This work focuses on vibration alleviation and energy harvesting in a dynamical system of a spring-pendulum. The structure of the pendulum is modified using an independent electromagnetic harvesting system. The harvesting depends on the oscillation of a magnet in a coil. An endeavor has been made to get both the energy harvesting and mitigation of vibration efficacy of the harvester. The governing kinematics equations are derived using Lagrange’s equations and are solved asymptotically using the multiple scales method to achieve the intended outcome as new and precise results. The resonance states are classified, and the influence of various parameters of the studied system is analyzed. Fixed points at steady states are categorized into stable and unstable. The time behavior of the solutions, the modified amplitudes, and phases are examined and interpreted in the light of their graphical plots. Zones of stability and instability are concerned, in which the system’s behavior is stable for a wide range of used parameters. This model has become essential in recent times as it uses control sensors in industrial applications, buildings, infrastructure, automobiles, and transportation.

https://journals.sagepub.com/doi/pdf/10.1177/14613484221077474

Controlling the kinematics of a spring-pendulum system using an energy harvesting device

This paper presents the motion of a harmonically excited dynamical system with three degrees of freedom (3-DOF) in which it consists of a connected rigid body with a damped spring pendulum whose suspension point moves in a Lissajous curve path. Multiple scales method is utilized to obtain the asymptotic solutions of the equations of motion up to third approximation. Some types of resonances and the conditions of solvability for the steady state solutions have been clarified in light of the achieved modulation equations. The temporal representation of the achieved solutions and resonance curves are presented in some plots to show the good effect of the distinct parameters on the dynamical motion of the investigated system. The numerical solutions of the governing system of motion are gained utilizing the Runge-Kutta method from fourth order. The comparison between these solutions and the analytical ones reflects the good accuracy of the analytical solutions and the used perturbation techniques.

On the motion of a damped rigid body near resonances under the influence of harmonically external force and moments

This work demonstrates the 3D motion of 6-DOF rigid body around the main inertial axes relative to a fixed point. The novelty of this work is to examine the influence of the gyrostatic moment vector and the magnetic field on the gyrostatic motion besides the action of the Newtonian field in accordance with the Bobylev-Steklov conditions. This motion is characterized by the possibility of placing an acceptable constraint on the governing system of motion in the form of imposing a large value of the projection of the angular velocity around the axial axis of the body. The importance of this restriction is that the governing system in addition to the first-integrals can be reduced to an appropriate system of two semi-linear differential equations from second-order and one first integral. The new analytic approximate solutions of the governing equations are achieved using the small parameter technique of Poincaré. Euler’s angles are determined to evaluate the motion at any instance. The realized solutions are represented graphically for different values ​​of the gyrostatic moment and the point charge causing the magnetic field to estimate the impact of these values ​​on the rotational motion. Moreover, the phase plane plots which illustrate the stability behavior of the body are portrayed. The significance of this work goes back to its several applications in practical life and in engineering application like satellites, submarines, and aircraft. 

Modelling of the rotational motion of 6-DOF rigid body according to the Bobylev-Steklov conditions

This article sheds light on the rotatory motion of a symmetric gyrostat around a fixed point under the effectiveness of both a magnetic field and a Newtonian one besides the action of a gyrostatic moment around the principal inertia’s axes. The gyrostat is assumed to have initially high angular velocity around the principal axis of dynamic symmetry. The controlling system of motion and its first integrals are significantly decreased to a new autonomous system and only one first integral. Poincare’s method of a small parameter is utilized to achieve the asymptotic solutions of the controlling system. Euler’s angles are evaluated to perceive the motion at every blink. The achieved solutions are graphically portrayed through certain plots for various values of the charge causing the magnetic field and gyrostatic projection's values to estimate the output of these values on the gyrostatic motion. Moreover, the graphs of phase planes for these solutions are sketched to provide a complete explanation of the stability of considered motion. Many applications such as submarines, aircraft, and satellites, are of great interest to the topic concerned.

Modelling and analyzing the rotatory motion of a symmetric gyrostat subjected to a Newtonian and magnetic fields

This paper attempts to explore the general rotatory 3D motion of a magnetic heavy solid body around one fixed point under the impact of a gyrostatic moment when the body awarded initially high angular velocity about one of its inertia principal axes. The asymptotic technique of Krylov-Bogoliubov-Mitropolski and its adjustment are applied to achieve new asymptotic periodic solutions of the controlling equations of motion. Discussion of these solutions is presented taking into consideration their graphs representations. The graphical depictions of such solutions and their corresponding phase planes are plotted to give an induction about its behaviour during the interval time of motion and to reveal the good effect of the different applied forces and moments on the body's motion. The application of the achieved results of the present work can be widely found in gyroscopic devices especially that used inertia reference systems like in satellites, airplanes, and missiles. Moreover, for devices that are responsible for the stability of motion for these applications.

The 3D motion of a charged solid body using the asymptotic technique of KBM

The dynamical motion of a gyrostat for the irrational frequency case

This paper investigates the motion of a dynamically symmetric rigid body about one of its fixed points in which it is incurred to Newtonian force field, electromagnetic field and external torques. These torques acted in the direction of the principal axes of the body and are represented by perturbing and gyrostatic torques. The body rotates initially about the symmetry dynamic axis with high angular velocity. Therefore, a small parameter can be introduced. The averaging method is applied to convert the governing system of motion into another convenient averaging one in terms of a small parameter. The asymptotic analytical solutions of the latter system are obtained for some distinct applications depending upon the perturbing torques and the restoring one. The good effect of the body parameters on the motion is clarified in the graphical representations of the obtained solutions of the considered applications. The significance of this work is due to its great implementation of practical life like submarines, devices that maintain the stability and measure the orientations of airplanes and for several physics and engineering applications.

The asymptotic solutions of the governing system of a charged symmetric body under the influence of external torques

The free motion of a rigid body carrying a rotating mass without change of the centroid (this system may be called one-rotor gyrostat) is discussed. Equations of motion are derived: first integrals as a vectorial equation which contained the right vector of an angular velocity of the given rotor with respect to the carrier body and the turn-tensor of this body; a scalar relation between rotation angle of the given rotor with respect to the carrier body and the angular velocity of the carrier body. Only two of these parameters are independent variables. To get equations and to exclude the singular points in the solutions, it is necessary to determine the turn-tensor of the carrier body in the most suitable form. To this end the representation theorem of the turn-tensor and some additional arguments are used. As a final result, we enabled to get two complicated differential equations of the first order. In particular case, the exact solution is represented. Excluding the singular points numerical solutions are determined.

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The Study on Motion of a Rigid Body Carrying a Rotating Mass


Purpose
The purpose of this paper is to study the dynamical properties of a rotating rigid body (RB) containing a viscous incompressible fluid.


Design/methodology/approach
The Reynolds number is assumed to be small so that the governing equations can be easily obtained, and the asymptotic technique is used to solve the problem.


Findings
The effects of the various body parameter values on the motion’s behavior are theoretically elucidated, which can be used for optimization of the charged RB.


Originality/value
This paper finds the missing piece of the puzzle when it comes to the rotating RB containing a viscous fluid; it clearly elucidates graphically how the body parameters affect its dynamical properties.


H. F. Elkafly

Papers

The dynamical motion of a gyrostat for the irrational frequency case

The asymptotic solutions of the governing system of a charged symmetric body under the influence of external torques

The 3D motion of a charged solid body using the asymptotic technique of KBM

The Study on Motion of a Rigid Body Carrying a Rotating Mass

Dynamical analysis of a rotating rigid body containing a viscous incompressible fluid