scholarly journals Thermally Induced Principal Parametric Resonance in Circular Plates

2002 ◽  
Vol 9 (3) ◽  
pp. 143-150 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Waleed Faris
Keyword(s):  
Parametric Resonance ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Temperature Fields ◽  
Circular Plates ◽  
Thermally Induced ◽  
Principal Parametric Resonance ◽  
Large Amplitude Vibrations ◽  
External Heat Flux

We consider the problem of large-amplitude vibrations of a simply supported circular flat plate subjected to harmonically varying temperature fields arising from an external heat flux (aeroheating for example). The plate is modeled using the von Karman equations. We used the method of multiple scales to determine an approximate solution for the case in which the frequency of the thermal variations is approximately twice the fundamental natural frequency of the plate; that is, the case of principal parametric resonance. The results show that such thermal loads produce large-amplitude vibrations, with associated multi-valued responses and subcritical instabilities.

Frequency Response of Parametric Resonance of Electrostatically Actuated Circular Plates Under Hard Excitations

2019 ◽  
Author(s):  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Frequency Response ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Reduced Order Model ◽  
Circular Plates ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

Abstract In this paper, the Method of Multiple Scales, and the Reduced Order Model method of two modes of vibration are used to investigate the amplitude-frequency response of parametric resonance of electrostatically actuated circular plates under hard excitations. Results show that the Method of Multiple Scales is accurate for low voltages. However, it starts to separate from the Reduced Order Model results as the voltage values are larger. The Method of Multiple Scales is good for low amplitudes and weak non-linearities. Furthermore the Reduced Order Model running with AUTO 07p is validated and calibrated using the 2 Term ROM time responses.

Voltage Response of Parametric Resonance of MEMS Circular Plates Under Hard Excitations

2019 ◽  
Author(s):  
Julio S. Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Voltage Response ◽  
Circular Plates ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

Abstract This work deals with the voltage response of parametric resonance of electrostatically actuated microelectromechanical (MEMS) circular plates under hard excitations. Method of Multiple Scales (MMS) and Reduced Order Model (ROM) method using two modes of vibration are used to predict the voltage-amplitude response of the MEMS circular plates. ROM is solved using AUTO 07p, a software package for continuation and bifurcation. MMS used in this paper has one term in the electrostatic force being considered significant. This is the way MMS is used to model hard excitations. MMS shows results similar to those of ROM at lower amplitudes and lower voltages. The differences between the two methods, MMS and ROM, are significant in high amplitudes for all voltages, and the differences are significant in all amplitudes for larger voltages. Significant differences can be noted in the effect of different parameters such as the detuning frequency and damping on the voltage response. ROM AUTO 07p is calibrated using ROM time responses in which the ROM is solved using the solver ode15s in Matlab.

Analysis of Parametric Resonances in In-Plane Vibrations of Electrostrictive Hyperelastic Plates

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Electric Field ◽  
Parametric Resonance ◽  
Mechanical Response ◽  
Element Model ◽  
Mode Shapes ◽  
Principal Parametric Resonance ◽  
Parametric Resonances ◽  
Second Mode ◽  
Electrostrictive Polymer ◽  
Large Amplitude Vibrations

Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

Principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law

2005 ◽  
Vol 461 (2061) ◽  
pp. 2701-2720 ◽  
Author(s):  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Constitutive Law ◽  
Transverse Motion ◽  
Viscoelastic Parameters ◽  
Principal Parametric Resonance ◽  
Linearized Stability ◽  
Speed Fluctuation ◽  
Axial Speed

The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) non-trivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial-speed fluctuation amplitude, and the viscoelastic parameters.

Subcritical Large-Amplitude Vibrations of Imperfect Rotating Shafts

1989 ◽  
Vol 42 (11S) ◽  
pp. S157-S160
Author(s):  
C. E. N. Mazzilli
Keyword(s):  
Dry Friction ◽  
Large Amplitude ◽  
Critical Speed ◽  
Multiple Scales ◽  
Viscous Damping ◽  
Rotating Shaft ◽  
Geometrical Imperfection ◽  
Rotating Shafts ◽  
Large Amplitude Vibrations ◽  
Practical Implications

The effect of a geometrical imperfection, such as the axis flexural deformation, on the large-amplitude vibrations of a horizontal rotating shaft is analyzed with the aid of the Multiple Scales Method. Internal viscous damping and linear elasticity are assumed in the model. It is then seen that no matter how small the imperfection is, a “critical” speed of the order of half the classical critical speed arises, with relevant practical implications. A number of reported large-amplitude cases may eventually be explained this way. It is possible that events such as this will not appear in systems with high dry friction.

Parametric Resonance of Electrostatically Actuated MEMS Cantilever Resonators Using Method of Multiple Scales and Homotopy Perturbation Method

2017 ◽  
Author(s):  
Christopher Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Perturbation Method ◽  
Parametric Resonance ◽  
Homotopy Perturbation Method ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Plate Electrode ◽  
Electrostatically Actuated ◽  
Homotopy Perturbation ◽  
Ground Plate

This paper investigates the dynamics governing the behavior of electrostatically actuated MEMS cantilever resonators. The cantilever is held parallel to a ground plate (electrode) with an AC voltage between the plate and the electrode causing the electrostatic actuation (excitation). For the purposes of this paper this is soft excitation. The frequency of the excitation is near the natural frequency of the cantilever leading to what is known as parametric resonance. The electrostatic force in the problem investigated throughout the paper is nonlinear in nature and includes the fringe effect. Two methods are used in investigating this problem: the method of multiple scales (MMS) and the homotopy perturbation method (HPM). The two methods work well for small non-linearities and small amplitudes. The influence of voltage, fringe, damping, Casimir, and Van der Waals parameters will be investigated in this paper using MMS and HPM as a means of verifying the results obtained.

Nonlinear Free Vibrations of a Timoshenko Beam Using Multiple Scales Method

Volume 2 ◽  
2004 ◽  
Author(s):  
Asghar Ramezani ◽  
Mehrdaad Ghorashi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Vibration ◽  
Timoshenko Beam ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Free Vibrations ◽  
Partial Differential ◽  
Cantilevered Beam

In this paper, the large amplitude free vibration of a cantilever Timoshenko beam is considered. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the beam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the thick-cantilevered beam.

On Nonlinear Dynamics of Nanotweezers

2016 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Bin Liu
Keyword(s):  
Nonlinear Dynamics ◽  
Frequency Response ◽  
Parametric Resonance ◽  
Taylor Series ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Van Der Waals ◽  
Van Der Waals Forces ◽  
The Other ◽  
Amplitude Frequency Response

This paper deals with amplitude-frequency response of electrostatic nanotube nanotweezer device system. Soft alternating current (AC) of frequency near natural frequency actuates the nanotubes. This leads the system into parametric resonance. The Method of Multiple Scales (MMS) in which the nonlinear electrostatic and van der Waals forces are expanded in Taylor series is used to compare two expansions, one up to third power and the other up to fifth power. The frequency response of the system is reported and the effects of van der Waals forces, electrostatic forces, and damping forces on the frequency response are investigated.

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