Discontinuous Galerkin Method and Dual-SLERP for Time Integration of Flexible Multibody Dynamics

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Discontinuous Galerkin ◽  
Multibody Dynamics ◽  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Differential Algebraic Equations ◽  
Body Motion ◽  
Motion Field ◽  
Algebraic Equations ◽  
Time Step ◽  
Flexible Multibody

Abstract A novel time-discontinuous Galerkin (DG) method is introduced for the time integration of the differential-algebraic equations governing the dynamic response of flexible multibody systems. In contrast to traditional Galerkin methods, the rigid-body motion field is interpolated using the dual spherical linear scheme. Furthermore, the jumps inherent to time-DG methods are expressed in terms of a parameterization of the relative motion from one time-step to the next. The proposed scheme is third-order accurate for initial value problems of both rigid and flexible multibody dynamics.

Parallel Time-Integration of Flexible Multibody Dynamics Based on Newton-Waveform Method

2017 ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Multibody Dynamics ◽  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Small Scale ◽  
Time Step ◽  
Current Time ◽  
Parallel Time ◽  
Previous Time ◽  
Flexible Multibody ◽  
Multi Body

Traditionally, the time integration algorithms for multibody dynamics are in sequential. The predictions of previous time steps are necessary to get the solutions at current time step. This time-marching character impedes the application of parallel processor implementation. In this paper, the idea of computing a number of time steps concurrently is applied to flexible multi-body dynamics, which makes parallel time-integration possible. In the present method, the solution at the current time step is computed before accurate values at previous time step are available. This method is suitable for small-scale parallel analysis of flexible multibody systems.

Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Peter Betsch ◽  
Christian Hesch ◽  
Nicolas Sänger ◽  
Stefan Uhlar
Keyword(s):  
Multibody Dynamics ◽  
Flexible Multibody Dynamics ◽  
Discrete Systems ◽  
Differential Algebraic Equations ◽  
Nonlinear Finite Element ◽  
Variational Integrators ◽  
Algebraic Equations ◽  
Variational Integrator ◽  
Nonlinear Finite Element Methods ◽  
Flexible Multibody

This work contains a comparison between variational integrators and energy-momentum schemes for flexible multibody dynamics. In this connection, a specific “rotationless” formulation of flexible multibody dynamics is employed. Flexible components such as continuum bodies and geometrically exact beams and shells are discretized in space by using nonlinear finite element methods. The motion of the resulting discrete systems are governed by a uniform set of differential-algebraic equations (DAEs). This makes possible the application and comparison of previously developed structure-preserving methods for the numerical integration of the DAEs. In particular, we apply a specific variational integrator and an energy-momentum scheme. The performance of both integrators is assessed in the context of three representative numerical examples.

Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates

1993 ◽  
Vol 115 (2) ◽  
pp. 294-299 ◽  
Author(s):  
N. Vukasovic ◽  
J. T. Celigu¨eta ◽  
J. Garci´a de Jalo´n ◽  
E. Bayo
Keyword(s):  
Multibody Dynamics ◽  
Reference Frames ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Body Motion ◽  
Cartesian System ◽  
Flexible Multibody ◽  
Local Reference ◽  
Rigid Multibody

In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

Numerical Solution and Experiment Researches on Flexible Multibody Dynamics

1994 ◽  
Author(s):  
Liu Hongzhao ◽  
Cao Weiqing
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Multibody Dynamics ◽  
Lagrange Multipliers ◽  
Flexible Multibody Dynamics ◽  
Differential Algebraic Equations ◽  
Measuring System ◽  
Linkage Test ◽  
Test Bed ◽  
Flexible Multibody

Abstract In this paper, a numerical solution method and an experiment research on flexible multibody dynamics are presented. The augment method is employed to transfer the differential-algebraic equations into differential equations, both generalized coordinates and lagrange multipliers are treated as unknowns, and the second-order self-correcting algorithm is introduced to guarantee convergence. The lagrange multipliers are eliminated analytically, and the half implicit Runge-Kutta formula is taken for solving the differential equations. The whole procedure is developed into a FORTRAN program. An elastic linkage test bed and a measuring system are made for verification. The computing results are in agreement with the measuring data.

A TUTORIAL PRESENTATION OF ALTERNATIVE SOLUTIONS TO THE FLEXIBLE BEAM ON RIGID CART PROBLEM

2005 ◽  
Vol 29 (3) ◽  
pp. 357-373 ◽  
Author(s):  
R. G. Langlois ◽  
R. J. Anderson
Keyword(s):  
Rigid Body ◽  
Multibody Dynamics ◽  
Flexible Multibody Dynamics ◽  
Beam Theory ◽  
General Purpose ◽  
Rigid Body Dynamics ◽  
Body Motion ◽  
Flexible Beam ◽  
Euler Beam ◽  
Flexible Multibody

A classical planar problem in forward flexible multibody dynamics is thoroughly investigated. The system consists of a damped flexible beam cantilevered to a rigid translating cart. The problem is solved using three distinctly different conventional approaches presented in roughly the chronological order in which they have been applied to flexible dynamic systems. First, a modal superposition formulation based on Bernoulli-Euler beam theory is developed. Second, an alternative solution is developed drawing exclusively on methods for rigid body dynamics combined with a knowledge of the theoretical modal behaviour of continuous beams. Third, a formulation based on the conventional finite element method using four-degree-of-freedom planar beam elements is adapted to include the rigid body motion of the cart. The relative merits of the three formulations are discussed and numerical simulation results generated using each of the three formulations are compared with each other and with a solution from a general-purpose flexible multibody dynamics formulation that is briefly outlined. The relative accuracy and efficiency of the methods and the challenges associated with generalizing each formulation are discussed.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

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