Modeling and Stability Analysis for the Vibrating Motion of Three Degrees-of-Freedom Dynamical System Near Resonance

: The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using Lagrange’s equations, which is then solved analytically up to the third approximation employing the approach of multiple scales (AMS). The emerging cases of resonance are categorized according to the solvability requirements wherein the modulation equations (ME) have been found. The stability areas and the instability ones are examined utilizing the Routh–Hurwitz criteria (RHC) and analyzed in line with the solutions at the steady state. The obtained results, resonance responses, and stability regions are addressed and graphically depicted to explore the positive inﬂuence of the various inputs of the physical parameters on the rheological behavior of the inspected system. The signiﬁcance of the present work stems from its numerous applications in theoretical physics and engineering.


Introduction
In the last two decades, some researchers have produced numerous works trying to solve the problems of excessive vibrations of mechanical systems, including the use of absorbers to treat and absorb active and the passive vibrations, e.g., [1][2][3][4][5][6].
The motion of a pendulum vibration absorber (PVA) with a spin base is investigated in [2] to deal with vertical excitation. By altering the rotational motion, the distinctive frequency of the pendulum absorber can be modified dynamically over a large range. A longitudinal absorber is used in [3] to stabilize and regulate vibrations of a spring pendulum, with non-linear stiffness, expressing ship roll motion. To achieve a semi-closed solution for the approximation from second order, the authors used the approach of multiple scales (AMS) [7], investigating the response of the considered model near resonance cases. They applied the influence of an additional transvers absorber to generalize this problem as in [4] and [5]. It is demonstrated in [6] how to autonomously modify the rotating speed of PVA, with two degrees of freedom (DOF), by identifying the phase between the PVA and primary vibrations. The pivot's motion of a simple pendulum with rigid arm that connected with a longitudinal absorber on an elliptic trajectory is examined in [8]. All resonance cases are generally grouped, and the case of two concurrent basic external resonances is examined. The generalization of this work is found in [9] for the case of a damped elastic pendulum instead of the un-stretched one. The ME are obtained and solved numerically to check the stability and instability regions in view of RHC.

Description of the Vibrating System
A vibrational system with 3DOF is described in Figure 1, in which a damped elastic pendulum with normal length l 0 , non-linear stiffness K 1 and K 2 , and pendulum mass m 1 is considered. A pivot point O 1 of the pendulum is limited to move in an elliptic route with stationary angular velocity Ω, while the pendulum's other end is attached to a linear absorber of mass m 2 , a normal length l 10 , and linear stiffness K 3 . According to the sketch of Figure 1, we may write the corresponding coordinates of O 1 to the axes OX and OY in the forms a cos(Ωt) and b sin(Ωt), respectively. Here, the ellipse's minor and major axes are represented by a and b, respectively. On the supplementary circle b, the equivalent point of O 1 will be assigned by Q.

Description of the Vibrating System
A vibrational system with 3DOF is described in Figure 1, in which a damped elastic pendulum with normal length 0 l , non-linear stiffness 1 K and 2 K , and pendulum mass 1 m is considered. A pivot point 1 O of the pendulum is limited to move in an elliptic route with stationary angular velocity , while the pendulum's other end is attached to a linear absorber of mass 2 m , a normal length 10 l , and linear stiffness 3 K .
According to the sketch of Figure 1, we may write the corresponding coordinates of 1 O to the axes OX and OY in the forms cos( ) a t  and sin( ) b t  , respectively. Here, the ellipse's minor and major axes are represented by a and b , respectively. On the supplementary circle b , the equivalent point of 1 O will be assigned by Q .
The system's motion is addressed to be under the influence of applied harmonic force 1 1 ( ) cos( ) F t F t   at the spring's radial direction, as well as a harmonic moment  , respectively. Furthermore, 1 2 , , C C and 3 C are thought to indicate the coefficients of viscous damping for the spring's longitudinal, swing oscillations, and the absorber's elongation, respectively.  The system's motion is addressed to be under the influence of applied harmonic force F(t) = F 1 cos(Ω 1 t) at the spring's radial direction, as well as a harmonic moment M(t) = M 0 cos(Ω 2 t) at O 1 in the anticlockwise direction. Here, Ω 1 , Ω 2 and F 1 , M 0 are the frequencies and amplitudes of F(t) and M(t). The extensions of the spring and absorber are supposed to be Z(t) and ξ(t), respectively. Furthermore, C 1 , C 2 , and C 3 are thought to indicate the coefficients of viscous damping for the spring's longitudinal, swing oscillations, and the absorber's elongation, respectively.
According to Equation (1), Lagrange's function L = T − V can be determined, and then, the controlling system of motion can be obtained using the next equations of Lagrange [30] Here, (Q Z , Q Φ , Q ξ ) are the system's non-conservative generalized forces, while (Z, Φ, ξ) and (Z , Φ , ξ ) are the generalized coordinates and velocities, respectively. The forms of Q Z , Q Φ , and Q ξ are Substituting from (1) and (3) into (2), the dimensionless form of the controlling system of EOM has the following form ..
where 11, 11943 5 of 40 Denote the parameters that are dimensionless, and the dots represent the differentiation regarding to τ, in which the generalized coordinates and their corresponding first derivatives have the following initial conditions

The Desired Solutions
In this part of the present research work, we use the AMS to acquire the approximate analytic solutions of the EOM (4), categorize the different cases of resonance, and extract both the solvability criteria and the ME. Then, we look at the oscillations of the system close to the static equilibrium position [31]. To accomplish this target, we approximate the functions sin Φ and cos Φ up to the third-order as follows The damping coefficients, force's amplitudes, moment, elliptical semi-axes, and other parameters can then be represented in terms of a small parameter 0 < ε << 1 as follows As a result, we can express the functions z, Φ, and η in terms of ε and the new functions z, Φ, and η as follows According to the procedure of AMS, we can write these functions as follows where τ n = ε n τ (n = 0, 1, 2) denote new time dependent scales on τ, in which τ 0 and τ k (k = 1, 2) denote a fast and slow time scales, respectively. Because of the smallness of ε, the orders ε 3 and higher have been excluded. In light of the supposed solutions (8), we need to transform the time derivatives regarding τ into additional ones with respect to the scales τ 0 , τ 1 , and τ 2 . Therefore, we consider the following differential operators where D n = ∂ ∂τ n ; n = 1, 2. When (5)-(9) are substituted into (4), a family of partial differential equations (PDE) regarding ε arises. Equating the coefficients of the like powers of ε on both sides of each one of the families of PDE to gain the below groups of PDE gives as a result Order of ε Order of ε 2 Order of ε 3 Based on the preceding groups of Equations (10)-(12), we can solve them sequentially. Accordingly, we will start with the general solutions of the system of Equation (10) which take the following forms Here, B j (j = 1, 2, 3) represent functions of τ k (k = 1, 2) and B j denote their complex conjugates.
Making use of the above solutions (13) into the second group of PDE (11) yields secular terms, the removal of which demands that As a result, the second-order solutions are as follows: Here, c.c. signifies the complex conjugate of the prior terms. Substituting (13)- (15) into the third group of PDE (12) and removing terms that produce the secular one to gain the requirements of solvability of the third order of approximation results as follows Based on the foregoing, we can phrase the third-order solutions as in the forms where q s (s = 1, 2, 3, . . . , 33) are given in Appendix A. In light of the removal conditions (14) and (16) of secular terms, we can estimate the functions B j (j = 1, 2, 3). We can easily acquire the desired approximate analytical expressions of z, Φ, and η up to the third approximation in view of the uses of the hypothesis (7), solutions (8), and the attained solutions (13), (15), and (17).

Resonance Categorizes and Modulation Equations (ME)
In this section, we look at how to categorize the various cases of resonance based on the aforementioned solutions, which are legitimate as long as their dominators are not zero [8]. As these dominators go closer to zero, resonance cases emerge. As a result, these cases can be categorized into the fundamental external case of resonance which is met at p 1 = 1, p 2 = w 1 , p 2 = w 2 ; the internal case of resonance which is found at w 1 = 1, w 2 = 1, = 1, w 1 = 2, 2 w 1 = 1, w 2 = 2, = w 1 , = w 2 , w 1 = w 2 , 3w 1 = w 2 , 2w 1 = w 2 , w 1 = 0, w 2 = 0, = 0; and the combined resonance case which is encountered It should be emphasized that if any of the prior resonance cases is achieved, the behavior of the examined system would be difficult. As a result, the methods employed would have to be modified.
We will look at two fundamental external resonances and one internal resonance that are carried out simultaneously to handle this situation. As a consequence, we take into account the occurrence of all three of the following cases at the same time: These relations (18) indicate how close p 1 , p 2 , and 3w 1 are to 1 , w 1 , and w 2 , respectively. To do this, the dimensionless values defined by the parameters σ j (j = 1, 2, 3) of detuning (which characterize the distance between the oscillations and the rigorous resonance) can be inserted as follows: Therefore, the order of σ j can thus be inferred as follows: Substituting (19) and (20) into (11) and (12), and then removing the generated secular terms, we get the relevant solvability requirements that are based on the approximated equations as follows: According to a careful inspection of the foregoing solvability conditions, we have a system of six non-linear PDE in terms of the functions B j (j = 1, 2, 3) that are dependent on the slow scales τ k (k = 1, 2). Then, we can introduce the following polar form of these functions: where ψ j and a j denote real functions of the phases and their amplitudes of z, Φ, and η. Based on the above analysis, the first-order derivative of operators B j (j = 1, 2, 3) can be stated as follows: Therefore, we can convert the PDE (21) into ordinary differential equations (ODE) according to the uses of (22), (23), and the next modified phases, into the requirement of solvability (21). Partitioning the real parts and the imaginary ones yields the next system of six first-order ODE in terms of a j and θ j (j = 1, 2, 3) This system reveals the ME for both a j and θ j (j = 1, 2, 3) of the studied three cases of resonances. For the following selected values of the physical parameters of the considered model, the solutions of the system are graphically displayed in distinct plots as drawn in  Figure 5 and ω 1 = 3.354 as in Figure 6.
When the amplitudes a j and the adjusted phases θ j are varied with time τ for distinct values of the damping coefficients c j (j = 1, 2, 3) and the frequencies ω k (k = 1, 2), we can predict that the above system (25) has a good influence according to these values.
According to the plotted curves in Figure 2, we observe that when c 1 has various values, the time histories of the amplitude a 1 and the adjusted phase θ 1 behave as decaying waves until reaching a stationary behavior at the end of the investigated period of time, as seen in Figure 2a,b. The fluctuations of a 2 and θ 2 waves with time τ become clear in the first quarter of the period of time and become stationary after that as explored in Figure 2c,d. On the other hand, a sharp descent of the curves describing the waves of a 3 and θ 3 is observed in the curves of Figure 2e,f, which is due to the last two equations of the system (25). There is no variation of the a 2 , a 3 and θ 2 , θ 3 curves with the change of c 1 values due to the formulations of the equations of these curves.      The change of the various values of c 2 is evident in the curves describing the time histories of the amplitude a 2 and the modified phase θ 2 because the third and fourth equations of system (25) are dependent on c 2 , as seen in Figure 3c,d, while the other equations do not depend explicitly on c 2 . Therefore, there is no variation, to some extent, of the curves describing a 1 , a 3 and θ 1 , θ 3 as drawn in the other parts of Figure 3.
Since the last two equations of system (25) depend on c 3 , an observed variation of the curves describes the modified phase θ 3 , as seen in Figure 4f. There is no observed variation in the curves of the other variables because the first four equations of system (25) do not depend on c 3 explicitly as indicated in the other parts of Figure 4.
An examination of the system of equations (25) shows that these equations are dependent on ω 1 and ω 2 . Therefore, we expect a good impact of these parameters on the time histories of a j (j = 1, 2, 3) and θ j which met with the plotted curves of Figures 5 and 6. These curves describing the waves of these variables oscillate in a decaying manner as drawn in parts (a)-(d) of these figures or monotonically decrease with time as seen in parts (e) and (f) of the same figures. Based on this analysis, we come to the conclusion that the behavior of the system of equations (25) is stable and free of chaos.
Figures 7-11 present the phase plane diagrams of a j (j = 1, 2, 3) and θ j when c j and ω k (k = 1, 2) have various values. An inspection of the curves of these figures shows that we have spirals curves that are directed to one point, which gives an impression of the steady motion of these amplitudes and phases.   Figures 10 and 11, respectively. Therefore, we can say that these curves have a spiral form from the outside to the inside, and it is directed toward a single point for each curve, which means that all values of ( 1,2) k k   have a significant impact on the curves of these planes. The reason goes back to the equations of system (25) that depend explicitly on    Parts of Figures 7-9 are drawn when c 1 , c 2 , and c 3 have different values, respectively, to reveal the variation of curves of the phase planes a j θ j (j = 1, 2, 3) with these values, while Figures 10 and 11 describe the change of these planes that happened at different values of ω 1 and ω 2 , respectively.
According to the curves of these figures and the system of equations (25), we observe that the plane a 1 θ 1 is impacted by the various values of c 1 , as seen in Figure 7a, while there is no variation of the curves of planes a 2 θ 2 and a 3 θ 3 as indicated in Figure 7b,c. The curves of the phase planes a 2 θ 2 and a 3 θ 3 have been impacted with the variation of c 2 values of as seen in Figure 8b,c. On the other hand, there is no observed variation of the curves drawn in plane a 1 θ 1 when c 2 changes, as noticed in Figure 8a. According to the plotted curves in Figure 9, we can see that the curves shown in parts (a) and (b) that describe the phase planes a 1 θ 1 and a 2 θ 2 respectively, have no variation with various values of c 3 . The good impact of the values of c 3 is observed in parts (c), (d), and (e) of Figure 9 for the phase plane a 3 θ 3 .
The influence of the frequencies ω 1 (=3.316, 3.354, 3.391) and ω 2 (=3.212, 3.131, 3.084) on the phase plane diagrams a j θ j (j = 1, 2, 3) is observed from the curves of Figures 10 and 11, respectively. Therefore, we can say that these curves have a spiral form from the outside to the inside, and it is directed toward a single point for each curve, which means that all values of ω k (k = 1, 2) have a significant impact on the curves of these planes. The reason goes back to the equations of system (25) that depend explicitly on ω k (k = 1, 2).
It is important to remember that the obtained approximate solutions z, Φ, and η describe the spring's elongation, the rotation angle at the point O 1 , and the elongation of the transverse absorber, respectively.    0 1000 .      0 1000 .

Steady state Solutions
The major objective of the present section is to study the oscillations of the examined system in the case of steady state. From the equations of system (25)  Based on the sketched curves of the solutions z, Φ, and η, we observe that the waves describing these solutions have a periodic manner in which the number of oscillations and their wavelengths remain stationary to some extent with the variation of c j (j = 1, 2, 3) values. Parts (a) of these figures have an explicitly periodic form for the wave of the solution z. It is notable from parts (b) of these figures that each period of the wave contains a constant number of vibrations that are repeated for each period. This is due to the analytical form of the rotation angle Φ, in which its behavior has a spinning form. On the other hand, the wave describing spring's elongation η experiences rapid oscillations at the beginning of the motion due to the absorber's effect and damping impact on the investigated dynamical system, in which it settles down after that and vanishes at the end of time interval, as seen in parts (c) of these figures.
According to the calculations of Figures 15 and 16, we get to the conclusion that the change of the ω k (k = 1, 2) values has a considerable impact on the attitude of the describing waves of the attained solutions. Regardless of the fact that the wave's behavior of the solutions is periodic, we observe that the amplitudes of these waves increase and decrease with the increasing of ω k as seen in Figures 15a-f and 16a-f, respectively.

Steady State Solutions
The major objective of the present section is to study the oscillations of the examined system in the case of steady state. From the equations of system (25), we can obtain both of the modified phases θ j (j = 1, 2, 3) and amplitudes a j in the steady state case. Alternatively, the zero values of the left-hand sides of the equations of this system are considered. Therefore, we consider da j dτ = 0 and dθ j dτ = 0 [32], to obtain the next algebraic system of six equations of the functions θ j and a j .
Now, we can remove the adjusted phases θ j from the preceding system to produce the following three non-linear algebraic equations between longitudinal amplitude a 1 , the swing oscillations a 2 , and the frequency represented by the detuning parameters σ j and the absorber's amplitude a 3 .
Stability testing is considered a crucial aspect of the vibrations in the steady state case. To explore such a circumstance, the behavior of the system in a domain relatively near to fixed points is investigated. Therefore, the substitutions listed below are employed in (25) to achieve this purpose. a 1 = a 10 + a 11 , θ 1 = θ 10 + θ 11 , a 2 = a 20 + a 21 , θ 2 = θ 20 + θ 21 , a 3 = a 30 + a 31 , Here, a j0 and θ j0 (j = 1, 2, 3) denote the steady state solutions, whereas a j1 and θ j1 represent relatively small disturbances in relation to a j0 and θ j0 . As a result of linearization and the reality of the fixed points of (25), we get da 11 dτ = 1 2 ( f 1 θ 11 cos θ 10 − c 1 a 11 ), a 10 1 a 20 (a 20 a 11 + 2a 10 a 21 ) + 1 2 f 1 θ 11 sin θ 10 , Since a j1 and θ j1 (j = 1, 2, 3) are perturbed functions of amplitudes and phases of the aforementioned linear system. Then, the linear function k s e λτ (s = 1, 2, 3, 4, 5, 6) of the exponential form can be used to express about their solutions, where k s and λ are constants and the perturbation's eigenvalue, respectively. The real parts of the roots of the next characteristic equation of (29) should be negative if the steady state solutions are asymptotically stable [33,34].
where Γ s (s = 1, 2, . . . , 6) are functions of a j0 , θ j0 , and c j (see Appendix B). The required and sufficient requirements of stability for certain solutions at steady state can be expressed as follows

The Stability Analysis
In this section, we investigate the model's stability as well as its non-linear evolution using the Routh-Hurwitz non-linear stability approach. It must be remembered that a damped spring connected with a transverse absorber under the action of F(t) and M(t). Some factors have been revealed to play a substantial impact in the stability processes such as damping's constants c j (j = 1, 2, 3), frequencies ω k (k = 1, 2), and parameters of detuning σ j . To gain the stability plots of system (25), a specific process with various parameters of the system has been used. The adjusted amplitudes a j (j = 1, 2, 3) plotted via time are for various parametrical regions, in addition to the graphical representations of their characteristics through the phase plane paths. Curves of frequency responses of a j versus σ 2 and the system's fixed points have been portrayed in Figures 17-31, in which the flowing data have been taken into account besides the previous ones.         It is obvious that c 1 has a bigger role on the curves of plane a 1 σ 2 than on the frequency response curves of planes a 2 σ 2 and a 3 σ 2 , which is due to the formulation of system (25). It is noted that all parts of these figures have only one critical fixed point, and this means that we have one only area for both stability and instability. Stable fixed points have been detected in the area −0.5 ≤ σ 2 ≤ −0.04 while the unstable fixed points have been found in the area −0.04 < σ 2 ≤ 0.5, in which the stable points and the unstable ones have been presented by solid and dashed curves, respectively.

Non-Linear Interpretations
This section focuses on elucidating the non-linear amplitude's characteristics of system (25) as well as evaluating their stability. As a result, the following transformations are taken into account [31,35] First, (32) were substituted into (25), and then, the real and imaginary parts were separated to produce  Figures 26-28, we conclude that there exists one peak point with different locations, and each curve has just one essential fixed point. It is clear that ω 1 has a significant impact on the frequency response curves because the equations of system (25) depend directly on the frequencies parameters. Moreover, the stable and unstable regions of the fixed points are calculated as in Table 1.
The above remarks can be applied to the curves of Figures 29-31 when the frequency values ω 2 (=3.212, 3.131, 3.084) are considered. The ranges of stable fixed points and unstable ones have been given in Table 2.

Non-Linear Interpretations
This section focuses on elucidating the non-linear amplitude's characteristics of system (25) as well as evaluating their stability. As a result, the following transformations are taken into account [31,35] First, (32) were substituted into (25), and then, the real and imaginary parts were separated to produce .
where U j = ε u j , V j = ε v j ; (j = 1, 2, 3). The adjusted amplitudes were then modified throughout time in various parametric zones and the amplitude's properties were depicted in phase-plane curves. Then, the parameters prior values are taken into account to plot  Figures 35 and 36 when ω k (k = 1, 2) has distinct values. Moreover, the curves in planes u j v j , represented in parts (c), (f), and (i), behave as spiral curves and move toward a single point to confirm that the motion is free of chaos.
A closer look to Figures 32-34 reveals that the new parameters u 1 and v 1 besides the phase plane curves u 1 v 1 have been impacted with the change of c 1 more than the others values of c 2 and c 3 . The time histories of u 2 , v 2 and u 3 , v 3 in addition to the curves of the planes u 2 v 2 and u 3 v 3 have been impacted with the change of the damping parameters c 2 and c 3 , respectively. The principal reason goes back to the nature formulation of the system of equations (33). The spiral curves are directed from the outside to the inside, describing the stability of the studied system.   Figures 35 and 36. It is noted that these waves have been impacted with the various values of the frequency parameters, in which decay curves of the time histories are obtained and spiral ones of the phase plane toward one single point are plotted, indicating that the motion is smooth, steady and without disorder.

Conclusions
The non-linear motion of a damped spring pendulum with an attached linear damped transverse absorber in the direction of the spring has been investigated. Under the impact of a harmonic force and moment, the motion of the pendulum's hanging point has been constrained to an elliptic path. The EOM have been derived applying Lagrange's equations from the second kind. The AMS has been used to achieve the approximate solutions up to the third order. Based on the solvability requirements, the ME The good effectiveness of the change of the frequency parameters ω 1 and ω 2 on the dynamical behavior of considered system (33) has been shown in parts of Figures 35 and 36. Time histories curves of the new parameters u j (j = 1, 2, 3) and v j are plotted in parts (a), (d), and (g) and (b), (e), and (h), respectively. Whereas the plane curves u j v j have been drawn in parts (c), (f), and (i) of Figures 35 and 36. It is noted that these waves have been impacted with the various values of the frequency parameters, in which decay curves of the time histories are obtained and spiral ones of the phase plane toward one single point are plotted, indicating that the motion is smooth, steady and without disorder.

Conclusions
The non-linear motion of a damped spring pendulum with an attached linear damped transverse absorber in the direction of the spring has been investigated. Under the impact of a harmonic force and moment, the motion of the pendulum's hanging point has been constrained to an elliptic path. The EOM have been derived applying Lagrange's equations from the second kind. The AMS has been used to achieve the approximate solutions up to the third order. Based on the solvability requirements, the ME have been recognized. Three resonance cases of primary external and internal resonance were investigated simultaneously. The RHC was used to investigate and evaluate the stability of fixed points' locations. The time histories of the achieved solutions, resonance responses, and the stability and instability zones at the steady state case were drawn and analyzed. The impact of various inputs of physical parameters on the performance of the system under investigation was examined. This system carries a lot of weight due to its use in engineering vibrational control applications.