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Research Papers

# Dynamic Behavior of Flexible Multiple Links Captured Inside a Closed Space

[+] Author and Article Information
A. M. Shafei

Department of Mechanical Engineering,
Shahid Bahonar University of Kerman,
Kerman 76188-68366, Iran
e-mail: shafei@uk.ac.ir

H. R. Shafei

Department of Mechanical Engineering,
Shahid Bahonar University of Kerman,
Kerman 76188-68366, Iran

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 1, 2015; final manuscript received December 12, 2015; published online February 25, 2016. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 11(5), 051016 (Feb 25, 2016) (13 pages) Paper No: CND-15-1320; doi: 10.1115/1.4032388 History: Received October 01, 2015; Revised December 12, 2015

## Abstract

This work presents a systematic method for the dynamic modeling of flexible multiple links that are confined within a closed environment. The behavior of such a system can be completely formulated by two different mathematical models. Highly coupled differential equations are employed to model the confined multilink system when it has no impact with the surrounding walls; and algebraic equations are exploited whenever this open kinematic chain system collides with the confining surfaces. Here, to avoid using the 4 × 4 transformation matrices, which suffers from high computational complexities for deriving the governing equations of flexible multiple links, 3 × 3 rotational matrices based on the recursive Gibbs-Appell formulation has been utilized. In fact, the main aspect of this paper is the automatic approach, which is used to switch from the differential equations to the algebraic equations when this multilink chain collides with the surrounding walls. In this study, the flexible links are modeled according to the Euler–Bernoulli beam theory (EBBT) and the assumed mode method. Moreover, in deriving the motion equations, the manipulators are not limited to have only planar motions. In fact, for systematic modeling of the motion of a multiflexible-link system in 3D space, two imaginary links are added to the $n$ real links of a manipulator in order to model the spatial rotations of the system. Finally, two case studies are simulated to demonstrate the correctness of the proposed approach.

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## Figures

Fig. 1

An open kinematic chain of flexible links floating through the space

Fig. 2

Two imaginary links connected with n real links to model spatial motions

Fig. 3

Inertia matrix of the whole system in the flight phase

Fig. 4

Right hand side vector of governing equations for the flight phase

Fig. 5

A multiflexible-link chin in impact phase

Fig. 6

A two-flexible-link planar robotic manipulator confined within a square

Fig. 7

Angular positions of the joints

Fig. 8

Angular velocities of the joints

Fig. 9

Modal generalized coordinates of the links

Fig. 10

Modal generalized velocities of the links

Fig. 11

Positions of the joints in the refX1 direction

Fig. 12

Absolute velocities of the joints in the refX1 direction

Fig. 13

Positions of the joints in the refX2 direction

Fig. 14

Absolute velocities of the joints in the refX2 direction

Fig. 15

(a)(b) Configurations of the system at different times

Fig. 16

A single flying link confined inside a closed cube

Fig. 17

Angular positions of joints

Fig. 18

Angular velocities of the joints

Fig. 19

(a)–(b) Modal generalized coordinate and its derivative with respect to time

Fig. 20

(a)(c) Positions of the joints in refX1, refX2, and refX3 directions

Fig. 21

(a)(c) Absolute velocities of the joints in refX1, refX2, and refX3 directions

Fig. 22

Configurations of the system at impact moments

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