An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Differential Algebraic Equations ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Generalized Coordinates ◽  
Axial Forces ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable-Car Systems—Part I: Methodology

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
W. D. Zhu ◽  
H. Ren
Keyword(s):  
Differential Algebraic Equations ◽  
Dynamic Responses ◽  
Distributed Parameter ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Substructure Method ◽  
Axial Forces ◽  
Geometric Matching ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to accurately calculate dynamic responses of one-dimensional structural systems, which consist of length-variant distributed-parameter components, such as strings, rods, and beams, and lumped-parameter components, such as point masses and rigid bodies. The dependent variable of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge if the dependent variables are smooth enough. Spatial derivatives of the dependent variables, which are related to internal forces/moments of the distributed-parameter components, such as axial forces, bending moments, and shear forces, can be accurately calculated. Combining component equations that are derived from Lagrange's equations and geometric matching conditions that arise from continuity relations leads to a system of differential algebraic equations (DAEs). When the geometric matching conditions are linear, the DAEs can be transformed to a system of ordinary differential equations (ODEs), which can be solved by an ODE solver. The methodology is applied to several moving elevator cable-car systems in Part II of this work.

Null Space Method of Differential Equation Type for Motion Analysis of Multibody Systems

2017 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yusaku Yamashita
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the constraint Jacobian. In previous reports, one of the authors presented methods which use the null space matrix. In the procedure to obtain the null space matrix, the inverse of a matrix whose regularity may not be always guaranteed. In this report, a new method is proposed in which the null space matrix is obtained by solving differential equations that can be always defined by using the QR decomposition, even if the constraints are redundant. Examples of numerical analysis are shown to validate the proposed method.

Continuous Null Space Method for Constrained Mechanical Systems

2011 ◽  
Author(s):  
Keisuke Kamiya ◽  
Makoto Sawada ◽  
Yuji Furusawa
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations for multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. It is desirable for efficient and accurate analysis to eliminate the Lagrange multipliers and dependent variables. As a method to solve the DAEs by eliminating the Lagrange multipliers, there is a method called the null space method. In this report, first, it is shown that using the null space matrix one can eliminate the Lagrange multipliers and reduce the number of velocities to that of the independent ones. Then, a new method to obtain the continuous null space matrix is presented. Finally, the presented method is applied to four-bar linkages.

Globally Independent Coordinates for Real-Time Vehicle Simulation

1998 ◽  
Vol 122 (4) ◽  
pp. 575-582 ◽  
Author(s):  
Radu Serban ◽  
Edward J. Haug
Keyword(s):  
State Space ◽  
Real Time ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Vehicle Simulation ◽  
Generalized Coordinates ◽  
Dynamics Of Multibody Systems ◽  
Improved Performance ◽  
Control Forces

Models of the dynamics of multibody systems generally result in a set of differential-algebraic equations (DAE). State-space methods for solving the DAE of motion are based on reduction of the DAE to ordinary differential equations (ODE), by means of local parameterizations of the constraint manifold that must be often modified during a simulation. In this paper it is shown that, for vehicle multibody systems, generalized coordinates that are dual to suspension and/or control forces in the model are independent for the entire range of motion of the system. Therefore, these additional coordinates, together with Cartesian coordinates describing the position and orientation of the chassis, form a set of globally independent coordinates. In addition to the immediate advantage of avoiding the computationally expensive redefinition of local parameterization in a state-space formulation, the existence of globally independent coordinates leads to efficient algorithms for recovery of dependent generalized coordinates. A topology based approach to identify efficient computational sequences is presented. Numerical examples with realistic vehicle handling models demonstrate the improved performance of the proposed approach, relative to the conventional Cartesian coordinate formulation, yielding real-time for vehicle simulation. [S1050-0472(00)00404-9]

Ideal Compliant Joints and Integration of Computer Aided Design and Analysis

2014 ◽  
Author(s):  
Ashraf M. Hamed ◽  
Paramsothy Jayakumar ◽  
Michael D. Letherwood ◽  
David J. Gorsich ◽  
Antonio M. Recuero ◽  
...  
Keyword(s):  
Computer Aided Design ◽  
Large Scale ◽  
Algebraic Equations ◽  
B Spline ◽  
Constraint Equations ◽  
Compliant Joints ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Computer Aided ◽  
Aided Design

This paper discusses fundamental issues related to the integration of computer aided design and analysis (I-CAD-A) by introducing a new class of ideal compliant joints that account for the distributed inertia and elasticity. The absolute nodal coordinate formulation (ANCF) degrees of freedom are used in order to capture modes of deformation that cannot be captured using existing formulations. The ideal compliant joints developed can be formulated, for the most part, using linear algebraic equations, allowing for the elimination of the dependent variables at a preprocessing stage, thereby significantly reducing the problem dimension and array storage needed. Furthermore, the constraint equations are automatically satisfied at the position, velocity, and acceleration levels. When using the proposed approach to model large scale chain systems, differences in computational efficiency between the augmented formulation and the recursive methods are eliminated, and the CPU times resulting from the use of the two formulations become similar regardless of the complexity of the system. The elimination of the joint constraint equations and the associated dependent variables also contribute to the solution of a fundamental singularity problem encountered in the analysis of closed loop chains and mechanisms by eliminating the need to repeatedly change the chain or mechanism independent coordinates. It is shown that the concept of the knot multiplicity used in computational geometry methods, such as B-spline and NURBS (Non-Uniform Rational B-Spline), to control the degree of continuity at the breakpoints is not suited for the formulation of many ideal compliant joints. As explained in this paper, this issue is closely related to the inability of B-spline and NURBS to model structural discontinuities. Another contribution of this paper is demonstrating that large deformation ANCF finite elements can be effective, in some MBS application, in solving small deformation problems. This is demonstrated using a heavily constrained tracked vehicle with flexible link chains. Without using the proposed approach, modeling such a complex system with flexible links can be very challenging. The analysis presented in this paper also demonstrates that adding significant model details does not necessarily imply increasing the complexity of the MBS algorithm.

Transient Response of Distributed Parameter Systems

1997 ◽  
Author(s):  
Jianping Zhou ◽  
Zhigang Feng
Keyword(s):  
Differential Equations ◽  
Transient Response ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
One Dimensional ◽  
Linear Algebraic Equations ◽  
Global Equilibrium ◽  
Distributed Transfer Function

Abstract A semi-analytic method is presented for the analysis of transient response of distributed parameter systems which are consist of one dimensional subsystems. The system is first divided into one dimensional sub-systems. Within each subsystem, replacing differentials on time t by finite difference, the governing partial differential equations are reduced to difference-differential equations. The solution of derived ordinary differential equations is obtained in an exact and closed form by distributed transfer function method and represented in nodal displacement parameters. Assemling global equilibrium equations at each nodes according to displacement continuity and force equilibrium requirements gives simutaneous linear algebraic equations. Numerical results are illustrated and compared with that of finite element method.

Rosenbrock-Nystrom Integrator for SSODE of Mechanical Systems

2003 ◽  
Author(s):  
Adrian Sandu ◽  
Dan Negrut ◽  
Corina Sandu ◽  
Edward J. Haug ◽  
Florian A. Potra
Keyword(s):  
Mechanical System ◽  
Time Evolution ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Generalized Coordinates ◽  
System A ◽  
Analytical Derivatives

When performing dynamic analysis of a constrained mechanical system, a set of index 3 Differential-Algebraic Equations (DAE) describes the time evolution of the system. The paper presents a state-space based method for the numerical solution of the resulting DAE. A subset of so called independent generalized coordinates, equal in number to the number of degrees of freedom of the mechanical system, is used to express the time evolution of the mechanical system. The second order statespace ordinary differential equations (SSODE) that describe the time variation of independent coordinates are numerically integrated using a Rosenbrock type formula. For stiff mechanical systems, the proposed algorithm is shown to significantly reduce simulation times when compared to state of the art existent algorithms. The better efficiency is due to the use of an L-stable integrator and a rigorous and general approach to providing analytical derivatives required by it.

The Newmark Integration Method for Simulation of Multibody Systems: Analytical Considerations

2005 ◽  
Author(s):  
B. Gavrea ◽  
D. Negrut ◽  
F. A. Potra
Keyword(s):  
Convergence Analysis ◽  
Size Control ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Generalized Coordinates ◽  
Numerical Integrator ◽  
Space Reduction ◽  
System Solution ◽  
Computationally Intensive

When simulating the behavior of a mechanical system, the time evolution of the generalized coordinates used to represent the configuration of the model is computed as the solution of a combined set of ordinary differential and algebraic equations (DAEs). There are several ways in which the numerical solution of the resulting index 3 DAE problem can be approached. The most well-known and time-honored algorithms are the direct discretization approach, and the state-space reduction approach, respectively. In the latter, the problem is reduced to a minimal set of potentially new generalized coordinates in which the problem assumes the form of a pure second order set of Ordinary Differential Equations (ODE). This approach is very accurate, but computationally intensive, especially when dealing with large mechanical systems that contain flexible parts, stiff components, and contact/impact. The direct discretization approach is less but nevertheless sufficiently accurate yet significantly faster, and it is the approach that is considered in this paper. In the context of direct discretization methods, approaches based on the Backward Differentiation Formulas (BDF) have been the traditional choice for more than 20 years. This paper proposes a new approach in which BDF methods are replaced by the Newmark formulas. Local convergence analysis is carried out for the proposed method, and step-size control, error estimation, and nonlinear system solution related issues are discussed in detail. A series of two simple models are used to validate the method. The global convergence analysis and a computational-efficiency comparison with the most widely used numerical integrator available in the MSC.ADAMS commercial simulation package are forthcoming. The new method has been implemented successfully for industrial strength Dynamic Analysis simulations in the 2005 version of the MSC.ADAMS software and used very effectively for the simulation of systems with more than 15,000 differential-algebraic equations.

scholarly journals Regularization of Operator DAEs

2015 ◽  
Author(s):  
Robert Altmann
Keyword(s):  
General Framework ◽  
Stokes Equations ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Navier Stokes Equations ◽  
First Order ◽  
Index 2 ◽  
Differentiation Index

A general framework for the regularization of constrained PDEs, also called operator differential-algebraic equations (DAEs), is presented. For this, we consider semi-explicit systems of first order which includes the Navier-Stokes equations. The proposed reformulation is consistent in the sense that the solution of the PDE remains untouched. However, one can observe improved numerical properties in terms of the sensitivity to perturbations and the fact that a spatial discretization leads to a DAE of lower index, i.e., of differentiation index $1$ instead of differentiation index 2.

State-Space Based Implicit Integration of the Differential-Algebraic Equations of Multibody Dynamics

1999 ◽  
Author(s):  
Dan Negrut ◽  
Edward J. Haug
Keyword(s):  
State Space ◽  
Mechanical System ◽  
Multibody Dynamics ◽  
Reference Frames ◽  
Differential Algebraic Equations ◽  
The State ◽  
Algebraic Equations ◽  
Implicit Integration ◽  
Euler Parameters ◽  
Generalized Coordinates

Abstract Three methods for the state-space based implicit integration of differential-algebraic equations of multibody dynamics are summarized and numerically compared. In the state-space approach, the time evolution of a mechanical system is characterized using a number of generalized coordinates equal with the number of degrees of freedom of the system. In this paper these independent generalized coordinates are a subset of the Cartesian position coordinates and orientation Euler parameters of body centroidal reference frames. Depending on the method, the independent generalized coordinates are implicitly integrated and dependent quantities (including Lagrange multipliers) are determined to satisfy constraint equations at position, velocity, and acceleration levels. Five computational algorithms based on the proposed methods are used to simulate the motion of a stiff 14-body vehicle model. Results show that the proposed methods deal effectively with challenges posed by stiff mechanical system simulation. A comparison with a state-space based explicit algorithm for the simulation of the same model indicates a speed-up of approximately two orders of magnitude.

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