The Response and Instability of Cross-Rope Suspension Towers Under Harmonic Excitation

2017 ◽  
Vol 17 (10) ◽  
pp. 1750124 ◽  
Author(s):  
Zhitao Yan ◽  
Yu Zhu ◽  
Yi You ◽  
Jing Wang
Keyword(s):  
Transmission Lines ◽  
Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Steady State Solution ◽  
Harmonic Excitation ◽  
Wind Load ◽  
Dynamic Force ◽  
Harmonic Load ◽  
Straight Beam

The galloping or vortex-induced vibration of transmission lines will lead to a periodic excitations to the masts of the cross-rope suspension tower (CRST). The mast of the CRST is modeled as a straight beam with an elastic support subjected to a pulsating axial force on the top, which will change the stiffness of the mast, thereby resulting in produce harmonic excitation and instability. The dynamic characteristics of the system are investigated, which show that the bending frequency of the CRST decreases linearly with increase in axial static load, while it increases nonlinearly with the increase in boundary stiffness. Then, the method of multiple scales is adopted to analyze the vibration. It is found that the wind load on the mast brings primary resonance, but has no effects on instability. In addition, the steady state solution of the primary resonance is obtained by the polar form of the reduced amplitude modulation equations (RAMEs), with the effects of the following parameters on the vibration amplitude of the mast studied: the prestressing load in the guy, magnitude of the dynamic force, detuning parameter and wind load. Finally, the instability regions of two cases ([Formula: see text] near [Formula: see text] and [Formula: see text] near [Formula: see text]) are studied by the Cartesian form of the RAMEs, with focus on the influence of the axial harmonic load produced by the galloping of the transmission lines on the instability area. It is observed that the magnitude of excitation frequency of the dynamic force in the range of instability region becomes larger until the spring stiffness is increased up to a certain value.

Resonance Analysis of Fractional-Order Mathieu Oscillator

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Jiangchuan Niu ◽  
Hector Gutierrez ◽  
Bin Ren
Keyword(s):  
Fractional Order ◽  
Order Term ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Steady State Solution ◽  
Harmonic Excitation ◽  
Resonance Analysis ◽  
Approximate Analytical Solutions ◽  
Resonance Response ◽  
Resonant Behavior

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

2009 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis ◽  
Fadi M. Alsaleem ◽  
Ronald Miles ◽  
Weili Cui
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Harmonic Load ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Model Equations

In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.

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.

NONLINEAR BEHAVIOR OF TUNED ROTOR-AMB SYSTEM WITH TIME VARYING STIFFNESS

2011 ◽  
Vol 21 (01) ◽  
pp. 195-207 ◽  
Author(s):  
M. EISSA ◽  
M. KAMEL ◽  
H. S. BAUOMY
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Steady State Solution ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Practical Case ◽  
Time Varying ◽  
Amb System

A rotor-active magnetic bearing (AMB) system with a periodically time-varying stiffness subjected to tuned and external excitations is studied and solved. The tuned excitation represents an imposed noise on the external excitation to simulate the practical case. The method of multiple scales is applied to analyze the response of the system two modes near the simultaneous combined and primary resonance cases. The stability of the steady state solution near this resonance case is studied applying Lyapunov's first method. The system exhibits many typical nonlinear behaviors including multiple-valued solutions, jump phenomenon, softening nonlinearity and saturation. The presence of the tuned excitation increased the steady state amplitudes and produced a chaotic system. The effects of the different parameters on the steady state solutions are investigated and discussed. Comparison with previous work is reported.

Primary Resonance Voltage Response of Electrostatically Actuated M/NEMS Circular Plate Resonators

2014 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Reynaldo Oyervides
Keyword(s):  
Casimir Force ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Voltage Response ◽  
Circular Plates ◽  
Amplitude Response ◽  
Weakly Nonlinear ◽  
Electrostatically Actuated

This paper investigates the voltage-amplitude response of soft AC electrostatically actuated M/NEMS clamped circular plates. AC frequency is near half natural frequency of the plate. This results in primary resonance. The system is analytically modeled using the Method of Multiple Scales (MMS). The system is assumed weakly nonlinear. The behavior of the system including pull-in instability as the AC voltage is swept up and down while the excitation frequency is constant is reported. The effects of detuning frequency, damping, Casimir force, and van der Waals force are reported as well.

Multimodal Resonances in Suspended Cables via a Direct Perturbation Approach

1997 ◽  
Author(s):  
Giuseppe Rega ◽  
Walter Lacarbonara ◽  
Ali H. Nayfeh ◽  
Char-Ming Chin
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
A Priori ◽  
Three Dimensional ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Finite Amplitude ◽  
Perturbation Approach ◽  
Plane Symmetric ◽  
Complex Valued

Abstract We analyze the nonlinear three–dimensional response of an elastic suspended cable with small sag-to-span ratio to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one–to–one internal resonance with the first antisymmetric planar and nonplanar modes and a two–to–one internal resonance with the first symmetric nonplanar mode. We apply the method of multiple scales directly to the governing two integro–partial–differential equations and associated boundary conditions with no a priori assumption on the shape of the motion. The result is a system of four coupled nonlinear complex–valued equations describing the modulation of the amplitudes and phases of the four interacting modes. The spatial-temporal corrections to the displacement field at higher orders show that the solution is not separable in space and time. Prelimary comparisons with a companion Galerkin-type discretized model show that the latter must be used with some care in studying finite–amplitude motions of cables.

The Nonlinear Response of a Simply Supported Rectangular Metallic Plate to Transverse Harmonic Excitation

2000 ◽  
Vol 67 (3) ◽  
pp. 621-626 ◽  
Author(s):  
O. Elbeyli and ◽  
G. Anlas
Keyword(s):  
Critical Points ◽  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Stability Of Solutions ◽  
Metallic Plate ◽  
Simply Supported

In this study, the nonlinear response of a simply supported metallic rectangular plate subject to transverse harmonic excitations is analyzed using the method of multiple scales. Stability of solutions, critical points, types of bifurcation in the presence of a one-to-one internal resonance, together with primary resonance, are determined. [S0021-8936(00)00603-6]

Nonlinear Forced Vibration Analysis of Fluid Conveying Nanotubes Under Electromagnetic Actuation

2014 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Governing Equation ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Fluid Velocity ◽  
Fluid Motion ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Electromagnetic Excitation ◽  
Electromagnetic Actuation ◽  
Nonlinear Forced Vibration

Nonlinear forced vibration of fluid-conveying nanotubes based on Euler-Bernoulli beam theory under electromagnetic actuation is studied. The nanotube is modeled as cantilever type beam and the effects of fluid motion and external harmonic excitation are considered in the governing equation of the structure vibration. The Galerkin procedure is applied in order to discretize the governing equation of vibration of the system. The well-known multiple scales method is utilized to investigate the primary resonance in the forced vibration of nanotubes. The effects of various parameters, namely, fluid velocity, position of applied force, aspect ratio and electromagnetic excitation on the primary resonance of the system are fully investigated. It is revealed that the electromagnetic excitation is highly influential on the frequency response of the considered system.

The Consistency between MLP Method and Modified Method of Multiple Scales for Strong Nonlinear Primary Resonance of Duffing Equation Subject to Harmonic Excitation in Complex Number Field

2014 ◽  
Vol 472 ◽  
pp. 62-68
Author(s):  
Zhi’an Yang ◽  
Xi Li ◽  
Jia Jia Meng
Keyword(s):  
Frequency Response ◽  
Number Field ◽  
Complex Number ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Duffing Equation ◽  
Modified Method ◽  
Amplitude Frequency Response ◽  
Complex Number Field

With changing strong nonlinear Duffing equation subject to harmonic excitation with damping in complex number field as an object, the amplitude frequency response equation of primary resonance of the system is obtained through parametric transformation with application of MLP method and modified method of multiple scales. In different approximation solution forms and different time scales, the two methods lead to the same amplitude frequency response equation. Thus, the two methods are mutually verifiable. Numerical analysis shows that for the strong nonlinear Duffing equation with damping in complex number field, the nonlinear stiffness coefficient is more than zero and the amplitude frequency response curve of primary resonance leans to the left, which is different from the weak nonlinear Duffing equation.Chinese books catalog: O321

Vibration of an axially moving beam supported by a slightly curved elastic foundation

2017 ◽  
Vol 24 (17) ◽  
pp. 4000-4009 ◽  
Author(s):  
Aalim M Mustafa ◽  
Muhammad A Hawwa ◽  
David E Hardt
Keyword(s):  
Numerical Solutions ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Period Doubling ◽  
Elastic Support ◽  
Axially Moving Beam ◽  
Decomposition Approach ◽  
Straight Beam ◽  
Moving Beam ◽  
Axially Moving

The vibration of an axially moving beam following a slightly curved path was studied. The simply supported beam was travelling axially on a curved frictionless foundation with nonlinear elastic characteristics. The main objective of this work was to investigate the effect of the moving beam path curvature on its vibration, and the effect of different parameters on the system’s dynamic response. These parameters include axial speed, applied tension, and stiffness of the supporting foundation. A Galerkin decomposition approach with four-term truncation accuracy was used to realize a mathematical model that describes the dynamic behavior of the axially moving beam on a slightly curved foundation. Numerical solutions showed that the natural frequency of the axially moving beam travelling on the curved elastic support was higher than that of an axially moving straight beam for all cases considered of different path curvatures and different degrees of support stiffness. Forced vibrations of an axially moving beam on a curved elastic support were also considered under harmonic excitation. Bifurcation diagrams were obtained for the primary resonance excitation using the excitation amplitude as a controlling parameter, while keeping the excitation frequency fixed. It was found that the amplitude–frequency diagram for the axially moving beam on the curved path exhibited many types of bifurcations, including period doubling bifurcation, period four bifurcation and many jumps, compared to that of an axially moving beam resting on a straight elastic support.

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