scholarly journals The Combined Internal and Principal Parametric Resonances on Continuum Stator System of Asynchronous Machine

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Baizhou Li ◽  
Qichang Zhang
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Defense Industry ◽  
Response Curves ◽  
Shell System ◽  
State Response ◽  
Noise Characteristics ◽  
Runge Kutta Method ◽  
Nonlinear Dynamic Equations ◽  
Parametric Resonances

With the increasing requirement of quiet electrical machines in the civil and defense industry, it is very significant and necessary to predict the vibration and noise characteristics of stator and rotor in the early conceptual phase. Therefore, the combined internal and principal parametric resonances of a stator system excited by radial electromagnetic force are presented in this paper. The stator structure is modeled as a continuum double-shell system which is loaded by a varying distributed electromagnetic load. The nonlinear dynamic equations are derived and solved by the method of multiple scales. The influences of mechanical and electromagnetic parameters on resonance characteristics are illustrated by the frequency-response curves. Furthermore, the Runge-Kutta method is adopted to numerically analyze steady-state response for the further understanding of the resonance characteristics with different parameters.

Analytical solution for primary resonances of a rotating shaft with stretching non-linearity

2008 ◽  
Vol 222 (9) ◽  
pp. 1655-1664 ◽  
Author(s):  
S A A Hosseini ◽  
S E Khadem
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Complex Form ◽  
Steady State Response ◽  
Response Curves ◽  
Rotating Shaft ◽  
Simply Supported ◽  
State Response ◽  
Gyroscopic Effects

In this paper, primary resonances of a simply supported rotating shaft with stretching non-linearity are studied. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The equations of motion are derived with the aid of Hamilton's principle and then transformed to the complex form. To analyse the primary resonances, the method of multiple scales is directly applied to the partial differential equation of motion. The frequency—response curves are plotted for the first two modes. It is shown that these resonance curves are of the hardening type. The effects of eccentricity and the damping coefficient are investigated on the steady-state response of the rotating shaft.

Response of a Stationary, Damped, Circular Plate Under a Rotating Slider Bearing System

1993 ◽  
Vol 115 (1) ◽  
pp. 65-69 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Circular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Slider Bearing ◽  
Numerical Examples ◽  
Elastic Circular Plate ◽  
Parametric Resonances ◽  
Supercritical Speed ◽  
Bearing System

This paper is to demonstrate that axisymmetric plate damping will suppress unbounded response of a stationary, elastic, circular plate excited by a rotating slider. Use of the method of multiple scales shows that the axisymmetric plate damping will suppress parametric resonances excited by slider stiffness and slider inertia at supercritical speed. In addition, the plate damping will increase the onset speed above which slider damping destabilizes the elastic circular plate. Moreover, numerical examples show that the plate damping could stabilize the plate/slider system at discrete rotation speeds above the onset speed.

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.

Primary Resonance of the Current-Carrying Beam in Thermal-Magneto-Elasticity Field

2010 ◽  
Vol 29-32 ◽  
pp. 16-21 ◽  
Author(s):  
Xiao Yan Xi ◽  
Zhian Yang ◽  
Li Li Meng ◽  
Chang Jian Zhu
Keyword(s):  
Response Curve ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Response Curves ◽  
Bending Vibration ◽  
Approximation Solution ◽  
System Parameters ◽  
Practical Engineering

On base of the electro-magneto-elastic theory and the theory of the bending vibration of the electric beam, nonlinear vibration equation of current-carrying beam subjected to thermal-magneto-elasticity field is studied. The Lorentz force and thermal force on the beam are derived. According to the method of multiple scales for nonlinear vibrations the approximation solution of the primary resonance of the system is obtained. Numerical analysis results show that the amplitude changed with the system parameters. With the decrease of magnetic intensity, the amplitude increases rapidly. The response curve occurs bending phenomenon and soft features is increased gradually. Increasing current, the amplitudes increase. With the decrease of temperature, the peak of response curves decrease. With the increase of temperature, natural frequency decreased. It is useful in practical engineering.

Analysis of Transverse Vibration of Axially Moving Viscoelastic Sandwich Beam with Time-Dependent Velocity

2011 ◽  
Vol 338 ◽  
pp. 487-490 ◽  
Author(s):  
Hai Wei Lv ◽  
Ying Hui Li ◽  
Qi Kuan Liu ◽  
Liang Li
Keyword(s):  
Transverse Vibration ◽  
Multiple Scales ◽  
Sandwich Beam ◽  
Core Layer ◽  
Response Elimination ◽  
Response Curves ◽  
Mean Velocity ◽  
State Response ◽  
Axially Moving ◽  
The Stability

Transverse vibration of an axially moving viscoelastic sandwich beam is investigated in this paper. Based on the Kelvin constitutive equation, transverse controlling equation is established. First of all, the multiple scales method is applied to obtained steady-state response. Elimination of scales terms will give us the amplitude of vibrations. Additionally, the stability conditions of trivial and non-trivial solutions are analyzed using Routh-Hurwitz criterion. Eventually, numerical results are obtained to show the thickness of core layer, mean velocity, the amplitude of fluctuation effects on natural frequencies and response curves.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

scholarly journals Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification

2012 ◽  
Vol 19 (4) ◽  
pp. 527-543 ◽  
Author(s):  
Li-Qun Chen ◽  
Hu Ding ◽  
C.W. Lim
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Steady State Response ◽  
Governing Equations ◽  
State Response ◽  
Non Linear ◽  
Axial Speed

Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude.

Nonlinear Vibrations of a Beam-Spring-Mass System

1994 ◽  
Vol 116 (4) ◽  
pp. 433-439 ◽  
Author(s):  
M. Pakdemirli ◽  
A. H. Nayfeh
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Primary Resonance ◽  
Fast Time ◽  
Lagrange Equations ◽  
Response Curves ◽  
Mass System

The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

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.

On Internal Resonance of Nonlinear Nonuniform Beams

2009 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Steady State Response ◽  
Amplitude Equations ◽  
Bending Vibrations ◽  
State Response ◽  
Phase Amplitude

This paper reports the case of internal resonance three-to-one with frequency of excitation near natural frequency in the case of bending vibrations of nonuniform cantilever with small damping. The case of nonlinear curvature, moderately large amplitudes, is considered. The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions. The phase-amplitude equations are analytically determined. Steady-state response is reported.

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