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Research Papers

Reference Conditions and Substructuring Techniques in Flexible Multibody System Dynamics

[+] Author and Article Information
James J. O'Shea

Department of Mechanical
and Industrial Engineering,
University of Illinois at Chicago,
842 West Taylor Street,
Chicago, IL 60607
e-mail: joshea2@uic.edu

Paramsothy Jayakumar

U.S. Army TARDEC,
6501 E. 11 Mile Road,
Warren, MI 48397-5000
e-mail: paramsothy.jayakumar.civ@mail.mil

Dave Mechergui

U.S. Army TARDEC,
6501 E. 11 Mile Road,
Warren, MI 48397-5000
e-mail: dave.mechergui.civ@mail.mil

Ahmed A. Shabana

Department of Mechanical
and Industrial Engineering,
University of Illinois at Chicago,
842 West Taylor Street,
Chicago, IL 60607
e-mail: shabana@uic.edu

Liang Wang

Department of Mechanical
and Industrial Engineering,
University of Illinois at Chicago,
842 West Taylor Street,
Chicago, IL 60607
e-mail: lwang69@uic.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 28, 2017; final manuscript received December 21, 2017; published online February 28, 2018. Assoc. Editor: Bala Balachandran.This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Comput. Nonlinear Dynam 13(4), 041007 (Feb 28, 2018) (12 pages) Paper No: CND-17-1136; doi: 10.1115/1.4039059 History: Received March 28, 2017; Revised December 21, 2017

The floating frame of reference (FFR) formulation is widely used in multibody system (MBS) simulations for the deformation analysis. Nonetheless, the use of elastic degrees-of-freedom (DOF) in the deformation analysis can increase significantly the problem dimension. For this reason, modal reduction techniques have been proposed in order to define a proper set of assumed body deformation modes. Crucial to the proper definition of these modes when the finite element (FE) FFR formulation is used is the concept of the reference conditions, which define the nature of the deformable body coordinate system. Substructuring techniques, such as the Craig–Bampton (CB) method, on the other hand, have been proposed for developing efficient models using an assembly of their lower order substructure models. In this study, the appropriateness and generality of using the CB method in MBS algorithms are discussed. It is shown that, when a set of reference conditions are not applied, the CB transformation leads to the free–free deformation modes. Because a square CB transformation is equivalent to a similarity transformation that does not alter the problem to be solved, the motivation of using the CB method in MBS codes to improve the solution is examined. This paper demonstrates that free–free deformation modes cannot be used in all applications, shedding light on the importance of the concept of the FE/FFR reference conditions. It is demonstrated numerically that a unique model resonance frequency is achieved using different modes associated with different reference conditions if the shapes are similar.

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References

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Figures

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Fig. 1

Simply supported beam

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Fig. 2

Slider–crank mechanism with elastic connecting rod

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Fig. 3

Simply supported, body-fixed, and mean-axis reference conditions

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Fig. 4

Euler–Bernoulli beam nodal coordinates ( node, undeformed element)

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Fig. 5

Euler–Bernoulli beam left node shape functions ( original nodal position, displaced nodal position, overlapped original/displaced node) ( undeformed element, -------- deformed element)

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Fig. 6

Euler–Bernoulli beam right node shape functions ( original nodal position, displaced nodal position) ( undeformed element, -------- deformed element)

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Fig. 7

Euler–Bernoulli beam transverse rigid body mode ( original nodal position, displaced nodal position) ( undeformed element, -------- deformed element)

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Fig. 8

Simply supported reference conditions in the FFR formulation ( original nodal position, displaced nodal position) ( undeformed element, -------- deformed element) (body reference XiYiZi, IECS reference XiijYiijZiij, element reference XijYijZij)

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Fig. 9

Right nodal coordinates defined in body reference frame ( original nodal position, displaced nodal position, overlapped original/displaced node) ( undeformed element, -------- deformed element)

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Fig. 10

Simply supported FE beam under actuated load ( original nodal position, revolute joint, -------- undeformed beam)

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Fig. 11

Free–free modes ( undeformed beam axis, -------- deformed element) ( displaced nodal position)

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Fig. 12

Simply supported modes ( undeformed beam axis, -------- deformed shape) ( displaced nodal position)

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Fig. 13

Maximum deflection under actuated load using free–free reference conditions. ( simply supported beam, free–free beam).

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Fig. 14

Constraint modes ( undeformed beam axis, -------- deformed shape) ( displaced nodal position)

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Fig. 15

Fixed-interface normal modes. ( undeformed beam axis, -------- deformed shape) ( displaced nodal position).

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Fig. 16

Eigenvectors of the Craig–Bampton representation ( undeformed beam axis, -------- deformed shape) ( displaced nodal position)

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Fig. 17

Resulting free–free modes from Craig–Bampton method ( undeformed beam axis, -------- deformed shape) ( displaced nodal position)

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Fig. 18

Slider–crank mechanism

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Fig. 19

Dimensionless deflection of connecting rod center (-------- MBS-REF, MBS-CB, analytical)

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Fig. 20

Extended slider–crank mechanism

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Fig. 21

Simply supported extended connecting rod modes ( undeformed beam axis, -------- deformed shape) ( boundary node, interior node)

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Fig. 22

Craig–Bampton extended connecting rod modes ( undeformed beam axis, -------- deformed shape) ( boundary node, interior node)

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Fig. 23

Amplification of the first simply supported extended connecting rod mode

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Fig. 24

Amplification of the first free–free extended connecting rod mode

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Fig. 25

Dimensionless deflection of the connecting rod pin–pin center (-------- MBS-REF, MBS-FF, MBS-CB)

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Fig. 26

Axial dimensionless deformation of the connecting rod pin–pin center ( MBS-FF, MBS-CB)

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