Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator

We study the dynamical response of an asymmetric forced, dampedHelmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of elliptic balance, and Fourier series. By assuming that the modulus of the elliptic functions is slowly varying as a function of time and by considering the primary resonance response of theHelmholtz-Duffingoscillator, we derived an approximate solution that provides the time-dependent amplitude-frequency response curves.The accuracy of the derived approximate solution is evaluated by studying the evolution of the response curves of an asymmetric Duffing oscillator that describes the motion of a damped, forced system supported symmetrically by simple shear springs on a smooth inclined bearing surface. We also use the percentage overshoot value to study the influence of damping and nonlinearity on the transient and steady-state oscillatory amplitudes.


Introduction
Since most of the nonlinear differential equations that characterized the motion of several physical systems do not have closed-form solution, we have to use numerical or perturbation techniques to study the dynamical response of these systems.However, most of the perturbation techniques such as multiple scales, averaging, and harmonic balance, to say a few, focus on only the determination of steady-state approximate solutions because of the complexity involved in finding transient solutions of nonlinear differential equations [1,2].
The aim of this paper is to investigate the influence on the dynamical behavior of the transient and steady-state solutions of the system ẍ + 2] ẋ +  +  2 +  3 =  cos (  ) (1) close to the primary resonance region.Here,  denotes the displacement of the system,  is the natural frequency, ] is the damping coefficient,  is a dimensionless nonlinear parameter,   is the driving frequency,  is the running time,  is the amplitude of the driving force, and  is a system parameter.To solve the homogeneous Helmholtz-Duffing oscillator for which  = 0 in (1), Hu used the harmonic balance method to calculate the first-order approximations to the periodic solutions of this equation [3].Belhaq and Lakrad used the harmonic balance method involving Jacobian elliptic functions to obtain the approximate solution of (1) by taking ] = 0 and   = 0 [4].Tamura [5] and Hu [6] developed the exact solution of a quadratic nonlinear oscillator that is part of (1) by using an elliptic function.Cao and coworkers investigated in [7] the various symmetry breaking phenomena associated with the Helmholtz-Duffing oscillator (1) in the case for which  = 1 −  and for different values of the so-called symmetric parameter .They also used the secondorder averaging method to investigate its local bifurcation behavior.By considering a rational form elliptic solution to (1) when  = 0, Elías-Zúñiga derived its analytical solution which is similar in form to that of its exact solution when ] = 0 [8].
Recently, Kovacic and coworkers studied the primary resonance response of (1) by applying the harmonic balance method and derived nonlinear algebraic equations for the steady-state system response [9], while Jeyakumari et al. analyzed how the potential well of an asymmetric Duffing oscillator affects the vibrational resonance response [10].
Here the approximate solution of ( 1) is derived by taking into account the transient and the steady-state responses without the simplifications regarding undamped and unforced system included in previously developed solutions such as [4,6].The approximate solution is based on trigonometric and Jacobi elliptic functions with slowly varying parameters that will help us to obtain amplitudefrequency response curves that evolve with time.Then, the influence of the nonlinear transient responses is investigated since recent studies show that transient vibrations can not only provide additional information to fully predict the system stable behavior [11], but also can be used to predict the system overshoot value [12,13].The determination of this value is of practical interest in understanding the importance of time in controlling the dynamical behavior of oscillatory systems [14,15] therefore, the percentage overshoot value will be computed by considering the influence of the system parameters such as nonlinear and damping effects.
In the next section, the approximate general solution of (1) is derived by using Jacobi elliptic functions.

Approximate Solution
In order to obtain the general solution of (1) in the region of the primary resonance, we will consider that this equation can be written in equivalent form as where cn(  ,  2  ) is the Jacobian elliptic function that has a period in    equal to 4( 2  ), and ( 2  ) is the complete elliptic integral of the first kind for the modulus   [16,17].Note that 0 ≤ | 2  | < 1 must hold, and when the modulus of cn(  ,  2  ) is zero, then cn(  ,  2  ) and the trigonometric function cos(  ) coincide and, thus, (2) reduces to (1).Now, we make the assumption that (2) has an approximate general solution of the form where  1 ,  1 (), (),  1 (),  2 (),  1 (), and   () need to be determined.For the sake of simplicity, it is assumed that (),  1 (),  1 (),  2 (),  1 (), and   () are slowly varying as a function of time so that ḃ , ω 1 , Ḋ 1 , Ḋ 2 , k 1 , and k  or higher order are small enough to be ignored [18].Also, for simplicity, we will use the following notation cn  ≡ cn  = cn(  ,  2  ), and cn 1 ≡ cn 1 = cn( 1  +  1 ,  2  1 ) with similar notation for the functions sn and dn.Thus, substitution of (3) into (2) gives in which the following identities sn 2 + cn 2 = 1 and dn 2 +  2  sn 2 = 1 have been used [19].
Notice that (4) depends on elliptic functions with different modulus.To simplify (4), we first compute its average with respect to 4  while keeping terms with period 4 1 constant; this yields the following equation: We now take the average of ( 5) with respect to 4 1 , to get where the   terms are defined in the appendix.Next, the average of ( 4) is computed with respect to 4 1 while terms with period 4  are held constant; this yields Then, the average of ( 7) is computed with respect to 4  , to get the following expression: To determine the unknown parameters  1 ,  1 (), (),  1 (),  2 (),  1 (), and   (), Fourier series are applied to (7) to transform the terms sn dn and sn cn 2 into sn, cn sn and cn dn into cn Jacobian elliptic functions, respectively [20][21][22].Therefore, (7) where  1 ,  2 , and  3 are defined in the appendix.If we now use the amplitude for Jacobi elliptic functions so that cos   = cn(  ;  2  ) = cn  , and sin   = sn(  ;  2  ) = sn  , then ( 5) and ( 9) become Since our original system (1) is subjected to a driving force of sinusoidal type, then the modulus   ≡ 0 and the following identities hold: Mathematical Problems in Engineering and  2 = 1.If we also assume that 0 < | 2 1 | < 1/2, then, from (A.1),  1 ≃ 1/2 and (6) becomes exactly the same as (8).This provides the following simplifications for (10): We then follow the harmonic balance procedure and ignore higher harmonics terms in (11), to find the following expressions: ] + ċ = 0.
Notice that the variable  may be determined by integration of ( 16); this yields where  is a constant of integration.Substitution of ( 17) into (8), as well as ( 12)-( 15), yields Then, for given parameters values of ], , , , , and   , the unknown parameters  1 ,  1 , ,  1 ,  2 , , and  1 can be found from ( 18)-( 22) and from the initial conditions (I.C.) (0) =  10 , ẋ (0) = ẋ 10 .If we assume that the value of the initial velocity ẋ (0) = ẋ 10 is such that  1 = 0 in (3), then our approximate solution for (2) can be written as Since (0) =  10 at  = 0, we get from (23) that We next use (20) and ( 21) to find () and  2 () as where the sign in ( 25) and ( 26) must be appropriately chosen to ensure that the higher elliptic terms in (23) have small amplitudes relative to the leading ones [23].Substituting ( 25) and ( 26) into ( 18), we can easily prove, after some algebraic computations, that the values of  1 () are determined from a ninth-order polynomial equation.Once the values of  1 () are known, we may use (19) to determine  1 ().Notice that when the running time  increases, the term  −] approaches to zero, and thus ( 18), (19), and ( 22)-(26) become timeindependent.In this case, the values of ,  1 ,  2 ,  1 , and  1 in (23) will remain constant.This condition provides the steady-state solution of (1).We will next discuss the accuracy of our derived approximate solution by studying the dynamical response of an asymmetric Duffing oscillator by plotting the amplitudefrequency and the percentage overshoot response curves and show how the evolution of time influences the systems behavior.

Numerical Simulations
In order to verify the accuracy of our proposed solution described in (23), we study the dynamical response of a rigid body supported symmetrically by a simple shear spring that is sliding over a smooth inclined bearing surface [24].The system under consideration contains the following critical conditions: (a) finite amplitude forced vibration and (b) damping, demonstrating the general nature of the proposed solution for the resonance response of an asymmetric Duffing oscillator.In accordance with Elías-Zúñiga and Beatty [24], the equation for the damped motion of the load is a nonlinear, ordinary differential equation of the forced Duffing type with a constant static shift term of the form T i m e , t D r iv in g f r e q u e n c y ,  f where the dot denotes the derivative with respect to ,  represents the amount of simple shear deformation,   denotes the amount of static shear deflection of the load, ] is the damping ratio,  is a dimensionless driving force, and  is the dimensionless running time.If we transform (28) relative to   by using  =  −   , then (28) becomes similar to the Helmholtz-Duffing equation (1) where the parameters  and  are described by To find the amplitude-frequency response curves of (1), we select the following parameter values:  = 0.02,   = 0.497,  = 0.1, ] = 0.01,  0 = 1, and ẋ 0 = 0.022, and plot these curves at the values of  = 1, 15, 30, 80, and 100. Figure 1 shows the evolution of these amplitude-frequency curves with time.In the curves showed in Figure 1, we have plotted   () = |() +  1 () +  2 ()| versus   .These curves characterize the evolution of the general motion of the system for the selected set of system parameter values.In these curves, the blue dots represent unstable system behavior determined by following the bifurcation analysis described in [9,24].Notice from Figure 1, that when the running time  increases, that is,  ≈ 100, the unstable regions in the amplitude-frequency response curves become smaller since the influence of time in (18), (19), and ( 22)-( 26) is almost negligible.In Figure 1, the brown dots represent the stable steady-state amplitude-frequency response curve that characterizes the damped, forced nonlinear motion of a body supported by viscohyperelastic shear mountings [24].Figure 2 illustrates the amplitude versus time response curve by considering the parameters values of   = 0.497,  = 0.02, ] = 0.01,  = 0.1, and   = 1.1 and the initial conditions  0 = 0 and ẋ 0 = 0.0663 for which  1 = 0. There, the solid lines represent the numerical integration solution of (28) while the dashed lines represent the approximate solution provided by (23).In this case, the values of , ,  1 ,  2 ,  1 , and  1 were obtained from ( 18), ( 19), ( 22), ( 24), (25), and (26).In fact, at  = 0, we get that (0) = −0.0086, = 0.5496,  1 (0) = −0.541, 2 (0) = 0.0653,  1 (0) = 0.05418, ẋ (0) = 0.0663, and  1 (0) = 1.0144.As shown in Figures 2(a) and 2(c), the highest vibration amplitudes occur during the transient oscillations.This transient oscillatory behavior provides information to ensure that the system design constraints are not exceeded, even if the shear suspension system performs well during its steady-state behavior.as illustrated in Figure 2(c).When the system reaches its steady-state behavior, the system phase portrait, depicted in Figure 2(d), exhibits stable behavior about the steady-state equilibria condition.
We next investigate the influence of the transient solution on the dynamics response of (28) by introducing the percentage of overshoot which is defined as where  trans and  st represent the peak transient and steadystate amplitudes of the system, respectively, that can be estimated by setting to zero the time derivative of (23). Figure 3 shows the calculated system overshoot plotted against the system angular frequency,   , by considering the following system parameter values:   = 0.497,  = 0.1, ] = 0.01, and  0 = 0.As we can see from Figure 3, the influence of the transient solution on the system response is evident since the percentage of overshoot values is close to a 100% at the approximate frequency values of 0.1, 0.5, 1.5, and 2.Here the blue, the red, and the purple dashed lines were calculated by considering the nonlinear parameter values of  = 0.02, 0.1, and 0.25, respectively.It is interesting to observe in Figure 3 the influence of the nonlinearity of the system in the overshoot qualitative and quantitative curves behavior.
In fact, one can conclude from Figure 3 that the smallest percentage values of the overshoot occur with  = 0.02 and near to the resonance frequency.Furthermore, the predicted overshoot curves oscillate about the value of 92% on the interval 1.5 ≤   ≤ 2.7 that corresponds to the region depicted in Figure 4 at which the ratio of the transient and the steady-state amplitudes has almost the constant value of 52%.
To examine damping effects on the system overshoot, we next use the parameter values of   = 0.497,  = 0.1,  = 0.02, and  0 = 0 and consider the damping ratio values of ] = 0.01, 0.05, and 0.1.The influence of the transient response on the peak overshoot values for the curve in which  = 0.02, ] = 0.01,   = 0.5 and 1.5 is illustrated in Figures 5(a) and 5(b).Figure 5(c) shows the percentage overshoot curves plotted against the driving frequency   .As me can see from Figure 5(c) and for a damping value of ] = 0.1, the overshoot values do not exceed 73%.It is also observed that for increasing damping values, the percentage of the system overshoot becomes smaller.Figures 5(d) and 5(e) illustrate, by using the Continuous Wavelet Transform, the influence of the transient and the driving frequencies on the nonlinear behavior of the system.It is evident from Figures 5(d) and 5(e) that at the beginning of the system motion, the transient frequency  1 influences the fast increase of the system oscillations.Then, its effects on the system non-linear motion decrease when the running time exceeds the value of  ≈ 200.Based on these findings, it is concluded that transient amplitude values are bigger than those of the steady-state ones as showed in Figures 3  and 5. Therefore, if one wants to reduce the impact of the transient oscillations amplitudes, we need, when possible, a fast tuning of the system to its operating bandwidth frequency region which can be achieved by using suitable control algorithms.

Conclusions
In this paper, we have used elliptic functions to obtain the transient and the steady-state solutions of an asymmetric Duffing oscillator and studied its influence on the region of the primary resonance response.The theoretical predictions of our derived solution given by (23) are based on the assumption that the physical parameters  1 (),  1 (),  2 (),  1 (), and () are slowly varying as a function of time.To obtain computational tractable expressions to determine the aforementioned parameters, we have computed the average of the corresponding equations with respect to the complete elliptic integral of the first kind for the moduli  1 and   .Then, we have used Fourier series and followed the harmonic balance method to balance the lowest harmonic terms.These steps plus the assumption that 0 < | 2  1 | < 1/2 provide us with the time-dependent equations to determine the unknown parameters.The effect of the running time on the shapes of the amplitude-frequency response curves becomes evident during the study of the dynamical oscillator response with no linear stiffness term and with hardening characteristics.We have also shown that when the running time increases, the frequency range at which unstable motion occurs becomes smaller.Furthermore, we have obtained the percentage overshoot charts to address the influence of the nonlinear and damping terms on the transient and steady-state system responses when the system is suddenly switched on.We have found that when the nonlinearity is small, the percentage overshoot values becomes smaller at the overshoot bandwidth regions closer to the resonance frequency with increasing values of the steady-state and the transient amplitudes ratio.The usage of Continuous Wavelet Transform helps to identify the influence of the transient response signal on the system behaviour as time evolve.(A.2) Here   = am(  ;  2  ) is called the amplitude for Jacobi elliptic functions.