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.

scholarly journals STUDY OF SOLUTIONS TO A FOURTH ORDER PARABOLIC EQUATION∗

2016 ◽  
Vol 21 (1) ◽  
pp. 1-15
Author(s):  
Bo Liang ◽  
Xiting Peng ◽  
Huiying Shen
Keyword(s):  
Steady State ◽  
Parabolic Equation ◽  
Fourth Order ◽  
Steady State Solution ◽  
State Solution ◽  
Discrete Problem ◽  
Bounded Solutions ◽  
Order Parabolic Equation ◽  
Fourth Order Parabolic Equation ◽  
Iteration Technique

This paper studies a fourth-order parabolic equation ut + ε(unuxxx)x − δ|uxx|muxx = 0 with the boundary conditions uxx = 0, u = l and the initial condition u(x, 0) = u0(x). The existence of solutions is obtained from the semidiscretization method. When the initial function is close to a constant steady state solution, the uniqueness of the bounded solutions is obtained. Finally, by the iteration technique from its semi-discrete problem, the solution exponentially converges to a constant steady state solution as the time tends to infinity.

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.

Deflection of a Very Flexible Spinning Disk Due to a Stationary Transverse Load

1978 ◽  
Vol 45 (3) ◽  
pp. 636-642 ◽  
Author(s):  
R. C. Benson ◽  
D. B. Bogy
Keyword(s):  
Standing Waves ◽  
Outer Region ◽  
Transverse Load ◽  
Fourier Series Expansion ◽  
Stiffness Parameter ◽  
Membrane Theory ◽  
Concentrated Load ◽  
Transverse Loads ◽  
Spinning Disk ◽  
Load Location

This paper addresses the problem of steady deflection of a very flexible spinning disk due to transverse loads that are fixed in space. We first approach this problem within the context of membrane theory. The membrane differential operator is classified and shown to be hyperbolic in the outer region and elliptic in the inner portion. The characteristics are studied and qualitatively compared with some experimentally observed standing waves in the outer region of a membrane-like disk. The eigenvalue problem is examined and the membrane operator is found to be singular. We therefore conclude that the problem of interest cannot be solved within the context of membrane theory. Finally, the problem is formulated with bending stiffness retained. The concentrated load problem is solved by use of a Fourier series expansion in the angular direction in conjunction with a numerical solution for the radial modes. Graphical results are presented for various values of the stiffness parameter and load location.

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.

An airfoil theory of bifurcating laminar separation from thin obstacles

1990 ◽  
Vol 216 ◽  
pp. 255-284 ◽  
Author(s):  
C. J. Lee ◽  
H. K. Cheng
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Steady State ◽  
Steady State Solution ◽  
Equation System ◽  
Branch Points ◽  
Laminar Separation ◽  
Kutta Condition ◽  
Numerical Studies ◽  
Steady State Solutions

Global interaction of the boundary layer separating from an obstacle with resulting open/closed wakes is studied for a thin airfoil in a steady flow. Replacing the Kutta condition of the classical theory is the breakaway criterion of the laminar triple-deck interaction (Sychev 1972; Smith 1977), which, together with the assumption of a uniform wake/eddy pressure, leads to a nonlinear equation system for the breakaway location and wake shape. The solutions depend on a Reynolds numberReand an airfoil thickness ratio or incidence τ and, in the domain$Re^{\frac{1}{16}}\tau = O(1)$considered, the separation locations are found to be far removed from the classical Brillouin–Villat point for the breakaway from a smooth shape. Bifurcations of the steady-state solution are found among examples of symmetrical and asymmetrical flows, allowing open and closed wakes, as well as symmetry breaking in an otherwise symmetrical flow. Accordingly, the influence of thickness and incidence, as well as Reynolds number is critical in the vicinity of branch points and cut-off points where steady-state solutions can/must change branches/types. The study suggests a correspondence of this bifurcation feature with the lift hysteresis and other aerodynamic anomalies observed from wind-tunnel and numerical studies in subcritical and high-subcriticalReflows.

Dependency of Unsteady Time-Linearized Flutter Investigations on the Steady State Flow Field

2011 ◽  
Author(s):  
Michael Blocher ◽  
Markus May ◽  
Harald Schoenenborn
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Elastic Stability ◽  
Compressor Cascade ◽  
Steady State Flow ◽  
State Flow ◽  
Flow Parameters ◽  
Pressure Distributions ◽  
German Aerospace ◽  
École Polytechnique

The influence of the steady state flow solution on the aero-elastic stability behaviour of an annular compressor cascade shall be studied in order to determine sensitivities of the aero-dynamic damping with respect to characteristic flow parameters. In this context two different flow regimes — a subsonic and a transonic case — are subject to the analysis. The pressure distributions, steady as well as unsteady, on the blade surface of the NACA3506 profile are compared to experimental data that has been gained by the Institute of Aeroelasticity of the German Aerospace Center (DLR) during several wind tunnel tests at the annular compressor cascade facility RGP-400 of the Ecole Polytechnique Fe´de´rale de Lausanne (EPFL). Whereas a certain robustness of the unsteady CFD results can be stated for the subsonic flow regime, the transonic regime proves to be very sensitive with respect to the steady state solution.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

Dynamic Stability of an Impact System Connected With Rock Drilling

1969 ◽  
Vol 36 (4) ◽  
pp. 743-749 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Computer Simulation ◽  
Exact Solution ◽  
Steady State ◽  
Asymptotic Stability ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
External Excitation ◽  
Rock Drilling ◽  
Impact System ◽  
Number Of Cycles

This paper deals with asymptotic stability of an analytically derived, synchronous as well as nonsynchronous, steady-state solution of an impact system which exhibits piecewise linear characteristics connected with rock drilling. The exact solution, which assumes one impact for a given number of cycles of the external excitation, is derived, its asymptotic stability is examined, and ranges of parameters are determined for which asymptotic stability is assured. The theoretically predicted stability or instability is verified by a digital computer simulation.

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