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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 [

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 [

In this section, a practical case of nonlinear oscillation system of SDOF in Case

First, we consider the system shown in Figure

Model for the bulking of a column [

Neglecting the weight of springs and columns shows that the governing equation for the motion of

The model of two-mass system with three springs is shown in Figure

. Model of the two-mass system with three springs [

The mathematical model of the system is [

Introducing the new variables

Transforming (

From (

Substituting (

Setting

Note that the case of

Similarly, the model of system with one spring is shown in Figure

Model of the two-mass system with spring [

The generalized coordinates of the system are

Similar to the previous section, to simplify these equations, we apply the variables that was introduced in (

Solving (

Substituting (

As mentioned, these models can be transformed to a cubic nonlinear differential equation in general form with different values

In order to use the PEM, we rewrite the general form of Duffing equation in the following form [

Substituting (

Considering the initial conditions

For achieving the secular term, we use Fourier expansion series as follows:

Substituting (

For avoiding secular term, we have

Setting

Substituting (

In order to use the PEM, we rewrite (

Substituting (

Considering the initial conditions

It is possible to perform the following Fourier series expansion:

Substituting (

No secular term in

Setting

Substituting (

From (

The periodic solution of (

Substituting (

From (

Taking into account that

To determine the second-order approximate solution, it is necessary to substitute (

Substituting

It is possible to do the following Fourier series expansion:

Substituting (

The secular term in the solution for

Solving (

On the other hand, From (

Then, we can obtain

From (

Comparing the right hands of (

In this section, we present the first and second approximate frequency and period values of (

Also, we can obtain the first and second-order approximations solutions for Case

Similarly, for

To illustrate and verify accuracy of PEM, comparisons with the exact solution are given in Tables

Comparison of approximate and exact periods for Case 1.

Constant parameters | Approximate solutions | Exact solution | ||||||||

1 | 1 | 1 | 10 | 5 | 1 | 1.96254 | 1.96451 | 1.96451 | 0.101% | 0.000% |

5 | 1.5 | 5 | 5 | 6 | 3 | 3.23743 | 3.32518 | 3.32368 | 2.664% | 0.045% |

10 | 10 | 10 | 10 | 50 | 10 | 0.32418 | 0.33145 | 0.33143 | 2.210% | 0.031% |

50 | 25 | 40 | 30 | 100 | 20 | 0.25640 | 0.26216 | 0.26208 | 2.216% | 0.031% |

70 | 20 | −30 | 50 | 100 | 10 | 0.60486 | 0.61827 | 0.61809 | 2.187% | 0.030% |

100 | 50 | 150 | 70 | 20 | 100 | 0.16221 | 0.16586 | 0.16580 | 2.218% | 0.031% |

500 | 150 | 220 | 120 | 500 | 0.5 | 9.67637 | 9.71682 | 9.71672 | 0.417% | 0.001% |

1000 | 500 | 1000 | 500 | 500 | 1 | 6.73241 | 6.75877 | 6.75871 | 0.391% | 0.001% |

Comparison of approximate and “Exact” frequencies for Case 2.

Constant parameters | Approximate solutions | Exact solution | ||||||||

1 | 1 | 1 | 1 | 5 | 1 | 5.1962 | 5.1068 | 5.1078 | 1.73% | 0.0185% |

2 | 1 | 3 | 5 | 8 | 10 | 4.3012 | 4.2401 | 4.2406 | 1.43% | 0.0185% |

5 | 10 | 20 | 30 | −10 | 10 | 60.08328 | 58.7677 | 58.7856 | 2.21% | 0.0305% |

10 | 50 | 70 | 90 | 20 | −40 | 220.4972 | 215.6448 | 215.7113 | 2.22% | 0.0308% |

10 | 25 | 20 | 0.5 | −10 | 10 | 6.0415 | 5.9533 | 5.9541 | 1.47% | 0.0132% |

100 | 200 | 300 | 400 | −50 | 50 | 244.9653 | 239.5715 | 239.6455 | 2.22% | 0.0309% |

Comparison of approximate and “Exact” frequencies for Case 3.

Constant parameters | Approximate solutions | Exact solution | ||||||||

1 | 2 | 5 | 1 | −4 | 1 | 5.9687 | 5.8885 | 5.8892 | 1.35% | 0.011% |

3 | 5 | 2 | 5 | 5 | −5 | 14.1798 | 13.8710 | 13.8752 | 2.20% | 0.011% |

1 | 5 | 5 | 1 | 5 | −5 | 9.7980 | 9.6096 | 9.6119 | 1.94% | 0.023% |

10 | 5 | 10 | 10 | 20 | 30 | 15.0997 | 14.7763 | 14.7806 | 2.16% | 0.029% |

5 | 10 | 50 | −0.01 | −20 | 40 | 2.6268 | 2.5452 | 2.5468 | 3.14% | 0.064% |

100 | 1 | 10 | 5 | 20 | 25 | 10.2366 | 10.0545 | 10.0564 | 1.79% | 0.020% |

50 | 100 | 50 | 100 | 100 | 25 | 112.5067 | 110.0293 | 110.0633 | 2.22% | 0.031% |

1000 | 100 | 200 | 300 | 400 | 200 | 314.6461 | 307.7164 | 307.8115 | 2.17% | 0.031% |

Substituting

Substituting

Using (

The first- and second-order analytical approximation for

Similarly, substituting

After obtaining

It should be noted that

To illustrate and verify accuracy of this analytical approach, comparisons of analytical and exact results for the practical cases are presented in Tables

Comparison of approximate periodic solutions of Bucking of a Column equation (Case 1) with the exact one for

Comparison of approximate periodic solutions of Bucking of a Column equation (Case 1) with the exact one for

Comparison of approximate periodic solutions of Bucking of a Column equation (Case 1) with the exact one for

Comparison of the first- and second-order analytical approximate solutions with the exact solution for

Comparison of the first- and second-order analytical approximate solutions with the exact solution for

Comparison of the first- and second-order analytical approximate solutions with the exact solution for

Comparison of the first- and second-order analytical approximate solutions with the exact solution for

Figures

Figures

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

The exact solution of cubic nonlinear differential equation can be obtained by integrating the governing differential equation as follows:

Equating (

Integrating (

Substituting

The exact frequency