## 1 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 transverse 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. Different trajectories for a pivot point of linear and non-linear elastic pendulums with various DOF are studied in numerous works, e.g., [10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. The case of fixed pivot point of 3 DOF rigid body pendulum (RBP) is examined in [10], in which three resonance cases are studied simultaneously from the perspective of the AMS. In [11,12], the authors dealt with a damped pendulum motion under the influence of various forces taking into account the movement of the pivot point on a circle with a stationary angular velocity. In [13, 14, 15], Lissajous paths are considered for the motion of the spring's suspension point as constrained motion. All possible fixed points are graphed in [13] and categorized in terms of being stable or not, while in [14], the authors verified the obtained analytic solutions for the elastic RBP through comparison with the numerical results. The approximate solutions and their stability regions for a non-linear damped RBP are examined in [15] using the AMS and RHC, respectively. Many scientific works have recently studied the motion of the pendulum's suspension point on elliptic pathways for a variety of models such as [16, 17, 18, 19]. The linear and non-linear dampings of a spring RBP on an elliptic route are inspected in [16,17], respectively. The double pendulum motion on an elliptic pathway was studied in [18]; one of them is a simple rigid pendulum while the other one is an elastically damped. The motion was investigated when a longitudinal harmonic force and a rotating moment were applied to the pendulum end point and to the suspension point, respectively. The effect of two external perpendicular forces besides an applied moment on a spring pendulum motion movement that follows the same route is examined in [19]. Two systems were considered in [20]: the first one is a subsystem that produces a swing motion, and the other is an absorber that moves linearly with respect to it, which can be used in various fields of engineering. The harmonic balance method (HBM) [7] is applied in [21] to examine the vibrational response of a connected mass with a damped oscillator, in addition to an excited pendulum hinged to this mass. The averaging approach is employed in [22] to obtain the analytic results of the 2 DOF non-linear vibrating dynamical system consisting of a non-linear spring with attached damper.