Transient Vibration of Gyroscopic Systems With Unsteady Superposed Motion

1995 ◽  
Author(s):  
J. A. Wickert
Keyword(s):  
Equations Of Motion ◽  
Mathematical Structure ◽  
Time Dependent ◽  
Distributed Parameter ◽  
Rotating Shaft ◽  
Transient Vibration ◽  
Modal Amplitude ◽  
Technical Interest ◽  
Gyroscopic Systems ◽  
Two Degree Of Freedom

Abstract The equations of motion for a gyroscopic system with unsteady superposed motion are derived for the prototypical problem in which motion of an oscillating particle is measured relative to a non-inertial frame. The resulting coefficient matrices are time-dependent, and skew-symmetric acceleration terms are present both as Coriolis acceleration and as a component of net stiffness. Such mathematical structure is also demonstrated in the context of other unsteady gyroscopic systems, including flexible media that translate with time-dependent speed. Following the asymptotic approach of Krylov, Bogoliubov and Mitropolsky, a perturbation method is developed for the case in which the superposed motion varies slowly when viewed on the time scale of the natural periods of oscillation. First-order approximations for the modal amplitude and phase are obtained in closed form. The method is illustrated through two examples of technical interest: a two degree-of-freedom model of a rotating shaft, and a distributed parameter model of a moving tape.

Modal Estimation of Gyroscopic Effect in Rotordynamic Systems

2001 ◽  
Author(s):  
Jerzy T. Sawicki
Keyword(s):  
Decay Rate ◽  
Natural Frequency ◽  
Controller Design ◽  
Equations Of Motion ◽  
Viscous Damping ◽  
Equivalent Viscous Damping ◽  
New Approach ◽  
Technical Interest ◽  
Gyroscopic Effects ◽  
Gyroscopic Systems

Abstract A new approach for uncoupling the equations of motion typical for rotordynamical systems is presented. The method does not neglect the speed dependent effects, like gyroscopic effects, and can be particularly valuable in the controller design of actively controlled rotors. In the presence of hysteretic type of damping, the resulting uncoupled gyroscopic systems come with an equivalent viscous damping, equivalent in a sense of the same natural frequency and decay rate. The approach is illustrated through the example of technical interest. The generated results demonstrate that the developed approach is correct and straightforward.

Simulation of a MEMS RF Filter

2005 ◽  
Author(s):  
Eihab M. Abdel-Rahman ◽  
Bashar K. Hammad ◽  
Ali H. Nayfeh
Keyword(s):  
Equations Of Motion ◽  
Distributed Parameter ◽  
Parameter System ◽  
Transmission Characteristics ◽  
Coupled Resonators ◽  
Lagrange Equations ◽  
Filter Model ◽  
Rf Filter ◽  
Two Degree Of Freedom ◽  
Filter Response

We simulate the motions in a MEMS bandpass Radio-Frequency (RF) filter. The filter model is obtained by discretizing the Lagrangian of the distributed-parameter system using a Galerkin procedure. The Euler-Lagrange equations are then used to obtain a two-degree-of-freedom model consisting of two non-linearly coupled ordinary-differential equations of motion. We use the model to study the transmission characteristics of a bandpass filter made up of two coupled resonators. Three distinct response regimes, separated by two critical amplification levels Vcr1 and Vcr2, are identified in the filter response. For amplification levels up to Vcr1, the pass signal is artifact free. Two types of artifacts due to the filter dynamics appear and distort the signal for amplification levels beyond Vcr1.

Modeling and Boundary Control of a Hanging Cable Immersed in Water

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Michael Böhm ◽  
Miroslav Krstic ◽  
Sebastian Küchler ◽  
Oliver Sawodny
Keyword(s):  
Differential Equation ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Distributed Parameter ◽  
Stream Velocity ◽  
Water Stream ◽  
Varying Coefficients ◽  
Feedforward Controller ◽  
Spatially Varying ◽  
Two Degree Of Freedom

A nonlinear distributed parameter system model governing the motion of a cable with an attached payload immersed in water is derived. The payload is subject to a drag force due to a constant water stream velocity. Such a system is found, for example, in deep sea oil exploration, where a crane mounted on a ship is used for construction and thus positioning of underwater parts of an offshore drilling platform. The equations of motion are linearized, resulting in two coupled, one-dimensional wave equations with spatially varying coefficients and dynamic boundary conditions of second order in time. The wave equations model the normal and tangential displacements of cable elements, respectively. A two degree of freedom controller is designed for this system with a Dirichlet input at the boundary opposite to the payload. A feedforward controller is designed by inverting the system using a Taylor-series, which is then truncated. The coupling is ignored for the feedback design, allowing for a separate design for each direction of motion. Transformations are introduced, in order to transform the system into a cascade of a partial differential equation (PDE) and an ordinary differential equation (ODE), and PDE backstepping is applied. Closed-loop stability is proven. This is supported by simulation results for different cable lengths and payload masses. These simulations also illustrate the performance of the feedforward controller.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

2021 ◽  
pp. 107754632199358
Author(s):  
Ali Fasihi ◽  
Majid Shahgholi ◽  
Saeed Ghahremani
Keyword(s):  
Energy Harvester ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Energy Sink ◽  
Piezoelectric Energy ◽  
Energy Sink ◽  
Two Degree Of Freedom ◽  
Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

Reduction of Noise Inside a Cavity by Piezoelectric Actuators

2003 ◽  
Vol 125 (1) ◽  
pp. 12-17 ◽  
Author(s):  
I. Hagiwara ◽  
D. W. Wang ◽  
Q. Z. Shi ◽  
R. S. Rao
Keyword(s):  
Sound Pressure ◽  
Control Mechanism ◽  
Equations Of Motion ◽  
Piezoelectric Actuators ◽  
Governing Equations ◽  
Control Laws ◽  
Modal Coupling ◽  
Modal Amplitude ◽  
Global And Local ◽  
Fully Coupled

A new analytical model is developed for the reduction of noise inside a cavity using distributed piezoelectric actuators. A modal coupling method is used to establish the governing equations of motion of the fully coupled acoustics-structure-piezoelectric patch system. Two performance functions relating “global” and “local” optimal control of sound pressure levels (SPL) respectively are applied to obtain the control laws. The discussions on associated control mechanism show that both the mechanisms of modal amplitude suppression and modal rearrangement may sometimes coexist in the implementation of optimal noise control.

Stability of Gyroscopic Systems Near Vanishing Eigenvalues

1998 ◽  
Vol 65 (4) ◽  
pp. 1062-1064 ◽  
Author(s):  
A. A. Renshaw
Keyword(s):  
System Parameter ◽  
Degree Of Freedom ◽  
Independent System ◽  
Physical Systems ◽  
Gyroscopic Systems ◽  
Two Degree Of Freedom

Renshaw and Mote (1996) proposed a conjecture concerning the growth of vibrating eigensolutions of gyroscopic systems in the neighborhood of a vanishing eigenvalue when the system operators depend on an independent system parameter. Although the conjecture was not proved, it was supported by several examples drawn from well-known continuous physical systems. Lancaster and Kliem (1997), however, recently presented three two-degree-of-freedom counter examples. Unlike the examples tested by Renshaw and Mote (1996), these counter examples lack a definiteness property that is usually found in models derived from physical systems which appears to be essential to the conjecture. This Brief Note revises the original conjecture to include this definiteness criterion and proves the conjecture for general two-degree-of-freedom systems.

scholarly journals Boussinesq modeling of surface waves due to underwater landslides

2013 ◽  
Vol 20 (3) ◽  
pp. 267-285 ◽  
Author(s):  
D. Dutykh ◽  
H. Kalisch
Keyword(s):  
Surface Waves ◽  
Shallow Water ◽  
Bottom Topography ◽  
Equations Of Motion ◽  
Boussinesq Equations ◽  
Time Dependent ◽  
Viscous Drag ◽  
Underwater Landslide ◽  
Landslide Mass ◽  
Run Up

Abstract. Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion that govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced that is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved simultaneously. The numerical solver for the Boussinesq equations can also be restricted to implement a shallow-water solver, and the shallow-water and Boussinesq configurations are compared. A particular bathymetry is chosen to illustrate the general method, and it is found that the Boussinesq system predicts larger wave run-up than the shallow-water theory in the example treated in this paper. It is also found that the finite fluid domain has a significant impact on the behavior of the wave run-up.

scholarly journals Symmetry analysis of rotating fluid

2005 ◽  
Vol 47 (1) ◽  
pp. 65-74 ◽  
Author(s):  
K. Fakhar ◽  
Zu-Chi Chen ◽  
Xiaoda Ji
Keyword(s):  
Steady State ◽  
Infinite Number ◽  
Special Function ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Lie Theory ◽  
Rotating Fluid ◽  
Type Solution ◽  
Function Type ◽  
Basic Equations

AbstractThe machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

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