Analytical and numerical study of a vibrating magnetic inverted pendulum

Abstract

The current study investigates the stability structure of the base periodic motion of an inverted pendulum (IP). A uniform magnetic field affects the motion in the direction of the plane configuration. Furthermore, a non-conservative force as one that dampens air is considered. Its underlying equation of motion is derived from traditional analytical mechanics. The mathematical analysis is made simpler by substituting the Taylor theory in order to expand the restoring forces. The modified Homotopy perturbation method (HPM) is employed to achieve a roughly adequate regular result. To support the prior result, a numerical method based on the fourth-order Runge-Kutta method (RK4) is employed. The graphs for both the analytic and numerical solutions are highly consistent with one another, which indicates that the perturbation strategy is accurate. The solution time history curve exhibits a decaying performance and indicates that it is steady and without chaos. The resonance and non-resonance cases are found through the stability study by using the time scale method. In all perturbation approaches, the methodology of multiple time scales is actually regarded as a further standard approach. The time history is used to create a collection of graphs. Some graphical representations are used to illustrate how the typical physical values affect the behavior of the discovered solution. It has been discovered that the statically unstable IP can have its instability reduced by raising the spring torsional constant stiffness as well as the damped coefficient. Moreover, the magnetic field has a significant role in the stability configuration, which explains that at higher values of this field, the decaying waves take much more time than the smaller values of this field. Accordingly, it can be employed in various engineering devices that need a certain period of time to be more stable.

Inspection of a Time-Delayed Excited Damping Duffing Oscillator

“…Therefore, the adaptation of the HPM is urgent. The coming exponential factor was previously provided [20].…”

“…However, the linearized stability may be examined around the equilibrium points. Consequently, the subsequent steady-state solution may be assumed [20,25]:…”

Stability analysis of an acted asymmetric rigid body by a gyrostatic moment and a constant body-fixed torque

“…p and η ¼ tan À1 ½x 3 ð0Þ=x 2 ð0Þ are known, respectively, by the amplitude and the phase angle of the solutions (16), in which jηj ≤ π and we have…”

“…15 and Ref. 16 to treat with the vibrating movements of dynamical systems, is regarded as a significant perturbation method that may be utilized to the solutions of numerous differential equations. 17 These solutions are contrasted with the numerical ones to show the precise accuracy of this method.…”

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

“…In 2008, He employed the HPM to solve boundary value problems [20]. In 2007, Javidi and Golbabai used a revised version of the HPM to solve non-linear Fredholm integral equations [21].Recently, HPM with small variations has been applied to study fractal duffing oscillator problems under arbitrary conditions [22], modified HPM for nonlinear oscillators Anjum and He [23], attachment oscillator arising in nanotechnology [24], conservative nonlinear oscillators [25], non-linear oscillator problems in a fractal space [26] and HPM including Aboodh transformation to solve fractional calculus Tao et al [27], vibrating magnetic inverted pendulum Moatimid et al [28], Symmetry-breaking and pull-down motion for the helmholtz-duffing oscillator Niu et al [29], nonlinear fractional Drinfeld-Sokolov-Wilson Equation Nadeem and Alsayaad [30], trajectory analysis of a zero-pitch-angle e-Sail Niccolai et al [31], natural convection between two concentric horizontal circular cylinders Abdulameer and Ali Al-Saif [32], nonlocal initialboundary value problems for parabolic and hyperbolic Al-Hayani and Younis [33], multi-step iterative methods for solving nonlinear equations Saeed et al [34], telegraph equation Moazzzam et al [35], triangular linear diophantine fuzzy system of equations Shams et al [36], condensing coagulation model and Lifshitz-Slyzov equation Arora et al [37], singular nonlinear system of boundary value problems Pathak et al [38], rikitake-yype system Ene and Pop [39], heat and mass transfer with 2D unsteady squeezing viscous flow problem Abdul-Ameer and Ali Al-Saif [40], variable Speed Wind Turbine Control Shalbafian and Ganjefar [41], radial thrust problem Niccolai et al [42], special third grade fluid flow with viscous dissipation effect over a stretching sheet Swain et al [43], and the frequency-amplitude relationship of a nonlinear oscillator with cubic and quintic nonlinearities He et al [44]. The HPM has become a widely-used technique to solve a large variety of problems in different fields and many research papers have been published each year using this method as evidenced by a simple search on Google Scholar.…”