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The response of a nonlinear multidegrees of freedom (M-DOF) for a nature dynamical system represented by a spring pendulum which moves in an elliptic path is investigated. Lagrange’s equations are used in order to derive the governing equations of motion. One of the important perturbation techniques MS (multiple scales) is utilized to achieve the approximate analytical solutions of these equations and to identify the resonances of the system. Besides, the amplitude and the phase variables are renowned to study the steady-state solutions and to recognize their stability conditions. The time history for the attained solutions and the projections of the phase plane are presented to interpret the behavior of the dynamical system. The mentioned model is considered one of the important scientific applications like in instrumentation, addressing the oscillations occurring in sawing buildings and the most of various applications of pendulum dampers.

Dynamical system is considered a collection of particles in motion with finite numbers of DOF and can be determined through some processes during a period of time. Chaos theory studies the behavior of dynamical systems that are oversensitive to initial conditions and is considered one of the most important subjects in applied mathematics, physics, and engineering fields. It has several applications ranging from weather forecasts, technology, and physical and life sciences. The kinematical nonlinear systems are of great interest for many outstanding researchers during the last three decades. In [

In the current work, we extend the previous work in [

Let us consider a dynamical system which consists of a mass

Description of the motion.

To gain over the governing system of motion, we consider the planar motion of our model. So after time

Use the following Lagrange’s equations of the second type to obtain the equations of motion:

In addition to the influence of the kinematic excitation on the examined system, the moments

According to the nonconservative forces, the generalized forces take the form

Substituting (

The aim of this section is obtain the analytic solutions of the previous equations of motion using the MS method and to get the modulation equations in order to obtain the resonances conditions. So it is necessary to start with the approximations of

It is customary to define both of the damping coefficients and the amplitudes of external forces, in order to achieve the analysis procedure as

The solutions of (

Substituting (

As we developed previously, substitute (

The main task of this section is to determine the resonances cases. For this purpose, we note that solutions (

Principal (primary) external resonances occurring at

The spring’s resonance arising from the kinematic excitation at

The pendulum resonance produced from kinematic excitation at

Internal resonance occurring at

Combined resonances at

The behavior of the system will be very complicated if the natural frequencies satisfy the above resonance cases. Let us examine the first case of resonance categories

Substituting (

Now, we can determine the unknown functions

Insert the following definition of modified phases:

It is worthwhile to notice that definition (

Moreover, the previous system of (

The graphical representations of the numerical solutions for the original system of (

Figures ^{2}·s^{−1}], ^{−1}], ^{−1}], ^{−1}],

Time history for the solution

Time history for the solution

The scope of this section is to obtain the steady-state solutions of both the amplitudes and modified phases corresponding to the zero values of the previous system (

If two resonances occur at the same time, (

The graphs displayed in Figures ^{−1}], ^{−1}], and ^{−1}], respectively. The dashed lines represent the locus of the roots of (^{−1}], ^{−1}], ^{2}·s^{−1}], and ^{2}·s^{−1}].

Steady-state solution when ^{−1}] and ^{−1}].

Steady-state solution when ^{−1}].

Steady-state solution when ^{−1}] and ^{−1}].

This elucidates that the desired possible solutions can be expressed as functions of the oscillating system’s parameters. Moreover, the lowest number of possible solutions is one and the maximum is may be up to seven.

One of the important factors for the mentioned problem of the steady-state oscillations is to investigate their stability. For this task, we analyze the manner of the system in a region that is very close to the fixed points.

To discuss the stability for the particular solution of the steady state, we introduce the substitutions

Take into consideration that the small perturbations

However, according to the Routh-Hurwitz criterion [

The used analysis to check the stability of the proposed solutions that are plotted in Figures ^{−1}].

Projection on the phase plane when ^{−1}], ^{−1}], and ^{−1}].

Projection on the phase plane when ^{−1}], ^{−1}], and ^{−1}].

Diagram illustrating the variation of ^{−1}], ^{−1}], and ^{−1}].

Diagram illustrating the variation of ^{−1}], ^{−1}], and ^{−1}].

Diagram displaying the portrait of ^{−1}], ^{−1}], and ^{−1}].

Diagram displaying the portrait of ^{−1}], ^{−1}], and ^{−1}].

This section is devoted to provide some examples about the motion of the considered model, especially when the axes of the ellipse become equal; that is,

In this case, we observe that the suspended point ^{−1}], ^{−1}], ^{−1}],

In this case the modulations of amplitudes for both solutions

Parts (a), (b), (c), and (d) represent the portraits of time history, while parts (e) and (f) symbolize the projections of the phase planes at

In this case we shall concentrate on the solutions when the supported point will be fixed. Figure ^{−1}], ^{−1}], ^{−1}], ^{−1}] and ^{−1}], respectively. The variation of the solution

Plots (a), (b), (c), and (d) preform the portraits of time history, while plots (e) and (f) display the projections of the phase plane when

The task of this case is to interpret the motion of the dynamical system when the pivot point moves horizontally. So the dynamical motion will be in a horizontal axis of length ^{−1}], ^{−1}], ^{−1}], ^{−1}], and

Parts (a) and (b) show the portraits of time history, while part (c) represents the variation of the solution

One of the important concepts is to study the dynamical motion when the supported point ^{−1}], ^{−1}], ^{−1}], ^{−1}], and

It must be notice that the steady-state solutions at

Parts (a) and (b) illustrate the portraits of time history, while part (c) displays the variation of the attained solutions

The analytical solutions of the derived original system (represents a nonlinear 2-DOF equations (

The authors declare that they have no competing interests.