A finite extensibility nonlinear oscillator

M. FEBBO

APPLIED MATHEMATICS AND COMPUTATION

ELSEVIER SCIENCE INC

Año: 2011 vol. 217 p. 6464 - 6475

0096-3003

The dynamics of a finite extensibility nonlinear oscillator (FENO) is studied analytically bymeans of two different approaches: a generalized decomposition method (GDM) and a linearizedharmonic balance procedure (LHB). From both approaches, analytical approximationsto the frequency of oscillation and periodic solutions are obtained, which are validfor a large range of amplitudes of oscillation. Within the generalized decompositionmethod, two different versions are presented, which provide different kinds of approximateanalytical solutions. In the first version, it is shown that the truncation of the perturbationsolution up to the third order provides a remarkable degree of accuracy for almostthe whole range of amplitudes. The second version, which expands the nonlinear term inTaylor’s series around the equilibrium point, exhibits a little lower degree of accuracy, butit supplies an infinite series as the approximate solution. On the other hand, a linearizedharmonic balance method is also employed, and the comparison between the approximateperiod and the exact one (numerically calculated) is slightly better than that obtained byboth versions of the GDM. In general, the agreement between the results obtained bythe three methods and the exact solution (numerically integrated) for amplitudes (A)between 0 < A 6 0.9 is very good both for the period and the amplitude of oscillation. Forthe rest of the amplitude range (0.9 < A < 1), an exponentially large L2 error demonstratesthat all three approximations do not represent a good description for the FENO, and higherorder perturbation solutions are needed instead. As a complement, very accurate asymptoticrepresentations of the period are provided for the whole range of amplitudes ofoscillation.