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 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.

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.

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.

Dynamics of Flexible Body Negotiating a Curve

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Huailong Shi ◽  
Liang Wang ◽  
Ahmed A. Shabana
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Flexible Body ◽  
Motion Trajectory ◽  
Centrifugal Forces ◽  
Coriolis Forces ◽  
Body Dynamics ◽  
Body Reference

When a rigid body negotiates a curve, the centrifugal force takes a simple form which is function of the body mass, forward velocity, and the radius of curvature of the curve. In this simple case of rigid body dynamics, curve negotiation does not lead to Coriolis forces. In the case of a flexible body negotiating a curve, on the other hand, the inertia of the body becomes function of the deformation, curve negotiations lead to Coriolis forces, and the expression for the deformation-dependent centrifugal forces becomes more complex. In this paper, the nonlinear constrained dynamic equations of motion of a flexible body negotiating a circular curve are used to develop a systematic procedure for the calculation of the centrifugal forces during curve negotiations. The floating frame of reference (FFR) formulation is used to describe the body deformation and define the nonlinear centrifugal and Coriolis forces. The algebraic constraint equations which define the motion trajectory along the curve are formulated in terms of the body reference and elastic coordinates. It is shown in this paper how these algebraic motion trajectory constraint equations can be used to define the constraint forces that lead to a systematic definition of the resultant centrifugal force in the case of curve negotiations. The embedding technique is used to eliminate the dependent variables and define the equations of motion in terms of the system degrees of freedom. As demonstrated in this paper, the motion trajectory constraints lead to constant generalized forces associated with the elastic coordinates, and as a consequence, the elastic velocities and accelerations approach zero in the steady state. It is also shown that if the motion trajectory constraints are imposed on the coordinates of a flexible body reference that satisfies the mean-axis conditions, the centrifugal forces take the same form as in the case of rigid body dynamics. The resulting flexible body dynamic equations can be solved numerically in order to obtain the body coordinates and evaluate numerically the constraint and centrifugal forces. The results obtained for a flexible body negotiating a circular curve are compared with the results obtained for the rigid body in order to have a better understanding of the effect of the deformation on the centrifugal forces and the overall dynamics of the body.

scholarly journals Multi-Flexible Body Dynamics Capturing Motion-Induced Stiffness

1991 ◽  
Vol 58 (3) ◽  
pp. 766-775 ◽  
Author(s):  
A. K. Banerjee ◽  
M. E. Lemak
Keyword(s):  
Equations Of Motion ◽  
The Body ◽  
Flexible Body ◽  
Dynamical Equations ◽  
Large Rotation ◽  
Geometric Stiffness ◽  
Body Dynamics ◽  
Flexible Body Dynamics ◽  
Inertia Forces ◽  
Modal Coordinates

This paper presents a multi-flexible-body dynamics formulation incorporating a recently developed theory for capturing motion-induced stiffness for an arbitrary structure undergoing large rotation and translation accompanied by small vibrations. In essence, the method consists of correcting dynamical equations for an arbitrary flexible body, unavoidably linearized prematurely in modal coordinates, with generalized active forces due to geometric stiffness corresponding to a system of 12 inertia forces and 9 inertia couples distributed over the body. Computation of geometric stiffness in this way does not require any iterative update. Equations of motion are derived by means of Kane’s method. A treatment is given for handling prescribed motions and calculating interaction forces. Results of simulations of motions of three flexible spacecraft, involving stiffening during spinup motion, dynamic buckling, and a slewing maneuver, demonstrate the validity and generality of the theory.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

Numerical modelling of swirling momentumless turbulent wake dynamics

2018 ◽  
pp. 37-48
Author(s):  
Андрей Геннадьевич Деменков ◽  
Геннадий Георгиевич Черных
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Mathematical Models ◽  
Equations Of Motion ◽  
Turbulent Wake ◽  
The Body ◽  
Jet Flows ◽  
Reynolds Stresses ◽  
Bodies Of Revolution ◽  
Averaged Equations

С применением математической модели, включающей осредненные уравнения движения и дифференциальные уравнения переноса нормальных рейнольдсовых напряжений и скорости диссипации, выполнено численное моделирование эволюции безымпульсного закрученного турбулентного следа с ненулевым моментом количества движения за телом вращения. Получено, что начиная с расстояний порядка 1000 диаметров от тела течение становится автомодельным. На основе анализа результатов численных экспериментов построены упрощенные математические модели дальнего следа. Swirling turbulent jet flows are of interest in connection with the design and development of various energy and chemical-technological devices as well as both study of flow around bodies and solving problems of environmental hydrodynamics, etc. An interesting example of such a flow is a swirling turbulent wake behind bodies of revolution. Analysis of the known works on the numerical simulation of swirling turbulent wakes behind bodies of revolution indicates lack of knowledge on the dynamics of the momentumless swirling turbulent wake. A special case of the motion of a body with a propulsor whose thrust compensates the swirl is studied, but there is a nonzero integral swirl in the flow. In previous works with the participation of the authors, a numerical simulation of the initial stage of the evolution of a swirling momentumless turbulent wake based on a hierarchy of second-order mathematical models was performed. It is shown that a satisfactory agreement of the results of calculations with the available experimental data is possible only with the use of a mathematical model that includes the averaged equations of motion and differential equations for the transfer of normal Reynolds stresses along the rate of dissipation. In the present work, based on the above mentioned mathematical model, a numerical simulation of the evolution of a far momentumless swirling turbulent wake with a nonzero angular momentum behind the body of revolution is performed. It is shown that starting from distances of the order of 1000 diameters from the body the flow becomes self-similar. Based on the analysis of the results of numerical experiments, simplified mathematical models of the far wake are constructed. The authors dedicate this work to the blessed memory of Vladimir Alekseevich Kostomakha.

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.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

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.

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