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Research Papers

An Algorithm to Study the Elastodynamics of Parallel Kinematic Machines With Lower Kinematic Pairs

[+] Author and Article Information
Alessandro Cammarata

e-mail: acamma@diim.unict.it

Rosario Sinatra

Department of Industrial Engineering,
University of Catania,
Viale A. Doria 6,
95125 Catania, Italy

Hereafter, the letter i will be referred to bodies, while the letter j to joints.

Notice that stiffness units change according to the corresponding degrees of freedom, as instance: position coordinates, slope coordinates and so on.

If the base platform is considered fixed, the expressions of case (a) are simplified deleting all terms containing the nodal-displacement array of the base.

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received February 10, 2011; final manuscript received September 11, 2012; published online October 19, 2012. Assoc. Editor: Vijay Kumar.

J. Mechanisms Robotics 5(1), 011004 (Jan 19, 2012) (9 pages) Paper No: JMR-11-1016; doi: 10.1115/1.4007705 History: Received February 10, 2011; Revised September 11, 2012

In this paper, an algorithm to help designers to integrate the elastodynamics analysis along with the inverse positioning and orienting problems of a parallel kinematic machine (PKM) into a single package is conceived. The proposed algorithm applies concepts from the matrix structural analysis (MSA) and finite element analysis (FEA) to determine the generalized stiffness matrix and the linearized elastodynamics equations of a PKM with only lower kinematic pairs. A PKM is modeled as a system of flexible links and rigid bodies connected by means of joints. Three cases are analyzed to consider the combinations between flexible and rigid bodies in order to find the local stiffness matrices. The latter are combined to obtain the limb matrices and, then, the global stiffness matrix of the whole robotic system. The same nodes coming from the links discretization are considered to include joint masses/inertias into the model. Finally, a case study is proposed to show some feasible applications and to compare results to commercial software for validation.

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Figures

Grahic Jump Location
Fig. 1

Notation of the algorithm

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Fig. 3

Application to a kinematic chain

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Fig. 2

Possible combinations of bodies and nodes inside a limb: (a) rigid body–flexible body; (b) flexible–flexible; and (c) flexible body–rigid body

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Fig. 4

CAD model of tripteron: undeformed pose

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Fig. 5

The first nine modes and the associated level of strain energy V

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Fig. 6

Relative error versus mode number of the algorithm compared to FEA commercial software results

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