Dynamics and Stability of Non-Planar Rigid Rotor Equipped with Two Ball-Spring Autobalancers

2019 ◽  
Vol 19 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Mousa Rezaee ◽  
Mir Mohammad Ettefagh ◽  
Reza Fathi
Keyword(s):  
Mathematical Model ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Rigid Rotor ◽  
Out Of Plane ◽  
New Type ◽  
Nonlinear Equations Of Motion ◽  
The Mathematical Model ◽  
System Time

Recently, a new type of automatic ball balancer (ABB), called the ball-spring autobalancer (AB), has been proposed, which substantially eliminates the drawbacks of the traditional ABBs. In previous studies, the dynamics of the Jeffcott planar rotor equipped with ball-spring AB has been investigated. In the Jeffcott model, it is assumed that the ABB is located on the plane of the unbalance disk. However, for the non-planar rigid rotor with distributed imbalances, out-of-plane motions may occur, and the Jeffcott model becomes unreliable as the tilting motion cannot be explained. To this end, the aim of this paper is to analyze the capability of the ball-spring AB in balancing non-planar rotors and to reconfirm its pre-claimed advantages over the traditional ABBs for balancing non-planar rotors. To start, the mathematical model of the rigid rotor with two ball-spring ABs is established, based on which the nonlinear equations of motion are derived. Then, the system time responses are computed numerically and the balanced stable regions are acquired by the Lyapunov’s first method. The results of this study show that the ball-spring ABs can balance the non-planar rotors and the tilting motion does not impair the pre-claimed advantages of the ball-spring AB.

Theoretical and Experimental Study of the Dynamics of the Tripod-Ball Sliding Joint With Clearance

2001 ◽  
Author(s):  
Jia Xiaohong ◽  
Ji Linhong ◽  
Jin Dewen ◽  
Zhang Jichuan
Keyword(s):  
Mathematical Model ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Active System ◽  
New Concepts ◽  
Sliding Joint ◽  
New Type ◽  
Set Up ◽  
The Mathematical Model ◽  
Kane Method

Abstract Clearance is inevitable in the kinematic joints of mechanisms. In this paper the dynamic behavior of a crank-slider mechanism with clearance in its tripod-ball sliding joint is investigated theoretically and experimentally. The mathematical model of this new-type joint is established, and the new concepts of basal system and active system are put forward. Based on the mode-change criterion established in this paper, the consistent equations of motion in full-scale are derived by using Kane method. The experimental rig was set up to measure the effects of the clearance on the dynamic response. Corresponding experimental studies verify the theoretical results satisfactorily. In addition, due to the nonlinear elements in the improved mathematical model of the joint with clearance, the chaotic responses are found in numerical simulation.

Low-Speed Controllability of Ships in Wind

1968 ◽  
Vol 12 (03) ◽  
pp. 181-200
Author(s):  
H. Eda
Keyword(s):  
Mathematical Model ◽  
Nonlinear Equations ◽  
Digital Computer ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equilibrium Conditions ◽  
Region Of Stability ◽  
Nonlinear Equations Of Motion ◽  
Hydrodynamic Data ◽  
Relative Wind

Aerodynamic and hydrodynamic data for the Manner-class vessel, gathered in earlier experiments, were used to formulate a mathematical model representing the dynamic behavior of ships in wind. A digital computer was used to solve the eigenvalues of the system. Perturbation equations were linearized, with respect to equilibrium conditions, from nonlinear equations of motion. In addition, ship trajectory in certain wind conditions was examined by means of numerical solutions of the nonlinear equations of motion. Results indicate that the ship in bow wind tends, even without an autopilot system, to maintain its original course-with perturbation in yaw inducing yaw oscillations, the convergence of which depends upon the magnitude of relative wind velocity. It is shown that beam wind creates greater difficulties, although the use of an adequate autopilot increases the region of stability in wind of certain velocities (except in some conditions of relatively strong beam wind). An increase in rudder size is shown to improve controllability in wind significantly. Computations with and without the assumption of constant longitudinal speed indicate that the effect of surge motion on yaw and sway responses in wind is important, especially in beam wind.

Study on the working state of the five hundred-meter aperture spherical telescope using step-wise assignment method

2014 ◽  
Vol 229 (2) ◽  
pp. 316-324 ◽  
Author(s):  
Yu Liu ◽  
Feng Gao
Keyword(s):  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Driving Mechanism ◽  
Initial Value ◽  
Unknown Parameters ◽  
Support Structure ◽  
Assignment Method ◽  
New Type ◽  
Set Up ◽  
The Mathematical Model

The working state of the five hundred-meter aperture spherical telescope (FAST) is solved using the step-wise assignment method. In this paper, the mathematical model of the cable-net support structure of the FAST is set up by the catenary equation. There are a large number of nonlinear equations and unknown parameters of the model. The nonlinear equations are solved by using the step-wise assignment method. The method is using the analytical solutions of the cable-net equations of one working state as the initial value for the next working state, from which the analytical solutions of the nonlinear equations of the cable-net for each working state of the FAST and the tension and length of each driving cable can be obtained. The suggested algorithm is quite practically well suited to study the working state of the cable-net structures of the FAST. Also, the working state analysis result of the cable-net support structure of a reduced model of the cable-net structure reflector for the FAST is given to verify the reliability of the method. In order to show the validity of the method, comparisons with another algorithm to set the initial value are presented. This method has an important guiding significance to the further study on the control of the new type of flexible cable driving mechanism, especially the FAST.

Nonlinear Analysis of a Rigid Rotor on Magnetic Bearings

1995 ◽  
Author(s):  
C. Nataraj
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Magnetic Bearings ◽  
Rigid Rotor ◽  
Stability Criteria ◽  
Safe Operation ◽  
Qualitative And Quantitative ◽  
Quantitative Results ◽  
Nonlinear Equations Of Motion

A simple model of a rigid rotor supported on magnetic bearings is considered. A proportional control architecture is assumed, the nonlinear equations of motion are derived and some essential nondimensional parameters are identified. The free and forced response of the system is analyzed using techniques of nonlinear analysis. Both qualitative and quantitative results are obtained and stability criteria are derived for safe operation of the system.

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