Research Article | Open Access
Alex Elías-Zúñiga, Oscar Martínez-Romero, "Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator", Mathematical Problems in Engineering, vol. 2013, Article ID 574696, 9 pages, 2013. https://doi.org/10.1155/2013/574696
Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator
We study the dynamical response of an asymmetric forced, damped Helmholtz-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 the Helmholtz-Duffing oscillator, 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.
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 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 in (1), Hu used the harmonic balance method to calculate the first-order approximations to the periodic solutions of this equation . Belhaq and Lakrad used the harmonic balance method involving Jacobian elliptic functions to obtain the approximate solution of (1) by taking and . Tamura  and Hu  developed the exact solution of a quadratic nonlinear oscillator that is part of (1) by using an elliptic function. Cao and coworkers investigated in  the various symmetry breaking phenomena associated with the Helmholtz-Duffing oscillator (1) in the case for which and for different values of the so-called symmetric parameter . They also used the second-order averaging method to investigate its local bifurcation behavior. By considering a rational form elliptic solution to (1) when , Elías-Zúñiga derived its analytical solution which is similar in form to that of its exact solution when .
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 , while Jeyakumari et al. analyzed how the potential well of an asymmetric Duffing oscillator affects the vibrational resonance response .
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 amplitude-frequency 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 , 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.
2. 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 is the Jacobian elliptic function that has a period in equal to , and is the complete elliptic integral of the first kind for the modulus [16, 17]. Note that must hold, and when the modulus of is zero, then and the trigonometric function coincide and, thus, (2) reduces to (1). Now, we make the assumption that (2) has an approximate general solution of the form where , , , , , , and need to be determined. For the sake of simplicity, it is assumed that , , , , , and are slowly varying as a function of time so that , , , , , and or higher order are small enough to be ignored . Also, for simplicity, we will use the following notation , and with similar notation for the functions and . Thus, substitution of (3) into (2) gives in which the following identities and have been used .
Notice that (4) depends on elliptic functions with different modulus. To simplify (4), we first compute its average with respect to while keeping terms with period constant; this yields the following equation: We now take the average of (5) with respect to , to get where the terms are defined in the appendix.
To determine the unknown parameters , , , , , , and , Fourier series are applied to (7) to transform the terms and into , and into Jacobian elliptic functions, respectively [20–22]. Therefore, (7) simplifies to where , , and are defined in the appendix. If we now use the amplitude for Jacobi elliptic functions so that , and , then (5) and (9) become Since our original system (1) is subjected to a driving force of sinusoidal type, then the modulus and the following identities hold: , , , , and . If we also assume that , then, from , 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: 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 , , , , , , and can be found from (18)–(22) and from the initial conditions (I. C.) , . If we assume that the value of the initial velocity is such that in (3), then our approximate solution for (2) can be written as Since at , we get from (23) that We next use (20) and (21) to find and 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 . Substituting (25) and (26) into (18), we can easily prove, after some algebraic computations, that the values of are determined from a ninth-order polynomial equation. Once the values of are known, we may use (19) to determine . Notice that when the running time increases, the term approaches to zero, and thus (18), (19), and (22)–(26) become time-independent. In this case, the values of , , , , and 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 amplitude-frequency and the percentage overshoot response curves and show how the evolution of time influences the systems behavior.
3. 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 . 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 , 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 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: , , , , , and , and plot these curves at the values of , , , , and . Figure 1 shows the evolution of these amplitude-frequency curves with time. In the curves showed in Figure 1, we have plotted 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, , 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 . Figure 2 illustrates the amplitude versus time response curve by considering the parameters values of , , , , and and the initial conditions and for which . 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 , , , , , and were obtained from (18), (19), (22), (24), (25), and (26). In fact, at , we get that , , , , , , and . 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. Figure 2(b) shows the phase plane of the transient oscillatory motion on the time interval plotted by considering the following initial conditions of and . It is clear from Figure 2(b) that the transient oscillations experience a fast increase in their amplitude values followed by a decrease when , 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 and represent the peak transient and steady-state 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: , , , and . 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 , , , and . Here the blue, the red, and the purple dashed lines were calculated by considering the nonlinear parameter values of , , and , 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 and near to the resonance frequency. Furthermore, the predicted overshoot curves oscillate about the value of 92% on the interval 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 , , , and and consider the damping ratio values of , , and . The influence of the transient response on the peak overshoot values for the curve in which , , and is illustrated in Figure 5(a) and 5(b). Figure 5(c) shows the percentage overshoot curves plotted against the driving frequency . As me can see from Figures 5(c) and for a damping value of , 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 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 . 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.
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 , , , , 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 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 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. We also found that by adding damping to the system, the percentage of overshoot in the system does not exceed, in the case of , 52% if the system is tuned into the overshoot bandwidth region of .
Based on these findings, we can conclude that if one really wants to have a better understanding of the system dynamical behavior, transient oscillations amplitudes must be estimated to avoid undesirable system effects such as peak transient amplitude values that could violate design constraints or unstable system behavior when the excitation is suddenly switched on. In this case, the percentage overshoot charts could be used to identify the frequency bandwidth region at which the system can be tuned to have a reliable dynamical response.
The terms , , and in (9) are defined as Here is called the amplitude for Jacobi elliptic functions.
This work was partially funded by the Tecnológico de Monterrey, Campus Monterrey, through the Research Chairs in Nanotechnology and Intelligent Machines. Additional support was provided by the project FOMIX NL-2010-C30-145429.
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Copyright © 2013 Alex Elías-Zúñiga and Oscar Martínez-Romero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.