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Research Papers

Generalized Component Mode Synthesis for the Spatial Motion of Flexible Bodies With Large Rotations About One Axis1

[+] Author and Article Information
Pascal Ziegler

Institute of Engineering and
Computational Mechanics,
University of Stuttgart,
Stuttgart 70569, Germany
e-mail: pascal.ziegler@itm.uni-stuttgart.de

Alexander Humer

Institute of Technical Mechanics,
Johannes Kepler University Linz,
Linz 4040, Austria
e-mail: alexander.humer@jku.at

Astrid Pechstein

Institute of Technical Mechanics,
Johannes Kepler University Linz,
Linz 4040, Austria
e-mail: astrid.pechstein@jku.at

Johannes Gerstmayr

Institute of Mechatronics,
University of Innsbruck,
Innsbruck 6020, Austria
e-mail: johannes.gerstmayr@uibk.ac.at

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 28, 2015; final manuscript received November 26, 2015; published online May 12, 2016. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 11(4), 041018 (May 12, 2016) (10 pages) Paper No: CND-15-1145; doi: 10.1115/1.4032160 History: Received May 28, 2015; Revised November 26, 2015

In industrial practice, the floating frame of reference formulation (FFRF)—often combined with the component mode synthesis (CMS) in order to reduce the number of flexible degrees-of-freedom—is the common approach to describe arbitrarily shaped bodies in flexible multibody systems. Owed to the relative formulation of the flexible deformation with respect to the reference frame, the equations of motion show state-dependent nonconstant inertia terms. Such relative description, however, comes along with considerable numerical costs, since both the mass matrix and gyroscopic forces, i.e., the quadratic velocity vector, need to be evaluated in every integration step. The state dependency of the inertia terms can be avoided by employing an alternative formulation based on the mode shapes as in the classical CMS approach. In this approach, which is referred to as generalized component mode synthesis (GCMS), the total (absolute) displacements are approximated directly. Consequently, the mass matrix is constant, no quadratic velocity vector appears, and the stiffness matrix is a corotated but otherwise constant matrix. In order to represent the same flexible deformation as in the classical FFRF-based CMS, however, a comparatively large number of degrees-of-freedom is required. The approach described in the present paper makes use of the fact that a majority of components in technical systems are constrained to motions showing large rotations only about a single spatially fixed axis. For this reason, the GCMS is adapted for multibody systems that are subjected to small flexible deformations and undergo a rigid body motion showing large translations, large rotations about one axis, but small rotations otherwise. Thereby, the number of shape functions representing the flexible deformation is reduced, which further increases numerical efficiency compared to the original GCMS formulation for arbitrary rotations.

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Figures

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Fig. 1

Decomposition of the total deformation into rigid body motion and flexible displacement

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Fig. 2

One-cylinder piston engine with three flexible components

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Fig. 3

Angular velocity of the crankshaft during run-up

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Fig. 4

Deflection in the middle of the connecting rod during run-up

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Fig. 5

Convergence of the deflection of the connecting rod for different time step sizes and number of modes

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