Pendulum Equation Research Articles

On The Power Series Solution to The Nonlinear Pendulum

Summary The exact solution to the simple pendulum problem has long been known in terms of Jacobi elliptic functions, of which an efficient numerical evaluation is standard in most scientific computing software packages. Alternatively, and as done in V. Fairén, López and L. Conde (Power series approximation to solutions of nonlinear systems of differential equation, Am. J. Phys. 56 (1988) 57–61], the pendulum equation can be analytically solved exactly by the power series solution method. Although recursive formulae for the series coefficients were provided in V. Fairén et al., the series itself—as well as the optimal location about which an expansion should be chosen to maximize the domain of convergence—has not yet been examined, and this is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the power series solution in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. We provide an exact resummation of the pendulum series that accelerates the series’ convergence uniformly from the top to the bottom of the trajectory. We also provide the formulae needed to extend the pendulum series for all time via symmetry. The pendulum problem, in its relative simplicity, provides an explicit demonstration of the effect of singularity structure and initial condition location on convergence properties of power series solutions.

Uniqueness result for a weighted pendulum equation modeling domain walls in notched ferromagnetic nanowires

Uniqueness result for a weighted pendulum equation modeling domain walls in notched ferromagnetic nanowires

Solution for a rotational pendulum system by the Rach–Adomian–Meyers decomposition method

Abstract In this article, we report for the first time the application of a novel and extremely valuable methodology called the Rach–Adomian–Meyers decomposition method (MDM) to obtain numerical solutions to the rotational pendulum equation. MDM is a tool for solving nonlinear differential equations that combines both series solution and the Adomian decomposition method efficiently. We present a simple and highly accurate MDM-based algorithm and its numerical implementation via a one-step recurrence approach for obtaining periodic solutions to the rotational pendulum equation. Finally, numerical simulations are performed to demonstrate the efficiency and accuracy of the proposed technique for both large and small amplitudes of oscillation.

The Differential Equations of Gravity-free Double Pendulum: Lauricella Hypergeometric Solutions and Their Inversion

This paper solves in closed form the system of ODEs ruling the 2D motion of a gravity free double pendulum (GFDP), not subjected to any force. In such a way its movement is governed by the initial conditions only. The relevant strongly non linear ODEs have been put back to hyperelliptic quadratures which, through the Integral Representation Theorem (IRT), are driven to the Lauricella hypergeometric functions.
 We compute time laws and trajectories of both point masses forming the GFDP in explicit closed form. Suitable sample problems are carried out in order to prove the method effectiveness.

Spin-Torque-Driven Terahertz Auto-Oscillations in Noncollinear Coplanar Antiferromagnets

We theoretically and numerically study the terahertz auto-oscillations, or self-oscillations, in thin-film metallic noncollinear coplanar antiferromagnets (AFMs), such as ${\mathrm{Mn}}_{3}\mathrm{Sn}$ and ${\mathrm{Mn}}_{3}\mathrm{Ir}$, under the effect of antidamping spin torque with spin polarization perpendicular to the plane of the film. To obtain the order parameter dynamics in these AFMs, we solve three Landau-Lifshitz-Gilbert equations coupled by exchange interactions assuming both single- and multidomain (micromagnetics) dynamical processes. In the limit of a strong exchange interaction, the oscillatory dynamics of the order parameter in these AFMs, which have opposite chiralities, could be mapped to that of two damped-driven pendulums with significant differences in the magnitude of the threshold currents and the range of frequency of operation. The theoretical framework allows us to identify the input current requirements as a function of the material and geometry parameters for exciting an oscillatory response. We also obtain a closed-form approximate solution of the oscillation frequency for large input currents in the case of both ${\mathrm{Mn}}_{3}\mathrm{Ir}$ and ${\mathrm{Mn}}_{3}\mathrm{Sn}$. Our analytical predictions of threshold current and oscillation frequency agree well with the numerical results and thus can be used as compact models to design and optimize the auto-oscillator. Employing a circuit model, based on the principle of tunnel anisotropy magnetoresistance, we present detailed models of the output power and efficiency versus oscillation frequency of the auto-oscillator. Finally, we explore the spiking dynamics of two unidirectional as well as bidirectional coupled AFM oscillators using nonlinear damped-driven pendulum equations. Our results could be a starting point for building experimental setups to demonstrate auto-oscillations in metallic AFMs, which have potential applications in terahertz sensing, imaging, and neuromorphic computing based on oscillatory or spiking neurons.

Nonlinear Modeling of Series-Type Pendulum Tuned Mass Damper-Tuned Liquid Damper

Abstract The nonlinear response of a series-type pendulum tuned mass damper-tuned liquid damper (TMD-TLD) system is investigated in this study. The TLD is mounted on the pendulum TMD in series to remove the need for costly viscous damping elements. Since the response of the TMD is greater than that of the primary structure, the TLD experiences a significant base motion, leading to a highly nonlinear response that is difficult to model. The nonlinear pendulum TMD equation of motion is modeled without linearizing assumptions. The TLD is represented by an incompressible smoothed particle hydrodynamics (SPH) model that can capture large sloshing responses. The nonlinear model results are compared to shake table testing for a TMD-TLD system and a linear equivalent mechanical model. Four system configurations are considered. The nonlinear model shows good agreement with the experimental data for the TMD displacement and TLD wave heights in both time and frequency domains. The nonlinear model shows improved agreement compared to the linear model for all cases studied, especially for the TLD wave heights. The impact of simplifying the pendulum TMD equation of motion by the small-angle assumption is investigated for two cases. The results indicate that the simplified pendulum equation does not properly capture the frequency of the TMD in the TMD-TLD system and results in a reduction in calculated TLD wave heights compared to the fully nonlinear equation. It is therefore critical to consider the fully nonlinear pendulum TMD response to capture the TMD-TLD behavior.

High order Melnikov method: Pendulums

In this paper, we present an explicit integral formula for the high order Melnikov function D1(t0) for periodically perturbed pendulum equations. The acquired integral formula is then applied to a specific pendulum equation to offer an example of high frequency perturbation, of which the classical Melnikov function D0(t0) fails to dominate the higher order term εD1(t0).

Novel Analytical and Numerical Approximations to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Method

In this work, some novel approximate analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation and some relation equations of motion on the pivot vertically for arbitrary angles are obtained. The analytical approximation is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical approximations, the Chebyshev collocation numerical method is introduced for analyzing the equation of motion. Moreover, the analytical approximation and numerical approximation using the Chebyshev collocation numerical method and the MATHEMATICA command Fit are compared with the Runge–Kutta (RK) numerical solution. Also, the maximum distance error to all obtained approximations is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

Some Novel Solutions to a Quadratically Damped Pendulum Oscillator: Analytical and Numerical Approximations

In this paper, some novel analytical and numerical techniques are introduced for solving and analyzing nonlinear second‐order ordinary differential equations (ODEs) that are associated to some strongly nonlinear oscillators such as a quadratically damped pendulum equation. Two different analytical approximations are obtained: for the first approximation, the ansatz method with the help of Chebyshev approximate polynomial is employed to derive an approximation in the form of trigonometric functions. For the second analytical approximation, a novel hybrid homotopy with Krylov–Bogoliubov–Mitropolsky method (HKBMM) is introduced for the first time for analyzing the evolution equation. For the numerical approximation, both the finite difference method (FDM) and Galerkin method (GM) are presented for analyzing the strong nonlinear quadratically damped pendulum equation that arises in real life, such as nonlinear phenomena in plasma physics, engineering, and so on. Several examples are discussed and compared to the Runge–Kutta (RK) numerical approximation to investigate and examine the accuracy of the obtained approximations. Moreover, the accuracy of all obtained approximations is checked by estimating the maximum residual and distance errors.

An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

Structural Design, Simulation and Experiment of Quadruped Robot

This paper carried out a series of designs, simulations and implementations by using the physical-like mechanism of a bionic quadruped robot dog as a vehicle. Through an investigation of the walking mechanisms of quadrupeds, a bionic structure is proposed that is capable of omnidirectional movements and smooth motions. Furthermore, the kinematic and inverse kinematic solutions based on the DH method are explored to lay the foundation for the gait algorithm. Afterward, a classical compound pendulum equation is applied as the foot-end trajectory and inverse kinematic solutions are combined to complete the gait planning. With appropriate foot–ground contact modeling, MATLAB and ADAMS are used to simulate the dynamic behavior and the diagonal trot gait of the quadruped robot. Finally, the physical prototype is constructed, designed and debugged, and its performance is measured through real-world experiments. Results show that the quadruped robot is able to balance itself during trot motion, for both its pitch and roll attitude. The goal of this work is to provide an affordable yet comprehensive platform for novice researchers in the field to study the dynamics, contact modeling, gait planning and attitude control of quadruped robots.

Drift Resonance Between Particles and Compressional Toroidal ULF Waves in Dipole Magnetic Field

AbstractWe examine the drift‐resonant particle dynamics for toroidal ultralow frequency (ULF) waves in a pure dipole background geomagnetic field. We confirm that the resonant condition originally believed to apply only for poloidal ULF waves, , also applies for compressional toroidal waves. The predicted phase relationships have been confirmed from Van Allen Probes observations. Their good agreement provides the first observational evidence for the drift resonance condition controlled by the compressional toroidal ULF wave. Moreover, we extend the drift resonance theory into a nonlinear regime. The resulting particle motion can be described by a modified pendulum equation with solutions depending on the wave number . For high‐ toroidal waves, the resonant islands become asymmetric to perturb the particle trajectories within each potential well and consequently increase the trapping widths in both energy and L‐shell. We further carry out test‐particle simulations to show the evolution of electron distribution functions when they interact with either toroidal or poloidal waves. These findings demonstrate that toroidal ULF waves, like their poloidal counterparts, play an important role in magnetospheric particle dynamics.

Solving 2nd Order Nonlinear Differential Equations Using Piecewise Analytic Method (Pendulum Equations)

In this paper, piecewise analytic method (PAM) is used for solving highly nonlinear 2nd  order differential equation (pendulum equations) which is a big problem for engineers and scientists. PAM is used for showing the nonlinear dynamics of the solution with and without linearizion. The error and accuracy of the solution are controlled easily according to our needs.

Theoretical model of the nonlinear resonant interaction of whistler-mode waves and field-aligned electrons

The nonlinear resonant interaction of intense whistler-mode waves and energetic electrons in the Earth's radiation belts is traditionally described by theoretical models based on the consideration of slow–fast resonant systems. Such models reduce the electron dynamics around the resonance to the single pendulum equation that provides solutions for the electron nonlinear scattering (phase bunching) and phase trapping. Applicability of this approach is limited to not-too-small electron pitch-angles (i.e., sufficiently large electron magnetic moments), whereas model predictions contradict to the test particle results for small pitch-angle electrons. This study is focused on such field-aligned (small pitch-angle) electron resonances. We show that the nonlinear resonant interaction can be described by the slow–fast Hamiltonian system with the separatrix crossing. For the first cyclotron resonance, this interaction results in the electron pitch-angle increase for all resonant electrons, contrast to the pitch-angle decrease predicted by the pendulum equation for scattered electrons. We derive the threshold value of the magnetic moment of the transition to a new regime of the nonlinear resonant interaction. For field-aligned electrons, the proposed model provides the magnitude of magnetic moment changes in the nonlinear resonance. This model supplements existing models for not-too-small pitch-angles and contributes to the theory of the nonlinear resonant electron interaction with intense whistler-mode waves.

A geometric interpretation of maximal Lyapunov exponent based on deviation curvature

In this study, we discuss a relationship between the behavior of nonlinear dynamical systems and geometry of a system of second-order differential equations based on the Jacobi stability analysis. We consider how a maximal Lyapunov exponent is related to the geometric quantities. As a result of a theoretical investigation, the maximal Lyapunov exponent can be represented by a nonlinear connection and a deviation curvature. Thus, this means that the Jacobi stability given by the sign of the deviation curvature affects the change of the maximal Lyapunov exponent. Additionally, for an equation of nonlinear pendulum, we numerically confirm the theoretical results. We observe that a change of the maximal Lyapunov exponent is related to a change of an average deviation curvature. These results indicate that the deviation curvature and Jacobi stability are essential for considering the change of maximal Lyapunov exponent.

On the Analytical Solutions of the Forced Damping Duffing Equation in the Form of Weierstrass Elliptic Function and its Applications

In this study, a novel analytical solution to the integrable undamping Duffing equation with constant forced term is obtained. Also, a new approximate analytical (semianalytical) solution for the nonintegrable linear damping Duffing oscillator with constant forced term is reported. The analytical solution is given in terms of the Weierstrass elliptic function with arbitrary initial conditions. With respect to it, the semianalytical solution is constructed depending on a new ansatz and the exact solution of the standard Duffing equation (in the absence of both damping and forced terms). A comparison between the obtained solutions and the Runge–Kutta fourth-order (RK4) is carried out. Moreover, some complicated oscillator equations such as the constant forced damping pendulum equation, forced damping cubic-quintic Duffing equation, and constant forced damping Helmholtz–Duffing equation are reduced to the forced damping Duffing oscillator, in which its solution is known. As a practical application, the proposed techniques are applied to investigate the characteristics behavior of the signal oscillations arising in the RLC circuit with externally applied voltage.

Generalized Bessel Functions and Their Use in Bremsstrahlung and Multi-Photon Processes

The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed at enhancing the harmonic content of the emitted radiation. The analysis is eventually extended to the generalization of the FEL pendulum equation to treat Free Electron Laser operating with multi-harmonic undulators. The paper aims at picking out those elements supporting the usefulness of the Generalized Bessel Functions in the elaboration of the theory underlying the study of the spectral properties of the bremsstrahlung radiation emitted by relativistic charges, along with the relevant flexibility in accounting for a large variety of apparently uncorrelated phenomenolgies, like multi-photon processes, including non linear Compton scattering.

Approximate analytical and numerical solutions to the damped pendulum oscillator: Newton–Raphson and moving boundary methods

In this work, some new approximate solutions to the damped pendulum equation are obtained. In addition, the Newton–Raphson method (NRM), moving boundary method, and fourth-order Runge Kutta forth-order (RK4) are introduced to analyze the problem under study numerically. With respect to the approximate analytic solutions, two schemes are devoted: in the first approach, we can solve our problem with specific values for the initial conditions (zero initial angle) and after that compare our analytic solution with numerical solutions and with some published solutions. Thereafter, some modifications and improvements for the analytic solution will be constructed in order to get high-accurate solutions. With respect to the second scheme, we can solve our problem with arbitrary initial conditions and then make a comparison between the obtained results and the mentioned numerical solutions. Moreover, the distance error for all obtained solutions is estimated with respect to the RK4 solution.

Review of approximate equations for the pendulum period

Precision measurements of the pendulum period as a function of the amplitude can now be made with a variety of instruments including MEMs gyro/accelerometers, and thus theoretical expressions are required for comparison. Unfortunately exact solution of the pendulum equation involves elliptic integrals, which cannot be expressed in terms of elementary functions, and therefore a wide variety of approximations have been published. These range from simple single-term formulae to more sophisticated equations, which apply to a wider range of amplitudes, to an iterative procedure for calculating the precise period. The published approximations are compared as Taylor series expansions, and graphically to indicate their accuracy and their regions of applicability.

Stability analysis of multi-term fractional-differential equations with three fractional derivatives

Necessary and sufficient stability and instability conditions are obtained for multi-term homogeneous linear fractional differential equations with three Caputo derivatives and constant coefficients. In both cases, fractional-order-dependent as well as fractional-order-independent characterisations of stability and instability properties are obtained, in terms of the coefficients of the multi-term fractional differential equation. The theoretical results are exemplified for the particular cases of the Basset and Bagley-Torvik equations, as well as for a multi-term fractional differential equation of an inextensible pendulum with fractional damping terms, and for a fractional harmonic oscillator.