Stability of Liquid-Filled Spinning Spheroids Via Liapunov’s Second Method

1979 ◽  
Vol 46 (2) ◽  
pp. 259-262 ◽  
Author(s):  
P. C. Parks
Keyword(s):  
Stability Analysis ◽  
Nonlinear Equations ◽  
Stability Theory ◽  
Equations Of Motion ◽  
Spin Axis ◽  
Liapunov Functions ◽  
The Stability ◽  
Nonlinear Equations Of Motion

In this paper the stability theory of Poincare´ and others, expounded in the last few pages of Sir Horace Lamb’s textbook Hydrodynamics, is extended by a stability analysis using Liapunov functions to cover the full nonlinear equations of motion of spinning liquid-filled spheroids. The linearized criterion is confirmed, that is for stability c/a < 1 or c/a > 3 where the semiaxes of the spheroid are a, a, c, the c-axis being also the spin axis. The unstable motion for 1 < c/a < 3 is examined, and a conjecture that viscosity will destabilize spinning spheroids with c/a > 3 is proposed.

Linear Stability Analysis of a Waveboard Multibody Model With a Minimal Set of Equations

2021 ◽  
Author(s):  
A. G. Agúndez ◽  
D. García-Vallejo ◽  
E. Freire ◽  
A. M. Mikkola
Keyword(s):  
Stability Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Multibody Model ◽  
Aspect Ratios ◽  
Holonomic And Nonholonomic Constraints ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, the skateboard consisting in two articulated platforms, coupled by a torsion bar and supported of two caster wheels, is analysed. The waveboard presents an interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms. The system is described using a multibody model with holonomic and nonholonomic constraints. To perform the stability analysis, the nonlinear equations of motion are linearized with respect to the forward upright motion with constant speed. The linearization is carried out resorting to a novel numerical linearization procedure, recently validated with a well-acknowledged bicycle benchmark, which allows the maximum possible reduction of the linearized equations of motion of multibody systems with holonomic and nonholonomic constraints. The approach allows the expression of the Jacobian matrix in terms of the main design parameters of the multibody system under study. This paper illustrates the use of this linearization approach with a complex multibody system as the waveboard. Furthermore, a sensitivity analysis of the eigenvalues considering different scenarios is performed, and the influence of the forward speed, the casters’ inclination angle and the tori aspect ratios of the toroidal wheels on the stability of the system is analysed.

scholarly journals Dynamic Analysis of a Pendulum Dynamic Automatic Balancer

2007 ◽  
Vol 14 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Jin-Seung Sohn ◽  
Jin Woo Lee ◽  
Eun-Hyoung Cho ◽  
No-Cheol Park ◽  
Young-Pil Park
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Critical Speed ◽  
Equations Of Motion ◽  
Polar Coordinates ◽  
Response Analysis ◽  
Rotating Speed ◽  
Time Response Analysis ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The automatic dynamic balancer is a device to reduce the vibration from unbalanced mass of rotors. Instead of considering prevailing ball automatic dynamic balancer, pendulum automatic dynamic balancer is analyzed. For the analysis of dynamic stability and behavior, the nonlinear equations of motion for a system are derived with respect to polar coordinates by the Lagrange's equations. The perturbation method is applied to investigate the dynamic behavior of the system around the equilibrium position. Based on the linearized equations, the dynamic stability of the system around the equilibrium positions is investigated by the eigenvalue analysis.The stability analysis provides the design requirements for the pendulum automatic dynamic balancer to achieve a balancing of the system. The efficiency of ball automatic dynamic balancer, for reducing the total vibration of the system, is better than one of pendulum automatic balancer, if the rotating speed is above critical speed. However, pendulum automatic dynamic balancer can achieve balancing even if the rotating speed is below critical speed.The time response analysis demonstrates the stability analysis from computing the radial displacement of the rotating system and the positions of pendulums. Furthermore, in order to confirm the theoretical analysis, various experiments are made on pendulum automatic dynamic balancer.

Stability analysis of rigid rotors supported by gas foil bearings coupled with electromagnetic actuators

2019 ◽  
Vol 234 (2) ◽  
pp. 427-443 ◽  
Author(s):  
Kamal Kumar Basumatary ◽  
Gaurav Kumar ◽  
Karuna Kalita ◽  
Sashindra K Kakoty
Keyword(s):  
Stability Analysis ◽  
Equations Of Motion ◽  
Electromagnetic Actuator ◽  
Electromagnetic Actuators ◽  
Foil Bearings ◽  
Damping Characteristics ◽  
Foil Bearing ◽  
Model Combining ◽  
The Stability ◽  
Rigid Rotors

Rotors supported on gas foil bearings have low damping characteristics, which limits its application. A possible solution could be an integration of a gas foil bearing with an electromagnetic actuator. This paper discusses the effect of electromagnetic actuators on the stability of a rotor supported on gas foil bearings. A coupled dynamic model combining the dynamics of gas foil bearing and electromagnetic actuator has been developed. The fluid film forces from the gas foil bearings and the electromagnetic forces from the electromagnetic actuators are integrated into the equations of motion of the rotor. The sub-synchronous vibration present in case of conventional gas foil bearings is reduced and the stability band of the rotor is increased due to the implementation of electromagnetic actuator.

Stability Analysis on S Plane Control of Underwater Vehicles

2013 ◽  
Vol 437 ◽  
pp. 716-721 ◽  
Author(s):  
Yue Ming Li ◽  
Ying Hao Zhang ◽  
Guo Cheng Zhang ◽  
Zhong Hui Hu
Keyword(s):  
Stability Analysis ◽  
Lyapunov Stability ◽  
Stability Theory ◽  
Lyapunov Stability Theory ◽  
Underwater Vehicles ◽  
Velocity Control ◽  
Position Controller ◽  
The Stability

This paper addresses the stability analysis on S Plane Control in terms of both position and velocity control. Employing Lyapunov stability theory and T-passivity theory, this paper proves the stability of the position controller based on S Plane Control, and on this ground, the stability analysis of the velocity controller based on S Plane Control is done. Finally, the S Plane Control results obtained from the sea trials are given.

Analysis of the Nonlinear Dynamics of a Railway Vehicle Wheelset

1973 ◽  
Vol 95 (1) ◽  
pp. 28-35 ◽  
Author(s):  
E. Harry Law ◽  
R. S. Brand
Keyword(s):  
Lateral Displacement ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Linear Analysis ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Nonlinear Variation ◽  
Constraint Equations ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The nonlinear equations of motion for a railway vehicle wheelset having curved wheel profiles and wheel-flange/rail contact are presented. The dependence of axle roll and vertical displacement on lateral displacement and yaw is formulated by two holonomic constraint equations. The method of Krylov-Bogoliubov is used to derive expressions for the amplitudes of stationary oscillations. A perturbation analysis is then used to derive conditions for the stability characteristics of the stationary oscillations. The expressions for the amplitude and the stability conditions are shown to have a simple geometrical interpretation which facilitates the evaluation of the effects of design parameters on the motion. It is shown that flange clearance and the nonlinear variation of axle roll with lateral displacement significantly influence the motion of the wheelset. Stationary oscillations may occur at forward speeds both below and above the critical speed at which a linear analysis predicts the onset of instability.

Non-Linear Transient Stability Analysis of a Rigid Rotor Supported on Journal Bearings With Rectangular Dimples

2015 ◽  
Author(s):  
Ram Turaga
Keyword(s):  
Stability Analysis ◽  
Surface Texture ◽  
Transient Stability ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Pressure Curve ◽  
Rigid Rotor ◽  
Non Linear ◽  
The Stability ◽  
Transient Stability Analysis

The influence of deterministic surface texture on the sub-synchronous whirl stability of a rigid rotor has been studied. Non-linear transient stability analysis has been performed to study the stability of a rigid rotor supported on two symmetric journal bearings with a rectangular dimple of large aspect ratio. The surface texture parameters considered are dimple depth to minimum film thickness ratio and the location of the dimple on the bearing surface. Journal bearings of different Length to diameter ratios have been studied. The governing Reynolds equation for finite journal bearings with incompressible fluid has been solved using the Finite Element Method under isothermal conditions. The trajectories of the journal center have been obtained by solving the equations of motion of the journal center by the fourth-order Runge-Kutta method. When the dimple is located in the raising part of the pressure curve the positive rectangular dimple is seen to decrease the stability whereas the negative rectangular dimple is seen to improve the stability of the rigid rotor.

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