Resonance of a nonlinear forced system with two-frequency parametric and self-excitations
Introduction
There are many natural phenomena in which excited parametric and self-excited vibrations interact with one another. Examples are flow-induced vibrations and vibrations in forced rotor systems. However, some parametric excitations may contain multiple periodic components whose frequencies are not multiples of one another. It is worth to mention that complex dynamic behaviour has been observed when some of the excitation frequencies are closed to one another.
The responses of single-degree-of-freedom nonlinear excited systems to parametric excitations with single frequency have been investigated in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Yano [1] investigated some nonlinear models described by typical excitation mechanisms, in which the self-excitation was investigated by a nonlinear resistance of van der Pol type and parametric excitation contains quadratic and cubic nonlinearities. Both principal and fundamental parametric resonance cases were discussed in comparison to the analysis of linear modeling. Kotera and Yano [2] dealt with a van der Pol-Mathieu type equation with cubic nonlinearities in the restoring force, which described a beam subjected to a periodic axial force and simultaneously to a flow-induced vibration. Periodic solutions in the regions of principal and fundamental parametric resonance cases were approximated by the sum of two frequency components and a stability criterion of periodic solutions was established. Kojima et al. [3] considered second order superharmonic and one-half order subharmonic resonance cases in the vibration of a beam with a mass subjected to an alternating electromagnetic force. In Zavodney and Nayfeh [4], [5] analyzed the responses of fundamental and principal resonance cases for one-degree-of-freedom system with quadratic and cubic nonlinearities. In the work of Chen and Langford [6], a generic classification of steady-state responses for the nonlinear Mathieu equation was investigated by qualitative methods of catastrophe theory and equivalent singularity theory. In reference [7], the authors are studied the principal resonance of a nonlinear systems with two-frequency parametric and self-excitations and free form any external excitations. They carried out the results in the case in which the parametric excitation terms with close frequencies.
In references [8], [9], [10] Abdelhafez at first concerned with the solution of one-degree-of freedom system with quadratic and cubic nonlinearities to the amplitude modulated excitation whose carrier frequency is much higher than the natural frequency of the system. In second one, two methods (the multiple scales and the generalized synchronization) are used to investigate second-order approximate analytic solution of external and parametric excited one-degree-of-freedom system with quadratic, cubic and quratic nonlinearities. In addition, bifurcation diagram is constructed under the interaction of external and parametric excitations. The third paper dealt with certain forms of nonlinear oscillations of a vibratory system with quadratic, cubic and quartic nonlinearities subjected to a sinusoidal excitation that involves multiple frequencies.
In the monograph by Schmidt and Tondl [11] there are some discussions on the interactions between parametric and self-excitations for very similar nonlinear systems but these systems were free from any excitation forces. In [12] without considering any parametric excitations Nayfeh used the method of multiple scales to analyze the response of single-degree-of-freedom systems with cubic non-linearities to excitations that involve multiple frequencies.
The present paper deals with a more general case than given in by Lu [7]. That is, in this work we concerned with the solution of the second order differential equation describes a nonlinear system with two-frequency parametric and self-excitations but in addition this nonlinear system is under an external excitation force. The system we mean is governed by the following equationwhere ε is a small parameter, μ,ν≠0 are constants, α, β, Ω, Ω1, Ω2 are positive constants. x is the displacement of a unit mass from its equilibrium position.
Section snippets
Analysis
The method of multiple scales (Nayfeh and Mook [13]) can be used to determine a first-order of the solution of Eq. (1) in the formwhere Tn=εnt. In terms of T0 and T1, the time derivatives transform asSubstituting , , into Eq. (1) and then setting the coefficients of like powers of ε we obtainThe solution of Eq. (5) can be written in the form
Numerical results
Added to the analytic results given in unit 2, we also studied graphically, in Figs. 1–8. the response of one-degree-of-freedom parametric nonlinear system at different resonance cases. The response is present at the amplitude x against time t. The phase plane is presented as the amplitude against the velocity y=dx/dt. Mostly, the initial values of all cases are x(0)=0 and y(0)=1. It is worthy to note that any small variation of the value of the initial condition x(0) or y(0) does not carry any
Conclusion
The steady-state response of one-degree-of-freedom nonlinear excited system with two-frequency parametric and self-excitations is being investigated analytically and numerically. The response of the system is stable for suitable positive or negative magnitudes of the damping factor μ, however, the system response is stable only if ν≥0.
The approximate multiple scales method used to obtain the first-order ordinary differential equations for evolution of the steady-state amplitude and phase with
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