of multi-parametric nonlinear systems is presented which makes use of a parametric continuation strategy based on simple shooting method. Shooting method is used to determine the periodic solutions of the nonlinear system and multi-parametric continuation is then employed to trace the change in the system dynamics as the design parameters are varied. The information on the variation of system dynamics with the value of the parameter vector is then used to find out the exact parameter values for which the system attains the required response. This involves a multi-parametric optimisation procedure which is accomplished by the coupling of parameter continuation with different search algorithms. Genetic Algorithm as well as Gradient Search methods are coupled with parametric continuation to develop an optimisation scheme. Furthermore, in the coupling of continuation and Genetic Algorithm, a “norm-minimising” strategy is developed and made use of minimising the use of continuation. The optimisation procedure developed is applied to the Duffing oscillator for the minimisation of the system acceleration with nonlinear stiffness and damping coefficient as the parameters and the results are reported. It is also briefly indicated how the proposed method can be successfully used to tune nonlinear vibration absorbers.
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References
Ficken, F.: The continuation method for nonlinear functional equations. Comm. Pure Appl. Math. 4, 435–456 (1951)
Lahaye, M.E.: Solution of system of transcendental equations. Bull. Cl. Sci., Acad. R. Belg. 5, 805–822 (1948)
Davidenko, D.F.: On the approximate solution of systems of nonlinear equations. Ukr. Mat. Zh. 5, 196–206 (1953)
Morozov, N.F.: Nonlinear problems in theory of thin plates. Vestn. Leningr. Univ. 19, 100–124 (1958)
Kubicek, M., Marek, M.: Computational Methods in Bifurcation Theory and Dissipative Structures. Springer, New York (1983)
Seydel, R.: From Equilibrium to Chaos: Practical Stability and Bifurcation Analysis. Elsevier, New York (1988)
Grigolyuk, E.I., Shalashilin, V.I.: Problems of Nonlinear Deformation. Kluwer Academic, Dordrecht (1991)
Vannucci, P., Cochelin, B., Damil, N., Potier-Ferry, M.: An asymptotic numerical method to compute bifurcation branches. Int. J. Numer. Methods Eng. 41, 1365–1389 (1997)
Cochelin, B.: A path following technique via an asymptotic numerical method. Comput. Struct. 53, 1181–1192 (1994)
Noor, A.K., Peters, J.M.: Tracing post limit paths with reduced basis technique. Computer Methods in Applied Mechanics 28, 217–240 (1981)
Padmanabhan, C., Singh, R.: Analysis of periodically excited nonlinear systems by a parametric continuation method. J. Sound Vib. 184, 35–58 (1995)
Padmanabhan, C., Singh, R.: Dynamic analysis of a piecewise nonlinear system subjected to dual harmonic excitation using parametric continuation. J. Sound Vib. 184, 767–799 (1995)
Noah, S.T., Sundararajan, P.: Dynamics of forced nonlinear systems using shooting/arc length continuation method—application to rotor systems. ASME J. Vib. Acoust. 119, 9–20 (1997)
Shalashilin, V.I., Kuznetsov, E.B.: Parametric Continuation and Optimal Parameterization in Applied Mathematics and Mechanics. Kluwer Academic, Dordrecht (2003)
Burton, T.D.: Introduction to Dynamic System Analysis. McGraw-Hill, New York (1994)
Goldberg, D.E.: Genetic Algorithms in Search, Optimisation and Machine Learning. Addison-Wesley, Reading (1989)
Rao, S.S.: Engineering Optimization: Theory and Practice. New Age, New Delhi (1996)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Den Hartog, J.P.: Mechanical Vibrations. Dover, New York (1985)
Pipes, L.: Analysis of a nonlinear dynamic vibration absorber. J. Appl. Mech. 20, 515–518 (1953)
Hunt, J., Nissen, J.C.: The broadband dynamic vibration absorber. J. Sound Vib. 83, 573–578 (1982)
Asami, T., Nishihara, O.: Closed form exact solution to HINF optimization of dynamic vibration absorbers. J. Vib. Acoust. 125, 381–411 (2003)
Viguie, R., Kerschen, G.: Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology. J. Sound Vib. 326, 780–793 (2009)
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Balaram, B., Narayanan, M.D. & Rajendrakumar, P.K. Optimal design of multi-parametric nonlinear systems using a parametric continuation based Genetic Algorithm approach. Nonlinear Dyn 67, 2759–2777 (2012). https://doi.org/10.1007/s11071-011-0187-z
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DOI: https://doi.org/10.1007/s11071-011-0187-z