Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

Nikravesh, Parviz E.; Gim, Gwanghun

doi:10.1115/1.2919310

Cited by 57 publications

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“…It has been determined that numerical computations associated with rotational transformation matrices and their corresponding coordinate transformations between reference frames is time consuming and, therefore, if these computations are avoided more efficient codes may be developed (Nikravesh and Attia, 1994). The elimination of rotational coordinates can also be found very beneficial in design sensitivity analysis of multibody systems.…”

Section: Case Of An Open-chain or Closed-chainmentioning

confidence: 99%

“…Then, the equations of motion are transformed to a reduced set in terms of the relative joint variables. Another method uses initially the point coordinate formulation in which a dynamically equivalent constrained system of particles replaces the rigid bodies (De Jalon et al, 1986, Attia, 1993, Nikravesh and Attia, 1994, and Attia, 1998. The global motion of the constrained system of particles together with the constraints imposed upon them represent both the translational and rotational motions of the rigid body.…”

Section: Introductionmentioning

confidence: 99%

“…constrained system of particles discussed in (De Jalon et al, 1986, Attia, 1993, Nikravesh and Attia, 1994, Attia, 1998and Attia, 2004 with essential modifications and improvements. The concepts of the linear and angular momentums are used to formulate the rigid body dynamical equations.…”

Section: Introductionmentioning

confidence: 99%

“…For the resulting open-chain system, the equations of motion are generated recursively along the serial chains instead of the matrix formulation derived in (De Jalon et al, 1986, Attia, 1993, Nikravesh and Attia, 1994, and Attia, 1998. Most of the kinematical constraints due to the kinematical joints are automatically eliminated by properly locating the equivalent particles.…”

Section: Introductionmentioning

confidence: 99%

“…Some formulations are developed using a two-step transformation which leads to a simple and reduced system of equations. One method (Kim andVanderploeg, 1986 andNikravesh andGim, 1989) and uses initially the absolute coordinate formulation where the location of each rigid body in the system is described in terms of a set of translational and rotational coordinates. Then, the equations of motion are transformed to a reduced set in terms of the relative joint variables.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic simulation of constrained mechanical systems using recursive projection algorithm

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Section: Case Of An Open-chain or Closed-chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamic simulation of constrained mechanical systems using recursive projection algorithm

Attia¹

2006

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Unconditionally energy stable implicit time integration: application to multibody system analysis and design

Chen

Hansen

Tortorelli

2000

Int. J. Numer. Meth. Engng.

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SUMMARYThis paper focuses on the development of an unconditionally stable time-integration algorithm for multibody dynamics that does not artiÿcially dissipate energy. Unconditional stability is sought to alleviate any stability restrictions on the integration step size, while energy conservation is important for the accuracy of long-term simulations. In multibody system analysis, the time-integration scheme is complemented by a choice of coordinates that deÿne the kinematics of the system. As such, the current approach uses a non-dissipative implicit Newmark method to integrate the equations of motion deÿned in terms of the independent joint co-ordinates of the system. In order to extend the unconditional stability of the implicit Newmark method to non-linear dynamic systems, a discrete energy balance is enforced. This constraint, however, yields spurious oscillations in the computed accelerations and therefore, a new acceleration corrector is developed to eliminate these instabilities and hence retain unconditional stability in an energy sense. An additional beneÿt of employing the non-linearly implicit time-integration method is that it allows for an e cient design sensitivity analysis. In this paper, design sensitivities computed via the direct di erentiation method are used for mechanism performance optimization.

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Flexible multibody dynamics using joint coordinates and the Rayleigh‐Ritz approximation: The general framework behind and beyond Flex

Branlard¹

2019

Wind Energy

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This article presents a general framework to set up the equations of motion for flexible multibody system using joint coordinates and Rayleigh-Ritz approximation. The framework covers in particular the underlying formulation behind the aeroelastic code Flex developed by Øye. This formulation is strongly related to the one used in the module ElastoDyn of the code OpenFAST.The approach results in models with few degrees of freedom thanks to carefully selected shape functions and the direct account of the constraints. The specificities of the method for open-loop systems and thin beams are presented. The inclusion of torsion and couplings with finite element methods or superelements are discussed. Simple validations cases are used. A four degrees of freedom model of a two-bladed wind turbine is developed to illustrate the method. INTRODUCTIONThe conventional approach for the simulation of flexible multibody systems consists in discretizing the bodies in finite elements and solving the constraints equation together with the equations of motion. The discretization and the account of the constraints usually result in a large number of degrees of freedom (DOF). This approach is for instance used by the aeroelastic code Hawc2 1 for the simulation of wind turbines. Flex 2 is another widespread tool in the wind energy community, which uses a different formulation that results in a very limited number of DOF. The general framework leading to this formulation is the topic of this article. The ElastoDyn module of the aeroelastic code FAST (now OpenFAST) 3,4 uses a similar formulation, in particular, the early versions of both codes used the same DOFs to represent a wind turbine. The minor differences between the two will be highlighted in this article: ElastoDyn relies on Kane's method instead of the velocity transformation matrix, and ElastoDyn accounts directly for the radial shortening of beams.Flex was originally written by Øye and has been further developed by different wind turbine industries. The equations of motion are set using two main methods : the joint coordinate approach and the Rayleigh-Ritz (RR) approximation. The joint coordinate method was described for rigid bodies by Nikravesh 5 and flexible bodies by Book. 6 In this method, a minimal set of DOF is obtained by choosing as coordinates the ones necessary to describe the joint motions. The constraints equation are then automatically satisfied. The RR approximation, or assumed shape function method, consists in projecting the infinite number of DOF necessary to describe a flexible body onto a reduced and converging set of shape functions. The method is well documented in the book of Shabana. 7The current article relies on the mentioned references and the available documentations of Flex. 2,8 The general framework of the method is described in section 2 to ease the understanding of the specialized implementation of Flex. The recursive formulations that are obtained for open-loop systems are presented next. The possibility to couple the equations with finite ele...

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Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

Cited by 57 publications

References 0 publications

Dynamic simulation of constrained mechanical systems using recursive projection algorithm

Dynamic simulation of constrained mechanical systems using recursive projection algorithm

Unconditionally energy stable implicit time integration: application to multibody system analysis and design

Flexible multibody dynamics using joint coordinates and the Rayleigh‐Ritz approximation: The general framework behind and beyond Flex

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