This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems.In this solution procedure,no smal...This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems.In this solution procedure,no small parameter is assumed.The harmonic residue of balance equation is separated in two parts at each step.The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement.The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again.Three kinds of different differential equations involving general,fractional and delay ordinary differential systems are given as numerical examples respectively.Highly accurate limited cycle frequency and amplitude are captured.The results match well with the exact solutions or numerical solutions for a wide range of control parameters.Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems.Moreover,the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.展开更多
Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-di...Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-displacement relation is derived from the large deflection theory of thin-walled shells.A set of nonlinear differential equations describing the large deflection vibration are formulated by the Lagrange equation and the assumption of small strains.Steady-state responses of the system are predicted via the harmonic balance method with the arc length continuation,and their stabilities are determined via the modified sorting method.The effects of excitations on the steady-state responses are analyzed.The results reveal a crucial role played by the phase difference in the structural response,and the phase difference can effectively control the amplitude of vibration.展开更多
基金supported by the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2011AQ022 and ZR2012AL03)
文摘This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems.In this solution procedure,no small parameter is assumed.The harmonic residue of balance equation is separated in two parts at each step.The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement.The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again.Three kinds of different differential equations involving general,fractional and delay ordinary differential systems are given as numerical examples respectively.Highly accurate limited cycle frequency and amplitude are captured.The results match well with the exact solutions or numerical solutions for a wide range of control parameters.Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems.Moreover,the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11672069,11872145,11872159,12172086,and 12101106).
文摘Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-displacement relation is derived from the large deflection theory of thin-walled shells.A set of nonlinear differential equations describing the large deflection vibration are formulated by the Lagrange equation and the assumption of small strains.Steady-state responses of the system are predicted via the harmonic balance method with the arc length continuation,and their stabilities are determined via the modified sorting method.The effects of excitations on the steady-state responses are analyzed.The results reveal a crucial role played by the phase difference in the structural response,and the phase difference can effectively control the amplitude of vibration.
