Research Article | Open Access
Higher-Order Solutions of Coupled Systems Using the Parameter Expansion Method
We consider periodic solution for coupled systems of mass-spring. Three practical cases of these systems are explained and introduced. An analytical technique called Parameter Expansion Method (PEM) was applied to calculate approximations to the achieved nonlinear differential oscillation equations. Comparing with exact solutions, the first approximation to the frequency of oscillation produces tolerable error 3.14% as the maximum. By the second iteration the respective error became 1/5th, as it is 0.064%. So we conclude that the first approximation of PEM is so benefit when a quick answer is required, but the higher order approximation gives a convergent precise solution when an exact solution is required.
Nonlinear oscillators have been widely considered in physics and engineering. Surveys of literature with numerous references, and useful bibliographies, have been given by Mickens , Nayfeh and Mook , Agarwal et al. , and more recently by He . To solve governing nonlinear equations and because limitation of existing exact solutions is one of the most time consuming and difficult affairs, many approaches for approximating the solutions to nonlinear oscillatory systems were excogitated. The most widely studied approximation methods are perturbation methods . But these methods have a main shortcoming; there is no small parameter in the equation, and no approximation could be obtained.
Later, new analytical methods without depending on presence of small parameter in the equation were developed for solving these complicated nonlinear systems. These techniques include the Homotopy Perturbation [6–13], Modified Lindstedt-Poincaré , Parameter-Expanding [15–18], Parameterized Perturbation , Multiple Scale , Harmonic Balance [20, 21], Linearized Perturbation , Energy Balance [23–25], Variational Iteration [26, 27], Variational Approach [25, 28, 29], Iteration Perturbation , Variational Homotopy Perturbation  methods, and more . Among these methods, Parameter Perturbation Method (PEM) is considered to be one powerful method that capable to handle strongly nonlinear behaviors. For this sake, we apply PEM to analysis of three practical cases [2, 33, 34] of nonlinear oscillatory system. Unlike the past investigations, here, it had assumed that the spring's property is nonlinear. The TDOF oscillation systems were consist of two coupled nonhomogeneous ordinary differential equations. So, we attempted to transform the equations of motion of a mechanical system which associated with the linear and nonlinear springs into a set of differential algebraic equations by introducing new variables. The analytical solutions of practical cases based on the cubic oscillation are presented by means of PEM for two iterations. Comparisons between analytical and exact solutions show that PEM can converge to an accurate periodic solution for nonlinear systems.
2. The Models of Nonlinear Oscillation Systems
Case 1 1 (Model of a Bulking Column). First, we consider the system shown in Figure 1. The mass can move in the horizontal direction only. Using this model representing a column, we demonstrate how one can study its static stability by determining the nature of the singular point at of the dynamic equations. This “dynamic” approach is simpler to use, and arguments are more satisfying than the “static” approach . Vito  analyzed the stability of vibration of a particle in a plane constrained by identical springs.
Neglecting the weight of springs and columns shows that the governing equation for the motion of is  where , . The spring force is given by
Case 2 2 (Two-Mass System with Three Springs). The model of two-mass system with three springs is shown in Figure 2. In this system, two equal masses are connected with the fixed supports using spring . The connection between two masses makes a compact item which is a spring with nonlinear properties. The linear coefficient of spring elasticity is and of the cubic nonlinearity is . The system has two degrees of freedom. The generalized coordinates are and .
Introducing the new variables
Transforming (2.4) yields
From (2.6), we have
Setting , (2.9) can be written as
Note that the case of corresponds to a hardening spring while indicates a softening one.
Case 3 3 (Two-Mass System with a Connection Spring). Similarly, the model of system with one spring is shown in Figure 3. Two masses, and , are connected with a spring in which linear coefficient of rigidity is , and the nonlinear coefficient is . The system has two degrees of freedom.
The generalized coordinates of the system are and . The equation of motion of the system is described by :
Solving (2.12) for yields
As mentioned, these models can be transformed to a cubic nonlinear differential equation in general form with different values and . The general form of cubic nonlinear differential is as follows:
3. Basic Idea of PEM
In order to use the PEM, we rewrite the general form of Duffing equation in the following form : where includes the nonlinear term. Expanding the solution , as a coefficient of , and as a coefficient of , the series of can be introduced as follows:
For achieving the secular term, we use Fourier expansion series as follows:
For avoiding secular term, we have
Substituting (3.11) and (3.12) into (3.10), we will achieve the first-order approximation frequency (2.10). Note that, from (3.4) and (3.12), we can find that , for all In the following section we will describe the second-order of modified PEM solution in details for solving the cubic nonlinear differential equation.
4. Application of PEM to Cubic Equation
In order to use the PEM, we rewrite (2.16) as follows:
It is possible to perform the following Fourier series expansion:
No secular term in requires that
From (4.15), the coefficients (for ) can be written as follows:
Taking into account that , (4.14) yields
To determine the second-order approximate solution, it is necessary to substitute (4.14) into (4.4). Then secular term is eliminated, and parameter can be calculated. has an infinite series; however, to simplify the solution procedure, we can truncate the series expansion of (4.14) and (4.17) and write an approximate equation in the following form:
It is possible to do the following Fourier series expansion:
The secular term in the solution for can be eliminated if
Solving (4.22) gives
On the other hand, From (4.16), the following expression for the coefficient is obtained:
Then, we can obtain
5. Analytical Solution of Practical Cases
In this section, we present the first and second approximate frequency and period values of (2.16) for different values of and . Substituting and in (4.10), (4.11), (4.26), and (4.27) gives the following results for first- and second-order approximations of the model of nonlinear SDOF Bucking Column system in Case 1:
Similarly, for and , we obtain the following frequency and period values for Case 3:
6. Results and Discussions
To illustrate and verify accuracy of PEM, comparisons with the exact solution are given in Tables 1, 2, and 3. According to the appendix, the exact frequency, , of nonlinear differential equation in the cubic form is
Using (2.8) and , we can obtain
It should be noted that contains an integral which could only be solved numerically in general. The limitation of amplitude, , in the cubic oscillation equation satisfies ; the Duffing equation has a heteroclinic orbit with period . Hence, in order to avoid the heteroclinic orbit with period for the Duffing equation in (2.16), the value of in the first two cases and for the third case should, respectively, satisfy in (6.4), (6.5), and (6.6) where and .
To illustrate and verify accuracy of this analytical approach, comparisons of analytical and exact results for the practical cases are presented in Tables 1–3 and Figures 4–10. For this reason, we use the following specific parameter and initial values: Case 1: , , , , , , Case 2: , , , , , , and Case 3: , , , , , .
Figures 4–6, which are correspond to Case 1, indicate the comparison of this analytical method for different parameter with initial values , , , and , and , and , and , , , , and which are in an excellent agreement with exact solutions.
Figures 7 and 8 represent the and diagrams which obtained analytically and exactly solving of Case 2 with different parameter and initial values with , and , , , , , . Also, the corresponding diagrams (, ) of Case 3 are plotted in Figures 9 and 10. The different parameters and initial values that used in plotting diagrams of Case 3 are: , , , , , and , , , , , .
According to these tables and figures, the difference between analytical and exact solutions is negligible. In other words, the first-order approximate results of PEM are accurate, but we significantly improve the percentage error from lower-order to second-order analytical approximations. We did it using modified PEM in second iteration for different parameters and initial amplitudes. Hence, it is concluded and provides an excellent agreement with the exact solutions.
The parameter expansion method (PEM) has been used to obtain the first- and second-order approximate frequencies and periods for Single- and Two-Degrees-Of-Freedom (SDOF and TDOF) systems. Excellent agreements between approximate frequencies and the exact one have been demonstrated and discussed, and the discrepancy of the second-order approximate frequency with respect to the exact one is as low as 0.064%. In general, we conclude that this method is efficient for calculating periodic solutions for nonlinear oscillatory systems, and we think that the method has a great potential and could be applied to other strongly nonlinear oscillators.
The exact solution of cubic nonlinear differential equation can be obtained by integrating the governing differential equation as follows: where is a constant. Imposing initial conditions , yields
Integrating (A.4), the period of oscillation is
Substituting into (A.5) and integrating where
The exact frequency is also a function of and can be obtained from the period of the oscillation as
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