A Logarithmic Complexity Divide and Conquer Algorithm for Flexible Multibody Dynamics

2005 ◽  
Author(s):  
Rudranarayan Mukherjee ◽  
Kurt Anderson
Keyword(s):  
Elastic Deformation ◽  
Degrees Of Freedom ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Flexible Multibody Dynamics ◽  
Divide And Conquer ◽  
Element Analysis ◽  
Large Rotations ◽  
Nonlinear Equations Of Motion

This paper presents an efficient algorithm for parallel implementation of multi-flexible-body dynamics systems simulation and analysis. The effective overall computational cost of the algorithm is logarithmic when implemented with a processor optimal O(n) processors. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to small elastic deformation together with large rotations and translations. The large rotations or translations are modeled as rigid body degree of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modeled through the use of admissible shape functions generated using standard finite element analysis software or otherwise. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative and applicable to generalized multi-flexible chain and free topologies.

A Logarithmic Complexity Divide-and-Conquer Algorithm for Multi-flexible Articulated Body Dynamics

2006 ◽  
Vol 2 (1) ◽  
pp. 10-21 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson
Keyword(s):  
Degrees Of Freedom ◽  
Temporal Integration ◽  
Equations Of Motion ◽  
Turnaround Time ◽  
Divide And Conquer ◽  
Dynamics Simulation ◽  
Flexible Body ◽  
Integration Step ◽  
Large Rotations ◽  
Nonlinear Equations Of Motion

This paper presents an efficient algorithm for the dynamics simulation and analysis of multi-flexible-body systems. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to the limitations of modal superposition, and body substructuring, with arbitrarily large rotations and translations. The large rotations or translations are modelled as rigid body degrees of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modelled through the use of modal coordinates and associated admissible shape functions. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative, and applicable to generalized multi-flexible chain and tree topologies. In its basic form, the algorithm is both time and processor optimal in its treatment of the nb joint variables, providing O(log(nb)) turnaround time per temporal integration step, achieved with O(nb) processors. The actual cost associated with the parallel treatment of the nf flexible degrees of freedom depends on the specific parallel method chosen for dealing with the individual coefficient matrices which are associated locally with each flexible body.

scholarly journals Nonlinear Structural Dynamic Finite Element Analysis Using Ritz Vector Reduced Basis Method

1996 ◽  
Vol 3 (4) ◽  
pp. 259-268 ◽  
Author(s):  
M.S. Yao
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Finite Amplitude ◽  
Basis Vector ◽  
Reduced Basis ◽  
Ritz Vector ◽  
Element Analysis ◽  
Structural Dynamic ◽  
Nonlinear Equations Of Motion

The large number of unknown variables in a finite element idealization for dynamic structural analysis is represented by a very small number of generalized variables, each associating with a generalized Ritz vector known as a basis vector. The large system of equations of motion is thereby reduced to a very small set by this transformation and computational cost of the analysis can be greatly reduced. In this article nonlinear equations of motion and their transformation are formulated in detail. A convenient way of selection of the generalized basis vector and its limitations are described. Some illustrative examples are given to demonstrate the speed and validity of the method. The method, within its limitations, may be applied to dynamic problems where the response is global in nature with finite amplitude.

Divide-and-Conquer-Based Large Deformation Formulations for Multi-Flexible Body Systems

2013 ◽  
Author(s):  
Imad M. Khan ◽  
Kurt S. Anderson
Keyword(s):  
Modeling And Simulation ◽  
Role Model ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Divide And Conquer ◽  
Model Complexity ◽  
Flexible Body ◽  
Elastic Bodies ◽  
Large Rotations ◽  
High Computational Cost

In the dynamic modeling and simulation of multi-flexible-body systems, large deformations and rotations has been a focus of keen interest. The reason is a wide variety of application area where highly elastic components play important role. Model complexity and high computational cost of simulations are the factors that contribute to the difficulty associated with these systems. As such, an efficient algorithm for modeling and simulation of systems undergoing large rotations and large deflections may be of great importance. We investigate the use of absolute nodal coordinate formulation (ANCF) for modeling articulated flexible bodies in a divide-and-conquer (DCA) framework. It is demonstrated that the equations of motion for individual finite elements or elastic bodies, as obtained by the ANCF, may be assembled and solved using a DCA type method. The current discussion is limited to planar problems but may easily be extended to spatial applications. Using numerical examples, we show that the present algorithm provides an efficient and robust method to model multibody systems employing highly elastic bodies.

Toward a Minimal Dynamic Model of a 6-DOF Parallel Robot

1997 ◽  
Author(s):  
L. Beji ◽  
M. Pascal ◽  
P. Joli
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Parallel Robot ◽  
Form Solution ◽  
Lagrangian Formalism ◽  
Inverse Kinematic Problem ◽  
Geometrical Properties

Abstract In this paper, an architecture of a six degrees of freedom (dof) parallel robot and three limbs is described. The robot is called Space Manipulator (SM). In a first step, the inverse kinematic problem for the robot is solved in closed form solution. Further, we need to inverse only a 3 × 3 passive jacobian matrix to solve the direct kinematic problem. In a second step, the dynamic equations are derived by using the Lagrangian formalism where the coordinates are the passive and active joint coordinates. Based on geometrical properties of the robot, the equations of motion are derived in terms of only 9 coordinates related by 3 kinematic constraints. The computational cost of the obtained dynamic model is reduced by using a minimum set of base inertial parameters.

Dynamic Response Analysis of Large Latticed Space Structures

1989 ◽  
Vol 4 (1) ◽  
pp. 25-42 ◽  
Author(s):  
A.R. Kukreti ◽  
N.D. Uchil
Keyword(s):  
Dynamic Response ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Time ◽  
Space Structures ◽  
Response Analysis ◽  
Actual System ◽  
Large Space ◽  
Element Analysis ◽  
Dynamic Response Analysis

In this paper an alternative method for dynamic response analysis of large space structures is presented, for which conventional finite element analysis would require excessive computer storage and computational time. Latticed structures in which the height is very small in comparison to its overall length and width are considered. The method is based on the assumption that the structure can be embedded in its continuum, in which any fiber can translate and rotate without deforming. An appropriate kinematically admissable series function is constructed to descrbe the deformation of the middle plane of this continuum. The unknown coefficients in this function are called the degree-of-freedom of the continuum, which is given the name “super element.” Transformation matrices are developed to express the equations of motion of the actual systems in terms of the degrees-of-freedom of the super element. Thus, by changing the number of terms in the assumed function, the degrees-of-freedom of the super element can be increased or decreased. The super element response results are transformed back to obtain the desired response results of the actual system. The method is demonstrated for a structure woven in the shape of an Archimedian spiral.

Detection of Cracks in Mistuned Bladed Disks Using Reduced-Order Models and Vibration Data

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Chulwoo Jung ◽  
Akira Saito ◽  
Bogdan I. Epureanu
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Savings ◽  
Forced Response ◽  
Response Analysis ◽  
Reduced Order Models ◽  
Reduced Order

A novel methodology to detect the presence of a crack and to predict the nonlinear forced response of mistuned turbine engine rotors with a cracked blade and mistuning is developed. The combined effects of the crack and mistuning are modeled. First, a hybrid-interface method based on component mode synthesis is employed to develop reduced-order models (ROMs) of the tuned system with a cracked blade. Constraint modes are added to model the displacements due to the intermittent contact between the crack surfaces. The degrees of freedom (DOFs) on the crack surfaces are retained as active DOFs so that the physical forces due to the contact/interaction (in the three-dimensional space) can be accurately modeled. Next, the presence of mistuning in the tuned system with a cracked blade is modeled. Component mode mistuning is used to account for mistuning present in the uncracked blades while the cracked blade is considered as a reference (with no mistuning). Next, the resulting (reduced-order) nonlinear equations of motion are solved by applying an alternating frequency/time-domain method. Using these efficient ROMs in a forced response analysis, it is found that the new modeling approach provides significant computational cost savings, while ensuring good accuracy relative to full-order finite element analyses. Furthermore, the effects of the cracked blade on the mistuned system are investigated and used to detect statistically the presence of a crack and to identify which blade of a full bladed disk is cracked. In particular, it is shown that cracks can be distinguished from mistuning.

Divide-and-Conquer Based Adaptive Coarse Grained Simulation of RNA

2010 ◽  
Author(s):  
Mohammad Poursina ◽  
Kishor Bhalerao ◽  
Kurt Anderson
Keyword(s):  
Molecular Modeling ◽  
Degrees Of Freedom ◽  
Traditional Approach ◽  
Equations Of Motion ◽  
Critical Role ◽  
Divide And Conquer ◽  
Coarse Grained ◽  
Biomolecular Systems ◽  
Spatial And Temporal Scales ◽  
O 10

Molecular modeling has gained increasing importance in recent years for predicting important structural properties of large biomolecular systems such as RNA which play a critical role in various biological processes. Given the complexity of biopolymers and their interactions within living organisms, efficient and adaptive multi-scale modeling approaches are necessary if one is to reasonably perform computational studies of interest. These studies nominally involve multiple important physical phenomena occurring at different spatial and temporal scales. These systems are typically characterized by large number of degrees of freedom O(103) – O(107). The temporal domains range from sub-femto seconds (O(10−16)) associated with the small high frequency oscillations of individual tightly bonded atoms to milliseconds (O(10−3)) or greater for the larger scale conformational motion. The traditional approach for molecular modeling involved fully atomistic models which results in fully decoupled equations of motion. The problems with this approach are well documented in literature.

High-Speed Vehicle Models Based on a New Concept of Vehicle/Structure Interaction Component: Part II—Algorithmic Treatment and Results for Multispan Guideways

1993 ◽  
Vol 115 (1) ◽  
pp. 148-155 ◽  
Author(s):  
L. Vu-Quoc ◽  
M. Olsson
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Structural Equations ◽  
Component Part ◽  
Vehicle Speed ◽  
Vehicle Structure ◽  
Algorithmic Treatment ◽  
Nonlinear Equations Of Motion ◽  
High Speed Vehicle

The predictor structural equations for the vehicle models developed in Part I are derived here for use with a new class of predictor/corrector algorithms to solve the mildly nonlinear equations of motion of the vehicle/structure models. Having all accelerations of the vehicle component eliminated, and with the aid of further simplifying approximations, the predictor structural equations are linear with respect to the structural degrees of freedom. In the algorithms, the predictor structural equations are different from the corrector structural equations; the proposed algorithmic treatment has been proved (elsewhere) to yield accurate energy balance. Results obtained for both continuous and discontinuous guideways are discussed, and optimal guideway configurations suggested. Effects of high-speed vehicle braking on a flexible guideway are analyzed using the vehicle models and the proposed algorithmic treatment. The influence of the guideway flexibility on the vehicle speed, an important feature of the present formulation, is clearly demonstrated.

An Efficient Algorithm for Nonlinear Analysis of Vehicle/Track Interaction Problems

2016 ◽  
Vol 16 (08) ◽  
pp. 1550040 ◽  
Author(s):  
J. Sadeghi ◽  
A. Khajehdezfuly ◽  
M. Esmaeili ◽  
D. Poorveis
Keyword(s):  
Time Domain ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Dynamic Interaction ◽  
Integration Scheme ◽  
Equilibrium Equations ◽  
Track System ◽  
Newton Raphson ◽  
Interaction Problem

In this paper, a new algorithm for solving the vehicle/track dynamic interaction problem is developed, aimed at reducing the computational cost. The algorithm called Advanced Solver Algorithm (ASA) uses the full Newton–Raphson incremental-iterative method in conjunction with the Newmark integration scheme to solve the equilibrium equations of the coupled vehicle/track system in time domain. Considering the track as a beam resting on a viscoelastic foundation and each vehicle as a wagon with ten degrees of freedom, the governing differential equations of motion of the vehicle/track system were derived. The wheel/rail contact was considered as a nonlinear Hertz spring and consequently the vehicle/track nonlinear dynamic interaction problem was solved. A comparison between the results of the ASA and those of the most advanced algorithm available was made to evaluate the efficiency of the ASA. It is confirmed that using the ASA can result in 40–70 % of reduction in computational cost.

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