Dynamics of a single-degree-of-freedom structure with quadratic, cubic and quartic non-linearities to a harmonic resonance

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Abstract

The solution of a non-linear equation having often encountered in the study of structural dynamically and in which both parametric and external excitations are present, have been obtained analytically and numerically. The only considered case is when both the parametric and external excitations have the same frequency. The method of multiple time scale is applied to determine the equations that describe up to and including the second order both the modulation of the amplitude and phase. These equations are used to determine the fixed points and their stability. Frequency response curves are presented graphically. Discussion of the results is given. Numerical solutions are presented for the effects of the different equation parameters on system stability, response and chaos. Different cases of resonance are investigated and reported.

Introduction

In this paper we considered dynamical systems governed by equations of motion having the formÜ+ω20U+μU̇+α2U2+2FUcosΩt+α3U3+α4U4=2HcosΩt,where the dots indicate differentiation with respect to time, where ω0, μ, α2, α3, α4, F, H and Ω are constants. Equations such as Eq. (1) can arise when the deflection of a structure is expressed as an expansion in terms of its linear, free vibration modes.

The following brief discussion of representative examples provides a background of the present paper. For a comprehensive review of the response of single and multi-degree-of-freedom systems, the reader is referred to textbooks by Tondl [1], Evan-Iwanowski [2] and Nayfeh and Mook [3]. Nayfeh [4] investigated the response of one-degree-of-freedom system with quadratic and cubic non-linearities to sub-harmonic excitation. Zavodney and Nayfeh [5] studied the response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric resonance. Zavodney et al. [6] analyzed the response of a single-degree-of-freedom system with quadratic and cubic non-linearities to principal parametric resonance. Elnaggar and Alhanadwah [7] studied parametric excitation of sub-harmonic oscillations. Agrawal et al. [8] investigated the non-linear control strategies for Duffing systems. HaQuang [9] and HaQuang et al. [10] investigated the response of a single and multi-degree-of-freedom systems with quadratic and cubic non-linearities to a harmonic external excitations, and parametric excitations and combined harmonic external and parametric excitations. Asmis and Tso [11] analyzed the response of two-degree-of-freedom systems with cubic non-linearities to a combination parametric resonance in the presence of one-to-one internal resonance. Tso and Asmis [12] analyzed the response of two-degree-of-freedom systems with cubic non-linearities to a harmonic parametric excitation in the absence of internal resonance. Elnaggar and El-Bassiouny [13], [14] studied the response of self-excited two-degree-and three-degree-of-freedom systems to multi-frequency excitations. They investigated harmonic, sub-harmonic, super-harmonic, simultaneous sub/super harmonic and combination resonance. Elnaggar and El-Bassiouny [15] investigated harmonic resonance of non-linear system of rods to a harmonic excitation. Eissa [16] studied the different resonance cases of mechanical oscillators subject to both parametric and external excitation with cubic non-linearities.

In this paper, an investigation of harmonic resonance for one-degree-of-freedom system with quadratic, cubic and quartic non-linearities under the interaction of external and parametric excitations is presented. The method of multiple time scale is applied to determine the equations that describe to second order the modulation of the amplitude and phase. Steady state solutions (periodic solutions) and their stability are obtained. Frequency response curves are presented graphically. Numerical solutions are determined applying the bisection method. Also, the solution is obtained by numerical integration of the original governing equation to give the time history of the steady state response and the phase-plane as a stability criterion.

Section snippets

Perturbation solution

An approximate analytical solution of Eq. (1) is obtained assuming that the coefficients of F, μ, α2, α3, α4, and H are small. Thus, we re-write Eq. (1) asÜ+ω20U+ε[α2U2+α4U4+2FUcosΩt]+ε2[2μU̇+α3U3−2HcosΩt]=0,whereμ=ε2μ,α2=εα2,α3=ε2α3,α4=εα4,F=εF,andH=ε2H,where ε is a small perturbation parameter.

Following the method of multiple scales [3], [6], one assumes that solution of Eq. (2) can be approximated by an expansion in the formU(t,ε)=U0(T0,T1,T2)+εU1(T0,T1,T2)+ε2U2(T0,T1,T2)+⋯,where T0=t is a

Numerical results and discussion

In this section the numerical solution of the frequency response equation (22) and the equation of motion (2) are studied. Frequency response equation Eq. (22) is a non-linear algebraic equation in the amplitude “a”. This equation and stability conditions , are solved numerically. The results are plotted in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, which present the variation of amplitude (a) against the detuning parameter σ at specific given values of

Numerical simulation

The solution of Eq. (2) is obtained numerically applying Runge–Kutta method. The stability of this solution is investigated using phase-plane trajectories. Now we will discuss the effects of the different parameters on system response, chaos and stability. In all numerical results ε has been selected at the value of 0.2.

Some resonance cases

Although the main system have been studied under the main incident resonance case, where the external excitation frequency equals the parametric one, more investigations have been carried out to study other resonance cases, which have been deduced from the proposed solution. These cases will be discussed in the following.

Discussion and conclusions

The method of multiple time scale has been applied to a non-linear mechanical oscillator subject to resonant parametric and external excitations. Both response and stability are determined for the considered system. The effects of the different parameters on both response and stability are investigated. From the study of the effects of the different parameters as damping coefficient, non-linear parameters, excitation frequency, natural frequency of the system and the amplitudes of both

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Present address: Department of Mathematics, Jubail College of Education for girls, Industrial Jubail 31961, P.O. Box 12020, Saudi Arabia.

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