A nonlinear frequency-amplitude relation is developed to investigate the vibrational amplitude effect on the dynamic pull-in instability of double-sided-actuated nano-torsional switches. The governing equation of a na...A nonlinear frequency-amplitude relation is developed to investigate the vibrational amplitude effect on the dynamic pull-in instability of double-sided-actuated nano-torsional switches. The governing equation of a nano-electro-mechanical system pre-deformed by an electric field contains the quintic nonlinear term. The influences of basic parameters on the pull-in instability and natural frequency are investigated using a powerful analytical approach called the homotopy perturbation method. It is demonstrated that two terms in series expansion are sufficient to produce an acceptable solution. The numerical results obtained have verified the soundness of the asymptotic procedure. The phase portraits of the double-sided nano-torsionalactuator exhibit periodic, homoclinic and heteroclinic orbits.展开更多
The(un)forced(un)damped parametric pendulum oscillator(PPO)is analyzed analytically and numerically using some simple,effective,and more accurate techniques.In the first technique,the ansatz method is employed for ana...The(un)forced(un)damped parametric pendulum oscillator(PPO)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 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 is applied for deriving some approximations to the(un)damped PPO.In the forth approach,He’s homotopy technique 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,is introduced for deriving an approximation to the forced damped PPO.In the final approach,the hybrid Padé-finite difference method 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.展开更多
In this paper, the theory of the free wobble of the triaxial Earth is developed and new conclusions are drawn: the Euler period should be actually expressed by the first kind of complete elliptic integral; the trace o...In this paper, the theory of the free wobble of the triaxial Earth is developed and new conclusions are drawn: the Euler period should be actually expressed by the first kind of complete elliptic integral; the trace of the free polar motion is elliptic and the orientations of its semi-minor and major axes are approximately parallel to the Earth's principal axes A and B, respectively. In addition, the present theory shows that there is a mechanism of frequency-amplitude modulation in the Chandler wobble, which might be a candidate for explaining the correlation between the amplitude and period of the Chandler wobble.展开更多
文摘A nonlinear frequency-amplitude relation is developed to investigate the vibrational amplitude effect on the dynamic pull-in instability of double-sided-actuated nano-torsional switches. The governing equation of a nano-electro-mechanical system pre-deformed by an electric field contains the quintic nonlinear term. The influences of basic parameters on the pull-in instability and natural frequency are investigated using a powerful analytical approach called the homotopy perturbation method. It is demonstrated that two terms in series expansion are sufficient to produce an acceptable solution. The numerical results obtained have verified the soundness of the asymptotic procedure. The phase portraits of the double-sided nano-torsionalactuator exhibit periodic, homoclinic and heteroclinic orbits.
基金The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project (Grant No. PNURSP2022R17)Taif University Researchers supporting project number (TURSP2020/275), Taif University, Taif, Saudi Arabia。
文摘The(un)forced(un)damped parametric pendulum oscillator(PPO)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 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 is applied for deriving some approximations to the(un)damped PPO.In the forth approach,He’s homotopy technique 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,is introduced for deriving an approximation to the forced damped PPO.In the final approach,the hybrid Padé-finite difference method 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.
基金Supported by the Special Project Fund of State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing (China)the Engagement Fund of Outstanding Doctoral Dissertation of Wuhan University (No.22)+1 种基金the Ph.D. Candidates Self-research (including 1+4) Program of Wu-han Unversity in 2008 (No.49)the Open Fund of Key Laboratory of Geospace Environment and Geodesy, Ministry of Education,China (No.08-02-02)
文摘In this paper, the theory of the free wobble of the triaxial Earth is developed and new conclusions are drawn: the Euler period should be actually expressed by the first kind of complete elliptic integral; the trace of the free polar motion is elliptic and the orientations of its semi-minor and major axes are approximately parallel to the Earth's principal axes A and B, respectively. In addition, the present theory shows that there is a mechanism of frequency-amplitude modulation in the Chandler wobble, which might be a candidate for explaining the correlation between the amplitude and period of the Chandler wobble.
