Analyzing the Stability for the Motion of an Unstretched Double Pendulum Near Resonance

: This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics. quirements of solvability of the solutions at the steady state have been obtained. The stability of the possible fixed points has been checked in line with the characteristic Equation (42) and the criterion of Routh–Hurwitz (44). Time histories sketches and resonance curves have been represented graphically. The impact of the various parameters on dynamical motion has been examined using the nonlinear stability approach. The analytical results have been compared with the numerical ones of the governing system of motion to show a high degree of consistency between them, which reflects the great accuracy of the used MST. The importance of the studied model is due to its significant applications in various fields such as physics and engineering applications that are based on vibrating systems such as shipbuilding and ships motion, structure vibration, rotor dynamics, pumps compressors, transportation devices and human or robotic walking analysis. Meth-odology, M.A.B.;


Introduction
A dynamical system is a system dependent on time. It describes the time dependency of a point in space such as a hanging clock and a current of water in a pipe [1]. It can be used in a variety of domains, including mathematics, physics, chemistry, engineering in the construction of swaying buildings, biology, economics and medicine [2,3]. Since applied mechanics is considered as a section of physical science that describes a response of the bodies' system, which started from rest or motion under the influence of external forces [4], it has been used in many fields of engineering, especially electrical and mechanical engineering, in the areas of engineering machines, rotor dynamics, pumps and compressors.
The nonlinear motion of a pendulum has piqued the interest of a number of researchers, e.g., [5][6][7][8][9]. Kyoung et al. [5] and Lee et al. [6] examined a chaotic response of a spring pendulum with two degrees of freedom (DOF). The multiple scales technique (MST) is utilized to solve the equations of motion (EOM). Moreover, they studied the bifurcation of the controlling system. In [7], the authors examined the response of the last pendulum using the MST. Additionally, the effects of internal and external resonance were studied. The impact of a higher-order approximation of an excited harmonically spring pendulum was examined in [8]. They compared the accuracy between the first and second approximation using the Lyapunov exponent method. In [9], the asymptotic and limiting phase trajectories (LPT) were applied to study the same problem in which MST and LPT were used to solve the governing system. The authors examined the impact of physical parameters on dynamical motion with the aid of LPT to describe any variation of the considered model. The analytical solutions were verified by comparing them with the numerical ones. From another perspective, the transversal tuned absorber of an exciting spring pendulum was used to control the system vibrations due to the existence of influential excitation forces, e.g., [10,11]. A numerical investigation of the positive impact of the absorber on the dynamical behaviour is presented. The asymptotic solutions of a nonlinear motion of a pendulum-type were investigated in [12] utilizing the MST, in which the trigonometric functions were approximated in the EOM using polynomial approximation. In [13], the authors explored a 2-DOF spring pendulum in an inviscid fluid flow. Approximate solutions were gained using the same approach as previously up to the second order of approximation. Resonance curves, steady-state solutions and stability of motion were examined and represented graphically. The dynamic response of a spring pendulum was investigated in [14,15], in which the MST was applied to gain the asymptotic solutions of the analyzed systems. A generalization of this work is found in [16], where the authors studied a damped spring motion that follows an elliptic route. An extension of this research was found in [17,18], when the suspension point was travelling in the same direction, with a constant angular velocity of a linear spring and nonlinear one, respectively. The case of the attached tuned absorber with the spring was investigated in [19]. The derived EOM were solved analytically, and the modulations equations were acquired in the line of the considered resonance cases. The time histories and the resonance curves were plotted to reveal the impact of the body's parameters on the motion. Another trajectory for the motion of the pivot point was considered in [20], in which the authors elaborated the dynamical motion of the 2-DOF damped elastic pendulum in a Lissajous curve. Conversely, the plane motion of the 3-DOF rigid body pendulum with a fixed suspension point was investigated in [21]. The latter two works are generalized in [22] when the supported point of the damped rigid body moved in a path similar to the Lissajous curve. The authors discussed all the possible amplitudes of the steady-state solutions for different parameters of the considered models in light of the perturbation approach used. All cases of resonance were categorized, and the case of simultaneously parametric and primary resonances was examined. A plane motion of a triple pendulum system was investigated in [23,24] in the presence of free conditions. The natural frequencies of this model are studied in [23], while the recurrence plot method was used in [24] to achieve the required solutions of the derived EOM.
This paper studied the dynamical motion of the 2-DOF double pendulum system with two different massless rods of constant lengths. It is assumed that its supported point moves along an elliptical path with a stationary angular speed. The governing EOM were derived using Lagrange's equations and were solved utilizing the MST up to the third order of approximation. The solvability conditions of the solutions at the steady state were determined. The resonance cases were obtained and categorized, in which the case of primary external resonance was examined. The curves of the time histories of the attained solutions and resonance ones were plotted with various selected values of the used parameters to reveal the good impact of these parameters on the dynamical behaviour of the investigated system. The analytical results were compared with the numerical calculations to reflect the high matching between them. The nonlinear stability analysis was used to investigate the characteristics of the dynamical motion. The importance of the model selected lies in its diverse set of applications in different fields, for example, in swaying buildings.

Dynamical Modelling
Nonlinear vibration of the 2-DOF dynamical paradigm consisting of a double pendulum is investigated in the present study. It is supposed that the supported point Q of the modelling system travels along an elliptic track with lengths 2b and 2a of its major and minor axes, respectively. This point is connected with the double pendulum with constant lengths 1  and 2  , see Figure 1. Consider N to be the compatibility point of Q on the spare circle b , in which it moves with a constant angular speed Ω . The kinematics Cartesian coordinates of Q are written in the following form: The kinetic and potential energies of the system can be expressed as follows: Using Lagrange's functions V T L − = , one can derive the governing EOM by using the following Lagrange's equations of the second type: Here, 1 θ and 2 θ are the system's generalized coordinates, and 1 Q θ and 2 Q θ are the generalized forces that have the following forms: Let us consider the following frequencies and parameters: (3) and (4), we can obtain the EOM in the following dimensionless forms:

Analysis of the Solution
In this section, the MST is applied to solve the EOM (5) and (6) analytically up to the third order of approximation, aiming to obtain the solvability resonance conditions and the modulation equations. To achieve this goal, the trigonometric functions sin , cos , sin , ϕ ϕ γ and cosγ can be approximated up to the third order using Taylor's series. Therefore, Equations (5) and (6) can be rewritten in the forms: As predicted, the amplitudes of all the oscillations are assumed to be small; hence, we may express them in terms of a small parameter ε , i.e., θ τ εφ τ ε θ τ εγ τ ε = = The approximate solutions φ and γ are sought in a power series of ε in the framework of the MST as follows: Here, τ τ are defined as the fast and slow time scales, respectively. Regarding these scales, the time derivatives will be transformed to these scales as follows: The higher terms of (11) are neglected. Suppose that the amplitudes of generalized forces, as well as some parameters, are relatively small, i.e., Substituting Expressions (9)-(12) into Equations (7) and (8), and then equalling the coefficients of the same powers of ε on both sides, we obtain the following groups of partial differential equations (PDE): Order of ( ) ε : Order of 2 ( ) ε : Order of 3 ( ) ε : A closer examination of the preceding six PDEs reveals that they can all be solved subsequently. According to the homogeneity of Equations (13) and (14), we can write the general solutions of them in the following forms: Here, ( 1,2) j j Α = are complex functions of the slow time scales 1 τ and 2 τ that will be determined later, while i Α refers to their complex conjugate.
By substituting the above first order solutions (19) and (20) into the second order equations of ε (15) and (16), and then eliminating the terms that yield the secular one, we have the elimination conditions in the following forms: Thus, we can write the solutions of the second order of approximation as follows: .
Here, CC represent the preceding complex conjugates terms.
Making use of (19), (20), (23) and (24) in Equations (17) and (18) yields secular terms. To eliminate these terms, it is required that Therefore, the solutions of the third order of approximation become With the use of the following initial conditions, we can determine the unknown functions of ( 1,2) j A j = from the system of the four differential equations that are represented by Equations (21), (23), (25) and (26):

Vibrations and Conditions of Resonance
In this section, we classify the arising different resonance cases, investigate two of them and obtain modulation equations in the framework of these cases. In general, resonance cases can be distinguished when the denominators of the above approximate solutions tend to zero [25]. Therefore, we can categorize them as a main (primary) external resonance that is found at If one of the resonance cases is satisfied, such as internal resonance, we can expect that the behaviour of the dynamical system to be challenging. The above asymptotic solutions observed in the last part are right if the oscillations escape from the resonances. If any of the preceding requirements are met, that indicates the necessity to regulate the method used. Now, we are going to examine two cases of main external resonances cases that occur simultaneously. Therefore, we consider the cases 1 then the effect of the resonance is then expressed in secular terms. The parameters of detuning are known as the distance gauge of the oscillations from the strict resonance [26].
As a result, they can be expressed in terms of ε as follows: By substituting (29) and (30) into (7) and (8), and focusing on the secular terms, the solvability conditions can be acquired from the elimination of these terms, which have the following forms: • For the second order approximation Referring to the last conditions, we can say that the observed conditions of solvability for the model constitute a nonlinear system of PDE with respect to the functions j A .
These functions depend only upon the time scale 2 τ . Therefore, we can write them as the following notations of polar forms: It is clear that j A is an independent function of 0 τ and 1 τ . Therefore, we can write the first order derivative as follows: Based on the above formula, we can turn the PDE Equations (32) and (33) into ordinary differential equations (ODE). Then, the modified phases can be represented according to 1 2 2 ( , ) ( ); ( 1, 2). By making use of (34)-(36) in (32) and (33), and then separating the real parts and imaginary ones, we can obtain the following:  Looking closely at the curves of Figure 6, we conclude that there is an excellent match between the analytical and numerical solutions in the comparison between them, suggesting high accuracy of the MST.
When we follow fluctuations of the studied dynamic system; Equation (37) allows us to see whether they are steady or not. The drawn trajectories of the amplitudes and the adjusted phases are thought to be the best way to characterize the behaviour of the system.

Steady-State Case
The major goal of this section is to investigate the oscillations of the studied model at the steady state. The modified phases and amplitudes at the steady state can be gained from Equation (37) when the value of the derivatives of their left-hand side is zero, i.e., To investigate the stability near fixed points and to examine the stability of the particular steady-state solutions, we consider 11 and ϕ 21 denote the perturbations that are supposed to be smaller.
By substituting (40) into (37), we can obtain the linearized equations in the following forms: Based on the Routh-Hurwitz criterion [27] and the following determinant:

Det
we can write the stability conditions at the steady-state solutions in the following forms: , as seen in Figure 10. Moreover, Figure 11 shows the modulation amplitudes versus the detuning parameter

Non-Linear Analysis
In this section, we will demonstrate the properties of nonlinear amplitude for the system of Equations (37) and investigate their stabilities [28][29][30] , Making use of (11) and (45) in (37), and separating the real parts and the imaginary ones yields the next system, which is as follows: In distinct parametric regions, the altered amplitudes were then analyzed across the used time interval and the attributes of amplitudes were displayed in the curves of the phase plane, as graphed in Figures 12-14, taking into account the prior data of the parameters that have been used.
The ω . It is observed that symmetric closed curves are portrayed to assert our prediction of the stability and the steady motion of the investigated system.   u v , (b) on the plane 2 2 u v .

Conclusions
The problem of the nonlinear dynamical motion of a 2-DOF double pendulum with different lengths has been investigated. It has been considered that its pivot point moves in an elliptic trajectory with a steady angular velocity. The governing EOM have been derived utilizing Lagrange's equations and have been solved up to the third order of approximation using the MST. All the arising resonances cases have been classified. The re-quirements of solvability of the solutions at the steady state have been obtained. The stability of the possible fixed points has been checked in line with the characteristic Equation (42) and the criterion of Routh-Hurwitz (44). Time histories sketches and resonance curves have been represented graphically. The impact of the various parameters on dynamical motion has been examined using the nonlinear stability approach. The analytical results have been compared with the numerical ones of the governing system of motion to show a high degree of consistency between them, which reflects the great accuracy of the used MST. The importance of the studied model is due to its significant applications in various fields such as physics and engineering applications that are based on vibrating systems such as shipbuilding and ships motion, structure vibration, rotor dynamics, pumps compressors, transportation devices and human or robotic walking analysis.