Destabilizing effect of Coulomb friction on vibration of a beam supported at an axially oscillating mount

Zaja̧czkowski, J.

doi:10.1016/0022-460x(81)90467-3

Cited by 7 publications

(5 citation statements)

References 3 publications

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“…The nonlinear oscillators can arise in several branches of science including the oscillations of a highamplitude physical pendulum, nonlinear electrical circuits, image processing, open states of DNA, the movement of satellites, Bose-Einstein condensates, oscillations in different plasma models, and many others phenomena [11][12][13][14][15]. Moreover, the simple pendulum has been used as a physical model to solve several natural problems related to bifurcations, oscillations, and chaos such as nonlinear plasma oscillations [16][17][18][19], and many other oscillations in different fields of science [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Alyousef

Alharthi²,

Salas³

et al. 2022

Commun. Theor. Phys.

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The (un)forced (un)damped parametric pendulum oscillator is analyzed analytically and numerically using some simple, effective, and more accurate techniques. In the first technique, the ansatz method is employed for analyzing the unforced damped parametric pendulum oscillator (PPO) and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions. In the second approach, some approximations to (un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method. In the third approach, He's Frequency-Amplitude principle (He's-FAP) is applied for deriving some approximations to the (un)damped PPO. In the forth approach, He's Homotopy technique (He's-HT) is employed for analyzing the forced (un)damped PPO numerically. In the fifth approach, the p-solution Method which is constructed based on Krylov-Bogoliúbov Mitropolsky method (KBMM) is introduced for deriving an approximation to the forced damped PPO. In the final approach, the hybrid Padé-finite difference method (Padé-FDM) is carried out for analyzing the damped PPO numerically. All proposed techniques are compared to the fourth-order Runge-Kutta (RK4) numerical solution. Moreover, the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations. The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science, e.g. waves and oscillations in plasma physics.

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Section: Introductionmentioning

confidence: 99%

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Alyousef

Alharthi²,

Salas³

et al. 2022

Commun. Theor. Phys.

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“…e pendulum oscillator and some related equation have been used as a physical model to solve several natural problems related to bifurcations, oscillations, and chaos such as nonlinear plasma oscillations [1][2][3][4][5][6][7][8][9], Du ng oscillators [10][11][12][13][14], and Helmholtz oscillations [12], and many other applications can be found in [15][16][17][18][19][20][21][22][23][24]. ere are few attempts for analyzing the equation of motion of the nonlinear damped pendulum taking the friction forces into account [25].…”

Section: Introductionmentioning

confidence: 99%

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

Alharthi

Salas

Albalawi

et al. 2022

Journal of Mathematics

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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.

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“…The pendulum oscillator and some related equation have been used as a physical model to solve several natural problems related to bifurcations, oscillations, and chaos such as nonlinear plasma oscillations [1][2][3][4][5], Du¤ing oscillators [6][7][8][9][10], Helmholtz oscillations [8], rigid plates that satisfy the Johanessen performance criteria [11], transverse vibrations nonlinear of a plate carrying a concentrated mass [12], a beam supported by a double periodic axial oscillating mount [13], cracks subjected to concentrated forces [14], surface waves in a plasma column [15], coupled modes of nonlinear bending vibrations of a circular ring [16], double spin spacecraft [17], motion of spacecraft over slowly rotating asteroids [18], nonlinear vibration of clasped beams [19], the nonlinear equation of wave [20], nonlinear mathematical models of DNA [21], the nonlinear Schrödinger equation etc. There are few attempts for analyzing the equation of motion of the nonlinear damped pendulum taking the friction forces into account [22].…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

Albalawi

Salas

El-Tantawy

2021

Preprint

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In this work, novel semi-analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation on the pivot vertically for arbitrary angles are obtained for the first time. The semi-analytical solution is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical analysis, the Chebyshev collocation numerical method is introduced for analyzing tthe forced damped parametric driven pendulum equation. Moreover, the semi-analytical solution and Chebyshev collocation numerical solution are compared with the Runge-Kutta (RK) numerical solution. Also, the maximum distance error to the obtained approximate solutions 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.

show abstract

Destabilizing effect of Coulomb friction on vibration of a beam supported at an axially oscillating mount

Cited by 7 publications

References 3 publications

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

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

Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

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