Separated-Form Equations of Motion of Controlled Flexible Multibody Systems

1994 ◽  
Vol 116 (4) ◽  
pp. 702-712 ◽  
Author(s):  
Junghsen Lieh
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Second Order ◽  
Active Suspensions ◽  
Physical Term ◽  
Moving Base ◽  
Matrix Expansion ◽  
Generalized Force ◽  
Coordinate Partitioning

This paper introduces a method leading to separated-form formulation of dynamic equations of multibody systems subject to control. The algorithm is derived from the virtual work principle and includes the moving base effects. The elastic members are treated as Euler-Bernoulli beams. Equations of motion are expanded using generalized coordinate partitioning and a Jacobian matrix expansion. The formulation of each physical term is separated, i.e., the inertia matrix, nonlinear coupling vector, generalized force vector and base motion-induced terms are established individually. The formulation is implemented on a workstation using MAPLE. Nonlinear and linearized equations with control are generated in FORTRAN format. The control design adopts second-order models directly. Several examples including a spin-up cantilever beam, an elastic vehicle with active suspensions and an elastic slider-crank mechanism are given. Numerical results for nonlinear and linear spin-up beam models are provided. Simulation for the active vehicle model using second-order control theory is presented.

A Closed-Form Formalism for Controlled and Hybrid Rigid/Elastic Multibody Systems: Part I — Formulation

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz Haque
Keyword(s):  
Closed Form ◽  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
New Formalism ◽  
Assumed Mode Method ◽  
Elastic Multibody Systems ◽  
Moving Base ◽  
Generalized Force

Abstract A new formalism leading to closed-form formulation of equations for controlled elastic multibody systems is presented. The method is derived from the virtual work principle and includes the effects of a moving base and rigid body dynamics. The elastic members are treated as Euler-Bernoulli beams and the assumed-mode method is adopted. The equations of motion are expanded in a closed form with a minimum set of variables using the generalized coordinate partitioning and a Jacobian matrix expansion. The inertia matrix, nonlinear coupling vector, generalized force vector and other terms containing the velocity and acceleration effects of a moving base are formulated separately. The formalism facilitates matrix computations and is very suitable for symbolic processing. The method is very systematic and general and can be applied to a multibody system subject to control and constraint conditions. Derivation of the formalism is presented in part I of the article, and symbolic implementation and application of the formalism to various elastic mechanical systems are presented in part II.

Modeling and Dynamic Analysis of a Slider-Crank Mechanism With Flexible Coupler and Joint

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz Haque
Keyword(s):  
Dynamic Analysis ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Joint Stiffness ◽  
Mechanism Model ◽  
Double Frequency ◽  
Varying Coefficients ◽  
Matrix Expansion ◽  
Time Varying Coefficients ◽  
Crank Mechanism

Abstract Modeling and dynamic analysis of a slider-crank mechanism with flexible joint and coupler is presented. The equations of motion of the mechanism model are formulated using a virtual work multibody formalism and cast in terms of a minimum set of generalized coordinates through a Jacobian matrix expansion. Numerical results show the influence of time-varying coefficients on the mechanism dynamic behavior due to a repeated task. The results illustrate that the joint motion and coupler deformation are highly coupled. The joint response is dominated by double frequency of input, however, the coupler deformation is influenced by the same frequency as that of excitation. Increase in joint stiffness tends to decrease the variations in coupler deformation.

On the Use of Lie Group Time Integrators in Multibody Dynamics

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Olivier Brüls ◽  
Alberto Cardona
Keyword(s):  
Lie Group ◽  
Dynamic Systems ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Second Order ◽  
Order Accuracy ◽  
Time Integrators ◽  
Group Time ◽  
Constrained Equations ◽  
New Algorithms

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-α method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

On the Use of Virtual Work in a Graph-Theoretic Formulation for Multibody Dynamics

1997 ◽  
Author(s):  
Pengfei Shi ◽  
John McPhee
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Flexible Multibody Systems ◽  
Theoretic Approach ◽  
Computer Implementation ◽  
Graph Theoretic ◽  
Graph Theoretic Approach ◽  
Flexible Multibody ◽  
New Formulation

Abstract In this paper, graph-theoretic and virtual work methods are combined in a new formulation of the equations of motion for rigid and flexible multibody systems. In addition to extending the theory for existing graph-theoretic approaches, this new formulation offers two distinct improvements. First, the set of differential-algebraic dynamic equations are smaller in number than those obtained using conventional formulations. Secondly, the equations of motion for rigid and flexible multibody systems can be generated using a consistent graph-theoretic approach, thereby leading to an efficient and modular computer implementation.

A Computational Method for Formulation and Solution of Dynamical Equations for Complex Mechanisms and Multibody Systems

2017 ◽  
Author(s):  
Kristopher Wehage ◽  
Bahram Ravani
Keyword(s):  
Multibody Systems ◽  
Sparse Matrix ◽  
Equations Of Motion ◽  
Schur Complement ◽  
Computational Method ◽  
Dynamical Equations ◽  
Multibody Dynamic ◽  
Sparse Matrix Techniques ◽  
Complex Mechanisms ◽  
Coordinate Partitioning

This paper presents a computational method for formulating and solving the dynamical equations of motion for complex mechanisms and multibody systems. The equations of motion are formulated in a preconditioned form using kinematic substructuring with a heuristic application of Generalized Coordinate Partitioning (GCP). This results in an optimal split of dependent and independent variables during run time. It also allows reliable handling of end-of-stroke conditions and bifurcations in mechanisms, thereby facilitating dynamic simulation of paradoxical linkages such as Bricard’s mechanism that has been known to cause problems with some multibody dynamic codes. The new Preconditioned Equations of Motion are then solved using a recursive formulation of the Schur Complement Method combined with Sparse Matrix Techniques. In this fashion the Preconditioned Equations of Motion are recursively uncoupled and solved one kinematic substructure at a time. The results are demonstrated using examples.

Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces

1999 ◽  
Vol 123 (2) ◽  
pp. 272-281 ◽  
Author(s):  
B. Fox ◽  
L. S. Jennings ◽  
A. Y. Zomaya
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Lagrange Equations ◽  
Constraint Equations

The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

Symbolic Closed-Form Modeling and Linearization of Multibody Systems Subject to Control

1991 ◽  
Vol 113 (2) ◽  
pp. 124-132 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz-ul Haque
Keyword(s):  
Closed Form ◽  
Multibody Systems ◽  
Space Form ◽  
Virtual Work ◽  
Dynamic Equations ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Active Suspensions ◽  
Linear Dynamic ◽  
Curved Track

Symbolic closed-form equation formulation and linearization for constrained multibody systems subject to control are presented. The formulation is based on the principle of virtual work. The algorithm is recursive, automatically eliminates the constraint forces and redundant coordinates, and generates the nonlinear or linear dynamic equations in closed-form. It is derived with respect to principal body coordinates and a moving reference frame that allows one to generate the dynamic equations for multibody systems moving along curved track or road. The output equations may be either in syntactically correct FORTRAN form or in the form as derived by hand. A procedure that simplifies the trigonometric expressions, linearizes the geometric nonlinearities, and converts the linearized equations in state-space form is included. Several examples have been used to validate the procedure. Included is a simulation using a seven-DOF automobile ride model with active suspensions.

Direct and Adjoint Sensitivity Analysis of Multibody Systems Using Maggi’s Equations

2013 ◽  
Author(s):  
Daniel Dopico ◽  
Yitao Zhu ◽  
Adrian Sandu ◽  
Corina Sandu
Keyword(s):  
Sensitivity Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Third Party ◽  
Adjoint Sensitivity Analysis ◽  
Adjoint Sensitivity ◽  
Dynamics Of Multibody Systems ◽  
Dynamical Optimization ◽  
Analytical Expressions ◽  
Coordinate Partitioning

The importance of the sensitivity analysis of multibody systems for several applications is well known, concretely design optimization based on the dynamics of multibody systems usually requires the sensitivity analysis of the equations of motion. A broad range of methods for the dynamics of multibody systems include the state space formulations based on Maggis equations, nullspace methods or coordinate partitioning. Dynamic sensitivities, when needed, are often calculated by means of finite differences but, depending of the number of parameters involved, this procedure can be very demanding in terms of CPU time and the accuracy obtained can be very poor in many cases. In this paper, several ways to perform the sensitivity analysis are explored and analytical expressions for the direct and adjoint sensitivity analysis of multibody systems are presented, all of them based on Maggi’s formulations. Moreover, two different approaches to the adjoint sensitivity analysis of multibody systems are presented. Although particularized to one formulation, the general expressions provided in the paper, are intended to be easily generalized and applied to any other formulation that can be expressed as an ODE-like system of equations, including penalty formulations. Besides, to check the validity and correctness of the proposed equations, the solutions of all the methods proposed are compared: 1) between them, 2) with the third party code FATODE and 3) with the numerical solution using real and complex perturbations. Finally, all the techniques proposed are applied to the dynamical optimization of a multibody system.

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

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