A two-loop sparse matrix numerical integration procedure for the solution of differential/algebraic equations: Application to multibody systems

2009 ◽  
Vol 327 (3-5) ◽  
pp. 557-563 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Bassam A. Hussein
Keyword(s):  
Numerical Integration ◽  
Multibody Systems ◽  
Sparse Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Integration Procedure ◽  
Numerical Integration Procedure

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.

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.

Control Constraint Realization for Multibody Systems

2009 ◽  
Author(s):  
Alessandro Fumagalli ◽  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Paolo Mantegazza
Keyword(s):  
Linear Systems ◽  
Multibody Systems ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Practical Observation ◽  
Alternative Condition ◽  
Deformable Bodies

This paper discusses the problem of control constraint realization applied to generic under-actuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that Differential-Algebraic Equations (DAE) need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

The use of gas-liquid chromatography to determine activity coefficients and second virial coefficients of mixtures I. Theory and verification of method of data analysis

1966 ◽  
Vol 295 (1442) ◽  
pp. 259-270 ◽  
Keyword(s):  
Numerical Integration ◽  
Retention Volume ◽  
Simple Extension ◽  
Gas Liquid Chromatography ◽  
Local Pressure ◽  
Integration Procedure ◽  
Carrier Gas ◽  
Wide Range ◽  
Analytical Integration ◽  
Numerical Integration Procedure

This paper reformulates the differential equation describing the local elution rate in a g. l. c. column in terms of the local pressure and the carrier gas outlet flow rate. Analytical integration for an ideal carrier gas suggests an accurate method for extrapolating a function of the retention volume linearly to zero pressure, where the intercept V ° N is simply related to the thermodynamic activity coefficient of the solute (1) in the stationary liquid (3) and the gradient β gives B 12 for the mixture solute + carrier gas (2). We argue that a simple extension of the method should apply also, with fair accuracy, to a non-ideal carrier gas. We support this argument with data obtained by a numerical integration procedure which gives retention volume in terms of specified V ° N and B for a range of inlet and outlet pressures. The reliability of the numerical integration procedure is established by comparing results for the ideal gas case with the results of analytical integration. The retention volumes obtained by numerical integration for a non-ideal carrier gas are then treated as ‘experimental’ observations, using in addition to our extrapolation procedure, two previously published procedures. Our procedures are consistently more successful than the others and recover accurately the V ° N originally specified over a wide range of flow conditions, even when the carrier gas shows large deviations from ideality. In the case of β , our method is significantly in error only when the carrier gas deviates largely from ideality in a low pressure column with large pressure drop. A simple refinement of our method is satisfactory for even this case.

On the Numerical Scheme to Solve a Realistic Chemical Vapor Infiltration Reactor Model

2007 ◽  
Author(s):  
John K. Kamel ◽  
Samuel Paolucci
Keyword(s):  
Mathematical Model ◽  
Numerical Integration ◽  
Chemical Vapor ◽  
Differential Algebraic Equations ◽  
Chemical Vapor Infiltration ◽  
Integration Scheme ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Reactor Model ◽  
Vapor Infiltration

We describe the general mathematical model as well as the numerical integration procedure arising in modeling a realistic chemical vapor infiltration process. The numerical solution of the model ultimately leads to the solution of a large system of stiff differential algebraic equations. An operator splitting algorithm is employed to overcome the stiffness associated with chemical reactions, whereas a projection method is employed to overcome the difficulty arising from having to solve a large coupled system for the velocity and pressure fields. The resulting mathematical model and the numerical integration scheme are used to explore temperature, velocity, and concentration fields inside a chemical vapor infiltration reactor used in the manufacturing of aircraft brakes.

The Dynamic Analysis of Multibody Systems with Uncertain Parameters Using Interval Method

2012 ◽  
Vol 152-154 ◽  
pp. 1555-1561 ◽  
Author(s):  
Jing Lai Wu ◽  
Yun Qing Zhang
Keyword(s):  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Uncertain Parameters ◽  
Algebraic Equations ◽  
Scanning Method ◽  
Inclusion Function ◽  
Bounded Interval ◽  
Interval Variables ◽  
Multibody Dynamic ◽  
Computational Aspects

The theoretical and computational aspects of interval methodology based on Chebyshev polynomials for modeling multibody dynamic systems in the presence of parametric uncertainties are proposed, where the uncertain parameters are modeled by uncertain-but-bounded interval variables which only need the bounds of uncertain parameters, not necessarily knowing the probabilistic distribution. The Chebyshev inclusion function which employs the truncated Chevbyshev series expansion to approximate the original function is proposed. Based on Chebyshev inclusion function, the algorithm for solving the nonlinear equations with interval parameters is proposed. Combining the HHT-I3 method, this algorithm is used to calculate the multibody systems dynamic response which is governed by differential algebraic equations (DAEs). A numerical example that is a slider-crank with uncertain parameters is presented, which shows that the novel methodology can control the overestimation effectively and is computationally faster than the scanning method.

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