Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system that correlates with two devices, namely a piezoelectric device and an electromagnetic one, to produce two novel models. These devices are connected to a nonlinear damping spring pendulum with two degrees of freedom. The damping spring pendulum is supported by a point moving in a circular orbit. Lagrange’s equations of the second kind were utilized to obtain the equations of motion. The asymptotic solutions of these equations were acquired up to the third approximation using the approach of multiple scales. The comparison between the approximate and the numerical solutions reveals high consistency between them. The steady-state solutions were investigated, and their stabilities were checked. The influences of excitation amplitudes, damping coefficients, and the different frequencies on energy-harvesting device outputs are examined and discussed. Finally, the nonlinear stability analysis of the modulation equations is discussed through the stability and instability ranges of the frequency response curves. The work is significant due to its real-life applications, such as a power supply of sensors, charging electronic devices, and medical applications.

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.

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.

Purpose: The motion of three degrees-of-freedom (DOF) of an automatic parametric pendulum attached with a damped system has been investigated. The kinematics equations of this system have been derived employing Lagrange’s equations in accordance to it’s the generalized coordinates. Methods: The method of multiple scales (MMS) has been used to obtain the solutions of the controlling equations up to the third-order of approximation. The solvability criteria and modulation equations for primary external resonance have been explored simultaneously. Results: The non-linear stability approach has been used to analyze the stability of the considered system according to its different parameters. Time histories of the amplitudes and the phases of this system have been graphed to characterize the motion of the system at any given occurrence. Conclusions: The different zones of stability and instability of this study have been checked and examined, in which the system's behavior has been revealed to be stable for various values of its variables.

Purpose: The planar dynamical motion of a double-rigid-body pendulum with two degrees-of-freedom close to resonance, in which its pivot point moves in a Lissajous curve has been addressed. In light of the generalized coordinates, equations of Lagrange have been used to construct the controlling equations of motion. Methods: New innovative analytic approximate solutions of the governing equations have been accomplished up to higher order of approximation utilizing the multiple scales method. Resonance cases have been classified and the solvability conditions of the steady-state solutions have been obtained. The fourth-order Runge–Kutta method has been utilized to gain the numerical solutions for the equations of the governing system. Results: The history timeline of the acquired solutions as well as the resonance curves have been graphically displayed to demonstrate the positive impact of the various parameters on the motion. The comparison between the analytical and numerical solutions revealed great consistency, which confirms and reinforces the accuracy of the achieved analytic solutions. Conclusions: The non-linear stability analysis of these solutions have been examined and discussed, in which the stability and instability areas have been portrayed. All resonance cases and a combination of them have been examined. The archived results are considered as generalization of some previous works that are related to one rigid body and for fixed pendulum’s pivot point.

Introduction: A hydraulic power injection machine is designed to use a driving screw to inject melted plastic into a specified mold. This machine can be found at an automotive spare parts factory. The cantilever-style heavy-duty screw injector is supported by one roller and secured at the end. An obvious need for vibration analysis on the roller support is essential. A mass spring damper model is proposed for deeply investigating the friction induced vibration mechanism for this injection system to well understand and analyze its vibration behavior. Purpose: A mechanical mode of two degrees-of-freedom (DOF) is designed to improve research on the dynamic features of the Plastic Hydraulic Injection System (PHIS) mechanism. Materials and methods: Experimental investigation and analysis of this mechanism are explored to obtain the instability speed and critical stick slip (SS) speed. The numerical imitation results of this work will help with the design and development of the PHIS mechanism. Conclusion: The stability of the system and SS behavior are next examined by determining the critical variability speediness and critical SS speed. A simulation study is carried out to evaluate the effect of various parameters of the system on its stability and on the behavior of the SS motion.