Modeling and analyzing the motion of a 2DOF dynamical tuned absorber system close to resonance

This work investigates the planar motion of a dynamical model with two degrees-of-freedom (DOF) consisting of a connected tuned absorber with a simple pendulum. It is taken into account that the pendulum’s pivot moves in a Lissajous trajectory with stationary angular velocity in the presence of a harmonic excitation moment. In terms of the model’s generalized coordinates, Lagrange’s equations are used to derive the motion’s controlling system. The approximate solutions of this system, up to a higher order of approximation, are achieved utilizing the approach of multiple scales (AMS). Resonance cases are all classified, in which two of them are examined simultaneously to gain the corresponding equations of modulation. The solutions at the steady-state are studied in terms of solvability conditions. According to the Routh-Hurwitz criteria, all potential fixed points at steady and unsteady states are determined and graphed. The dynamical behavior of the motion's time-histories and the curves of resonance are drawn. Regions of stability are examined by inspecting their graphs in order to assess the favorable impact of various parameters on the motion. The achieved outcomes are regarded as novel because the used methodology is applied to a specific dynamical system. The importance of this model under study can be seen from its numerous applications in disciplines like engineering and physics. Furthermore, pendulum vibration absorbers are commonly employed to reduce the vibrations in engineering constructions such as chimneys, bridges, television towers, high buildings, auto-balancing shafts, and antennas.


Introduction
It is well known that vibrations and dynamic chaos are undesirable phenomena, especially in structures. They have the potential to disrupt, annoy, injure, or damage the various dynamical systems or structures. Therefore, effort, time, and money are expended to eliminate or reduce vibrations, disorder, and noise. Consequently, controlling on the vibrations is considered as a current problem. Many methods have been developed to absorb these vibrations to achieve the stabilization cases of different dynamic systems. In addition, there are an excessive number of developed smart materials for this reason.
Mechanical structures must be developed to enable us for obtaining better performance under various types of loading, especially dynamic and temporal loads. One of the most common control methods is a vibration absorber. It is a supplementary system that is linked to a main one. As a result, the system gains a new degree of freedom. The absorber's parameters are selected in order to minimize or cancel the primary system's vibration. When the system is in main resonance, there is a good chance that it will be damaged; the absorber is often intended to control one mode or one frequency. In most circumstances when vibration must be decreased, a rectilinear motion is performed by the primary system. When a tuned absorber consists of a spring, a mass, and occasionally a damper, the motion can be rectilinear, but when dealing with a swinging pendulum, the rotating motion can be considered [1]. In [2], the authors investigated a dynamic absorber, in which it can be moved in a transverse direction or a longitudinal one, and is coupled to a simple pendulum that is externally triggered.
Many scientists have been interested in studying active vibration control [3][4][5][6][7][8][9]. In [3], the authors presented an approach for analyzing the structures using active and passive control systems, while a feedback of an optimal displacement control rule is constructed in [4] for a single DOF structure with an active tuned mass damper. In [5] and [6], the authors examined a basic pendulum connected with one tuned absorber in either the transverse or longitudinal directions. Negative velocity feedback or its square or cubic value is used to provide controllability to the system. Moreover, a feedback of reverse acceleration and angular displacement can be used besides their squares values or cubic ones to provide with an active control. Generalization of this work has been presented in [7] when a longitudinal absorber connected with the simple pendulum, in which the system's pivot point moves in a route of ellipse. Recently, the behavior of a dynamical system composed of a damped linear tuned-absorber in a transverse direction of a nonlinear damped-spring-pendulum whose hanging point rotates on an elliptic route is studied in [8]. Using the AMS, the approximate solutions for the controlling system are obtained up higher order of approximation. The conditions of Routh Hurwitz [10] have been used to investigate the stability and instability zones, and the results are compared with the steady-state solutions. A limited case from this work can be found in [9] for a fixed pivot point of an elastic pendulum attached with a transverse absorber.
Vibrational motion is used to treat severe damage that occurs in high-rise buildings, such as tall building which has a long period and low damping rate. This makes it more sensitive to long-range earthquakes and records near the fault, including long-period components due to resonance. Therefore, special attention should be paid to the seismic protection of tall buildings [11]. In [12,13], an adaptive-passive variable pendulum and a tuned mass damper (TMD) were used to control wind-induced vibrations in a tall building, while a revolutionary tuned mass damper that can retune its frequency and damping ratio in real-time to protect a structure over its entire life cycle is investigated in [14]. Recently, a high-rise structure is fitted in [15] with a passive adaptive pendulum stream eddy TMD, in the presence of four different soil conditions. In [16], the authors simplified a supported steel footbridge to be used as the test model with a single DOF system. More applications can be found in [17,18].
It must be noted that the motion of a damped linear or nonlinear spring pendulums in different paths has piqued the curiosity of many academics [19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The analytic approximate solution of the nonlinear controlling equations of an elastic pendulum whose suspension point oscillates vertically in investigated in [19]. Whereas, a mathematical pendulum's planar rotational motion, where its fulcrum point oscillates vertically and horizontally, is examined in [20]. The stability and performance of the obtained solutions are established. On the other hand, the rotatory movements of a suspension point of a spring pendulum on an elliptic trajectory and on a closed Lissajous curve near resonance cases are investigated in [21,22], respectively. Some prescribed motions as limited cases are presented. In [23], the authors examined the movement of a 3DOF dynamical system in a plane of a damped linear spring connected to a rigid body. The point of suspension is considered to be fixed. The generalizations of this problem are studied in [24] (for linear spring) and [25] (for nonlinear spring) when their pivot points rotate in elliptic trajectories. More descriptions of the rigid body pendulum can be found in [26] and [27]. The approximate solutions are obtained using the AMS, in which the corresponding equations of modulation are gained in light of the solvability requirements. The motion of a 2DOF autoparametric dynamical model consisting of a coupled rolling cylinder and a damped spring is examined in [28]. The nonlinear stability of this system is investigated in view of the Routh-Hurwitz Criteria. The rotational motion of a charged gyro around a fixed point is investigated in [29] when some external moments and forces are applied. The approximate solutions are obtained using the approach of small parameter and represented graphically. The planar movement of a heavy solid attached to a string from one end and the other end moves in a specified trajectory is investigated analytically and numerically in [30] using the approach of large parameter and [31] applying the Runge-Kutta algorithms, respectively. On the other hand, the motion of a spring in a plane is studied in [32] when its suspension point has a trajectory of a circle with sufficiently large radius. Computer codes are used to show the effect of the system's parameters on the examined motion. The nonlinear stability analysis for various motions of oscillating systems under harmonic external forces and torques is examined in [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].
In this study, the planar motion of a 2DOF dynamical model consisting of a coupled tuned longitudinal absorber with a simple pendulum is studied. This movement is devoted under the existence of an external harmonic excitation moment, in which the pendulum's pivot is constrained to move in a Lissajous trajectory with stationary angular velocity. Lagrange's equations are utilized to derive the governing EOM in light of the model's generalized coordinates. The AMS is used to estimate their solutions analytically till the third-order of approximation. Various classifications for resonance cases are presented, in which two of them are examined at the same time. As a result, the corresponding equations of modulation are obtained. The solvability criteria for the solutions at the steady-state are studied. The criteria of Routh-Hurwitz are used to find and graph all feasible fixed points at steady and unstable states. Modified phases and amplitudes, as well as the solutions' temporal histories, and resonance curves are graphed to investigate the dynamical behavior of the dynamical model at any instant. The impact of various parameters on the system's behavior is evaluated by checking the different stability regions. The significance of the investigated dynamical model can be observed in its several applications in practical life, especially in reducing vibrations that arise in various engineering structures.

Description of the problem
In this section, a full description of the examined model is presented, in which the EOM are derived and transformed to their dimensionless forms. Therefore, we consider the planar motion of a 2DOF dynamical model comprises of a simple pendulum of known mass M and with arm's length l besides its connection with a nonlinear absorber of mass m, which can only move in a longitudinal direction. It is considered that the pendulum's pivot point O follows a Lissajous curve, in an anticlockwise rotation, harmonically in two mutually perpendicular directions (see Fig. 1). As a result, the coordinates of this point are where R j ( j x, y) and j are well-known parameters. Let us consider O 1 Y 1 and O 1 X 1 to represent the vertical downward axis and the horizontal one, respectively, with the origin O 1 . Furthermore, let ϕ be the rotation angle (the angle between the pendulum arm O A and the vertical at O), u is the absorber's longitudinal elongation from the equilibrium location, l 0 is the natural length of the absorber, c 1 and c 2 are the damping coefficients of rotation and absorber, M(t) f cos( t) is the harmonic excitation rotation moment at O in which is its frequency, and k 1 and k 2 are the spring's linear and nonlinear stiffness, respectively.
The model's kinetic and potential energies T and V have the forms Here, the letter g denotes the acceleration due to gravity and dots denote the derivatives regarding time t. Based on the latter equations, Lagrange's function L T − V can be immediately obtained, and then one can derive the regulating system of the EOM according to the model's generalized coordinates ϕ and u utilizing the following Lagrange's equations [38] d dt To deal with the handled model, let us consider the following dimensionless forms The substitution of (2) and (4) into (3), yields the next dimensionless forms of the controlling system Here, the over dot represents the derivatives regarding τ . These equations represent a system of two nonlinear ordinary differential equations (ODE) of second order.

The perturbation methodology
The main purpose of this part is to acquire the approximate solutions of the governing system of EOM (5) and (6) utilizing the AMS up to the third-order of approximation in light of the exclusion of secular terms. To attain this objective, the trigonometric functions must be approximated using Taylor series up to the third-order in a way that is valid in a neighborhood region of static equilibrium. Therefore, introducing the following forms of sin ϕ and cos ϕ into Eqs. (5) and (6) to obtain the EOM in the forms It is worthy to note that, the functions u 1 and ϕ must be represented in terms of a small parameter 0 < ε << 1 in light of the new functions φ and Z according to Based on AMS, the approximate solutions of these functions can be represented as a power series of ε as follows [39] where τ 0 τ, τ 1 ετ, and τ 2 ε 2 τ are distinct time scales, in which τ 0 and τ 1 , τ 2 are known as the fast and slow time scales, respectively. Since we have a variety of scales, then the derivatives of τ can then be transformed to τ n (n 0, 1, 2) as follows d dτ The terms of O(ε 3 ) are obviously ignored in (12) due to the smallness of them. The damping parameters, amplitude of the force, and other parameters are assumed to be small according to the forms whereF,l 1 ,C j ,m 1 ,α,r x , andr y are parameters of order unity, Substituting (10)-(13) into (8)-(9), and then equating the different coefficients of powers of ε in each side to construct the next groups of partial differential equations (PDE).
Equations of order (ε) Equations of order (ε 2 ) Equations of order (ε 3 ) Our aim now is to solve the previous PDE (14)- (19), in which it can be solved successively. To accomplish this target, we start with the general solutions of the first group of Eqs. (14) and (15), which can be formulated as follows Here A j ( j 1, 2) and A j are known as complex functions and their conjugates, in which they depend on τ j .
Inserting these solutions into the second group of PDE (16)- (17) and then deleting terms that generate secular ones to find their removal criteria as below As a result of these criteria, we may write the solutions of the second group in the following forms where CC denotes the conjugates of the foregoing terms.
To obtain the criteria of solvability for the approximation of third-order, substitute (20), (21), (23), and (24) into the last set of PDE (18) and (19), and then remove terms that yield the secular ones. Therefore, we can writem In light of these criteria, the third-order approximations can be given in the forms The functions A j ( j 1, 2) can be estimated using the removal criteria (22), (25), and (16) for secular terms. Therefore, the approximate solutions ϕ and u 1 can be easily obtained using the above derived solutions of first to third orders and the postulates (10).

Resonance categorizations and modulating equations
The categorizations of resonance cases that may emerge in the acquired solutions, as well as the evaluation of one of these cases, are both significant aspects of this section. We infer that these solutions are collapsed when any of their denominators equal zero [40]. As a result, the following cases can be recognized.
(i) The primary external resonance case can be discovered at p ≈ 1 , (ii) The internal resonance case has been satisfied at ω 0, ω ±2, ω ±p x ,ω 1 ± p y , p y ±1, It is important to keep in mind that if any one of the previous resonance instances is fulfilled, the examined system's behavior will become convoluted. Consequently, we ought to modify the used approach. To deal with this circumstance, we will look at one of primary external resonance and one of internal resonance that operate together, i.e., p ≈ 1 and ω ≈ 2. These correlations show how closely p and ω to 1 and 2, respectively.
It is necessary to employ the dimensionless parameters known as detuning σ j ( j 1, 2), which measure the displacement from the oscillations to the tight resonance [39]. Then we can start writing Therefore, we can write them in terms of ε as follows The criteria of solvability can be gained through the substitution of (29) and (30) into the second and third groups of PDE (16)- (19). Elimination of terms that yield the secular terms gives the next conditions. Solvability criteria of the second group Solvability criteria of the third group This implies that the solvability criteria consist of four nonlinear PDE in which A j ( j 1, 2 ) depend only on τ 2 , which motivates us to depict them in the following polar form whereh j and ψ j are real functions of the amplitudes and phases of the solutions ϕ and u 1 . The below indicated modified phases can be used to adapt the preceding solvability criteria from PDE to ordinary differential equations (ODE).
Substituting (32) and (33) into (31), and then splitting the imaginary and real components to obtain the below four first-order modulation equations for the resonance cases under consideration The time histories of h 1 , θ 1 and h 2 , θ 2 are calculated when C 1 ( 0.0809848, 0.134975, 0.188964) and C 2 ( 0.102041, 0.153061, 0.255102), respectively. The planes θ 1 h 1 and θ 2 h 2 are graphed in accordance with the same values of C 1 and C 2 . The included curves in portions (a) and (b) of Fig. 2 oscillate rapidly around a fanciful horizontal symmetry axis at the first stage of the considered time interval and have a decay mode. Therefore, they have steady behavior till the end of the time interval. It is noticed that with the increasing of the values of damping coefficient C 1 , the solutions h 1 and θ 1 reach faster to their steady case than the other little values. It is found that h 1 and θ 1 have not any variation with the various values of C 2 . The reason is due to that their equations do not depend on C 2 .
On the other hand, the represented curves of h 2 (τ ) and θ 2 (τ ) are plotted in portions (a) and (b) of Fig. 3, which they are constitute with the solutions of the system of Eq. (34), in which they do not depend on C 1 . Therefore, we cannot expect any variation with C 1 . The corresponding phase plane curves of Figs. 2 and 3 are plotted in Fig. 4 with spiral curves and Fig. 5 with convex curves, respectively.
The time histories of the acquired approximate solutions (AS) ϕ(τ ) and u 1 (τ ) are drawn in portions (a) and (b) of Fig. 6 when C 1 and C 2 have different values. An assessment of the included curves in these portions reveals that periodic waves and decay ones are produced for the solutions ϕ and u 1 , respectively. Figure 7 depicts a comparison between the AS and numerical solutions (NS) of the governing system of motion, demonstrating their remarkable consistency. This indicates that the AS is precise with the used perturbation approach.
One of the important aspects of this research is to study the effect of various values of F, p x , and p y on the behavior of the amplitudes h j ( j 1, 2), phases θ j , and the approximate solutions ϕ and u 1 . Therefore, Figs Fig. 12. However, the behavior of ϕ has the form of standing periodic waves with some nodes, in which their amplitudes decrease with the increase in the values of F, as graphed in Fig. 12a. On the other hand, the variation of u 1 has a decaying pattern till the end of the investigated time interval. Therefore, these solutions have stable behaviors. As drawn in parts (c), (e) and (d), (f ), one can conclude that the variations of ϕ and u 1 become stationary (to some extent) and slightly with the variation of the values of p x and p y , respectively.

Steady-state solutions
The main goal of this part is to explore the vibrations of the investigated system in the case of steady-state, in which it will appear when the transitory processes fade away. As a result, the left-hand sides of Eq. (34) will be regarded as zero, i.e., dh j dτ dθ j dτ 0 ( j 1, 2) [23,27]. Then an algebraic system of four equations for the adjusted phases θ j and the amplitudes h j can be produced in the following form (2 − cos θ 2 ) 0, A closer look at the aforementioned system (36), we can find the elimination of its adjusted phases θ j yields the following functions of frequency response These equations are clearly implicit formulations of nonlinear algebraic equations between h j and σ j . It is well recognized that one of the main stages in the steady-state vibrations is to look into the stability analysis of fixed points. Consequently, the system's behavior in an area near the fixed points is investigated. To achieve this goal, inserting the next substitutions in (34) [40] h 1 h 10 + h 11 , θ 1 θ 10 + θ 11 , Here h j0 and θ j0 symbolize the solutions at the steady-state case of (34), while h j1 and θ j1 denote the perturbation that is very minor in relation to h j0 and θ j0 . Making use of (34) and (38) Taking into account that the functions θ j1 ( j 1, 2) and h j1 are unknown perturbed functions for the system of Eqs. (39). Then, their solutions can be represented as k s e λT (s 1, 2, 3, 4), where λ and k s are the eigenvalues and constants of these functions, respectively. When the steady-state solutions are asymptotically stable, then the real potions' roots of the below characteristic equation of system (39) should be negative [10,41] where s (s 1, 2, 3, 4) have the forms The criteria of Routh-Hurwitz [10] that outline the necessary and sufficient requirements of the solutions at the steady-state can be stated as follows (42) Equations of system (37) have been solved and drawn with different values of σ 1 , when σ 2 −0.0514914, C 1 0.134975, and C 2 0.153061 to yield the curves of Figs. 13, 14, 15, 16 and 17. The blue and red curves represent the first and second equations of system (37), respectively. Intersections of these curves produce fixed points which may be stable or not. The black and green circles express stable and unstable fixed points, respectively. Figures 13, 14, 15, 16, and 17 represent the variation of h 2 versus h 1 when σ 1 has the

Examination of the stability
The Routh-Hurwitz criteria are used in this part to assess the system's stability and to examine its nonlinear evolution, in which the motion of the model is studied in the presence of the external excitation harmonic moment M(t). Some elements, such as the coefficients of damping C j ( j 1, 2), the frequencies p, ω, and the parameters of detuning σ j have been found to have a significant effect on stability operations. A precise procedure involving several system parameters was employed to draw the stability plots of system (34). The modified amplitudes h j are displayed with σ j for different parametrical areas when the values F 0.140374,  The scrutiny of these figures detects that the stability domains of the fixed points are found in the area −3.0 ≤ σ 1 ≤ −0.2, while the unstable ones are located in the zone −0.2 < σ 1 ≤ 3.0. The continued curves and the dotted ones are used to represent the stable and unstable points, respectively. The locations of different peaks of fixed points (PFP) and the critical fixed points (CFP) of the response curves of these figures at the   Tables 1, 2, 3, 4, 5 and 6. The CFP is known as that point which is located between the stable and unstable areas, and then it may be stable or not. Moreover, each CFP represent PEP, while the inverse is not necessary.      Table 6 Highlights the destinations of the PFP and the CFP for the response curves of Fig

Conclusion
The planar motion of a 2DOF dynamical system consisting of a coupled tuned longitudinal absorber with a simple pendulum in the presence of a harmonic excitation moment has been examined. The pendulum's pivot has been forced to follow a Lissajous trajectory with constant angular velocity. The governing system of the EOM has been derived applying the equations of Lagrange of the second kind. One of the most accurate perturbation approaches, called AMS has been used to achieve the solutions of this system up to a greater order of approximation. The different types of resonance cases have been categorized, and two of them have been studied at the same time. The equations of modulation have been obtained in view of the system's solvability criteria. These equations have been reduced to an algebraic system of two nonlinear algebraic equations which are solved numerically and graphed to obtain all possible fixed points. The stability of these points at the steady-state has been examined based on the criteria of Routh-Hurwitz. The solutions' temporal histories, altered phases and amplitudes, and resonance curves have been drawn to analyze the dynamical manner at any moment. The effect of the various parameters on the system's behavior has been assessed through the examination of the various stability areas. The calculations have been performed using the computer's codes and Wolfram Mathematica software. The gained results can be considered novel, in which the used methodology has been applied to a specific dynamical system. The weight of the examined dynamical system can be seen from its numerous applications in real life, particularly in the reduction in vibrations in diverse engineering constructions.
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