Nonlinear Transverse Vibrations and Stability of Spinning Disks With Nonconstant Spinning Rate

1992 ◽  
Vol 114 (4) ◽  
pp. 506-513 ◽  
Author(s):  
T. H. Young
Keyword(s):  
Periodic Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Angular Speed ◽  
Equation Of Motion ◽  
Numerical Results ◽  
Transverse Vibrations ◽  
Small Periodic ◽  
Small Periodic Perturbation

This paper studies nonlinear transverse vibrations of spinning disks with nonconstant spinning rate. Here the angular speed of the disk is characterized as a small, periodic perturbation superimposed upon a constant speed. Due to this perturbation in angular speed, nonautonomous terms appear in the equation of motion, which results in the existence of parametric instability. In this paper, Galerkin’s method is first applied to yield a discretized system, and the method of multiple scales is used to obtain periodic solutions. All types of possible resonant combinations are investigated, and numerical results are shown for a simple harmonic speed perturbation.

Stability and Vibration of a Rotating Circular Plate Subjected to Stationary In-Plane Edge Loads

1996 ◽  
Vol 63 (1) ◽  
pp. 121-127 ◽  
Author(s):  
I. Y. Shen ◽  
Y. Song
Keyword(s):  
Circular Plate ◽  
Rotational Speed ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Single Mode ◽  
Equation Of Motion ◽  
Asymmetric Membrane ◽  
Parametric Resonances ◽  
Hill’S Equations

This paper predicts transverse vibration and stability of a rotating circular plate subjected to stationary, in-plane, concentrated edge loads. First of all, the equation of motion is discretized in a plate-based coordinate system resulting in a set of coupled Hill’s equations. Through use of the method of multiple scales, stability of the rotating plate is predicted in closed form in terms of the rotational speed and the in-plane edge loads. The asymmetric membrane stresses resulting from the stationary in-plane edge loads will transversely excite the rotating plates to single-mode parametric resonances as well as combination resonances at supercritical speed. In addition, introduction of plate damping will suppress the parametric instability when normalized edge loads are small. Moreover, the radial in-plane edge load dominates the rotational speed at which the instability occurs, and the tangential in-plane edge load dominates the width of the instability zones.

Dynamic Morphing of Elastic Plates via Principal Parametric Resonance

2020 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara
Keyword(s):  
Mechanical Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Equation Of Motion ◽  
Elastic Plates ◽  
Asymptotic Approach ◽  
Threshold Voltages ◽  
Parametric Resonances ◽  
Instability Regions

Abstract Principal parametric resonances of elastic plates actuated by periodic in-plane stresses effected by embedded piezoelectric wires are investigated to describe the morphing scenarios of flexible, ultra-lightweight panels. A mechanical model of elastic plate including geometric nonlinearities and the parametric actuation provided by the piezoelectric wires, is adopted to formulate the nonlinear equation of motion. A bifurcation analysis is carried out by means of an asymptotic approach based on the method of multiple scales leading to a comprehensive parametric study on the effect of the wires width on the morphing regions (i.e., parametric instability regions) associated with the principal parametric resonances. The threshold voltages triggering the onset of the principal parametric resonances of the lowest three symmetric modes are also calculated as a function of the wires size so as to determine the voltage requirements for the morphing activation.

Closed-Form Solutions of Axisymmetrical Transverse Vibrations of Concave Parabolic Thickness Annular Plates

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
Partial Differential ◽  
First Order ◽  
Annular Plates ◽  
Transverse Vibrations

This paper deals with transverse vibrations of axisymmetrical annular plates of concave parabolic thickness. A closed-form solution of the partial differential equation of motion is reported. An approach in which both method of multiple scales and method of factorization have been employed is presented. The method of multiple scales is used to reduce the partial differential equation of motion to two simpler partial differential equations that can be analytically solved. The solutions of the two differential equations are two levels of approximation of the exact solution of the problem. Using the factorization method for solving the first differential equation, which is homogeneous and includes a fourth-order spatial-dependent operator and second-order time-dependent operator, the general solution is obtained in terms of hypergeometric functions. The first diferential equation and the second differential equation (nonhomogeneous) along with the given boundary conditions give so-called zero-order and first-order approximations, respectively, of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Dynamic Stability of Disks With Periodically Varying Spin Rates Subjected to Stationary In-Plane Edge Loads

2004 ◽  
Vol 71 (4) ◽  
pp. 450-458 ◽  
Author(s):  
T. H. Young ◽  
M. Y. Wu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Stress Distributions ◽  
The Finite Element Method ◽  
Nodal Diameter ◽  
The Stability ◽  
Small Periodic ◽  
Small Periodic Perturbation

This paper presents an analysis of dynamic stability of an annular plate with a periodically varying spin rate subjected to a stationary in-plane edge load. The spin rate of the plate is characterized as the sum of a constant speed and a small, periodic perturbation. Due to this periodically varying spin rate, the plate may bring about parametric instability. In this work, the initial stress distributions caused by the periodically varying spin rate and the in-plane edge load are analyzed first. The finite element method is applied then to yield the discretized equations of motion. Finally, the method of multiple scales is adopted to determine the stability boundaries of the system. Numerical results show that combination resonances take place only between modes of the same nodal diameter if the stationary in-plane edge load is absent. However, there are additional combination resonances between modes of different nodal diameters if the stationary in-plane edge load is present.

scholarly journals Dynamic analysis of a parametrically excited golden Muga silk embedded pneumatic artificial muscle

2018 ◽  
Vol 211 ◽  
pp. 02008 ◽  
Author(s):  
Bhaben Kalita ◽  
S. K. Dwivedy
Keyword(s):  
Dynamic Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Quadratic Function ◽  
Artificial Muscle ◽  
Equation Of Motion ◽  
Time Response ◽  
Phase Portraits ◽  
Pneumatic Artificial Muscle ◽  
Pneumatic Muscle

In this work a novel pneumatic artificial muscle is fabricated using golden muga silk and silicon rubber. It is assumed that the muscle force is a quadratic function of pressure. Here a single degree of freedom system is considered where a mass is supported by a spring-damper-and pneumatically actuated muscle. While the spring-mass damper is a passive system, the addition of pneumatic muscle makes the system active. The dynamic analysis of this system is carried out by developing the equation of motion which contains multi-frequency excitations with both forced and parametric excitations. Using method of multiple scales the reduced equations are developed for simple and principal parametric resonance conditions. The time response obtained using method of multiple scales have been compared with those obtained by solving the original equation of motion numerically. Using both time response and phase portraits, variation of few systems parameters have been carried out. This work may find application in developing wearable device and robotic device for rehabilitation purpose.

Effects of Flexural Stiffness, Axial Velocity and Internal Support Locations on Translating Beam’s Natural Frequencies

2014 ◽  
Vol 532 ◽  
pp. 316-319 ◽  
Author(s):  
Ferid Köstekci
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Flexural Rigidity ◽  
Equations Of Motion ◽  
Flexural Stiffness ◽  
Transverse Vibrations ◽  
Internal Support ◽  
Simple Support ◽  
Moving Speed

The aim of this paper is to examine the natural frequencies of beams for different flexural stiffness, internal simple support locations and axial moving speed. In the present investigation, the linear transverse vibrations of an axially translating beam are considered based on Euler-Bernoulli model. The beam is passing through two frictionless guides and has an internal simple support between the guides. The governing differential equations of motion are derived using Hamiltons Principle for two regions of the beam. The method of multiple scales is employed to obtain approximate analytical solution. Some numerical calculations are conducted to present the effects of flexural rigidity, mean translating speed and different internal support locations on natural frequencies.

Dynamic Response of a Radial Beam With Nonconstant Angular Velocity

1987 ◽  
Vol 109 (2) ◽  
pp. 138-143 ◽  
Author(s):  
D. C. Kammer ◽  
A. L. Schlack
Keyword(s):  
Angular Velocity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Approximate Solutions ◽  
Euler Beam ◽  
Radial Beam ◽  
Kbm Method ◽  
Small Periodic ◽  
Small Periodic Perturbation

The effects of a nonconstant angular velocity upon the vibration of a rotating Euler beam are investigated. It is assumed that the angular velocity can be written as the sum of a steady-state value and a small periodic perturbation. The time-dependence of the angular velocity results in the appearance of terms in the equations of motion which cause the system to be nonautonomous. These terms result in the existence of regions of parametric instability within which the amplitude grows exponentially. A perturbation technique called the KBM method is used to derive approximate solutions and expressions for the boundaries between stable and unstable motion. A simple perturbation function is assumed to illustrate the use of the derived general equations.

Sensitivity of MEMS/NEMS Biosensors Near Twice Natural Frequency

2010 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin W. Knecht
Keyword(s):  
Natural Frequency ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Casimir Effect ◽  
Electrostatic Force ◽  
Equation Of Motion ◽  
Direct Approach ◽  
Mass Detection ◽  
Electrostatically Actuated ◽  
Phase Amplitude

Bio-MEMS/NEMS resonator sensors near twice natural frequency for mass detection are investigated. Electrostatic force along with fringe correction and Casimir effect are included in the model. They introduce parametric nonlinear terms in the system. The partial-differential equation of motion of the system is solved by using the method of multiple scales. A direct approach of the problem is then used. Two approximation problems resulting from the direct approach are solved. Phase-amplitude relationship is obtained. Numerical results for uniform electrostatically actuated micro resonator sensors are reported.

Nonlinear Dynamics of a Travelling Beam Subjected to Multi-Frequency Parametric Excitation

2014 ◽  
Vol 592-594 ◽  
pp. 2076-2080 ◽  
Author(s):  
Bamadev Sahoo ◽  
L.N. Panda ◽  
Goutam Pohit
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Reduced Equations ◽  
Stable Periodic ◽  
Bifurcation And Stability ◽  
First Order Differential Equations

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

Boundary Value Problem Method to Investigate Superharmonic Resonance of MEMS Resonators

2017 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Christian Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
The Other ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Amplitude Frequency Response ◽  
Problem Method ◽  
Mems Resonators

This paper deals with two different methods to analyze the amplitude frequency response of an electrostatically actuated micro resonator. The methods used in this paper are the method of multiple scales, which is an analytical method with one mode of vibration. The other method is based on system of odes which is derived using the partial differential equation of motion, as well as the boundary conditions. This system is then solved using a built in matlab function known as BVP4C. Results are then shown comparing the two methods, under a variety of parameters, including the influence of damping, voltage, and fringe.

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