Numerical Formulation of Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1995 ◽  
Author(s):  
Longxiang Yang ◽  
Stanley G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Numerical Formulation ◽  
Force Equilibrium

Abstract An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1998 ◽  
Vol 120 (2) ◽  
pp. 475-483 ◽  
Author(s):  
L. Yang ◽  
S. G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Thin Plate Theory ◽  
Force Equilibrium

An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

The Effect of Axisymmetric Nonflatness on the Oscillation Frequencies of a Rotating Disk

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Stress Function ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Governing Equations ◽  
Coupled Equations ◽  
Stationary Disk ◽  
Frequency Behavior

This study examines the frequency characteristics of thin rotating disks subjected to axisymmetric nonflatness. The equations of motion used are based on Von Karman’s plate theory. First, the eigenfunctions of the stationary disk problem corresponding to the stress function and transverse displacement are found. These eigenfunctions produce an equation that can be used in Galerkin’s method. The initial nonflatness is assumed to be a linear combination of the eigenfunctions of the transverse displacement of the stationary disk problem. Since the initial nonflatness is assumed to be axisymmetric, only eigenfunctions with no nodal diameters are considered to approximate the initial runout. It is supposed that the disk bending deflection is small compared with disk thickness, so we can ignore the second-order terms in the governing equations corresponding to the transverse displacement and the stress function. After simplifying and discretizing the governing equations of motion, we can obtain a set of coupled equations of motion, which takes the effect of the initial axisymmetric runout into account. These equations are then used to study the effect of the initial runout on the frequency behavior of the stationary disk. It is found that the initial runout increases the frequencies of the oscillations of a stationary disk. In the next step, we study the effect of the initial nonflatness on the critical speed behavior of a spinning disk.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

On the Frequency Response of an Axisymmetric Non-Flat Spinning Disk

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Stress Function ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Governing Equations ◽  
Coupled Equations ◽  
Stationary Disk ◽  
Spinning Disk

For rotating disks, the effect of axisymmetric runout is of interest. This study examines the frequency characteristics of thin rotating discs subjected to axisymmetric non-flatness. The equations of motion used are based on Von Karman’s plate theory. First, the eigenfunctions of the stationary disk problem corresponding to the stress function and transverse displacement are found. These eigenfunctions produce an equation that can be used in the Gelrkin’s method. The initial nonflatness is assumed to be a linear combination of the eigenfunctions of the transverse displacement of the stationary disk problem. Since the initial non-flatness is assumed to be axisymmetric, only eigenfunctions with no nodal diameters are considered to approximate the initial runout. It is supposed that the disk bending deflection is small compared to disk thickness, so we can ignore the second-order terms in the governing equations corresponding to transverse displacement and stress function. After simplifying and discretizing the governing equations of motion, we can obtain a set of coupled equations of motion which takes the effect of initial axisymmetric runout into account. These equations are then used to study the effect of initial runout on the frequency response of the stationary disk. It is found that the initial runout increases the frequencies of the oscillations of a stationary disk. In the next step, we study the effect of initial non-flatness on the critical speed behavior of a spinning disk.

On the Frequency Response of a Spinning Disk With Large Deformations

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Nonlinear Frequency ◽  
Response Characteristics ◽  
Geometrical Nonlinear ◽  
Oscillation Frequencies ◽  
Multi Mode ◽  
Different Levels

In this paper, the effect of geometrical nonlinear terms, caused by a space fixed point force, on the frequencies of oscillations of a rotating disk with clamped-free boundary conditions is investigated. The nonlinear geometrical equations of motion are based on Von Karman plate theory. Using the eigenfunctions of a stationary disk as approximating functions in Galerkin’s method, the equations of motion are transformed into a set of coupled nonlinear Ordinary Differential Equations (ODEs). These equations are then used to find the equilibrium positions of the disk at different discrete blade speeds. At any given speed, the governing equations are linearized about the equilibrium solution of the disk under the application of a space fixed external force. These linearized equations are then used to find the oscillation frequencies of the disk considering the effect of large deformation. Using multi mode approximation and different levels of nonlinearity, the frequency response of the disk considering the effect of geometrical nonlinear terms are studied. It is found that at the linear critical speed, the nonlinear frequency of the corresponding mode is not zero. Results are presented that illustrate the effect of the magnitude of disk displacement upon the frequency response characteristics. It is also found that for each mode, including the effect of the geometrical nonlinear terms due to the applied load causes a separation in the frequency responses of its backward and forward traveling waves when the disk is stationary. This effect is similar to the effect of a space fixed constraint in the linear problem. In order to verify the numerical results, experiments are conducted and the results are presented.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

scholarly journals Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers

Micromachines ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 575 ◽  
Author(s):  
Aymen Jallouli ◽  
Najib Kacem ◽  
Joseph Lardies
Keyword(s):  
Differential Equations ◽  
Dynamic Behavior ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Ultrasonic Transducers ◽  
Initial Deflection ◽  
Geometric Imperfections ◽  
Resonance Frequencies

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

scholarly journals Nonlinear Vibrations of Functionally Graded Cylindrical Shells: Effect of the Geometry

2012 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano ◽  
Antonio Zippo
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Linear Analysis ◽  
Finite Amplitude ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Displacement Fields

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. In the linear analysis, after spatial discretization, mass and stiff matrices are computed, natural frequencies and mode shapes of the shell are obtained. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions obtained by the linear analysis; specific modes are selected. The Lagrange equations reduce nonlinear partial differential equations to a set of ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response of the shell. A convergence analysis is carried out to determine the correct number of the modes to be used. The analysis is focused on determining the nonlinear character of the response as the geometry of the shell varies.

Nonlinear Vibrations of Plates in Axial Pulsating Flow

2014 ◽  
Author(s):  
E. Tubaldi ◽  
M. Amabili ◽  
F. Alijani
Keyword(s):  
Flow Velocity ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Fluid Model ◽  
Equations Of Motion ◽  
Transmural Pressure ◽  
Pulsating Flow ◽  
Pulsation Frequency ◽  
Rectangular Plates ◽  
Flow Perturbation

A theoretical approach is presented to study nonlinear vibrations of thin infinitely long rectangular plates subjected to pulsatile axial inviscid flow. The case of plates in axial uniform flow under the action of constant transmural pressure is also addressed for different flow velocities. The equations of motion are obtained based on the von Karman nonlinear plate theory retaining in-plane inertia via Lagrangian approach. The fluid model is based on potential flow theory and the Galerkin method is applied to determine the expression of the flow perturbation potential. The effect of different system parameters such as flow velocity, pulsation amplitude, pulsation frequency, and channel pressurization on the stability of the plate and its geometrically nonlinear response to pulsating flow are fully discussed. In case of zero uniform transmural pressure numerical results show hardening type behavior for the entire flow velocity range when the pulsation frequency is spanned in the neighbourhood of the plate’s fundamental frequency. Conversely, a softening type behavior is presented when a uniform transmural pressure is introduced.

Steady Laminar Convective Flow with Variable Properties Due to a Porous Rotating Disk

2005 ◽  
Vol 127 (12) ◽  
pp. 1406-1409 ◽  
Author(s):  
Kh. Abdul Maleque ◽  
Md. Abdus Sattar
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Polar Coordinates ◽  
Shooting Methods ◽  
Variable Properties ◽  
Relative Temperature ◽  
Flow And Heat Transfer ◽  
Steady Laminar

The present paper investigates the effects of variable properties (density (ρ), viscosity (μ), and thermal conductivity (κ)) on steady laminar flow and heat transfer for a viscous fluid due to an impulsively started rotating porous infinite disk. These properties ρ, μ and κ are taken to be the functions of temperature. The system of axisymmetric nonlinear partial differential equations governing the steady flow and heat transfer are written in cylindrical polar coordinates and are reduced to nonlinear ordinary differential equations by introducing suitable similarity parameters. The resulting steady equations are solved numerically by using Runge-Kutta and Shooting methods, and the effects of the relative temperature difference and suction/injection parameters are examined.

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