Effects of Load Parameters on the Natural Frequencies and Stability of a Flexible Spinning Disk With a Stationary Load System

1992 ◽  
Vol 59 (2S) ◽  
pp. S230-S235 ◽  
Author(s):  
Jen-San Chen ◽  
D. B. Bogy
Keyword(s):  
Friction Force ◽  
Centrifugal Force ◽  
Natural Frequencies ◽  
Numerical Solutions ◽  
Positive Definite ◽  
Transverse Mass ◽  
Gyroscopic System ◽  
Theoretical Understanding ◽  
Spinning Disk ◽  
Derivatives Of

The orthogonality properties among the eigenfunctions for a gyroscopic system are derived for a stiffness operator that is not positive definite. The derivatives of the eigenvalues with respect to certain parameters in the system are then obtained. The results are applied to a spinning disk in contact with a stationary load system, which contains such parameters as friction force, transverse mass, damping, stiffness, and the analogous pitching elements, to predict the effects of these parameters and the stiffening of the disk due to the centrifugal force on the natural frequencies and stability of the spinning disk. The results obtained provide a theoretical understanding for previously reported observations based on numerical solutions.

scholarly journals Mathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System

1992 ◽  
Vol 59 (2) ◽  
pp. 390-397 ◽  
Author(s):  
Jen-San Chen ◽  
D. B. Bogy
Keyword(s):  
Friction Force ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Mathematical Structure ◽  
Transverse Mass ◽  
Modal Interactions ◽  
Spinning Disk ◽  
Physical Insight ◽  
Insight Into

In a previous paper (Chen and Bogy, 1992) we studied the effects of various load parameters, such as friction force, transverse mass, damping, stiffness and the analogous pitching parameters, of a stationary load system in contact with the spinning disk on the natural frequencies and stability of the system when the original eigenvalues of interest are well separated. This paper is a follow-up investigation to deal with the situations in which two eigenvalues of the freely spinning disk are almost equal (degenerate) and strong modal interactions occur when the load parameters are introduced. After comparing an eigenfunction expansion with the finite element numerical results, we find that for each of the transverse and pitching load parameters, a properly chosen two-mode approximation can exhibit all the important features of the eigenvalue changes. Based on this two-mode approximation we study the mathematical structure of the eigenvalues in the neighborhood of degenerate points in the natural frequency-rotation speed plane. In the case of friction force, however, it is found that at least a four-mode approximation is required to reproduce the eigenvalue structure. The observations and analyses presented provide physical insight into the modal interactions induced by various load parameters in a spinning disk-stationary load system.

Implications of Using Dual Number Derivatives With a Numerical Solution

2013 ◽  
Author(s):  
Suryanarayana R. Pakalapati ◽  
Hayri Sezer ◽  
Ismail B. Celik
Keyword(s):  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Computer Code ◽  
Model Parameters ◽  
Model Parameter ◽  
Systematic Assessment ◽  
Dual Number ◽  
Advection Diffusion Equation ◽  
Model Equations ◽  
Derivatives Of

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.

Eigenvalue Inclusion Principles for Distributed Gyroscopic Systems

1992 ◽  
Vol 59 (3) ◽  
pp. 650-656 ◽  
Author(s):  
B. Yang
Keyword(s):  
Transfer Function ◽  
Discrete System ◽  
Natural Frequencies ◽  
Gyroscopic System ◽  
Positive Real ◽  
Constrained System ◽  
Function Formulation ◽  
Pointwise Constraints ◽  
Gyroscopic Systems ◽  
Vibrating Systems

In his famous treatise The Theory of Sound, Rayleigh enunciated an eigenvalue inclusion principle for the discrete, self-adjoint vibrating system under a constraint. According to this principle, the natural frequencies of the discrete system without and with the constraint are alternately located along the positive real axis. Although it is commonly believed that the same rule also applied for distributed vibrating systems, no proof has been given for the distributed gyroscopic system. This paper presents several eigenvalue inclusion principles for a class of distributed gyroscopic systems under pointwise constraints. A transfer function formulation is proposed to describe the constrained system. Five types of nondissipative constraints and their effects on the system natural frequencies are studied. It is shown that the transfer function formulation is a systematic and convenient way to handle constraint problems for the distributed gyroscopic system.

Flow Driven by a Finite, Shrouded Spinning Disk With Suction

1985 ◽  
Vol 52 (4) ◽  
pp. 766-770 ◽  
Author(s):  
J. M. Hyun
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Ekman Layer ◽  
Initial State ◽  
Disk Model ◽  
Stationary Disk ◽  
Spinning Disk ◽  
Real Flow ◽  
Finite Configuration

Numerical solutions are presented for the flow driven by a spinning disk which forms an endwall of a finite, closed cylinder. The effects of imposing a uniform suction (or blowing) through the spinning disk in finite configuration are investigated. The Reynolds number is large and the cylinder aspect ratio is 0(1). Finite-difference techniques are employed to integrate the time-dependent Navier-Stokes equations. The initial state is taken to be a uniform axial motion. Integration is performed until an approximate steady state is attained. When there is no suction, the infinite disk model is shown to provide a qualitatively representative approximation to the flow in the central core region. As a suction (blowing) is imposed, the core rotation rate in the case of finite configuration becomes smaller (larger) than that for the case of no suction, which is in disagreement with the predictions of the infinite disk model. These significant discrepancies point to a fundamental difficulty of the infinite disk model to adequately describe the real flow infinite geometry when there is a mass flux across the system boundary. Plots showing the meridional stream function at various times are constructed. Details of the flow structure in the approximate steady state are analyzed. When there is a suction, a strong Ekman layer is present on the spinning disk but the Ekman layer on the stationary disk fades. When there is a blowing, a strong Ekman layer forms on the stationary disk. It is shown that the dynamic effects influencing the character of the flow are confined to these Ekman layers.

The Maximum Transient Resonance Response of Rotating Blades With Regard to Centrifugal Force and Non-Linear Damping Effects

1997 ◽  
Author(s):  
Horst D. Irretier
Keyword(s):  
Centrifugal Force ◽  
Natural Frequencies ◽  
Linear Change ◽  
Rotating Blades ◽  
Operation Process ◽  
Non Linear ◽  
Linear Damping ◽  
Pass Through ◽  
Run Up ◽  
Approximate Formulas

During the operation process in many types of fluid flow machines the rotating blades pass through various resonances e.g. during run-up or run-down or other transient conditions. Therefore, for the high cycle fatigue problem of the blades it might be important to consider the transient vibratory response of the blades during these passages through resonance and to get knowledge about the occuring maximum vibratory stresses. In the paper, approximate formulas are presented which allow the estimation of the maximum transient response of the blades. Thereby, the influence of the change of the natural frequencies due to the increasing or decreasing centrifugal force field during the run-up or run-down, respectively, is taken into consideration. Basically, the approximate formulas are based on a linear change of the natural frequencies versus time and on a linear viscous type of damping. Extensions to account for parabolic changes which are more realistic for centrifugal effects and for non-linear damping models e.g. friction damping or turbulence damping are discussed.

Approximation of 3D convection diffusion equation using DQM based on modified cubic trigonometric B-splines

2020 ◽  
pp. 1-10
Author(s):  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
F.S. Gill ◽  
Robin
Keyword(s):  
Diffusion Equation ◽  
Numerical Solutions ◽  
Basis Functions ◽  
B Spline ◽  
Convection Diffusion ◽  
B Splines ◽  
Convection Diffusion Equation ◽  
First Order ◽  
The Stability ◽  
Derivatives Of

This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions are used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method is validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.

Exact Solutions of the Vibration Problem of Beam Structures

2013 ◽  
Vol 390 ◽  
pp. 242-245 ◽  
Author(s):  
Alexander V. Chekanin
Keyword(s):  
Mathematical Model ◽  
Exact Solutions ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Displacement Method ◽  
Actual Problem ◽  
Beam Structures ◽  
The Mathematical Model

The article deals with the actual problem of improving the accuracy of determining the dynamic characteristics of beam structures. To solve such problems the displacement method is used. Defining matrices are calculated with the Godunovs scheme. Numerical solutions in this case can be obtained practically with any accuracy within accepted hypotheses of the mathematical model of the calculated object. This suggests that the resulting solutions are standard. The examples of determining the natural frequencies of vibrations of beam structures that demonstrate an extremely high accuracy of the proposed algorithm are given.

Effect of Centrifugal Force on Frequency Characteristic of Rotating Hollow Cylinders Using a Newly Designed Cylindrical Super Element

2006 ◽  
Author(s):  
M. Bonakdar ◽  
M. T. Ahmadian
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Centrifugal Force ◽  
Natural Frequencies ◽  
Rotating Cylinder ◽  
Finite Cylinder ◽  
Vibration Frequencies ◽  
Hollow Cylinders ◽  
Special Case

A sixteen node cylindrical super element is presented for evaluating the free vibration characteristics of a rotating laminated cylinder with conventional boundary conditions. It is shown that the natural frequencies are affected considerably when the centrifugal force is also taken into account. The vibration frequencies of rotating finite cylinder, obtained by conventional finite element are used to evaluate the accuracy of this approach. The special case of a stationary cylinder with zero spinning velocity is also considered as a check on this method. Results indicate only few number of cylindrical super elements are capable of predicting the natural frequency of the rotating cylinder within the same limit as many elements used in the conventional finite element method.

Forced Response Prediction of Blades With Loosely Assembled Dovetail Attachment by HBM

2009 ◽  
Author(s):  
Shangguan Bo ◽  
Zili Xu ◽  
Qilin Wu ◽  
XianDing Zhou ◽  
ShouHong Cao
Keyword(s):  
Friction Force ◽  
Centrifugal Force ◽  
Dry Friction ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Balance Method ◽  
Force Model ◽  
Equivalent Stiffness ◽  
Equivalent Damping

To understand the mechanism of interfacial damping of axial loosely assembled dovetail to suppress blade vibration, a dry friction force model is presented by the Coulomb friction law and the macroslip model, and the mathematical expression of the friction force is derived. The nonlinear friction force is linearized as an equivalent stiffness and an equivalent damping through the one-term harmonic balance method. The effect of centrifugal force on the equivalent stiffness and the equivalent damping is studied. The forced response of one simplified blade with loosely assembled dovetail attachment is predicted by the harmonic balance method, in which the blade is described by the lumped mass and spring model, and the friction contact joints is simplified as a ideal friction damper. The results show that the equivalent stiffness of loosely assembled dovetail attachment increases with blade centrifugal force, gradually reaches a certain value, and there exists the maximum value for the equivalent stiffness. The equivalent damping increases at the beginning and then decreases with blade centrifugal force increasing, there exists a maximum too. The resonant frequency of blade rises with blade centrifugal force, but it no longer increases when the centrifugal force exceed a certain value. There exists a special centrifugal force on which the effect of dry friction damping is the best.

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