SYMMETRY, GENERIC BIFURCATIONS, AND MODE INTERACTION IN NONLINEAR RAILWAY DYNAMICS

1999 ◽  
Vol 09 (07) ◽  
pp. 1321-1331 ◽  
Author(s):  
CARSTEN NORDSTRØM JENSEN ◽  
MARTIN GOLUBITSKY ◽  
HANS TRUE
Keyword(s):  
Mathematical Model ◽  
Control Parameter ◽  
Mode Interaction ◽  
Hopf Bifurcations ◽  
Additional Parameter ◽  
Railway Dynamics ◽  
Mode Interactions ◽  
Symmetric Periodic Solution ◽  
Pitchfork Bifurcations ◽  
Degenerate Bifurcation

We investigate Cooperrider's complex bogie, a mathematical model of a railway bogie running on an ideal straight track. The speed of the bogie v is the control parameter. Taking symmetry into account, we find that the generic bifurcations from a symmetric periodic solution of the model are Hopf bifurcations for maps (or Neimark bifurcations), saddle-node bifurcations, and pitchfork bifurcations. The last ones are symmetry-breaking bifurcations. By variation of an additional parameter, bifurcations of higher degeneracy are possible. In particular, we consider mode interactions near a degenerate bifurcation. The bifurcation analysis and path-finding are done numerically.

Detection of Tertiary Hopf Bifurcations Arising from Mode Interactions

1997 ◽  
Vol 07 (07) ◽  
pp. 1691-1698 ◽  
Author(s):  
F. Amdjadi ◽  
P. J. Aston
Keyword(s):  
Hopf Bifurcation ◽  
Geometric Structure ◽  
Trivial Solution ◽  
Mixed Mode ◽  
Mode Interaction ◽  
Hopf Bifurcations ◽  
Bifurcation Points ◽  
Mode Interactions ◽  
The Stability ◽  
Secondary Bifurcations

In the unfolding of a mode interaction, in addition to the primary bifurcations, there are also secondary bifurcations which occur on the primary branches giving rise to mixed mode solutions. A further tertiary Hopf bifurcation arises in some cases from the mixed mode solutions. The detection of Hopf bifurcation points is a numerically expensive procedure and so we consider whether it is possible to predict the existence of the tertiary Hopf bifurcation by considering only the geometric structure of the primary and secondary branches. We show that in some cases, it is possible to show that no Hopf bifurcation exists while in other cases, more information in the form of the stability of the trivial solution is required to determine whether or not the Hopf bifurcation exists. An algorithm for determining the existence of the Hopf bifurcation is given.

Inelastic buckling mode interactions and resonance instabilities in thin-walled columns due to the fatigue damage

2020 ◽  
Vol 37 (8) ◽  
pp. 2819-2845 ◽  
Author(s):  
Dragan D. Milašinović ◽  
Petar Marić ◽  
Žarko Živanov ◽  
Miroslav Hajduković
Keyword(s):  
Fatigue Damage ◽  
Great Influence ◽  
Mode Interaction ◽  
Buckling Mode ◽  
Inelastic Buckling ◽  
Content Type ◽  
Effective Stresses ◽  
Stiffness Reduction ◽  
Mode Interactions ◽  
Thin Walled

Purpose The problems of inelastic instability (buckling) and dynamic instability (resonance) have been the subject of extensive investigation and have received wide attention from the structural mechanics community. This paper aims to tackle these problems in thin-walled structures, taking into account geometrical and/or material non-linearity. Design/methodology/approach The inelastic buckling mode interactions and resonance instabilities of prismatic thin-walled columns are analysed by implementing the semi-analytical finite strip method (FSM). A scalar damage parameter is implemented in conjunction with a material modelling named rheological-dynamical analogy to address stiffness reduction induced by the fatigue damage. Findings Inelastic buckling stresses lag behind the elastic buckling stresses across all modes, which is a consequence of the viscoelastic behaviour of materials. Because of the lag, the same column length does not always correspond to the same mode at the elastic and inelastic critical stress. Originality/value This paper presents the influence of mode interactions on the effective stresses and resonance instabilities in thin-walled columns due to the fatigue damage. These mode interactions have a great influence on damage variables because of the fatigue and effective stresses around mode transitions. In its usual semi-analytical form, the FSM cannot be used to solve the mode interaction problem explained in this paper, because this technique ignores the important influence of interaction of the buckling modes when applied only for undamaged state of structure

scholarly journals The Influence of Slowly Varying Mass on Severity of Dynamics Nonlinearity of Bearing-Rotor Systems with Pedestal Looseness

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Mian Jiang ◽  
Jigang Wu ◽  
Shuangqi Liu
Keyword(s):  
Mathematical Model ◽  
Control Parameter ◽  
Rotor System ◽  
Cosine Function ◽  
Nonlinear Mathematical Model ◽  
Rotor Systems ◽  
Disk Mass ◽  
Amplitude Coefficient ◽  
Varying Mass

Nonlinearity measure is proposed to investigate the influence of slowly varying mass on severity of dynamics nonlinearity of bearing-rotor systems with pedestal looseness. A nonlinear mathematical model including the effect of slowly varying disk mass is developed for a bearing-rotor system with pedestal looseness. The varying of equivalent disk mass is described by a cosine function, and the amplitude coefficient is used as a control parameter. Then, nonlinearity measure is employed to quantify the severity of dynamics nonlinearity of bearing-rotor systems. With the increasing of looseness clearances, the curves that denote the trend of nonlinearity degree are plotted for each amplitude coefficient of mass varying. It can be concluded that larger amplitude coefficients of the disk mass varying will have more influence on the severity of dynamics nonlinearity and generation of chaotic behaviors in rotor systems with pedestal looseness.

Improved Positioning of Pneumatic Cylinder by Using Flexible Coupling between the Piston and Rod

1992 ◽  
Vol 206 (6) ◽  
pp. 431-435 ◽  
Author(s):  
G V Krejnin ◽  
I L Krivz ◽  
L A Smelov
Keyword(s):  
Mathematical Model ◽  
Computer Simulation ◽  
Parameter Optimization ◽  
Degrees Of Freedom ◽  
Control Parameter ◽  
Pneumatic Cylinder ◽  
Output Link ◽  
Positioning Accuracy ◽  
Flexible Coupling ◽  
And Control

Positioning accuracy of a pneumatic piston drive with flexible coupling between the piston and rod is considered. Improved positioning was expected due to the fact that the rod friction is usually considerably less than the piston friction. When the piston stops under the action of its friction force the rod continues the motion, providing the precision positioning of the output link. A mathematical model of a positioning pneumatic piston drive with two degrees of freedom was generated. Computer simulation of the performance of short and long strokes showed the feasibility of the improved positioning which provided design and control parameter optimization.

On an Optimal Linear Control of a Chaotic Non-Ideal Duffing System

2011 ◽  
Vol 138-139 ◽  
pp. 50-55 ◽  
Author(s):  
Fábio Roverto Chavarette
Keyword(s):  
Mathematical Model ◽  
Control Strategy ◽  
Control Parameter ◽  
Chaotic Behavior ◽  
Linear Control ◽  
Complete Agreement ◽  
Duffing System ◽  
Rotating Machine ◽  
Optimal Linear ◽  
Limited Power

In this work, we use a nonlinear control based on Optimal Linear Control. We used as mathematical model a Duffing equation to model a supporting structure for an unbalanced rotating machine with limited power (non-ideal motor). Numerical simulations are performed for a set control parameter (depending on the voltage of the motor, that is, in the static and dynamic characteristic of the motor) The interaction of the non-ideal excitation with the structure may lead to the occurrence of interesting phenomena during the forward passage through the several resonance states of the system. Chaotic behavior is obtained for values of the parameters. Then, the proposed control strategy is applied in order to regulate the chaotic behavior, in order to obtain a periodic orbit and to decrease its amplitude. Both methodologies were used in complete agreement between them. The purpose of the paper is to give suggestions and recommendations to designers and engineers on how to drive this kind of system through resonance.

Nonlinear regular dynamics of a single-degree robot model

Robotica ◽  
1996 ◽  
Vol 14 (4) ◽  
pp. 423-431 ◽  
Author(s):  
V. Paar ◽  
N. Pavin ◽  
N. Paar ◽  
B. Novaković
Keyword(s):  
Mathematical Model ◽  
Numerical Investigation ◽  
Control Parameter ◽  
Degree Of Freedom ◽  
Transient Chaos ◽  
Parameter Region ◽  
Single Degree ◽  
Engineering Significance ◽  
Arnold Tongues ◽  
Upper Boundary

SUMMARYThis paper presents a mathematical model of a robot with one degree of freedom and numerical investigation of its dynamics in a particular parameter scan which is close to the upper boundary of the estimates for the parameters of rigidity and friction, while the length parameter L is treated as a free control parameter. In this L-scan the quasiperiodic and frequency locked solutions, their pattern and order of appearance are studied in the interval from the parameter range of immediate engineering significance to the point of appearance of transient chaos. In particular, a fractaltype multiple splitting of Arnold tongues is found in the parameter region bordering the range of engineering significance.

Equivariant Hopf-Pitchfork Bifurcation of Symmetric Coupled Neural Network with Delay

2016 ◽  
Vol 26 (12) ◽  
pp. 1650205 ◽  
Author(s):  
Baodong Zheng ◽  
Haidong Yin ◽  
Chunrui Zhang
Keyword(s):  
Neural Network ◽  
Pitchfork Bifurcation ◽  
Stable Fixed Point ◽  
System Of Differential Equations ◽  
Codimension Two ◽  
Rich Variety ◽  
Mode Interactions ◽  
Generic Dynamics ◽  
Pitchfork Bifurcations ◽  
Two Parameter

This paper is concerned with how the symmetry and singularity of a system of differential equations affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem satisfying a [Formula: see text]-symmetry condition, the mode interactions in two-parameter bifurcations with a single zero and two pairs of imaginary roots are considered. The codimension two normal form with equivariant Hopf-pitchfork bifurcations are given. Through analyzing the unfolding structure, local classification in the neighborhood of equivariant Hopf-pitchfork bifurcations point for the [Formula: see text]-symmetric is undertaken. A rich variety of dynamical and bifurcation behaviors is pointed out. Beyond a stable fixed point or a pair of stable fixed points, some interesting phenomena are also found, such as the coexistence of two periodic solutions which are verified both theoretically and numerically.

scholarly journals Stabiizing The Steady State Solution of Lasser Fever: Problems and Prospect

2022 ◽  
Vol 05 (01) ◽  
Author(s):  
Innocent C. Eli ◽  
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Control Parameter ◽  
Steady State Solution ◽  
Lassa Fever ◽  
Sexual Contact ◽  
Opposite Sex ◽  
The Stability ◽  
And Control ◽  
Disease Free

The study of mathematical modeling of the stability analysis of Lassa fever was examined. A mathematical model for the spread and control of Lassa fever was formulated and analyzed. The model incorporates a control parameter, the use of condom to control human to human transmission through sexual contact with opposite sex. The disease free and endemic equilibrium states were analyzed.

scholarly journals MATHEMATICAL MODEL OF RAILWAY DYNAMICS AUTONOMOUS TRACTION MODULE

2021 ◽  
pp. 119-120
Author(s):  
Frantisek Bures
Keyword(s):  
Complex System ◽  
Mathematical Model ◽  
Mathematical Description ◽  
Railway Track ◽  
Railway Transport ◽  
Railway Dynamics ◽  
The Mathematical Model

In the report the author offers a mathematical description of the model of the dynamics of the railway autonomous traction module. The autonomous traction module is a multi-mass complex system moving on a railway track. The mathematical model takes into account the parameters and types of connections between the solids of the system, as well as takes into account the sliding forces between the wheels and rails. The mathematical model developed by the author can be applied at theoretical researches of innovative designs of autonomous traction means on railway transport.

Sign in / Sign up

Export Citation Format

Share Document