Abstract
The aim of this paper focuses on applying a nonlinearization method to transform forced, damped nonlinear equations of motion of oscillatory systems into the well-known forced, damped Duffing equation. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the amplitude-time, the phase portraits, and the continuous wavelet transform diagrams of the cubic-quintic Duffing equation, the generalized pendulum equation, the power-form elastic term oscillator, the Duffing equation with linear and cubic damped terms, and the pendulum equation with a cubic damped term.
1. Introduction
Here, in this paper, we illustrate how the nonlinearization approach can be used to obtain equivalent equations of motion of forced, damped nonlinear oscillators of the form where is the initial oscillation amplitude, and are the system conservative and nonconservative restoring forces, respectively, is a damping parameter, and is a periodic external force with driving frequency . In accordance with the nonlinear transformation approach [1], we first write the conservative force terms as a polynomial expression by using the Chebyshev polynomials of the first kind [2–4]: where and are the Chebyshev polynomials of the first kind. We can see that the usage of (3) could transform (2) into a fifth or higher order polynomial expression. In the case for which a fifth-order Chebyshev polynomial is used, the conservative force in (2) becomes where , and will be defined later on. Therefore, the equivalent representation form of (1) is given as By following the nonlinearization method, we now find the equivalent representation form of (1) as a function of a cubic-like polynomial equation. This procedure leads to where , , and are determined from in which Notice that in our proposed procedure we are assuming that the magnitude of the external force and its driving frequency remain constants during the transformation process. Thus, (1) can be written in equivalent form as We will next explore the applicability of our proposed approach and derive the equivalent representation form of some forced, damped nonlinear systems.
2. The Forced, Damped Cubic-Quintic Oscillator
The equation of motion that describes the dynamical response of the forced, damped cubic-quintic oscillator is given as where denotes the displacement of the system, is the damping coefficient, , , and are system constant parameters, is the magnitude of the external force, and is the driving frequency [6, 7]. We next use and write (11) as where , , , and . By following our proposed nonlinear method and by using (4) and (7)–(9), we obtain the equivalent representation form of (12) as where , , and can be determined from the following equations: Here, and are fitting parameters that satisfy (7)-(8). To examine the accuracy of (13), we next compare its solution with the one obtained from (12) by using the fourth-order Runge-Kutta numerical integration method. Let us consider the parameter values of , , , , , , and . In this case, the parameter values assigned to , , and provide a triple-well potential to the cubic-quintic Duffing oscillator that can have up to four resonance frequencies [8]. Figure 1 illustrates the comparison between the amplitude-time response curves of (12) and (13) obtained from their corresponding numerical integration solutions. As one can see form Figure 1, both solutions are almost the same. In fact, the computed root-mean-square error (RMSE) value does not exceed on with , , , , and . The accuracy of the numerical simulations is surprisingly good if we consider that the potential of a cubic Duffing oscillator cannot have triple-well form.
As a second example, let us consider the parameter values of , , , , , , and . Figure 2 shows the amplitude-time response curves and the corresponding phase portraits, as well as the Morlet continuos wavelet transforms (CWT) obtained from the numerical integration solutions of (12) and (13). Here, the values of and were computed from (15) which provides good agreement between (12) and (13). Notice that the numerical integration solutions of (12) and (13) are almost the same. In this particular problem, the Morlet CWT was used to extract system dynamics effects such as the one shown at the system transient motion in which the transient frequency has strong influence on the system dynamic behavior. In fact on the time interval , the transient frequency dominates the system motion. When , the system oscillates at the driving frequency . Besides, we have computed the RMSE value between both numerical solutions and found that it has the value of . Here, , , and . Of course, we can consider other parameter values, as those shown in Table 1, to describe the dynamic response of (12) by using (13). Therefore, we can conclude that our nonlinear method leads to the derivation of an equivalent equation of motion that follows well the qualitative and quantitative numerical response of the original equation (12).
We next determine the equivalent representation form of the forced, damped general pendulum equation.
3. The Forced, Damped General Pendulum Equation
We now proceed to derive the equivalent representation form of the forced, damped pendulum equation where and represent system constant parameter values, is the damping coefficient, is the driving frequency, and is the external force magnitude [9]. If we introduce the transformation , then (16) can be rewritten as with . By applying our proposed transformation method to (17), we obtain the following expression: where and , , and are given by (14) and (15). Here, and are the first and second order Bessel functions of the first kind. To illustrate the degree of accuracy attained by our derived solution (18), let us consider the system parameter values of , , , , and with or and . One can notice from Figure 3 that the numerical integration solutions of (17) and (18) are almost the same. In this case, ,, , , , and , and the values of , and were fitted by using (15), since these expressions provide the best predictions with a RMSE value of . The same degree of accuracy was found by considering different system parameter values, as those illustrated in Table 1 in which the RMSE values are close to zero.
To further assess the applicability of our nonlinear cubication approach, we next derive the equivalent representation form of a forced, damped oscillator with a power-form elastic term.
4. A Generalized Forced, Damped Power-Form Elastic Term Oscillator
The equation of motion of this oscillator is given as where and are constant parameters and can take any nonnegative real value, such as odd, even, rational, or irrational, that is, [10]. As usual, let us use the following coordinate transformation and write (20) as, We next use Chebyshev polynomial expansion to write the restoring forces as a nonic polynomial expression where Notice that in (23)–(27) the terms represent the Euler gamma function. It is important to point out that in this particular problem we have used five Chebyshev expansion coefficient terms that provide, for the system restoring force, an equivalent representation form that is based on a ninth-order polynomial expression. This example illustrates the applicability of our procedure in using more than three terms in (4). We next follow our solution procedure and find, by using (7) and (8), that where To assess the accuracy of our derived equivalent representation form (28) of (21), we shall consider the following data values: , , , , , and with a driving frequency value of [11]. Figure 4 illustrates the amplitude-time response curves obtained by numerically integrating (21) and (28). As we can see from Figure 4, the numerical integration of (28) follows closely the amplitude-time response curve obtained from (21). In this case, the RMSE value of is obtained by using equations (30)–(32). Here, the red solid and black dashed lines represent, respectively, the numerical integration solution of (21) and (28). The computed parameter values are , , , , , , , , , and . Also, Figure 5 provides a comparison of the numerical solutions of (21) and (28) with respect to the approximate general solution of (28) derived by using Jacobi elliptic functions [5]. One can notice from Figure 5 that all solutions are almost the same. Therefore, we can conclude that our derived equivalent representation form (28) describes well the qualitative and quantitative behavior of (21). The amplitude-frequency response curve of (28) can be obtained by using, for instance, the approximate solutions developed in [5, 11]. As a second case, we now use our equivalent representation form (28) and consider the following parameter values of , , , , , , and in (21) and compute the corresponding amplitude-time response curve. We can see from Figure 6 that the amplitude-time curve obtained from (28) follows well the curve obtained from (21). In this case, the RMSE value is about for which the parameter values are , , , , , , , , , and . For illustrative purposes, we show in Table 2 some values of the exponent with their fitting parameter values of and that can be used to study the dynamical behavior of some nonlinear oscillator with a rational or irrational power restoring forces.
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We next develop the equivalent representation form of the Duffing equation with linear and cubic damped terms.
5. The Forced Duffing Equation with Linear and Cubic Damped Terms
We now explore the applicability of our method to derive the equivalent representation form of the following equation of motion which has a linear damped term, , and a cubic one, [12, 13]. Let ; then, (33) can be written as where , , and .
Since (34) has a damped nonlinear term of the cubic-type, we need to modify our nonlinear method to take into account its effects on the solution response of (34). Therefore, we now assume that (7) and (8) can be re-written as which yield the equivalent representation form of (34) as where Before we evaluate the accuracy achieved by our derived expression (37), we first recall that Trueba and coworkers in [13], by using Melnikov analysis, found an equivalent equation of motion for (34) given as where is defined as In what follows, we will use (37) and (39) to compare their numerical predictions with those provided by (34). First, let us consider the parameter values of , , , , , and with , and and use our derived expressions to compute the values of , , , and which are given as , , , , and , respectively. Figure 7 shows a comparison of the amplitude-time curves, the phase portrait plots, and the Morlet CWT diagram obtained from the numerical integrations of (34), (37), and (39). Notice from Figure 7, that our equivalent equation of motion (37) closely follows the numerical integration curve of (34). Here, the RMSE value is close to , while the numerical predictions obtained from (39) show some discrepancies in the amplitude-time curve at the time interval of . In this solution, the computed RMSE value is .
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As a second case, we now explore the accuracy of our equivalent representation form (37) by assuming that , , , , , and with and . As we can see from Figure 8, the numerical integration solutions of (37) and (39) agree well with the solution of (34). Furthermore, the Morlet CWT of (39) shown in Figure 8, exhibits the subharmonic effects that the original system (34) experiences at the frequencies values of , , and which correspond to the Morlet continuous wavelet transform scale values of , , and , respectively. In this case, the estimated RMSE values are for (37) and for (39). The values of , , , and are found to be , , , , and . Notice that in both examples the values of , , and remain unchanged. In Figures 7 and 8, the black and the red dashed lines describe the amplitude-time and phase portrait curves of (34) and (39), respectively.
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As a final example, we now derive the equivalent equation of the forced pendulum equation with a cubic damped term where is the initial oscillation amplitude, , , with . We first use Chebyshev polynomial expansion and then (35) and (36) to get that where and , , and are determined from By using Melnikov analysis, Trueba and coworkers [13] developed the equivalent representation form of (41) which is given as We next consider the parameter values of , , , , , , and , and plot the numerical integration solutions of (41), (42), and (45). The corresponding amplitude-time, phase portrait, and Morlet CWT plots are shown in Figure 9 for which the computed parameter values are , , , , , , , with , and . We can see from Figure 9 that our solution closely follows the numerical simulations of (41). In this case, the computed RMSE values from (42) and (45) are and , respectively. In Figure 9, the black, the purple, and the red solid lines represent, respectively, the numerical integration solutions of (41), (42), and (45).
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This confirms the usefulness of our proposed nonlinear method to obtain equivalent equations of motion of nonlinear oscillators.
6. Conclusions
In this paper, we have obtained the equivalent representation form of some driven, damped nonlinear oscillators by using a nonlinearization approach. We have found that in all cases, the numerical predictions obtained from the corresponding equivalent representation form of the cubic-quintic, the general pendulum, the power-form elastic term, and the cubic damped nonlinear oscillators describe well the qualitative and quantitative behavior of their original equations of motion.
During the solution processes of the forced Duffing equation with a cubic damped term, we have found that our equivalent solution (37) provides numerical estimates that are similar to those obtained from (39) which was derived from Melnikov analysis in [13]. Besides, the numerical predictions of (37) are closer to the numerical integration values of (34) than those obtained from (39). In this oscillator, we have found that the values of , , and could remain unchanged even at larger values of , , , , and . To further evaluate the accuracy of our proposed method, we have developed the solution of a pendulum equation with a cubic damped term and compared its numerical estimated values with those obtained from the original equation of motion and with respect to those of its equivalent representation form derived by Melnikov analysis [13]. We found that our derived equation (42) describes well the numerical estimated values of (41). Moreover, and based on the numerical simulations performed on the last example of this work, it is clear that the derived equivalent equation of motion, when compared to other solutions such as the one derived by using the well-established Melnikov analysis, exhibits good accuracy for a wide range of system parameter values [13]. Of course, one must be careful when using the equivalent transformation forms previously derived, since their degree of accuracy depends not only on the system parameter values but also on the method used to replace the corresponding restoring forces for equivalent ones of the cubic type.
Finally, it is evident that our proposed nonlinearization method can be used to derive equivalent representation forms of other nonlinear oscillators such as the ones examined by the authors in [14, 15] and references cited therein, in which rational or irrational restoring forces, as well as damping terms, are used to model the dynamics behavior of common problems that arise in the physical sciences and engineering fields.
Acknowledgments
This work was funded by the Tecnológico de Monterrey, Campus Monterrey —Campus Monterrey, through the Research Chair in Nanomaterials for Medical Devices and Research Chair in Intelligent Machines. Additional support was provided by the European Union Seventh Framework Programme (FP7-PEOPLE-2009) under the grant agreement IRSES no. 247476 and from Consejo Nacional de Ciencia y Tecnología (Conacyt), México.