Limit–cycle oscillations of a heavy whirling cable subject to aerodynamic drag

James David Clark; Wynstone Barrie Fraser; Christopher David Rahn; Arun Rajamani

doi:10.1098/rspa.2004.1387

Limit–cycle oscillations of a heavy whirling cable subject to aerodynamic drag

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2004.1387 ◽

2005 ◽

Vol 461 (2055) ◽

pp. 875-893 ◽

Cited By ~ 3

Author(s):

James David Clark ◽

Wynstone Barrie Fraser ◽

Christopher David Rahn ◽

Arun Rajamani

Keyword(s):

Limit Cycle ◽

Aerodynamic Drag ◽

Equations Of Motion ◽

Experimental System ◽

Angular Speed ◽

Full Time ◽

Limit Cycle Oscillations ◽

Rotating Disc ◽

Linearized Stability ◽

Fixed End

In this paper a simple experimental system consisting of a length of cable, fixed to the edge of a rotating disc at its upper end, and free at its lower end or with a point mass ( drogue ) attached there, is described. This system exhibits a rich variety of bifurcation behaviours as the length of cable, angular speed of the fixed end, mass of the drogue and elasticity of the cable is varied. Bifurcation diagrams for the quasistationary configurations (cable shapes that appear stationary with respect to the rotating reference frame) are described. Linearized stability analyses of these quasistationary balloons are compared with solutions to the full time–dependent equations of motion. It is shown that there is an exchange of stability at the turning points of the quasi–stationary bifurcation curves, and that Hopf bifurcations occur at otherwise undistinguished points of these curves. It is shown that limit–cycle oscillations of the system occur at angular speeds corresponding to points on the bifurcations curves in the neighbourhood of the Hopf bifurcation points. These oscillations have been observed experimentally.

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Analytical and Experimental Investigations of a Multi-Degree-of-Freedom Flexible Cam-Follower Mechanisma

Volume 2A: 24th Biennial Mechanisms Conference ◽

10.1115/96-detc/mech-1182 ◽

1996 ◽

Author(s):

Sukhwant S. Khanuja ◽

Andi I. Mahyuddin ◽

Ashok Midha

Keyword(s):

Degrees Of Freedom ◽

Equations Of Motion ◽

Experimental System ◽

Angular Speed ◽

Degree Of Freedom ◽

Mass Motion ◽

Experimental Investigations ◽

Axial Stiffness ◽

Cam Follower ◽

Return Spring

Abstract A multi-degree-of-freedom model is developed herein for prediction of response of an experimental cam-follower mechanism. Both transverse and axial flexibility of the follower rod and return spring, as well as transverse and torsional flexibility of the camshaft are included. The camshaft is assigned two rotational degrees of freedom, one at the cam and the other at the flywheel. The follower mass motion is also described by two degrees of freedom, one each in the axial and the transverse directions. The model takes into account the fluctuating camshaft angular speed and treats it as an input excitation. The governing second-order, nonlinear, nondimensionalized ordinary differential equations of motion, with time-periodic coefficients, are developed. In doing so, a comprehensive modeling of the kinematics of deformation of the flexible camshaft and follower system is considered, with the first inclusion of the transverse flexibility of the follower rod and return spring. With axial deflection, the transverse flexibility of the return spring gives rise to a phenomenon defined as moment stiffening. The transverse degree of freedom of the follower mass significantly influences the equivalent axial stiffness of the system. Its inclusion in the equation of motion yields a more accurate prediction of the experimental system behavior.

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Oscillations of a vapour cavity in a rotating cylindrical tank

Journal of Fluid Mechanics ◽

10.1017/s0022112068000145 ◽

1968 ◽

Vol 31 (2) ◽

pp. 249-271

Author(s):

Sui-Kwong P. Pao ◽

J. Siekmann

Keyword(s):

Differential Equation ◽

Large Scale ◽

Equations Of Motion ◽

Angular Speed ◽

Forced Oscillations ◽

Infinite Length ◽

Elliptic Type ◽

Cylindrical Tank ◽

Linearized Stability ◽

Small Force

The oscillations of a curved interface are considered, neglecting the effects of gravity. The system under consideration consists of a right, circular, cylindrical tank of finite length, partially filled with an inviscid, incompressible, wetting liquid. When the container spins about its axis of revolution, the large-scale vapour cavity takes an elongated spheroid-like shape, symmetric about the axis of rotation. The fluid—vapour interface will oscillate about the equilibrium configuration if disturbing forces are present. The case where the vapour cavity touches the walls of the tank is not included in this investigation.The equations of motion are linearized. However, the resulting eigenvalue problem is non-linear. Surface tension and rotation are taken into account only to the extent allowed by a linearized stability theory.The self-sustained oscillations are governed by a partial differential equation of elliptic type, the field equation of the perturbation pressure. According to the results obtained from theory, all eigenfrequencies for this case are greater than twice the angular speed of the tank. The first two eigenfrequencies can be computed with high accuracy. The relation between the bubble shape and the eigen-frequency is shown in a graph for a specific example.The governing differential equation is hyperbolic for forced oscillations induced by a small force field of constant magnitude and direction in an inertial frame of reference. A solution for this problem exists only in case of a cylindrical tank of infinite length. Discontinuities in the velocity components occur in the flow field. A numerical example has been carried out.

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Flutter/LCO suppression for high-aspect ratio wings

The Aeronautical Journal ◽

10.1017/s0001924000003079 ◽

2009 ◽

Vol 113 (1144) ◽

pp. 409-416 ◽

Cited By ~ 4

Author(s):

D. Tang ◽

E. H. Dowell

Keyword(s):

Aspect Ratio ◽

Limit Cycle ◽

High Aspect Ratio ◽

Equations Of Motion ◽

Wind Tunnel Test ◽

Beam Theory ◽

Control Device ◽

Structural Equations ◽

Limit Cycle Oscillations ◽

Tip Mass

Abstract An experimental high-aspect ratio wing aeroelastic model with a device to provide a controllable slender body tip mass distribution has been constructed and the model response due to flutter and limit cycle oscillations has been measured in a wind tunnel test. A theoretical model has also been developed and calculations made to correlate with the experimental data. Structural equations of motion based on nonlinear beam theory are combined with the ONERA aerodynamic stall model (an empirical extension of Theodorsen aerodynamic theory that accounts for flow separation). A dynamic perturbation analysis about a nonlinear static equilibrium is used to determine the small perturbation flutter boundary which is compared to the experimentally determined flutter velocity and flutter frequency. Time simulation is used to compute the limit cycle oscillations response when the flutter/LCO control system is ON or OFF. Theory and experiment are in good agreement for predicting the flutter/LCO suppression that can be achieved with the control device.

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Post-Flutter Bifurcation Behavior in Long-Span Suspension Bridges

Volume 7A: 9th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2013-12429 ◽

2013 ◽

Author(s):

Andrea Arena ◽

Walter Lacarbonara ◽

Pier Marzocca

Keyword(s):

Limit Cycle ◽

Equations Of Motion ◽

Suspension Bridge ◽

Sensitivity Analyses ◽

Design Parameters ◽

Suspension Bridges ◽

Limit Cycle Oscillations ◽

Stability Range ◽

Long Span ◽

Fold Bifurcation

A linearized parametric continuum model of a long-span suspension bridge is coupled with a nonlinear quasi-steady aerodynamic model giving the aeroelastic partial differential equations of motion reduced to the state-space ordinary differential form by adopting the Galerkin method. Numerical time-domain simulations are performed to investigate the limit cycle oscillations occurring in the range of post-flutter wind speeds. Continuation tools are thus employed to path follow the limit cycles past the flutter speed where the Hopf bifurcation occurs. The stable post-flutter behavior, which can significantly affect the bridge by fatigue, terminate at a fold bifurcation. This result represents an important assessment of the conducted aeroelastic investigations. The stability range of the limit cycle oscillations is evaluated by carrying out sensitivity analyses with respect to the main design parameters, such as the structural damping and the initial wind angle of attack.

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