Use of the Non-Inertial Coordinates in the Analysis of Train Longitudinal Forces

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
Ahmed A. Shabana ◽  
Ahmed K. Aboubakr ◽  
Lifen Ding
Keyword(s):  
Differential Equations ◽  
Virtual Work ◽  
Three Dimensional ◽  
Coupled System ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
State Equations ◽  
Car Body ◽  
Nonlinear Coupler ◽  
Inertia Forces

In this investigation, a new three-dimensional nonlinear train car coupler model that takes into account the geometric nonlinearity due to the coupler and car body displacements is developed. The proposed nonlinear coupler model allows for arbitrary three-dimensional motion of the car bodies and captures kinematic degrees of freedom that are not captured using existing simpler models. The coupler kinematic equations are expressed in terms of the car body coordinates, as well as the relative coordinates of the coupler with respect to the car body. The virtual work is used to obtain expressions for the generalized forces associated with the car body and coupler coordinates. By assuming the inertia of the coupler components negligible compared to the inertia of the car body, the system coordinates are partitioned into two distinct sets: inertial and noninertial coordinates. The inertial coordinates that describe the car motion have inertia forces associated with them. The noninertial coupler coordinates; on the other hand, describe the coupler kinematics and have no inertia forces associated with them. The use of the principle of virtual work leads to a coupled system of differential and algebraic equations expressed in terms of the inertial and noninertial coordinates. The differential equations, which depend on the coupler noninertial coordinates, govern the motion of the train cars; whereas the algebraic force equations are the result of the quasi-static equilibrium conditions of the massless coupler components. Given the inertial coordinates and velocities, the quasi-static coupler algebraic force equations are solved iteratively for the noninertial coordinates using a Newton–Raphson algorithm. This approach leads to significant reduction in the numbers of state equations, system inertial coordinates, and constraint equations; and allows avoiding a system of stiff differential equations that can arise because of the relatively small coupler mass. The use of the concept of the noninertial coordinates and the resulting differential/algebraic equations obtained in this study is demonstrated using the knuckle coupler, which is widely used in North America. Numerical results of simple train models are presented in order to demonstrate the use of the formulation developed in this paper.

Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

2018 ◽  
Author(s):  
J. P. Meijaard ◽  
V. van der Wijk
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Force Balance ◽  
Dynamic Force ◽  
Algebraic Equations ◽  
Principle Of Virtual Work ◽  
Principal Vector ◽  
Reaction Forces ◽  
Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

Numerical methods for differential algebraic equations

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 141-198 ◽  
Author(s):  
Roswitha März
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Image Position

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

Fluid-Coupled Vibrations of Immersed Spent Nuclear Racks: A Nonlinear Model Accounting for Squeeze-Film and Dissipative Phenomena

2004 ◽  
Author(s):  
Miguel Moreira ◽  
Jose´ Antunes
Keyword(s):  
Dynamical Behavior ◽  
Coupled System ◽  
Differential Algebraic Equations ◽  
Nonlinear Flow ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Spent Fuel ◽  
Coupled Modes ◽  
Small Water ◽  
Flow Effects

Fluid-coupling effects lead to a complex dynamical behavior of immersed spent fuel assembly storage racks. Predicting their responses under strong earthquakes is of prime importance for the safety of nuclear plant facilities. In the near-past we introduced a simplified linearized model for the vibrations of such systems, in which gap-averaged velocity and pressure fields were described analytically in terms of a single space-coordinate for each fluid inter-rack channel. Using such approach it was possible to generate and assemble a complete set of differential-algebraic equations describing the multi-rack fluid coupled system dynamics. Because of the linearization assumptions, we achieved computation of the flow-structure coupled modes, but also time-domain simulations of the system responses. However, nonlinear squeeze-film and dissipative flow effects, connected with very large amplitude responses and/or relatively small water gaps, cannot be properly accounted unless the linearization assumption is relaxed. Such is the aim of the present paper. Here, using a similar approach, we generalize our theoretical model to deal with nonlinear flow effects. Besides that the proposed methodology can be automatically implemented in a symbolic computational environment, it is much less computer-intensive than finite element formulations. Using the proposed technique, computations of basic flow-coupled rack configurations subjected to impulse excitations are presented, in order to highlight the essential features of such systems as well as the relevance of squeeze-film and dissipative effects. Finally, more realistic simulations of complex system responses to strong seismic excitations are presented and discussed.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

Numerical Computation of Differential-Algebraic Equations for the Approximation of Artificial Satellite Trajectories and Planetary Ephemerides

1999 ◽  
Vol 67 (3) ◽  
pp. 574-580 ◽  
Author(s):  
B. Fox ◽  
L. S. Jennings ◽  
A. Y. Zomaya
Keyword(s):  
Differential Equations ◽  
Virtual Work ◽  
Initial Conditions ◽  
Artificial Satellite ◽  
System Modeling ◽  
Differential Algebraic Equations ◽  
The Body ◽  
Time Interval ◽  
Differential Algebraic Equation ◽  
Generalized Coordinates

The principle of virtual work and Lagrange’s equations of motion are used to construct a system of differential equations for constrained spatial multibody system modeling. The differential equations are augmented with algebraic constraints representing the system being modeled. The resulting system is a high index differential-algebraic equation (DAE) which is cast as an ordinary differential equation (ODE) by differentiating the constraint equations twice. The initial conditions are the heliocentric rectangular equatorial generalized coordinates and their first time derivatives of the planets of the solar system and an artificial satellite. The ODE is computed using the integration subroutine LSODAR to generate the body generalized coordinates and time derivatives and hence produce the planetary ephemerides and satellite trajectories for a time interval. Computer simulation and graphical output indicate the satellite and planetary positions and the latter may be compared with those provided in the Astronomical Almanac. Constraint compliance is investigated to establish the accuracy of the computation. [S0021-8936(00)03403-6]

scholarly journals Analytical Modelling of Three-Dimensional Squeezing Nanofluid Flow in a Rotating Channel on a Lower Stretching Porous Wall

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Navid Freidoonimehr ◽  
Behnam Rostami ◽  
Mohammad Mehdi Rashidi ◽  
Ebrahim Momoniat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Porous Wall ◽  
Coupled System ◽  
Physical Parameters ◽  
Nonlinear Ordinary Differential Equations ◽  
Suction Parameter ◽  
Rotating Channel

A coupled system of nonlinear ordinary differential equations that models the three-dimensional flow of a nanofluid in a rotating channel on a lower permeable stretching porous wall is derived. The mathematical equations are derived from the Navier-Stokes equations where the governing equations are normalized by suitable similarity transformations. The fluid in the rotating channel is water that contains different nanoparticles: silver, copper, copper oxide, titanium oxide, and aluminum oxide. The differential transform method (DTM) is employed to solve the coupled system of nonlinear ordinary differential equations. The effects of the following physical parameters on the flow are investigated: characteristic parameter of the flow, rotation parameter, the magnetic parameter, nanoparticle volume fraction, the suction parameter, and different types of nanoparticles. Results are illustrated graphically and discussed in detail.

Combined cycle dynamics

2003 ◽  
Vol 217 (3) ◽  
pp. 247-258 ◽  
Author(s):  
F. M. Mansour ◽  
A. M. Abdul Aziz ◽  
S. M. Abdel-Ghany ◽  
H. M. El-shaer
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Combined Cycle ◽  
Stable Operation ◽  
Algebraic Equations ◽  
State Equations ◽  
Combined Cycle Plant ◽  
Fast Settling ◽  
Operational Data ◽  
Good Agreement

A mathematical model describing the dynamic behaviour of each major component of the combined cycle is presented. The formulae are deduced from continuity, momentum, energy, and state equations. Partial differential equations (PDEs) are discretized to algebraic equations by using the implicit backward-central finite difference scheme and then solved by iteration. Explicit-Euler's integration method is applied to other differential equations (DEs). A multi-element control system is implemented to investigate its effect on the combined cycle's dynamic response. The simulation results are compared with the design and steady-state operational data of the unit number 4 in Cairo South Combined Cycle Power Plant, showing good agreement. The dynamic results prove the effectiveness of the multi-element control strategy to control the combined cycle plant with fast settling time, neglected steady-state error, and moderate overshoot or undershoot while assuring a stable operation under sudden changes of load.

Review of Knee Models: 1996 Update

1996 ◽  
Vol 49 (10S) ◽  
pp. S187-S193 ◽  
Author(s):  
Mohamed Samir Hefzy ◽  
T. Derek V. Cooke
Keyword(s):  
Knee Joint ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Full Extension ◽  
Body Contact ◽  
Patellar Cartilage ◽  
Three Body

This paper is an update on our previous review of knee models (Hefzy and Grood [30]). We find that progress was made in the area of dynamic modeling. Since 1988, a technique was developed to solve the system of differential algebraic equations describing the three-dimensional dynamic behavior of the knee joint. This technique allows to solve complex problems. However, not a single dynamic comprehensive anatomically based mathematical three-dimensional model of the knee joint that includes both tibio-femoral and patello-femoral joints has yet been developed. In the area of meniscal modeling, we find that only one three-dimensional model of the knee joint that includes the menisci is available in the literature. This quasi-static model includes menisci, tibial, femoral and patellar cartilage layers, and ligamentous structures. This model is limited since it solves only for one position of the knee joint: full extension. From this updated survey, we conclude that the development of a comprehensive three-dimensional anatomically based mathematical model of the knee joint (that solves the three-body contact problem at the tibio-femoral joint between the menisci, tibia and femur, and includes the patello-femoral joint) continues to present major challenge.

A Novel Finite-Element– Numerical-Integration Model for Composite Laminates Supported on Opposite Edges

2007 ◽  
Vol 74 (6) ◽  
pp. 1114-1124 ◽  
Author(s):  
Tarun Kant ◽  
Sandeep S. Pendhari ◽  
Yogesh M. Desai
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Composite Laminates ◽  
Three Dimensional ◽  
Laminated Composite ◽  
Composite Plates ◽  
Coupled System ◽  
Integration Model ◽  
Numerical Integrators ◽  
Support Conditions

An attempt is made here to devise a new methodology for an integrated stress analysis of laminated composite plates wherein both in-plane and transverse stresses are evaluated simultaneously. The method is based on the governing three-dimensional (3D) partial differential equations (PDEs) of elasticity. A systematic procedure is developed for a case when one of the two in-plane dimensions of the laminate is considered infinitely long (y direction) with no changes in loading and boundary conditions in that direction. The laminate could then be considered in a two-dimensional (2D) state of plane strain in x-z plane. It is here that the governing 2D PDEs are transformed into a coupled system of first-order ordinary differential equations (ODEs) in transverse z direction by introducing partial discretization in the finite inplane direction x. The mathematical model thus reduces to solution of a boundary value problem (BVP) in the transverse z direction in ODEs. This BVP is then transformed into a set of initial value problems (IVPs) so as to use the available efficient and effective numerical integrators for them. Through thickness displacement and stress fields at the finite element discrete nodes are observed to be in excellent agreement with the elasticity solution. A few new results for cross-ply laminates under clamped support conditions are also presented for future reference and also to show the generality of the formulation.

Sign in / Sign up

Export Citation Format

Share Document