scholarly journals Formulation of Equations of Motion for a Simply Supported Bridge under a Moving Railway Freight Vehicle

2007 ◽  
Vol 14 (6) ◽  
pp. 429-446 ◽  
Author(s):  
Ping Lou ◽  
Qing-yuan Zeng
Keyword(s):  
Equations Of Motion ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Railway Track ◽  
Time Step ◽  
Bogie Frame ◽  
Simply Supported ◽  
Car Body ◽  
Railway Freight ◽  
Interaction System

Based on energy approach, the equations of motion in matrix form for the railway freight vehicle-bridge interaction system are derived, in which the dynamic contact forces between vehicle and bridge are considered as internal forces. The freight vehicle is modelled as a multi-rigid-body system, which comprises one car body, two bogie frames and four wheelsets. The bogie frame is linked with the car body through spring-dashpot suspension systems, and the bogie frame is rigidly linked with wheelsets. The bridge deck, together with railway track resting on bridge, is modelled as a simply supported Bernoulli-Euler beam and its deflection is described by superimposing modes. The direct time integration method is applied to obtain the dynamic response of the vehicle-bridge interaction system at each time step. A computer program has been developed for analyzing this system. The correctness of the proposed procedure is confirmed by one numerical example. The effect of different beam mode numbers and various surface irregularities of beam on the dynamic responses of the vehicle-bridge interaction system are investigated.

A COMPARISON OF DIFFERENT SOLUTION ALGORITHMS FOR THE NUMERICAL ANALYSIS OF VEHICLE–BRIDGE INTERACTION

2014 ◽  
Vol 14 (02) ◽  
pp. 1350065 ◽  
Author(s):  
K. LIU ◽  
N. ZHANG ◽  
H. XIA ◽  
G. DE ROECK
Keyword(s):  
Numerical Analysis ◽  
Time Domain ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Good Choice ◽  
Contact Forces ◽  
Time Dependent ◽  
Time Step ◽  
Solution Algorithms ◽  
Coupled Equation

The interaction between a bridge and a train moving on the bridge is a coupled dynamic problem. The equations of motion of the bridge and the vehicle are coupled by the time dependent contact forces. At each time step, the motion of the bridge influences the forces transferred to the vehicle and this, in turn, changes the forces acting on the bridge. In this paper, a comparison of three different time domain solution algorithms for the coupled equation of motion of the train–bridge system is presented. Guidelines are given for a good choice of the time step.

A Computational Investigation of the Disk-Housing Impacts of Accelerating Rotors Supported by Hydrodynamic Bearings

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Jaroslav Zapoměl ◽  
Petr Ferfecki
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Angular Speed ◽  
Contact Forces ◽  
Computational Time ◽  
Radial Clearance ◽  
Lagrange Equations ◽  
Time Step ◽  
Hydrodynamic Bearings ◽  
Nonlinear Force

As the radial clearance between disks and the casing of rotating machines is usually very narrow, excessive lateral vibration of accelerating rotors passing critical speeds can produce impacts between the disks and the housing. The computer modeling method is an important tool for investigating such events. In the developed procedure, the shaft is flexible and the disks are absolutely rigid. The hydrodynamic bearings and the impacts are implemented in the mathematical model by means of nonlinear force couplings. Most of the publications and computer codes from the field of rotor dynamics are referred only in the case when the rotor turns at a constant angular speed and in simple cases of disk-housing impacts. Moreover, if the disks turning at variable speed are investigated, the resulting form of the equations of motion derived by different authors slightly differs and the differences depend on the method used for their derivation. Therefore, particular emphasis in this article is given to the derivation of the motion equations of a continuous rotor turning with variable revolutions to explain the mentioned differences, to develop a computer algorithm enabling the investigation of cases when impacts between an arbitrary number of disks and the stationary part take place, and to analyze the mutual interaction between the impacts and the fluid film bearings. The Hertz theory is applied to determine the contact forces. Calculation of the hydrodynamic forces acting on the bearings is based on solving the Reynolds equation and taking cavitation into account. Lagrange equations of the second kind and the principle of virtual work are used to derive equations of motion. The Runge–Kutta method with an adaptive time step is applied for their solution. The applicability of the developed procedure was tested by computer simulations. The results show that it can be used for the modeling of complex rotor systems and that the short computational time enables carrying out calculations for a number of design and operation parameters.

scholarly journals Directional Control of a Driver-Heavy-Vehicle Closed-Loop System

2011 ◽  
Vol 2-3 ◽  
pp. 33-38
Author(s):  
Shao Hua Li ◽  
Shao Pu Yang ◽  
Na Chen
Keyword(s):  
Closed Loop ◽  
Single Point ◽  
Equations Of Motion ◽  
Loop System ◽  
Dynamic Responses ◽  
Heavy Vehicle ◽  
Closed Loop System ◽  
Time Step ◽  
Steering Angle ◽  
Directional Control

A two degree of freedom (DOF) lateral dynamic model for a three-axe heavy vehicle is set up and the vehicle ordinary differential equations of motion are derived. The nonlinear lateral tire forces are obtained by Gim model with vertical loads, slip angles and cornering performances of front and rear tires being input parameters. A revised closed-loop single-point preview method is proposed to model the driver’s directional control performance. In this proposed method, the steering angle of front wheels is calculated in real time according to the track error between a certain point ahead of the vehicle and the required route. Then the steering angle is input into the vehicle model to gain the dynamic responses and position of the vehicle in next time step. Thus the driver-heavy-vehicle closed-loop system is built. The dynamic responses of the system are simulated on the condition of double lane change and the effects of system parameters on the path following behavior of the vehicle are researched. Then the advice on how to improve the vehicle directional control ability can be brought forward.

Coupling OCREC Contact Code With ADAMS: Simulations of One Coach at 130mph

2009 ◽  
Author(s):  
J. P. Pascal ◽  
J. Berger ◽  
F. Bondon ◽  
C. Clerc ◽  
S. Teppe
Keyword(s):  
High Speed ◽  
Model Updating ◽  
Contact Forces ◽  
Railway Track ◽  
Frenet Frame ◽  
Time Step ◽  
Space Curves ◽  
Track System ◽  
Curvature Measurements ◽  
Multi Body

This paper presents the Online Calculation of Railway Elastic Contacts (OCREC), a dynamic railway calculation tool based on an advanced contact kernel, and its coupling with the MSC ADAMS multi-body commercial software. The OCREC contact kernel is used as a subroutine of multi-body codes in order to calculate contact forces between wheelsets and rails. The OCREC method is “online” as it not only redefines new contact parameters at each time step but also determines all simultaneous contacts on each wheel as allowed by Hertz Elasticity theory. From the normal forces and relative velocities given by the Hertz theory, Tangential Forces are calculated using Kalker’s FASTSIM (modified for elliptical pressure distribution). After a description of the OCREC theory, the paper presents the linkage between OCREC and MSC ADAMS software. OCREC calculates contact forces within a Frenet frame (oxyz) following the track layout where ox is tangent to the track; oy is horizontal and oz normal to oxy. As ADAMS calculates inside a different frame, and as it has no built-in track system, it was necessary to develop a program capable of connecting 3 different frames: the ‘dummy’ track frame, the Frenet frame and the fixed ADAMS frame. Note that the ‘dummy’ frame is directly calculated from railway track curvature measurements recorded in so-called ‘space curves’. The OCREC ADAMS link is first validated by a bogie rolling on a dummy track. With the equations of the OCRECYM code established directly within the “dummy” frame, the OCREC-ADAMS results are compared to a specific OCRECYM validation code. Then, the results from an actual railway case are presented: behavior of one coach is calculated on a real measured track including curves and defaults. During the following step, the OCREC-ADAMS results are compared to OCRECYM results. After some model updating for adjustment to physical properties of elastic joints (helicoidal springs), a good correlation is obtained between the codes. The analysis of the different force and displacement components proves this kind of numerical tool’s capabilities of assessing the railway vehicle’s dynamic behavior. Especially, the Y/Q safety ratio is well calculated. Thus, the OCREC contact kernel, which is powerful for complex contact topologies such as conformal contacts, and necessary for high speed safety calculation, can be used as a subroutine of standard multi-body software, giving it high capabilities for dynamic railway calculation.

A Versatile 3D Vehicle-Track-Bridge Element for Dynamic Analysis of the Railway Bridges under Moving Train Loads

2019 ◽  
Vol 19 (04) ◽  
pp. 1950050 ◽  
Author(s):  
Xiang Xiao ◽  
Wei-Xin Ren
Keyword(s):  
Virtual Work ◽  
Interaction Analysis ◽  
Equations Of Motion ◽  
Time Integration ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Railway Bridges ◽  
Beam Elements ◽  
Track Bridge ◽  
Moving Train

There has been a growing interest to carry out the vehicle–track–bridge (VTB) dynamic interaction analysis using 2D or 3D finite elements based on simplified wheel–rail relationships. The simplified or elastic wheel–rail contact relationships, however, cannot consider the lateral contact forces and geometric shapes of the wheel and rails, and even the occasional jump of wheels from the rails. This does not guarantee a reliable analysis for the safety running of trains over bridges. To consider the wheel–rail constraint and contact forces, this paper proposes a versatile 3D VTB element, consisting of a vehicle, eight rail beam elements, four bridge beam elements, and continuous springs as well as the dampers between the rail and bridge girder. With the 3D VTB element matrices formulated, a procedure for assembling the interaction matrices of the 3D VTB element is presented based on the virtual work principle. The global equations of motion of the VTB interaction system are established accordingly, which can be solved by time integration methods to obtain the dynamic responses of the vehicle, track and bridge, as well as the stability and safety indices of the moving train. Finally, an illustrative example is used to verify the proposed the versatile 3D VTB element for the dynamic interactive analysis of railway bridges under moving train loads.

Comparative Studies on the Dynamic Performances of High Speed Turbocharger Rotor Supported on Oil-Free Bearings Versus Conventional Floating Ring Systems

2017 ◽  
Author(s):  
Rajasekhara Reddy Mutra ◽  
J. Srinivas
Keyword(s):  
Dynamic Response ◽  
High Speed ◽  
Equations Of Motion ◽  
Dynamic Modelling ◽  
Dynamic Responses ◽  
Effect Of Temperature ◽  
Time Step ◽  
Foil Bearing ◽  
Dynamic Performances ◽  
Bearing System

Present work focuses on the dynamic modelling of the dual-disc rotor supported on oil-free bearings idealizing a turbocharger rotor bearing system. The equations of motion of the rotor system are formulated and solved by finite element method to obtain the dynamic response of the system. The gas-foil bearing forces obtained from finite-difference approach at each time-step of solution. The same rotor model is used with the conventional floating ring bearing system where, the bearing forces are provided as displacement dependent time-varying oil and floating ring forces. As a practical environmental condition, the effect of temperature on the viscosity is studied using Dowson equation. The dynamic responses are illustrated both for rotor supported on both gas-foil and floating-ring type bearings. The effects of changes in bearing clearances on the overall dynamic characteristics of the rotor are reported. In order to utilize the gas foil bearing model, an identification study is performed to predict the operating clearance and air viscosity using dynamic response data.

DERAILMENT RISK ANALYSIS OF A TILTING RAILWAY VEHICLE MOVING OVER IRREGULAR TRACKS UNDER WIND LOADS

2013 ◽  
Vol 13 (08) ◽  
pp. 1350038 ◽  
Author(s):  
YUNG-CHANG CHENG ◽  
CHENG-HAO HUANG ◽  
CHEN-MING KUO ◽  
CHERN-HWA CHEN
Keyword(s):  
Equations Of Motion ◽  
Wind Loads ◽  
Creep Model ◽  
Railway Vehicle ◽  
Nonlinear Creep ◽  
Lateral Direction ◽  
Bogie Frame ◽  
Car Body ◽  
Tilting Angle ◽  
Rail Irregularities

Based on the nonlinear creep model and Kalker's linear theory, this paper studies the governing differential equations of motion for a tilting railway vehicle moving over irregular curved tracks under wind loads. The tilting vehicle is modeled by a 24-degree-of-freedom (24-DOF) car system, considering the lateral, roll and yaw motions of each wheelset, the lateral, vertical, roll and yaw motions of each bogie frame and the car body. The derailment quotients of the tilting railway vehicle with the wheelsets moving over irregular rails in the lateral direction and the car body acted upon by the wind loads are investigated for various tilting angles. The analysis results show that in general, the derailment quotient of the wheelset increases as the tilting angle of the railway vehicle increases. When the railway vehicle moves at low speeds, the derailment quotient calculated for the case with rail irregularities is greater than that for the case with no rail irregularities. Moreover, the derailment quotient of a wheelset moving over curved tracks of various radii is presented. Finally, the derailment quotient computed for the case under wind loads is greater than that free of wind loads. As a result, the influence of rail irregularities and wind loads on the derailment risk of a tilting vehicle cannot be ignored.

A Methodology to Detect the Precise Instant of Contact in Multibody Dynamics

2011 ◽  
Author(s):  
Paulo Flores
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Time Step ◽  
Dynamic Simulations ◽  
Integration Algorithm ◽  
Current Time ◽  
Impact Time ◽  
System Equations ◽  
The Impact

The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step for the contact and non contact periods in multibody dynamics. The basic idea of the described methodology is to ensure that the first impact within a multibody system does not occur with a large value for relative bodies’ penetration in order to avoid the artificially large contact forces associated. The detection of the instant of contact takes place when the distance between two bodies change the sign between two discrete moments in time. In fact, in theory, the contact starts when this distance is zero, or a very small value to prevent the round-off errors. Thus, during the numerical solution of the system equations of motion if the first penetration is below this small value previously specified, then the current time is taken as the impact time. On the other hand, if the first penetration is larger than the specified tolerance, then the current time step is beyond the impact time. In this case, integration algorithm is forced to go back and take a smaller time step until a step can be taken within the acceptable tolerance. The main features of this approach are the easiness to implement and the good computational efficiency. In addition, it can easily deal with the transitions between non contact and contact cases in multibody dynamics. Finally, results obtained from dynamic simulations are presented and discussed to study the validity of the methodology proposed in this work.

Formulation of Multibody Dynamics as Complementarity Problems

2003 ◽  
Author(s):  
J. C. Trinkle
Keyword(s):  
Discrete Time ◽  
Complementarity Problem ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Contact Forces ◽  
Integration Scheme ◽  
Time Step ◽  
Stick Slip ◽  
Frictional Contacts ◽  
Time Formulation

Multibody systems with rigid bodies and unilateral contacts are difficult to simulate due to discontinuities associated with gaining and losing contacts and stick-slip transitions. Methods for simulating such systems fall into two categories: penalty methods and complementarity methods. The former calculate penetration depths of virtual rigid bodies at every time step and compute restoring forces to repair penetrations, while the latter assume that the bodies are truly rigid and compute contact forces that prevent penetration from occurring at all. In this paper, we are concerned with complementarity methods. We present an instantaneous formulation of the equations of motion of multi-rigid-body systems with frictional contacts as a complementarity problem. The unknowns in this formulation are accelerations and forces at the contacts. Since it is known that this model does not always admit a finite solution, it is problematic to use it directly in an integration scheme. This fact motivates the discrete-time formulation presented second. Although the discrete-time formulation also takes the form of a complementarity problem, it does not suffer from non-existence, and thus it is suitable for simulation. Numerical results are compared to the exact solution for a sphere initially sliding, then rolling, on a horizontal plane.

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