Stability and Stationary Response of a Skew Jeffcott Rotor With Geometric Uncertainty

2007 ◽  
Author(s):  
Nicolas Driot ◽  
Alain Berlioz ◽  
Claude Henri Lamarque
Keyword(s):  
Translational Motion ◽  
Dynamical Behavior ◽  
Random Parameter ◽  
Forced Response ◽  
Motion Equations ◽  
State Response ◽  
Geometric Uncertainty ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Original Motion

The dynamical behavior of an asymmetrical Jeffcott rotor subjected to a base translational motion is investigated. As the geometry of the skew disk is not well defined, we introduce some randomness. This uncertainty affects a particular parameter in the time-variant motion equations. Consequently, the amplitude of the parametric excitation is a random parameter which leads us to investigate the robustness of the dynamics. The stability is first studied by introducing a transformation of coordinates (feasible in this case) making the problem simpler. Then, far away from the unstable area, the random forced steady state response is computed from the original motion equations. The Taguchi’s method is used to provide statistical moments of the forced response.

Stability and Stationary Response of a Skew Jeffcott Rotor With Geometric Uncertainty

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Nicolas Driot ◽  
Alain Berlioz ◽  
Claude-Henri Lamarque
Keyword(s):  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Forced Response ◽  
Stochastic Methods ◽  
Steady State Response ◽  
State Response ◽  
Internal Excitation ◽  
Geometric Uncertainty ◽  
Jeffcott Rotor ◽  
Support Motion

The aim of this work is to apply stochastic methods to investigate uncertain parameters of rotating machines with constant speed of rotation subjected to a support motion. As the geometry of the skew disk is not well defined, randomness is introduced and affects the amplitude of the internal excitation in the time-variant equations of motion. This causes uncertainty in dynamical behavior, leading us to investigate its robustness. Stability under uncertainty is first studied by introducing a transformation of coordinates (feasible in this case) to make the problem simpler. Then, at a point far from the unstable area, the random forced steady state response is computed from the original equations of motion. An analytical method provides the probability of instability, whereas Taguchi’s method is used to provide statistical moments of the forced response.

scholarly journals Analytical Study of Nonlinear Vibration in a Rub-Impact Jeffcott Rotor

Energies ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 8298
Author(s):  
Nicolae Herisanu ◽  
Vasile Marinca
Keyword(s):  
Nonlinear Vibration ◽  
Friction Force ◽  
Analytical Study ◽  
Analytical Approximation ◽  
Approximation Solution ◽  
State Response ◽  
Second Stage ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Two Stages

The purpose of this work is to explore the nonlinear vibration of a rub-impact Jeffcott rotor. In the first stage, the motion is not affected by the friction force, but in the second stage, the motion is influenced by the normal force and the friction force. The governing equations of the rotor of this model are derived in this paper. In consequence, there appears a difference between the two stages. We establish an approximate analytical solution for nonlinear vibrations corresponding to two stages with the mention of the location of jumps. The obtained results are compared with the numerical integration results. The steady-state response and the stability of the solutions are analytically determined for the two stages. The stability of a full annular rub solution is studied with the help of the Routh–Hurwitz criterion. Effects of different parameters of the system, the saddle-node bifurcation (turning points) and the Hopf bifurcation are presented. The main contribution lies in the analytical approximation solution based on the Optimal Auxiliary Functions Method.

Modeling Analysis of the Wrist Dynamics via an Ellipsoidal Joint

2020 ◽  
Author(s):  
Jiamin Wang ◽  
Sunit K. Gupta ◽  
Oumar Barry
Keyword(s):  
Wrist Joint ◽  
Translational Motion ◽  
Dynamical Behavior ◽  
State Space Model ◽  
Regression Function ◽  
Multibody Modeling ◽  
3 Dimensional ◽  
Motion Constraints ◽  
Flexion Extension ◽  
The Stability

Abstract An accurate wrist model is crucial to the understanding of human wrist mechanics and the development of forearm rehabilitation devices. This paper studied the nonlinear dynamics of the wrist through an ellipsoidal joint model. Compared to many studies where a universal joint is used to model the wrist, the proposed ellipsoidal model intends to better approximate the human wrist biomechanics with the use of kinematic constraints. The constraint on the original 3-dimensional rotation of the wrist is realized based on a quaternion formulation, reducing the wrist kinematics to the coupled 2-degree-of-freedom motions of flexion-extension and radial-ulnar deviation. The ellipsoidal joint also introduces additional coupling from the translational motion constraints. The multibody modeling of the wrist model is then established. The stability and control of the model are analyzed based on a constrained state-space model. Numerical simulations validate the analytical results and demonstrate the coupled dynamical behavior of the wrist. The simulations also show that the proposed model constraint is an ideal base regression function for wrist joint parameter identification. Finally, with the involvement of nonlinear stiffness and damping, chaotic-like behaviors and limit cycles are observed. The approach in this study is also generally applicable to a family of ellipsoidal joint systems.

Dynamical Behavior of a Rotor Under Rotational Random Base Excitation

2007 ◽  
Author(s):  
Lucie Bachelet ◽  
Nicolas Driot ◽  
Guy Ferraris ◽  
Fabrice Poirion
Keyword(s):  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Gaussian White Noise ◽  
Forced Response ◽  
Power Spectrum Density ◽  
Largest Lyapunov Exponent ◽  
Base Excitation ◽  
Spectrum Density ◽  
The Stability ◽  
Truncated Gaussian

This paper investigates the dynamical behavior of a rotor under a random rotational base excitation, which is assumed to be a stationary and ergodic truncated Gaussian white noise. As the base motion is a rotation, the equations of motion present both internal and external random excitations. The stability of the rotor is then studied by computing the largest Lyapunov exponent with an iterative formula. Then, the power spectrum density of the stationary forced response is obtained from a Monte Carlo simulation. Finally, we perform a comparative analysis on the influence of the required number of modal shapes to describe accurately the response.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

Stability of Shear-Thickening Liquids Between Rotating Cylinders

2010 ◽  
Author(s):  
Nariman Ashrafi ◽  
Habib Karimi Haghighi
Keyword(s):  
Couette Flow ◽  
Dynamical Behavior ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Taylor Vortex ◽  
Shear Dependent Viscosity ◽  
Shear Thinning Fluids ◽  
Shear Thickening Fluids ◽  
The Stability ◽  
Dependent Viscosity

The effects of nonlinearities on the stability are explored for shear thickening fluids in the narrow-gap limit of the Taylor-Couette flow. It is assumed that shear-thickening fluids behave exactly as opposite of shear thinning ones. A dynamical system is obtained from the conservation of mass and momentum equations which include nonlinear terms in velocity components due to the shear-dependent viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of Couette flow becomes higher as the shear-thickening effects increases. Similar to the shear thinning case, the Taylor vortex structure emerges in the shear thickening flow, however they quickly disappear thus bringing the flow back to the purely azimuthal flow. Naturally, one expects shear thickening fluids to result in inverse dynamical behavior of shear thinning fluids. This study proves that this is not the case for every point on the bifurcation diagram.

Phase space correspondence between Jordan and Einstein frames

2021 ◽  
pp. 2150087
Author(s):  
Amin Salehi
Keyword(s):  
Phase Space ◽  
Critical Point ◽  
Critical Points ◽  
Dynamical Behavior ◽  
Cosmological Parameters ◽  
Jordan Frame ◽  
Field Equations ◽  
Theories Of Gravity ◽  
The Stability ◽  
Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

Influence of Interaction Between Torsion and Bending on Long-Term Nonlinear Rotor Dynamics

1999 ◽  
Author(s):  
Jiazhong Zhang ◽  
Bram de Kraker ◽  
Dick H. van Campen
Keyword(s):  
Critical Speed ◽  
Floquet Theory ◽  
Rotor Dynamics ◽  
Steady State Response ◽  
Bending Vibrations ◽  
Bending Vibration ◽  
State Response ◽  
Stability And Bifurcation ◽  
The Stability

Abstract An elementary system with gears and excited by unbalance mass has been constructed for analyzing the interaction between torsion and bending vibration in rotor dynamics. For this system, only the interaction caused primarily by unbalance mass has been investigated. The stability and bifurcation characteristics of the system have been studied by numerical computation based on Hopf bifurcation and Floquet theory. The results show that the interaction between torsion and bending vibrations can affect the stability and bifurcation of the unbalance response, in particular the onset speed of instability. In addition to the above, the interaction also affects the steady-state response. To investigate the influence of unbalance mass, the periodic solution and its stability have been studied near the first bending critical speed of the decoupled system. All the results show that the coupling of torsion and bending vibrations can have a significant influence on the nonlinear dynamics of the whole system.

Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

2021 ◽  
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Optical Fibers ◽  
Pulse Propagation ◽  
Dynamical Behavior ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Contour Plots ◽  
System Extension ◽  
Inverse Spectral Method ◽  
Nonlinear Pulse Propagation ◽  
The Stability

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

scholarly journals Turning Gait Planning Method for Humanoid Robots

2018 ◽  
Vol 8 (8) ◽  
pp. 1257 ◽  
Author(s):  
Tianqi Yang ◽  
Weimin Zhang ◽  
Xuechao Chen ◽  
Zhangguo Yu ◽  
Libo Meng ◽  
...  
Keyword(s):  
Inverted Pendulum ◽  
Center Of Mass ◽  
Translational Motion ◽  
Humanoid Robots ◽  
Inverted Pendulum Model ◽  
Straight Line ◽  
Pendulum Model ◽  
The Stability ◽  
And Control ◽  
Present Simulation

The most important feature of this paper is to transform the complex motion of robot turning into a simple translational motion, thus simplifying the dynamic model. Compared with the method that generates a center of mass (COM) trajectory directly by the inverted pendulum model, this method is more precise. The non-inertial reference is introduced in the turning walk. This method can translate the turning walk into a straight-line walk when the inertial forces act on the robot. The dynamics of the robot model, called linear inverted pendulum (LIP), are changed and improved dynamics are derived to make them apply to the turning walk model. Then, we expend the new LIP model and control the zero moment point (ZMP) to guarantee the stability of the unstable parts of this model in order to generate a stable COM trajectory. We present simulation results for the improved LIP dynamics and verify the stability of the robot turning.

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