Logarithmic Complexity Sensitivity Analysis of Flexible Multibody Systems

2009 ◽  
Author(s):  
Kishor D. Bhalerao ◽  
Mohammad Poursina ◽  
Kurt S. Anderson
Keyword(s):  
Sensitivity Analysis ◽  
Data Storage ◽  
Multibody Systems ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Flexible Multibody Systems ◽  
Divide And Conquer ◽  
Dynamics Simulation ◽  
Modal Shape ◽  
Flexible Multibody

This paper presents a recursive direct differentiation method for sensitivity analysis of flexible multibody systems. Large rotations and translations in the system are modeled as rigid body degrees of freedom while the deformation field within each body is approximated by superposition of modal shape functions. The equations of motion for the flexible members are differentiated at body level and the sensitivity information is generated via a recursive divide and conquer scheme. The number of differentiations required in this method is minimal. The method works concurrently with the forward dynamics simulation of the system and requires minimum data storage. The use of divide and conquer framework makes the method linear and logarithmic in complexity for serial and parallel implementation, respectively, and ideally suited for general topologies. The method is applied to a flexible two arm robotic manipulator to calculate sensitivity information and the results are compared with the finite difference approach.

Efficient Operational Space Sensitivity Analysis of Dynamic Multibody Systems

2011 ◽  
Author(s):  
Rudranarayan M. Mukherjee
Keyword(s):  
Sensitivity Analysis ◽  
Data Storage ◽  
Multibody Systems ◽  
Parallel Implementation ◽  
Divide And Conquer ◽  
Parametric Perturbation ◽  
Divide And Conquer Algorithm ◽  
Operational Space ◽  
Tree Topologies ◽  
Logarithmic Complexity

This paper presents a generalization of the divide and conquer algorithm for sensitivity analysis of dynamic multibody systems based on direct differentiation. While similar sensitivity analysis approach has been demonstrated for multi-rigid and multi-flexible systems in tree topologies and a limited set of kinematically closed loop topologies, this paper presents the generalization of these approaches to systems in generalized topologies including many coupled kinematically closed loops. This generalization retains the efficient complexity of the underlying formulations i.e. linear and logarithmic complexity in serial and parallel implementation. Other than the computational efficiency, the advantages of this method include concurrent sensitivity analysis with forward dynamics, no numerical artifacts arising from parametric perturbation and significantly reduced data storage compared to traditional methods. An interesting application of this work in control of multibody systems is discussed.

The Motion Formalism for Flexible Multibody Systems

2021 ◽  
Author(s):  
Olivier Bauchau ◽  
Valentin Sonneville
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Solution Process ◽  
Flexible Multibody Systems ◽  
Structural Elements ◽  
Governing Equations ◽  
Euclidean Group ◽  
Reduced Order ◽  
Flexible Multibody ◽  
Element Approach

Abstract This paper describes a finite element approach to the analysis of flexible multibody systems. It is based on the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) recognizes that these are members of the Special Euclidean group thereby coupling their displacement and rotation components, and (3) resolves all tensors components in local frames. The goal of this review paper is not to provide an in-depth derivation of all the elements found in typical multibody codes but rather to demonstrate how the motion formalism (1) provides a theoretical framework that unifies the formulation of all structural elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, and (4) prevents the occurrence of singularities.

Dynamics Study and Sensitivity Analysis of Flexible Multibody Systems With Interval Parameters

2016 ◽  
Author(s):  
Wang Zhe ◽  
Qiang Tian ◽  
Hiayan Hu
Keyword(s):  
Sensitivity Analysis ◽  
Surrogate Model ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Flexible Multibody Systems ◽  
Computational Time ◽  
Interval Parameters ◽  
Computation Efficiency ◽  
Sensitivity Equations ◽  
Flexible Multibody

The dynamics of flexible multibody systems with interval parameters is studied based on a non-intrusive computation methodology. The Absolute Nodal Coordinate Formulation (ANCF) is used to model the rigid-flexible multibody system, including the finite elements of the ANCF and the ANCF Reference Nodes (ANCF-RNs). The Chebyshev sampling methods including Chebyshev tensor product (CTP) sampling method and Chebyshev collocation method (CCM), are utilized to generate the Chebyshev surrogate model for Interval Differential Algebraic Equations (IDAEs). For purpose of preventing the interval explosion problem and maintaining computation efficiency, the interval bounds of the IDAEs are determined by scanning the deduced Chebyshev surrogate model. To further improve the computation efficiency, OpenMP directives are also used to parallelize the solving process of the Differential Algebraic Equations (DAEs) by fixing the uncertain interval parameter at the given sampling points. The sensitivity analysis of flexible multibody systems with interval parameters is initially performed by using the direct differentiation method. The direct differentiation method differentiates the dynamic equations with respect to the design variable, which yields the system sensitivity equations governed by DAEs. The generalized alpha method is introduced to integrate the sensitivity DAEs. The sensitivity equations of flexible multibody systems with interval parameters are also described by the IDAEs. Based on the continuum mechanics, the computational efficient analytical formulations for the derivative items of the system sensitivity equations are deduced. Three examples are studied to validate the proposed methodology, including the complicated spatial rigid-flexible multibody systems with a large number of uncertain interval parameters, the flexible system with uncertain interval clearance size joint, and the first order sensitivity analysis of flexible multibody systems with interval parameters. Firstly, the dynamics analysis of a six-arm space robot with six interval parameters is performed. For this case study, the interval dynamics cannot be obtained by directly scanning the IDAEs because extremely huge sets of DAEs with deterministic samples have to be solved. The estimated total computational time for solving the scanned IDAEs will be 1850 days! However, the computational time for solving the scanned Chebyshev surrogate model is 9796.97 seconds. It shows the effectiveness of the proposed computation methodology. Then, the nonlinear dynamics of a planar slider-crank mechanism with uncertain interval clearance size joint is studied in this work. The kinetics model of the revolute clearance joints is formulated under the ANCF-RN framework. Moreover, the influence of the LuGre and the modified Coulomb’s friction force models on the system’s dynamic response is investigated. By analyzing the bounds of dynamic response, the bifurcation diagrams are observed. It must be highlighted that with increasing the size of clearance, it does not automatically lead to unstable behaviors. Finally, the first order sensitivity analysis of flexible multibody systems with interval parameters is also studied in this work. The third one of a flexible mechanism with interval parameters is used to perform the sensitivity analysis.

Discrete Adjoint Method for the Sensitivity Analysis of Flexible Multibody Systems

2019 ◽  
Vol 14 (2) ◽  
Author(s):  
Alfonso Callejo ◽  
Valentin Sonneville ◽  
Olivier A. Bauchau
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Adjoint Method ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Flexible Multibody Systems ◽  
Design Parameters ◽  
Integration Scheme ◽  
Discrete Version ◽  
Flexible Multibody

The gradient-based design optimization of mechanical systems requires robust and efficient sensitivity analysis tools. The adjoint method is regarded as the most efficient semi-analytical method to evaluate sensitivity derivatives for problems involving numerous design parameters and relatively few objective functions. This paper presents a discrete version of the adjoint method based on the generalized-alpha time integration scheme, which is applied to the dynamic simulation of flexible multibody systems. Rather than using an ad hoc backward integration solver, the proposed approach leads to a straightforward algebraic procedure that provides design sensitivities evaluated to machine accuracy. The approach is based on an intrinsic representation of motion that does not require a global parameterization of rotation. Design parameters associated with rigid bodies, kinematic joints, and beam sectional properties are considered. Rigid and flexible mechanical systems are investigated to validate the proposed approach and demonstrate its accuracy, efficiency, and robustness.

scholarly journals Formulation of Shell Elements Based on the Motion Formalism

2021 ◽  
Vol 2 (4) ◽  
pp. 1009-1036
Author(s):  
Olivier Bauchau ◽  
Valentin Sonneville
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Solution Process ◽  
Flexible Multibody Systems ◽  
Governing Equations ◽  
Euclidean Group ◽  
Shell Elements ◽  
Reduced Order ◽  
Flexible Multibody ◽  
Plates And Shells

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. The developments are set within the framework of the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) couples their displacement and rotation components by recognizing that configuration and motion are members of the Special Euclidean group, and (3) resolves all tensors components in local frames. The formulation based on the motion formalism (1) provides a theoretical framework that streamlines the formulation of shell elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, (4) circumvents the shear locking phenomenon that plagues shell formulations based on classical kinematic descriptions, and (5) prevents the occurrence of singularities in the treatment of finite rotation. Numerical examples are presented to illustrate the advantageous features of the proposed formulation.

Semi-Symbolical Equations of Motion for Flexible Multibody Systems

1993 ◽  
Author(s):  
Frank Melzer
Keyword(s):  
Data Model ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
General Purpose ◽  
Flexible Multibody Systems ◽  
Dynamical Analysis ◽  
Lightweight Structures ◽  
Elastic Bodies ◽  
Flexible Multibody ◽  
Increasing Demand

Abstract The need for computer aided engineering in the analysis of machines and mechanisms led to a wide variety of general purpose programs for the dynamical analysis of multibody systems. In the past few years the incorporation of flexible bodies in this methodology has evolved to one of the major research topics in the field of multibody dynamics, due to the use of more lightweight structures and an increasing demand for high-precision mechanisms such as robots. This paper presents a formalism for flexible multibody systems based on a minimum set of generalized coordinates and symbolic computation. A standardized object-oriented data model is used for the system matrices, describing the elastodynamic behaviour of the flexible body. Consequently, the equations of motion are derived in a form independent of the chosen modelling technique for the elastic bodies.

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