Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems

1982 ◽  
Vol 104 (1) ◽  
pp. 247-255 ◽  
Author(s):  
R. A. Wehage ◽  
E. J. Haug
Keyword(s):  
Differential Equations ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Step Size ◽  
Integration Algorithm ◽  
Independent Variables ◽  
Dependent Variables

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, to facilitate the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matix relating variations in dependent and independent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, is developed that integrates for only the independent variables, yet effectively determines dependent variables. Numerical results are presented for planar motion of two tracked vehicular systems with 13 and 24 degrees of freedom. Computational efficiency of the algorithm is shown to be an order of magnitude better than previously employed algorithms.

Application of Euler Parameters to the Dynamic Analysis of Three-Dimensional Constrained Mechanical Systems

1982 ◽  
Vol 104 (4) ◽  
pp. 785-791 ◽  
Author(s):  
P. E. Nikravesh ◽  
I. S. Chung
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Dependent Variables

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are the Euler parameters. Euler parameters, in contrast to Euler angles or any other set of three rotational generalized coordinates, have no critical singular cases. The maximal set of generalized coordinates facilitates the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matrix relating variations in dependent and indpendent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.

Generalized Coordinate Partitioning for Analysis of Mechanical Systems with Nonholonomic Constraints

1983 ◽  
Vol 105 (3) ◽  
pp. 379-384 ◽  
Author(s):  
P. E. Nikravesh ◽  
E. J. Haug
Keyword(s):  
Differential Equations ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Variable Order ◽  
Step Size ◽  
Euler Parameters ◽  
Integration Algorithm ◽  
Generalized Coordinates

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems with holonomic and nonholonomic constraints. Holonomic and nonholonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are Euler parameters. The maximal set of generalized coordinates facilitates the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix and identifies independent coordinates and velocities. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

Development of the Finite Segment Method for Modeling Railroad Track Structures

2011 ◽  
Author(s):  
Martin B. Hamper ◽  
Khaled E. Zaazaa ◽  
Ahmed A. Shabana
Keyword(s):  
Sparse Matrix ◽  
Deformable Body ◽  
Contact Point ◽  
Equations Of Motion ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Finite Segment ◽  
Finite Segment Method ◽  
Railroad Vehicle

In the finite segment method, the dynamics of a deformable body is described using a set of rigid bodies that are connected by elastic force elements. This approach can be used, as demonstrated in this investigation, in the simulation of some rail movement scenarios. The purpose of this investigation is to develop a new track model that combines the absolute nodal coordinate formulation (ANCF) geometry and the finite segment method. The ANCF finite elements define the track geometry in the reference configuration as well as the change in the geometry due to the movement of the finite segments of the track. Using ANCF geometry and the finite segment kinematics, the location of the wheel/rail contact point is predicted online and used to update the creepage expressions due to the finite segment displacements and rotations. The location of the wheel/rail contact point and the updated creepage expressions are used to evaluate the creep forces. A three-dimensional elastic contact formulation (ECF-A), that allows for wheel/rail separation, is used in this investigation. The rail displacement due to the applied loads is modeled by a set of rigid finite segments that are connected by set of spring-damper elements. Each rail finite segment is assumed to have six rigid body degrees of freedom. The equations of motion of the finite segments are integrated with the railroad vehicle system equations of motion in a sparse matrix formulation. The resulting dynamic equations are solved using a predictor-corrector numerical integration method that has a variable order and variable step size. As shown in this paper, the finite segments may be used to model specific phenomena that occur in railroad vehicle applications, including rail rotations and gage widening. The procedure used in this investigation to implement the finite segment method in a general purpose multibody system (MBS) computer program is described. Four simple models are presented in order to demonstrate the implementation of the finite segment method in MBS algorithms. The limitations of using the finite segments approach for modeling the track structure and rail flexibility are also discussed.

scholarly journals Solving Duffing Type Differential Equations using a Three-Point Block Variable Order Step Size Method

2019 ◽  
Vol 1366 ◽  
pp. 012024
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Muhammad Asyraf Asbullah ◽  
Koo Lee Feng ◽  
Wong Tze Jin ◽  
...  
Keyword(s):  
Differential Equations ◽  
Variable Order ◽  
Step Size

scholarly journals How to calculate all point symmetries of linear and linearizable differential equations

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140685 ◽  
Author(s):  
Robert J. Gray
Keyword(s):  
Differential Equations ◽  
Finite Number ◽  
Point Symmetry ◽  
Systematic Method ◽  
Linear Change ◽  
Independent Variables ◽  
Dependent Variables ◽  
Lie Point Symmetry ◽  
Symmetry Generators ◽  
Linear Pdes

The systematic method for calculating all point symmetries of partial differential equations (PDEs) with non-trivial Lie point symmetry algebras does not currently apply to linear PDEs. Consider any given locally solvable system of linear homogeneous PDEs, each of order 2 or higher, of general Kovalevskaya form. Suppose that its Lie point symmetry generators' characteristics are linear in the dependent variables, while those that are homogeneous of degree 1 in the dependent variables contain a finite number of arbitrary parameters (this is common for systems that arise from applications). Herein, every point symmetry is shown to be projectable/fibre-preserving: the transformed independent variables depend only on the original independent variables. Moreover, two (conditional) Lie-algebraic characterizations of the generators of scalings and superpositions of solutions are derived. One of them holds if the system cannot be decoupled (with a smooth, locally invertible, linear change of dependent variables). If any such characterization holds, every point symmetry maps the dependent variables to functions that are linear in the original dependent variables. The aforementioned systematic method is adapted to calculate these symmetries. This version of the method also applies to linearizable systems.

scholarly journals Memoir on the integration of partial differential equations of the second order in three independent variables, when an intermediary integral does not exist in general

1898 ◽  
Vol 62 (379-387) ◽  
pp. 283-285
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
General Feature ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables

The general feature of most of the methods of integration of any partial differential equation is the construction of an appropriate subsidiary system and the establishment of the proper relations between integrals of this system and the solution of the original equation. Methods, which in this sense may be called complete, are possessed for partial differential equations of the first order in one dependent variable and any number of independent variables; for certain classes of equations of the first order in two independent variables and a number of dependent variables; and for equations of the second (and higher) orders in one dependent and two independent variables.

scholarly journals Two-Point Block variable order step size Multistep Method for Solving Higher Order Ordinary Differential Equations Directly

2021 ◽  
pp. 101376
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak ◽  
Wong Tze Jin ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Multistep Method ◽  
Variable Order ◽  
Step Size

scholarly journals A Numerical Algorithm for Solving Stiff Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computation Time ◽  
Test Problems ◽  
Variable Order ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
The Stability

An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

The numerical solution of hyperbolic systems of partial differential equations in three independent variables

1960 ◽  
Vol 255 (1281) ◽  
pp. 232-252 ◽  
Keyword(s):  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Original Method ◽  
Plane Motion ◽  
Step Size ◽  
Independent Variables ◽  
Step Procedure ◽  
Dependent Variables ◽  
Derivatives Of ◽  
Unknown Point

An original method of integration is described for quasi-linear hyperbolic equations in three independent variables. The solution is constructed by means of a step-by-step procedure, employing difference relations along four bicharacteristics and one time-like ordinary curve through each point. From these difference relations the derivatives of the dependent variables at the unknown point are eliminated. The solution at any point can then be com­puted, with an error proportional to the step size cubed, without referring to conditions outside its domain of dependence. The application of the method to the systems of equations governing unsteady plane motion and steady supersonic flow of an inviscid, non-conducting fluid is discussed in detail. As an example of the use of the method, the flow over a particular delta-shaped body has been computed.

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