RESUMEN. En este trabajo se obtiene la solución semianalítica aproximada de un sistema cúbico no-lineal de dos grados de libertad al reemplazar las fuerzas restauradoras del sistema original por fuerzas equivalentes utilizando el método de cubicación. Al final del artículo, se hace una comparación de la solución aproximada obtenida mediante la cubicación del sistema original con la integración numérica y con el método de homotopía. El grado de precisión obtenido se ilustra en algunos casos y se concluye que la cubicación de las ecuaciones originales permite obtener soluciones aproximadas con alto grado de precisión en sistemas con alto nivel de no-linealidad. ABSTRACT. In this paper we obtain the approximate solution of a two-degree of freedom system with cubic no-linearities by replacing the original system by and equivalent one obtained by applying the cubication procedure. At the end of the article, we compare the approximate solution obtained by elliptic balance method with those obtained by numerical integration and by the homotopy perturbation method. The degree of precision obtained is illustrated in a few cases in which it is evident that the cubication procedure provides highly accurate results even for strongly nonlinear systems. INTRODUCTION There are several perturbation techniques that are used to find approximate solution of nonlinear, multi-degree of freedom dynamical systems. However, when the nonlinear terms are higher in magnitude,
most of these perturbation techniques fail since their approximate solutions are based on the assumption that the nonlinear terms are small [1]-[5]. Of course, each of these diverse methods has its own advantages and exhibits its own analytical complexities to find the approximate solutions. To overcome the shortcoming associated with several of the perturbation techniques and to work out solutions in systems with strong nonlinearities, in this work we proposed to replace the original system of equations by an equivalent one in which the restoring forces are replaced by a cubic type polynomial. This procedure decoupled one of the modal differential equation which is replaced by a nonlinear Duffing expression whose exact solution is well-known. Then, we may determine the approximate solution of the system by solving the remaining equation by using the elliptic balance procedure [6-7]. This approach provides solutions that described well the dynamical system behavior even for large nonlinear terms. Thus, we believe that this method could represent an alternative way to deal with the solutions of multi-degree of freedom systems with strong non-linearities [8]. We shall next describe our proposed approach to solve strong nonlinear systems and focus on the solutions of two undamped homogeneous, two-degree-of-freedom systems. SOLUTION APPROACH Let us assume the general case of a two-
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degrees of freedom system whose equation of motion in modal coordinates is given by
21 11
22 22
3 2 2 31 1 2 1 2 3 1 2 4 2
3 2 2 35 1 6 1 2 7 1 2 8 2
0
0
,
n
n
u u
u u
u u u u u u
u u u u u u
(1)
with initial conditions given by 0(0)i iu uand 0(0) ,i iu u i = 1, 2. Equation (1) describes the dynamical behavior of many physical conservative nonlinear oscillators [1] – [6]. Note that in Eq. (1), the dots denote the time derivative, u₁ and u₂ are known as the system normal coordinates, ε is a non-linear parameter, ωn1, ωn2, and ₁ through ₈ are the corresponding system parameters. If we apply the cubication procedure, then the restoring force of Eq. (1)1 can be written in equivalent form as
21 11
22 22
32 1
3 2 2 35 1 6 1 2 7 1 2 8 2
0
0
,
n
n
u u
u u
A u
u u u u u u
(2)
where A2 is a parameter whose value depend on the cubication method [3,6]. Notice that the first equation in (2) depends on only the variable u1 i.e., Eq. (2)1 now is decoupled with respect to the modal coordinate u2 and now it represents the well-known undamped Duffing equation
2 31 1 1 2 1 0,nu u A u (2)
whose exact solution is given as
2 21 10 10
22 10
210
cn( 1 ; );
.2(1 )
u u t u k
uk
u
(3)
The next step in the solution processes
consist now in substituting Eq. (3) into Eq. (2) and use the Jacobi elliptic balance procedure described in [7] thus, we may have the corresponding approximate solution to Eq. (1). Therefore, it clear that the general approximate solution to Eq. (1) is written as a function of Jacobi elliptic functions. It is important to bear in mind that our approximate solution based on Eq. (3), depends on Jacobi elliptic functions and it represents an analytical approximate solution to Eq. (1). NUMERICAL RESULTS To assess the accuracy of our derived analytical solution to Eq. (1), we compare the amplitude-time response curves obtained from the conservative nonlinear oscillatory system showed in Figure 1, with those obtained by using the sub-interval homotopy perturbation method [9], and with the corresponding fourth order Runge-Kutta numerical integration.
Figure1. A nonlinear conservative system.
In Figures 2 and 3, we illustrate how our analytical solution works by assuming strong nonlinear oscillatory system i.e., = 0.5 and 1, respectively. In these Figures 2 and 3, the solid lines represent the numerical integration solutions, the dashed lines represent our analytical solution, the dotted orange lines described the homotopy solution. For the system parameters showed in the caption of Figures 2 and 3, we may see that the derived analytical solution based on Jacobi elliptic functions, compares well with the homotopy and numerical integration solutions with a relative error that do not exceed of 0.5%. As a second example, we now considered the coupled
m1 m2
k1, k2 k2
x1 x2
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two-degree-of-freedom-system with three nonlinear springs, showed in Figure 4 [8].
In this particular case, it is easy to see that the corresponding modal equations of motion are given by
2 3 21 1 1 1 1 3 1 2
2 2 32 2 2 6 1 2 8 2
0,
0,
n
n
u u u u u
u u u u u
(4)
where
2 4 41 1 3
2 2 4 42 6 8
2 31 2 1
; ; ;4 4
; ; ;4 4
22; ; .
n
n
k k
m mk k
m mkk k k
m m m
(5)
Let us consider the following parameter values: k1 = k2 = k3 = k4 =1, m = 1, and initial conditions x0 = 1, 0 0,x y0 = 0.5, and
0 0.y In this particular problem, Cveticanin in [8] considered only the case of small nonlinearities and showed that his approximate solution derived by using the Krylov-Bogolubov method fitted well with the numerical integration of Eq. (4). However, here by using our proposed cubication procedure, we may prove that our solution can fit well numerical simulations not only for small but also for strong nonlinearities. Figure 5 illustrates de case for which = 1. As we may see, our solution follows well the fourth order Runge-Kutta numerical integration of Eq. (4). As a final case, now let us consider the value of = 10. Notice in Fig. 6, that our proposed solution described well the qualitative behavior showed by the numerical integration solution of Eq. (4). In Figures 5 and 6, the black dashed lines represent the approximate solution, while the red solid lines described the numerical integration solutions of Eq. (4).
Therefore, we may conclude that the usage of the cubication method is suitable to study the dynamical behavior of strongly non-linear oscillators of the Duffing-type,
Figure 2. Amplitude-time response curves for each mass showed in Fig. 1. The parameter values are: k1 = 0.45, k2 = 1, = 0.5, 2 = 2n1 = 1.357, x1(0) = 1, and x2(0) = 0.
provided that the restoring forces can be equivalent represented by polynomial expressions of the cubic type. Of course, further improvement can be achieved if we increased the polynomial order approximation i.e., if we use the exact solution of the cubic-quintic Duffing oscillator derived in [10] and applied in [11] through the so-called “Quintication” method. We shall not elaborate details about this procedure since it is out of the scope of the present article.
CONCLUDING REMARKS We have showed that the usage of and equivalent form representation to the restoring forces of Eq. (1), by using the cubication method, decouple the
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original modal coordinate system in u1 or u2. Thus, the derivation of the corresponding solutions to the original system is obtained through the usage of Jacobi elliptic functions. We also have shown that this procedure provides approximate solutions with high degree of accuracy when compare to the numerical integration solutions. Therefore, we believe that this approach provides a new alternative way to obtain analytical solution of strongly nonlinear systems if we can replace the original restoring force by an equivalent one.
Figure 3. Amplitude-time response curves for each mass showed in Fig. 1. The parame-ter values are: k1 = 1.82, k2 = 4, = 1, 2 = 2n1 = 1.5674, x1(0) = 1, and x2(0) = 0.
Figure 4. A two degree of freedom system with three nonlinear springs.
Figure 5. Amplitude-time response curves for each mass showed in Fig. 4. The parame-ter values are: k1 = k2 = k3 = k4 =1, m = 1, = 1, x(0) = 1, and y(0) = 1.
Figure 6. Amplitude-time response curves for each mass showed in Fig. 4. The parame-ter values are: k1 = k2 = k3 = k4 =1, m = 1, = 10, x(0) = 1, and y(0) = 1.
m mk1, k4
k2, k3
x y
k1, k4
Time, t
Am
plit
ude,
xTime, t
Am
plit
ude,
y
Time, t
Am
plit
ude,
x
Time, t
Am
plit
ude,
y
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REFERENCES [1] A. H. Nayfeh, and D. T. Mook, Non-linear Oscillations. John Wiley, New York, 1973. [2] A. Belendez, M. L. Alvarez, E. Fernan-dez and I. Pascual. Cubication of conserva-tive nonlinear oscillators", European Journal of Physics. European Journal of Physics 30 (2009) 973-981. [3] A. Beléndez, D. I. Méndez, E. Fernán-dez, S. Marini, I. Pascual. An explicit ap-proximate solution to the Duffing-harmonic oscillator by a cubication method. Physics Letter A 373 (2009) 2805-2809. [4] A. Belendez, G. Bernabeu, J. Frances, D.I. Mendez and S. Marini. An accurate closed-form approximate solution for the quintic Duffing oscillator equation. Mathe-matical and Computer Modelling 52 (2010) 637-641. [5] J. Cai, X. Wu, Y. P. Li. An equivalent nonlinearization method for strongly nonlin-ear oscillations. Mechanics Research Com-munications 32 (2005) 553-560. [6] A. Elías-Zúñiga, O. Martínez-Romero, and R. Córdoba-Díaz. “Analytical solution for the Duffing-harmonic oscillator by using the cubication and the equivalent nonlinearization methods”. (2012) Submit-ted for Publication. [7] Elías-Zúñiga, A. A general solution of the Duffing equation. Nonlinear Dynamics Volume 45, Numbers 3-4 (2006), 227-235. [8] L. Cveticanin. Vibrations of a coupled two-degree-of-freedom system. Journal of Sound and vibration 247(2) (2001), 279-292 [9] F.I. Compeán, D. Olvera, F.J. Campa, L.N. López de Lacalle, A. Elías-Zúñiga, and C.A. Rodríguez (2012). Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method. International Journal of Machine Tools and Manufacture, Volume 57, June 2012, Pages 27–33. [10] A. Elías-Zúñiga. Exact solution of the cubic-quintic Duffing oscillator. Applied Mathematical Modelling http://dx.doi.org/10.1016/j.apm.2012.04.005
[11] A. Elías-Zúñiga. "Quintication" method to obtain approximate analytical solutions of non-linear oscillators. (2012) Submitted for publication.
