Direct Differentiation Methods for the Design Sensitivity of Multibody Dynamic Systems

1996 ◽  
Author(s):  
Radu Serban ◽  
Jeffrey S. Freeman
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Mechanism Analysis ◽  
Algebraic Equations ◽  
First Order ◽  
Direct Differentiation ◽  
Multibody Dynamic ◽  
Standard Techniques

Abstract Methods for formulating the first-order design sensitivity of multibody systems by direct differentiation are presented. These types of systems, when formulated by Euler-Lagrange techniques, are representable using differential-algebraic equations (DAE). The sensitivity analysis methods presented also result in systems of DAE’s which can be solved using standard techniques. Problems with previous direct differentiation sensitivity analysis derivations are highlighted, since they do not result in valid systems of DAE’s. This is shown using the simple pendulum example, which can be analyzed in both ODE and DAE form. Finally, a slider-crank example is used to show application of the method to mechanism analysis.

Computational Efficiency of First and Second Order Sensitivity Analysis Methods for Dynamic Systems

1993 ◽  
Author(s):  
Qiushi Cao ◽  
Prakash Krishnaswami
Keyword(s):  
Sensitivity Analysis ◽  
Finite Difference ◽  
Dynamic Systems ◽  
Computational Efficiency ◽  
Computing Time ◽  
Design Sensitivity ◽  
Second Order ◽  
First Order ◽  
Comparison Results ◽  
Finite Difference Solution

Abstract First and second order design sensitivity information have become very popular in engineering design, due to the recent development of appropriate methodology and computer technology. This paper presents an empirical study of the computing effort required for computing first and second order design sensitivity information for constrained dynamic mechanical systems and compares the relative efficiency of analytical and finite difference method. Four typical examples have been solved, with the computing time being recorded in each case. The time comparison results indicate that it is generally much more efficient to produce first order design sensitivity information by a direct analytical method than by finite differencing. For second order design sensitivity analysis, the results indicate that a purely analytical solution is usually more efficient than a pure finite difference solution. However, a hybrid scheme appears to be very competitive in terms of computational efficiency.

Design Sensitivity Analysis of Multibody Dynamic Systems for Parallel Processing

1989 ◽  
Author(s):  
T. Tak ◽  
S. S. Kim
Keyword(s):  
Sensitivity Analysis ◽  
Parallel Processing ◽  
Dynamic Systems ◽  
Large Scale ◽  
Design Sensitivity ◽  
Design Sensitivity Analysis ◽  
Parallel Computer ◽  
Sensitivity Equations ◽  
Multibody Dynamic ◽  
High Level

Abstract Design sensitivity analysis of large scale multibody systems is a computationally intensive process, which is well suited for implementation on a parallel computer. This paper presents a parallel processing oriented generalized design sensitivity analysis method for multibody dynamic systems. A direct differentiation method, which is more efficient than an adjoint variable method in a parallel processing environment due to the inherent parallelism, is applied to a recursive formulation for multibody dynamics to set up dynamic sensitivity equations. A high level of parallelism is achieved, exploiting the independence of each set of design sensitivity equations. To verify the formulation for design sensitivity analysis and to demonstrate the speedup on a parallel computer, an example is presented.

Subset Selection Methods for Design Sensitivity Analysis of Constrained Dynamic Systems

1990 ◽  
Author(s):  
Jun-Tien Twu ◽  
Prakash Krishnaswami ◽  
Rajiv Rampalli
Keyword(s):  
Dynamic Systems ◽  
Correct Choice ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Selection Methods ◽  
Constrained Motion ◽  
Differentiation Formula ◽  
Backward Differentiation Formula ◽  
Solution Methods

Abstract In the Lagrangian formulation of the constrained motion of mechanical systems, a system of Differential-Algebraic Equations is generally encountered. The popular Backward Differentiation Formula for the numerical solution of such problems leads to an over-determined system of equations. The correct choice of a proper exactly determined subset can greatly enhance the performance of a solution algorithm. In this paper, we discuss four solution methods with different choices of subsets. Three numerical examples are solved to compare the accuracy and efficiency of these methods.

Direct Sensitivity Analysis of Spatial Multibody Systems With Joint Friction Using Index-1 Formulation

2021 ◽  
Author(s):  
Adwait Verulkar ◽  
Corina Sandu ◽  
Daniel Dopico ◽  
Adrian Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Multibody Systems ◽  
Friction Model ◽  
Design Parameters ◽  
Adjoint Sensitivity ◽  
Joint Friction ◽  
Multibody Dynamic ◽  
Model Dynamics ◽  
Direct Sensitivity

Abstract Sensitivity analysis is one of the most prominent gradient based optimization techniques for mechanical systems. Model sensitivities are the derivatives of the generalized coordinates defining the motion of the system in time with respect to the system design parameters. These sensitivities can be calculated using finite differences, but the accuracy and computational inefficiency of this method limits its use. Hence, the methodologies of direct and adjoint sensitivity analysis have gained prominence. Recent research has presented computationally efficient methodologies for both direct and adjoint sensitivity analysis of complex multibody dynamic systems. The contribution of this article is in the development of the mathematical framework for conducting the direct sensitivity analysis of multibody dynamic systems with joint friction using the index-1 formulation. For modeling friction in multibody systems, the Brown and McPhee friction model has been used. This model incorporates the effects of both static and dynamic friction on the model dynamics. A case study has been conducted on a spatial slider-crank mechanism to illustrate the application of this methodology to real-world systems. Using computer models, with and without joint friction, effect of friction on the dynamics and model sensitivities has been demonstrated. The sensitivities of slider velocity have been computed with respect to the design parameters of crank length, rod length, and the parameters defining the friction model. Due to the highly non-linear nature of friction, the model dynamics are more sensitive during the transition phases, where the friction coefficient changes from static to dynamic and vice versa.

Design Sensitivity Analysis of Nonlinear Multidisciplinary Multibody Systems in the MIXEDMODELS Platform

2007 ◽  
Author(s):  
Shilpa A. Vaze ◽  
Prakash Krishnaswami ◽  
James DeVault
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
The State ◽  
Design Sensitivity Analysis ◽  
Design Parameters ◽  
State Variables ◽  
Sensitivity Coefficients ◽  
Design Variables

Most state-of-the-art multibody systems are multidisciplinary and encompass a wide range of components from various domains such as electrical, mechanical, hydraulic, pneumatic, etc. The design considerations and design parameters of the system can come from any of these domains or from a combination of these domains. In order to perform analytical design sensitivity analysis on a multidisciplinary system (MDS), we first need a uniform modeling approach for this class of systems to obtain a unified mathematical model of the system. Based on this model, we can derive a unified formulation for design sensitivity analysis. In this paper, we present a modeling and design sensitivity formulation for MDS that has been successfully implemented in the MIXEDMODELS (Multidisciplinary Integrated eXtensible Engine for Driving Metamodeling, Optimization and DEsign of Large-scale Systems) platform. MIXEDMODELS is a unified analysis and design tool for MDS that is based on a procedural, symbolic-numeric architecture. This architecture allows any engineer to add components in his/her domain of expertise to the platform in a modular fashion. The symbolic engine in the MIXEDMODELS platform synthesizes the system governing equations as a unified set of non-linear differential-algebraic equations (DAE’s). These equations can then be differentiated with respect to design to obtain an additional set of DAE’s in the sensitivity coefficients of the system state variables with respect to the system’s design variables. This combined set of DAE’s can be solved numerically to obtain the solution for the state variables and state sensitivity coefficients of the system. Finally, knowing the system performance functions, we can calculate the design sensitivity coefficients of these performance functions by using the values of the state variables and state sensitivity coefficients obtained from the DAE’s. In this work we use the direct differentiation approach for sensitivity analysis, as opposed to the adjoint variable approach, for ease in error control and software implementation. The capabilities and performance of the proposed design sensitivity analysis formulation are demonstrated through a numerical example consisting of an AC rectified DC power supply driving a slider crank mechanism. In this case, the performance functions and design variables come from both electrical and mechanical domains. The results obtained were verified by perturbation analysis, and the method was shown to be very accurate and computationally viable.

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