On the elliptic balance method

Mathematics and Mechanics of Solids

Volumn 8, Issue 3, 2003, Pages 263-279

  Elías Zúñiga, Alex ^a  

^a SCHOOL OF ENGINEERING AND SCIENCES   (Mexico)

Author keywords

Elliptic balance; Jacobian elliptic functions; Nonlinear systems; Perturbation methods; Simple shear

Indexed keywords

DAMPING; DIFFERENTIAL EQUATIONS; NONLINEAR SYSTEMS; PERTURBATION TECHNIQUES; PLATES (STRUCTURAL COMPONENTS);

ELLIPTIC BALANCE METHOD (EBM);

MECHANICAL ENGINEERING;

EID: 0038717296     PISSN: 10812865     EISSN: None     Source Type: Journal    
DOI: 10.1177/1081286503008003002     Document Type: Article

References (31)

1. Zúñiga, A.E. and Beatty, M.F.: On the application of the elliptic balance method to two degree of freedom, undamped, homogeneous systems having cubic nonlinearities. In preparation.
2. Hsu, C.S.: On the application of elliptic functions in nonlinear forced oscillations. Q. Appl. Math., 17, 393-407 (1960).
3. Iwan, WD.: On defining equivalent systems for certain ordinary nonlinear differential equations. Int. J. Nonlinear Mech., 4, 325-334 (1969).
4. Hsu, C.S.: Some simple exact periodic responses for a nonlinear system under parametric excitation. J. Appl. Mech. 1135-1137 (1974).
5. Zúñiga, A.E. and Beatty, M.F.: Método de Balanceo Elíptico Aplicado a la Solución de la Ecuación Amortiguada de Duffing con Término Forzante de Tipo Elíptico. Avances en Ingeniería Mecánica. Memoria del ler Congreso Internacional de Ingeniería Electromecánica y de Systemas, IPN México D.F., 228-234 (1996).
6. Detinko, F.M.: Applications of Jacobian elliptic functions to the analysis of forced vibrations of a nonlinear conservative system. Mech. Solids, 31(4), 5-8 (1996).
7. Barkham, P. G. D. and Soudack, A. C.: An extension to the method of Kryloff and Bogolinboff. Int. J. Control, 10, 377-392 (1969).
8. Soudack, A. C. and Barkham, P. G. D.: On the transient solution of the unforced Duffing equation with large damping. Int. J. Control, 13, 767-769 (1971).
9. Christopher, P. A. T.: An approximate solution to a strongly nonlinear, second order, differential equation. Int. J. Control, 17, 597-608 (1973).
10. Christopher, P. A. T. and Brocklehurst, A.: A generalized form of an approximate solution to a strongly nonlinear, second order, differential equation. Int. J. Control, 19, 831-839 (1974).
11. Bravo Yuste, S. and Diaz Bejarano, J.: Construction of approximate analytical solutions to a new class of nonlinear oscillator equations. J. Sound Vib., 110, 347-350 (1986).
12. Bravo Yuste, S. and Diaz Bejarano, J.: Amplitude decay of damped nonlinear oscillators studied with Jacobian elliptic functions. J. Sound Vib., 114, 33-44 (1987).
13. Bravo Yuste, S. and Diaz Bejarano, J.: Extension and improvement to the Krylov-Bogoliubov methods using elliptic functions. Int. J. Control, 49, 1127-1141 (1989).
14. 4)Ẋ = 0. Int. J. Nonlinear Mech., 25, 663-675 (1990).
15. Coppola, T. and Rand, R. H.: Averaging using elliptic functions: approximation of limit cycles. Acta Mech., 81, 125-142 (1990).
16. Bravo Yuste, S. and Diaz Bejarano, J.: Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions. J. Sound Vib., 139, 151-163 (1990).
17. Bravo Yuste, S.: Comments on the method of harmonic balance in which Jacobi elliptic functions are used. J. Sound Vib., 145, 381-390 (1991).
18. Bravo Yuste, S.: Quasi-pure-cubic oscillators studied using a Krylov-Bogoliubov Method. J. Sound Vib., 158, 267-275 (1992).
19. Bravo Yuste, S.: Cubication of nonlinear oscillators using the principle of harmonic balance. Int. J. Nonlinear Mech., 27, 347-356 (1992).
20. Chen, S.H. and Cheung, Y.K.: An elliptic perturbation method for certain strongly nonlinear oscillators. J. Sound Vib., 192, 453-464 (1996).
21. Chen, S.H. and Cheung, Y.K.: An elliptic Lindstedt-Poincaré method for analysis of certain strongly nonlinear oscillators. Nonlinear Dyn., 12, 199-213 (1997).
22. Chen, S.H., Yang, X.M., and Cheung, YK.: Periodic solutions of strongly quadratic nonlinear oscillators by the elliptic perturbation method. J. Sound Vib., 212, 771-780 (1998).
23. Cveticanin, L.: Analytical methods for solving strongly nonlinear differential equations. J. Sound Vib., 214, 325-338 (1998).
24. Bejarano, J.D. and Margallo, J.O.: The greatest number of limit cycles of the generalized Rayleigh-Lienard oscillator. J. Sound Vib., 221, 133-142 (1999).
25. Chen, S.H. and Yang, X.M.: Periodic solutions of strongly quadratic nonlinear oscillators by the elliptic Lindstedt-Poincaré method. J. Sound Vib., 227, 1109-1118 (1999).
26. Beatty, M.F.: Stability of a body supported by a simple vehicular shear suspension system. Int. J. Non-Linear Mech., 24, 65-77 (1989).
27. Zúñiga, A.E.: Absorber Control of the Finite Amplitude Nonlinear Vibrations of a Simple Shear Suspension System. PhD Dissertation, University of Nebraska-Lincoln, 1994.
28. Nayfeh, A.H. and Mook, D.T.: Nonlinear Oscillations, Wiley, New York (1973).
29. Sanders, J. A. and Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York (1985).
30. Byrd, P.F. and Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin, (1953).
31. Meirovitch, L.: Elements of Vibration Analysis, McGraw-Hill, New York, (1986).