Parallel Algorithm for Modeling Multi-Rigid Body System Dynamics With Nonholonomic Constraints

2013 ◽  
Author(s):  
Rudranarayan Mukherjee ◽  
Pawel Malczyk
Keyword(s):  
Rigid Body ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Numerical Test ◽  
Nonholonomic Constraints ◽  
Test Case ◽  
Central Idea ◽  
Acceleration Level ◽  
Disassembly Process ◽  
Kinematic Joint

This paper presents a new algorithm for serial or parallel implementation of computer simulations of the dynamics of multi-rigid body systems subject to nonholonomic and holonomic constraints. The algorithm presents an elegant approach for eliminating the nonholonomic constraints explicitly from the equations of motion and implicitly expressing them in terms of nonlinear coupling in the operational inertias of the bodies subject to these constraints. The resulting equations are in the same form as those of a body subject to kinematic joint constraints. This enables the nonholohomic constraints to be seamlessly treated in either a (i) recursive or (ii) hierarchic assembly-disassembly process for solving the equations of motion of generalized multi-rigid body systems in serial or parallel implementations. The algorithm is non-iterative and although the nonholonomic constraints are imposed at the acceleration level, constraint satisfaction is excellent as demonstrated by the numerical test case implemented to verify the algorithm. The paper presents procedures for handling both cases where the nonholonomic constraints are imposed between terminal bodies of a system and the environment as well as when the constraints are imposed between bodies in the interior of the system topology. The algorithm uses a mixed set of coordinates and is built on the central idea of eliminating either constraint loads or relative accelerations from the equations of motion by projecting the equations of motion into the motion subspaces or their orthogonal complements.

Parallel Algorithm for Modeling Constrained Multi-Flexible Body System Dynamics

2013 ◽  
Author(s):  
Rudranarayan Mukherjee ◽  
Jeremy Laflin
Keyword(s):  
Parallel Implementation ◽  
Equations Of Motion ◽  
Small Deformation ◽  
The Body ◽  
Deformation Field ◽  
Flexible Body ◽  
Disassembly Process ◽  
Joint Constraints ◽  
Kinematic Loop ◽  
Kinematic Joint

This paper presents an algorithm for modeling the dynamics of multi-flexible body systems in closed kinematic loop configurations where the component bodies are modeled using the large displacement small deformation formulation. The algorithm uses a hierarchic assembly disassembly process in parallel implementation and a recursive assembly disassembly process in serial implementation to achieve highly efficient simulation turn-around times. The operational inertias arising from the rigid body modes of motion at the joint locations on a component body are modified to account for the nonlinear inertial effects and body forces arising from the body based deformations. Traditional issues, such as motion induced stiffness and temporal invariance of deformation field related inertia terms, are robustly addressed in this algorithm. The algorithm uses a mixed set of coordinates viz. (i) absolute coordinates for expressing the equations of motion of a body fixed reference frame, (ii) relative or internal coordinates to express the kinematic joint constraints and (iii) body fixed coordinates to account for the body’s deformation field. The kinematic joint constraints and the closed loop constraints are treated alike through the formalism of relative coordinates, joint motion spaces and their orthogonal complements. Verification of the algorithm is demonstrated using the planar fourbar mechanism problem that has been traditionally used in literature.

Parallel Algorithm for Constrained Multi-Rigid Body System Dynamics in Generalized Topologies

2013 ◽  
Author(s):  
Rudranarayan Mukherjee
Keyword(s):  
Rigid Body ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Closed Loops ◽  
Efficient Simulation ◽  
Disassembly Process ◽  
Relative Coordinates ◽  
Kinematic Loops ◽  
Absolute Coordinates ◽  
The Individual

This paper presents an algorithm for modeling the dynamics of multi-rigid body systems in generalized topologies including topologies with closed kinematic loops that may or may not be coupled together. The algorithm uses a hierarchic assembly disassembly process in parallel implementation and a recursive assembly disassembly process in serial implementation to achieve highly efficient simulation turn-around times. The kinematic constraints are imposed using the formalism of kinematic joints that are modeled using motion spaces and their orthogonal complements. A mixed set of coordinates are used viz. absolute coordinates to develop the equations of motion and internal or relative coordinates to impose the constraints. The equations of motion are posed in terms of operational inertias to capture the nonlinear coupling between the dynamics of the individual components of the system. Einstein’s notation is used to explain the generality of the approach. Constraint impositions at the acceleration, velocity and configuration levels are discussed. The application of this algorithm in modeling a complex micro robot with multiple coupled closed loops is discussed.

Multigranular Molecular Dynamics Simulations of Polymer Melts Using Multibody Algorithms

2005 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson ◽  
John Ziegler
Keyword(s):  
Molecular Dynamics ◽  
Polymer Melts ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Polymer Melt ◽  
Rigid Bodies ◽  
Computational Time ◽  
Integration Time ◽  
Test Case

In a multigranular approach for modeling molecular dynamics of polymer melts, different sections of the simulation box are modeled at different levels of detail viz. as particles, flexible bodies or rigid bodies. This approach eliminates high frequency localized motion while maintaining low frequency global conformational motion. This allows for longer integration time steps thus decreasing computational time. In this paper, we discuss our efforts to develop a consortium of dynamics algorithms capable of efficiently generating and solving the equations of motion at all three levels of modeling on a common software platform. A bead spring model of the polymer melt moving under the influence of truncated Lennard-Jones potential under periodic boundary conditions is pursued. Implementation issues and results from a test case consisting of 32 polymer chains of 16 beads each are presented. The paper also discusses the parallel implementation of this problem using MPI.

Efficient Impulse Momentum Formulation for Multi-Flexible Body Systems

2009 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson
Keyword(s):  
Equations Of Motion ◽  
Divide And Conquer ◽  
Flexible Bodies ◽  
Disassembly Process ◽  
Closed Loop Systems ◽  
Bilateral Constraints ◽  
Impulsive Loads ◽  
Tree Map ◽  
Unilateral And Bilateral Constraints ◽  
Kinematic Joint

This paper presents the generalization of the divide and conquer impulse momentum formulation to systems with flexible bodies. The approach utilizes a hierarchic assembly-disassembly process by traversing the system topology in a binary tree map to solve for the jumps in the system generalized speeds and the constraint impulsive loads in linear and logarithmic cost in serial and parallel implementations, respectively. The coupling between the unilateral and bilateral constraints is handled efficiently through the use of kinematic joint definitions. The equations of motion for the system are produced in a hierarchic sub-structured form. The generalized impulse momenta equations of flexible bodies are derived using a projection method. The equations are then cast into a format amenable to be incorporated in the basic divide and conquer form. The solution of the equations using the hierarchic assembly disassembly process by using the boundary conditions are discussed for free floating, anchored trees and kinematically closed loop systems.

scholarly journals Identification of coexisting dynamics in boundary layer flows through proper orthogonal decomposition with weighting matrices

Meccanica ◽  
2021 ◽  
Author(s):  
Matteo Dellacasagrande ◽  
Dario Barsi ◽  
Patrizia Bagnerini ◽  
Davide Lengani ◽  
Daniele Simoni
Keyword(s):  
Experimental Data ◽  
Proper Orthogonal Decomposition ◽  
Fourier Transforms ◽  
Free Stream ◽  
Numerical Test ◽  
Separated Flow ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Test Case ◽  
Proper Orthogonal

AbstractA different version of the classic proper orthogonal decomposition (POD) procedure introducing spatial and temporal weighting matrices is proposed. Furthermore, a newly defined non-Euclidean (NE) inner product that retain similarities with the POD is introduced in the paper. The aim is to emphasize fluctuation events localized in spatio-temporal regions with low kinetic energy magnitude, which are not highlighted by the classic POD. The different variants proposed in this work are applied to numerical and experimental data, highlighting analogies and differences with respect to the classic and other normalized variants of POD available in the literature. The numerical test case provides a noise-free environment of the strongly organized vortex shedding behind a cylinder. Conversely, experimental data describing transitional boundary layers are used to test the capability of the procedures in strongly not uniform flows. By-pass and separated flow transition processes developing with high free-stream disturbances have been considered. In both cases streaky structures are expected to interact with other vortical structures (i.e. free-stream vortices in the by-pass case and Kelvin–Helmholtz rolls in the separated type) that carry a significant different amount of energy. Modes obtained by the non-Euclidean POD (NE-POD) procedure (where weighted projections are considered) are shown to better extract low energy events sparse in time and space with respect to modes extracted by other variants. Moreover, NE-POD modes are further decomposed as a combination of Fourier transforms of the related temporal coefficients and the normalized data ensemble to isolate the frequency content of each mode.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

Seismic Analysis of Structures With Coulomb Friction

1983 ◽  
Vol 105 (2) ◽  
pp. 171-178 ◽  
Author(s):  
V. N. Shah ◽  
C. B. Gilmore
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Seismic Event ◽  
Seismic Analysis ◽  
Equations Of Motion ◽  
Relative Displacement ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Superposition Method ◽  
Modal Superposition Method

A modal superposition method for the dynamic analysis of a structure with Coulomb friction is presented. The finite element method is used to derive the equations of motion, and the nonlinearities due to friction are represented by pseudo-force vector. A structure standing freely on the ground may slide during a seismic event. The relative displacement response may be divided into two parts: elastic deformation and rigid body motion. The presence of rigid body motion necessitates the inclusion of the higher modes in the transient analysis. Three single degree-of-freedom problems are solved to verify this method. In a fourth problem, the dynamic response of a platform standing freely on the ground is analyzed during a seismic event.

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