Primary resonance of double-curved nanocomposite shells using nonlinear theory and multi-scales method: Modeling and analytical solution

2021 ◽  
pp. 103816
Author(s):  
Avey Mahmure ◽  
A.H. Sofiyev ◽  
N. Fantuzzi ◽  
N. Kuruoglu
Keyword(s):  
Analytical Solution ◽  
Nonlinear Theory ◽  
Primary Resonance ◽  
Multi Scales

scholarly journals On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Saad Ilyas ◽  
Feras K. Alfosail ◽  
Mohammad I. Younis
Keyword(s):  
Analytical Solution ◽  
Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Equation Of Motion ◽  
Higher Order ◽  
Resonance Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Electrostatically Actuated

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

Primary Resonance of Dry-Friction Oscillator With Fractional-Order Proportional-Integral-Derivative Controller of Velocity Feedback

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Yongjun Shen ◽  
Jiangchuan Niu ◽  
Shaopu Yang ◽  
Sujuan Li
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Dry Friction ◽  
Primary Resonance ◽  
Proportional Integral Derivative ◽  
Velocity Feedback ◽  
Integer Order ◽  
Proportional Integral ◽  
Friction Oscillator ◽  
Dynamical Properties

The classical mass-on-moving-belt model describing friction-induced vibration is studied. The primary resonance of dry-friction oscillator with fractional-order PID (FOPID) controller of velocity feedback is investigated by Krylov–Bogoliubov–Mitropolsky (KBM) asymptotic method, and the approximately analytical solution is obtained. The effects of the parameters in FOPID controller on dynamical properties are characterized by five equivalent parameters. Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response. The effects of dry friction on the dynamical properties are characterized in the form of the equivalent linear damping and nonlinear damping. The amplitude-frequency equation for steady-state solution associated with the stability condition is also studied. A comparison of the analytical solution with the numerical results is fulfilled, and their satisfactory agreement verifies the correctness of the approximately analytical results. Finally, the effects of the coefficients and orders in FOPID controller on the amplitude-frequency curves are analyzed, and the control performances of FOPID and traditional integer-order proportional-integral-derivative (PID) controllers are compared. The comparison results show that FOPID controller is better than traditional integer-order PID controller for controlling the primary resonance of dry-friction oscillator, when the coefficients of the two controllers are the same. This presents theoretical basis for scholars and engineers to design similar fractional-order controlled system.

Primary Resonance of Computer Numerical Control Worktable with Clearance and Friction

2020 ◽  
Author(s):  
Jiangchuan Niu ◽  
Zhishuang Zhao ◽  
Yongjun Shen ◽  
Shaopu Yang
Keyword(s):  
Analytical Solution ◽  
Dynamic Characteristics ◽  
Analytical Solutions ◽  
Approximate Analytical Solution ◽  
Primary Resonance ◽  
Computer Numerical Control ◽  
Harmonic Excitation ◽  
Numerical Control ◽  
Stick Slip ◽  
Approximate Analytical Solutions

Abstract Computer numerical control (CNC) worktable is the most important part of CNC machines. The CNC worktable exhibits complex nonlinear dynamic behaviors in the milling process. The physical model and mathematical model of CNC worktable are presented, where the nonlinear factors such as clearance and friction are considered. The primary resonance of computer numerical control worktable with clearance and friction under harmonic excitation is investigated. The approximate analytical solution of primary resonance is obtained by using the averaging method. The stability condition of the steady-state solution is also exhibited. It is found that the clearance affects the dynamic characteristics of the system in the form of equivalent nonlinear stiffness, and the friction coefficient acts in the form of equivalent nonlinear damping. The correctness of the approximate analytical solutions is verified by comparing the numerical results with the approximate analytical solutions. The approximate analytical solution is in good agreement with its corresponding numerical solution. The effects of clearance and friction on the dynamic characteristics of the system are analyzed in detail. The stick-slip vibration induced by friction is also analyzed by phase portrait at low feed velocity of machine tool. The results can provide a reference for the dynamic analysis of CNC worktable.

Waves Run-Up and Overtopping Simulations Using Lagrangian Blocks

2009 ◽  
Author(s):  
Lai Wai Tan ◽  
Vincent H. Chu
Keyword(s):  
Experimental Data ◽  
Analytical Solution ◽  
Nonlinear Theory ◽  
Dam Break ◽  
Coastal Structures ◽  
Block Method ◽  
Wave Fronts ◽  
Oscillation Problem ◽  
Eulerian Mesh ◽  
Run Up

Wave run-up and overtopping of coastal structures are simulated using Lagrangian Blocks on Eulerian Mesh (LBEM). In the LBEM simulations, the blocks carry the mass and momentum. The movement of the blocks is calculated in a Lagrangian reference frame. The water depth defined by the volume blocks is non-negative. The wave fronts across the wet-and-dry interface are simulated by the block method without interruption by the oscillation problem that has limited the applicability of many existing computational methods. To evaluate the accuracy of the LBEM method in this paper, simulations are carried out for (i) the dam-break waves, (ii) the wave run-up on plane beach, and (iii) the overtopping of solitary waves over levee. The simulations of the dam-break wave have produced excellent agreement with the exact solutions by Ritter [1] and Stoker [2], and the semi-analytical solution by Sakkas and Strelkoff [3,4]. The simulations of the wave run-up on plane beach agree with the experimental data and the nonlinear theory of Synolakis [5]. The simulations of wave overtopping trapezoidal dike agree with the finite-volume simulations of Stansby [6]. The results have demonstrated the accuracy of the LBEM method and the versatility of the method for general wave simulations over variable terrain.

scholarly journals Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Jufeng Chen ◽  
Xianghong Li ◽  
Jianhua Tang ◽  
Yafeng Liu
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Approximate Analytical Solution ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Delayed Feedback ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Similar System ◽  
Forced Excitation

The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractional-order delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system.

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