The major objective of this research is to study the planar dynamical motion of 2DOF of an auto-parametric pendulum attached with a damped system. Using Lagrange’s equations in terms of generalized coordinates, the fundamental equations of motion (EOM) are derived. The method of multiple scales (MMS) is applied to obtain the approximate solutions of these equations up to the second order of approximation. Resonance cases are classified, in which the primary external and internal resonance are investigated simultaneously to establish both the solvability conditions and the modulation equations. In the context of the stability conditions of these solutions, the equilibrium points are obtained and graphically displayed to derive the probable steady-state solutions near the resonances. The temporal histories of the attained results, the amplitude, and the phases of the dynamical system are depicted in graphs to describe the motion of the system at any instance. The stability and instability zones of the system are explored, and it is discovered that the system’s performance is stable for a significant number of its variables.

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.

This paper presents an alternative approach to solving Euler-Poisson’s dynamical equations, which describe the governing equations of how a rigid body (RGB) rotates around a stationary point with the influence of a gyrostatic moment (GM). The RGB’s angular velocity vector components in our solution are different from those in the well-known cases. It is expected that the RGB’s center of mass lies in the meridional plane along its principal axis of inertia. Additionally, it is assumed that the main inertia moments correspond to a fundamental algebraic equality. Additionally, there is a constraint on the first condition selection. The analytical solution of the problem is provided and depicted graphically using a computer codes, allowing us to analyze the motion at any given time. The influence of distinct values of the GM on these solution are also presented. One could consider the solution to be an entirely novel iteration of Euler’s original case. Solving this equation is crucial due to its broad range of applications, including the design and development of stability enhancement systems in automobiles, dynamics-based sensors like gyroscopic sensors, the analysis of space objects and robotic systems, and gaining insight into the complex movements of RGBs.

The main purpose of this study is to explore the feasibility of achieving a fourth integral for a charged rigid body (RGB) rotary movement, considering the influence of gyrostatic torque (GT) and a uniform electromagnetic field. Six nonlinear first-order differential equations (DEs) and their three first integrals, pertaining to geometry, area, and energy, regulate this problem. The analytical solution to the issue necessitates approaching a crucial fourth integral. This paper presents a pivotal criterion for a function F , contingent on all RGB variables, to qualify such integral is outlined. The achieved outcomes are validated through comparing them with the renowned cases and then they can be considered a consolidation or mainstreaming of prior research efforts.

The 3D modeling analysis for the rotary motion of an asymmetric rigid body (RB) that gains a charge is presented. Under the effect of a gyrostatic moment (GM), an electromagnetic force field (EFF), time-varying body-fixed torques (TVBFTs), and constant axial torque (CAT), Euler’s equation of motion (EOM) is derived to describe the body’s EOM. The process is to derive the analytic solutions for the general attitude motion of the RB that is nearly symmetrical; therefore, a novel analytical solution for the angular velocities of the body has been approached. These new solutions are obtained by considering torques that vary over time and expressing them as integrals. Additionally, a novel closed-form evaluation mechanism for these integrals is offered. Specifically, the case of a constant torque around the spin axis and transverse torques represented by polynomial functions of time is explored. When dealing with an axisymmetric RB subject to a CAT, the solutions obtained from Euler’s EOM are exact. However, it is important to note that novel analytic solutions for the Eulerian angles are approximations, as they rely on the assumption of small angles. Nonetheless, these approximations have broad applicability to a wide range of practical problems. The method’s precision is demonstrated through the graphical simulation of the proposed solutions. Additionally, a computer program is utilized to create diagrams and phase plane curves, highlighting the contribution of various body parameters to the motion. These plots depict the contributions of various values regarding GM, charge, and CAT. Motion stability is also examined through phase diagrams. In addition to presenting novel solutions and outcomes for the problem, this study plays a vital role in multiple scientific and engineering fields as it has the potential to optimize mechanical systems, explain celestial motion, and improve spacecraft performance.