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Extension of Poincaré’s nonlinear oscillation theory to continuum mechanics. II. Applications. (English) Zbl 0657.73042

In part I [ibid. 8, 1-10 (1987; Zbl 0611.73033)] we suggested a method of direct perturbation of partial differential equation and weighted integration to calculate the periodic solution for continuum mechanics. In this paper, by using the above method we calculate the resonant and nonresonant periodic solutions for beams with fixed span and different boundary conditions and the resonant periodic solution of square plates under the action of concentrated periodic loads. Besides, the influences of non-principal modes upon the periodic solution and of static loads upon amplitude-frequency curves are given.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
35B20 Perturbations in context of PDEs
35B10 Periodic solutions to PDEs

Citations:

Zbl 0611.73033
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References:

[1] Huo Lin-chun and Li Li, Extension of Poincaré’s nonlinear oscillation theory to continuum mechanics (I)–Basic theory and method,Appl. Math. and Mech.,8, 1 (1987). · Zbl 0611.73033 · doi:10.1007/BF02014493
[2] Keller, J.B. and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations,Comm. Pure Appl. Math.,19 (1966). · Zbl 0284.35004
[3] Chien Wei-zang,Theory of Singular Perturbations and Its Applications in Mechanics, Science Press (1981). (in Chinese)
[4] Kauderer, G.,Nonlinear mechanics, IIL, Moscow (1961). (in Russian)
[5] Volomir, A.S.,Nonlinear Dynamics of Plates and Shells, Moscow (1972). (in Russian)
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