^{1}

^{2}

^{1}

^{2}

In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameter

The pendulum models have attracted scientists and researchers with many descriptions of motions and their analysis as important examples in physics and theoretical and applied dynamics. The most important pendulum motions come from the moving of a heavy particle suspended a light rod which is jointed pivotally at a point on the ^{nd} approximation. The resonance cases and the steady-state ones are studied. The stability procedure is given using the phase plane diagrams. In [

In [^{nd} order which is solved by the small parameter perturbed technique. Numerical considerations are obtained through the Matlab program. These considerations show the validity of both analytical and numerical solutions.

In [

In this article, we consider a heavy solid suspended in a string which suspended to an ellipse in a uniform vertical plane. The equations of motion for this mathematical pendulum in terms of the two degrees of freedom are obtained, and a large parameter depends on the model properties is assumed. By the definition of the large parameter, the approximated periodic solutions are obtained using the large parameter perturbed procedure. The accuracy of these solutions is investigated through a numerical technique and computerized programs.

Let us consider the coordinate system

The pendulum model.

We assume that the axes of the system

The coordinate of the point

The kinetic energy of the system is

The potential energy of the system can be expressed as follows:

The Lagrangian of the system is [

It can be written as follows:

Let us define the following parameters.

We introduce the variables

The new Lagrangian function of the system can be written as follows:

For a small value of

According to Lagrange’s equations [

Introducing the parameters

The expressions

Now, to find the perturbed solution of the nonresonance case up to the second approximation, we apply the method of the large parameter [

Substituting from (

Coefficient of

Coefficient of

Coefficient of

Coefficient of

Making use of the equations (

Neglecting the secular terms [

In this subsection, we give a parametric analysis of the obtained results for the behavior of the obtained analytical solutions

We note that the domain of the obtained solutions under the assumed conditions is as follows:

This means that the obtained solutions are treated in a new domain which is considered as a complement space for the previous work domains.

In what follow, we study the validity of both analytical and numerical solutions. Using one of the numerical methods, we obtain numerical solutions and give more analysis of the results. The graphical representations for both solutions are obtained through computer programming.

This section is devoted to ascertaining the accuracy of the solutions being considered in Sections 3 and 4. Computer programs are developed for the representation of the obtained analytical solutions

The analytical and numerical solutions

The analytical and numerical angular velocities

The analytical and numerical solutions

The analytical and numerical angular velocities

The analytical and numerical stabilities

The analytical and numerical stabilities

The analytical and numerical stabilities

The analytical and numerical stabilities

The analytical and numerical stabilities

The analytical and numerical stabilities

This motion is very important as a mathematical pendulum model for many problems in fluid materials and gas translation in big and small tanks and is considered as a generalization for the problem studied in [

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

The author declares that he has no competing interests.