Nonlinear Vibrations of a Circular Spinning Disk With a Transverse Load

2008 ◽  
Author(s):  
Xiao-Li Yang ◽  
Wei Zhang
Keyword(s):  
Parametric Resonance ◽  
Nonlinear Vibrations ◽  
Transverse Load ◽  
Rotating Speed ◽  
Coupled Equations ◽  
Nonlinear Responses ◽  
Spinning Disk ◽  
Averaged Equations ◽  
Rotating Flexible Disk ◽  
Flexible Disk

In this paper, we analyze the transverse nonlinear vibration of a rotating flexible disk with a periodically varying rotating speed, subjected to a rotating point force. Based on a small-stretch, moderate-rotation flexible disk theory of the Nowinski and the von Karman type field equations, the nonlinear governing equations of motion are derived for the rotating flexible disk, which are coupled equations among the radial, tangential and transverse displacements. According to the Galerkin approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes are derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, the stabilities of the steady state nonlinear responses are analyzed. Using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including multi-pulse type chaotic motions, periodic and period-n motions for the spinning disk with a varying rotating speed. It is also found that among all parameters the damping and excitation have important influence on the nonlinear responses of the spinning disk with a varying rotating speed.

Observations on the Steady-State Solution of an Extremely Flexible Spinning Disk With a Transverse Load

1983 ◽  
Vol 50 (3) ◽  
pp. 525-530 ◽  
Author(s):  
R. C. Benson
Keyword(s):  
Steady State ◽  
Hybrid System ◽  
Fourth Order ◽  
Steady State Solution ◽  
Transverse Load ◽  
System Of Differential Equations ◽  
Membrane Theory ◽  
Small Disk ◽  
Spinning Disk ◽  
Flexible Disk

The steady deflection of a transversely loaded, extremely flexible, spinning disk is studied. Membrane theory is used to predict the shapes and locations of waves that dominate the response. It is found that waves in disconnected regions are possible. Some results are presented to show how disk stiffness moderates the membrane waves, the most important result being an upper bound on the highest ordered wave of significant amplitude. A hybrid system of differential equations and boundary conditions is developed to replace the pure membrane formulation that is singular, and the full fourth-order plate formulation that is numerically sensitive. The hybrid formulation retains the salient features of the flexible disk response and facilitates calculations for very small disk stiffnesses.

Axisymmetric Solution of a Flexible Disk Rotating Near a Rigid Surface

1988 ◽  
Vol 110 (4) ◽  
pp. 674-677 ◽  
Author(s):  
M. Carpino ◽  
G. A. Domoto
Keyword(s):  
Plane Stress ◽  
Asymptotic Expansions ◽  
Reynolds Equation ◽  
Flat Surface ◽  
Inertial Effects ◽  
Rigid Surface ◽  
Axisymmetric Solution ◽  
Spinning Disk ◽  
Rotating Flexible Disk ◽  
Flexible Disk

A rotating flexible disk separated from a rigid flat surface by a gas film is addressed. The gas film between the disk and the plate is represented by an incompressible Reynolds equation. Inertial effects are included. The disk is treated as a membrane where the tension is found from the plane stress solution for a spinning disk. Two different methods for the axisymmetric solution of this system are developed. The first uses the method of matched asymptotic expansions. The second method is a mixed numerical/perturbation procedure.

Vibration of a Spinnng Annular Disk With Coupled Rigid Body Motion

1991 ◽  
Author(s):  
Shih-Ming Yang
Keyword(s):  
Rigid Body ◽  
Coupling Effect ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Transverse Load ◽  
Stable Operation ◽  
Annular Disk ◽  
Spinning Disk ◽  
Vibration Coupling ◽  
Flexible Disk

Abstract The vibration of a spinning annular disk with coupled translational and rotational rigid body motion is analyzed. The spinning disk, with one linear spring as transverse load, is free to translate and rotate relative to the shaft axis. The governing equation includes the rigid body translation, rigid body rotation, and flexible disk vibration. Coupling effect between rigid body motion and each annular disk vibration modes is identified. Because of the coupling effects, stable operation of the spinning disk beyond divergence (critical speed) is achieved, the disk loses its stability to flutter. This stability prediction is different from that of a spinning disk without rigid body motion where the disk is unstable at and right after divergence.

Vibrational response of a moving suspension-slider loading system exciting a rotating flexible disk

2012 ◽  
Vol 331 (16) ◽  
pp. 3762-3773 ◽  
Author(s):  
Yong-Chen Pei ◽  
Qing-Chang Tan ◽  
Xin Yang ◽  
Chris Chatwin
Keyword(s):  
Vibrational Response ◽  
Loading System ◽  
Rotating Flexible Disk ◽  
Flexible Disk

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

scholarly journals Fluid-Induced Vibration Elimination of a Rotor/Seal System with the Dynamic Vibration Absorber

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Qi Xu ◽  
Junkai Niu ◽  
Hongliang Yao ◽  
Lichao Zhao ◽  
Bangchun Wen
Keyword(s):  
Natural Frequency ◽  
Nonlinear Differential Equations ◽  
Damping Ratio ◽  
Force Model ◽  
Vibration Absorbers ◽  
Dynamic Vibration Absorber ◽  
Major Purpose ◽  
Rotating Speed ◽  
Dynamic Vibration ◽  
Nonlinear Responses

The dynamic vibration absorbers have been applied to attenuate the rotor unbalance and torsional vibrations. The major purpose of this paper is to research the elimination of the fluid-induced vibration in the rotor/seal system using the absorber. The simplified rotor model with the absorber is established, and the Muszynska fluid force model is employed for the nonlinear seal force. The numerical method is used for the solutions of the nonlinear differential equations. The nonlinear responses of the rotor/seal system without and with the absorber are obtained, and then the rotating speed ranges by which the fluid-induced instability can be eliminated completely and partially are presented, respectively. The absorber parameters ranges by which the instability vibration can be eliminated completely and partially are obtained. The results show that the natural frequency vibration due to the fluid-induced instability in the rotor/seal system can be eliminated efficiently using the absorber. The appropriate natural frequency and damping ratio of the absorber can extend the complete elimination region of the instability vibration and postpone the occurrence of the instability vibration.

Nonlinear Vibrations and Chaotic Dynamics of the Laminated Composite Piezoelectric Beam

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Minghui Yao ◽  
Wei Zhang ◽  
Zhigang Yao
Keyword(s):  
Numerical Simulation ◽  
Axial Load ◽  
Nonlinear Vibrations ◽  
Dynamical Behavior ◽  
Laminated Composite ◽  
Transverse Load ◽  
Nonlinear Phenomenon ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Piezoelectric Beam

This paper investigates the complicated dynamics behavior and the evolution law of the nonlinear vibrations of the simply supported laminated composite piezoelectric beam subjected to the axial load and the transverse load. Using the third-order shear deformation theory and the Hamilton's principle, the nonlinear governing equations of motion for the laminated composite piezoelectric beam are derived. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of principal parametric resonance and 1:9 internal resonance. From the averaged equations obtained, numerical simulation is performed to study nonlinear vibrations of the laminated composite piezoelectric beam. The axial load, the transverse load, and the piezoelectric parameter are selected as the controlling parameters to analyze the law of complicated nonlinear dynamics of the laminated composite piezoelectric beam. Based on the results of numerical simulation, it is found that there exists the complex nonlinear phenomenon in motions of the laminated composite piezoelectric beam. In summary, numerical studies suggest that periodic motions and chaotic motions exist in nonlinear vibrations of the laminated composite piezoelectric beam. In addition, it is observed that the axial load, the transverse load and the piezoelectric parameter have significant influence on the nonlinear dynamical behavior of the beam. We can control the response of the system from chaotic motions to periodic motions by changing these parameters.

A Study of the Influence of Bearing Clearance on Lateral Coupled Shaft/Disk Rotordynamics

1992 ◽  
Author(s):  
George T. Flowers ◽  
Fang Sheng Wu
Keyword(s):  
Dynamical Behavior ◽  
High Amplitude ◽  
Bladed Disk ◽  
Bearing Clearance ◽  
Shaft System ◽  
Coupling Dynamics ◽  
Bearing Loads ◽  
Rotating Flexible Disk ◽  
Flexible Disk ◽  
Clearance Nonlinearity

This study examines the influence of bearing clearance on the dynamical behavior of a rotating, flexible disk/shaft system. Most previous work in nonlinear rotordynamics has tended to concentrate separately on shaft vibration or on bladed disk vibration, neglecting the coupling dynamics between them. The current work examines the important rotordynamical behavior of coupled disk/shaft dynamics. A simplified nonlinear model is developed for lateral vibration of a rotor system with a bearing clearance nonlinearity. The steady-state dynamical behavior of this system is explored using numerical simulation and limit cycle analysis. It is demonstrated that bearing clearance effects can produce superharmonic vibration that may serve to excite high amplitude disk vibration. Such vibration could lead to significantly increased bearing loads and catastrophic failure of blades and disks. In addition, multi-valued responses and aperiodic behavior was observed.

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