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Research Papers

# Dynamics of Flexible Body Negotiating a Curve

[+] Author and Article Information
Huailong Shi

State Key Laboratory of Traction Power,
Southwest Jiaotong University,
Chengdu, Sichuan 610031, China

Liang Wang, Ahmed A. Shabana

Department of Mechanical Engineering,
University of Illinois at Chicago,
Chicago, IL 60607

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 8, 2016; final manuscript received March 22, 2016; published online May 12, 2016. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 11(4), 041020 (May 12, 2016) (11 pages) Paper No: CND-16-1008; doi: 10.1115/1.4033254 History: Received January 08, 2016; Revised March 22, 2016

## Abstract

When a rigid body negotiates a curve, the centrifugal force takes a simple form which is function of the body mass, forward velocity, and the radius of curvature of the curve. In this simple case of rigid body dynamics, curve negotiation does not lead to Coriolis forces. In the case of a flexible body negotiating a curve, on the other hand, the inertia of the body becomes function of the deformation, curve negotiations lead to Coriolis forces, and the expression for the deformation-dependent centrifugal forces becomes more complex. In this paper, the nonlinear constrained dynamic equations of motion of a flexible body negotiating a circular curve are used to develop a systematic procedure for the calculation of the centrifugal forces during curve negotiations. The floating frame of reference (FFR) formulation is used to describe the body deformation and define the nonlinear centrifugal and Coriolis forces. The algebraic constraint equations which define the motion trajectory along the curve are formulated in terms of the body reference and elastic coordinates. It is shown in this paper how these algebraic motion trajectory constraint equations can be used to define the constraint forces that lead to a systematic definition of the resultant centrifugal force in the case of curve negotiations. The embedding technique is used to eliminate the dependent variables and define the equations of motion in terms of the system degrees of freedom. As demonstrated in this paper, the motion trajectory constraints lead to constant generalized forces associated with the elastic coordinates, and as a consequence, the elastic velocities and accelerations approach zero in the steady state. It is also shown that if the motion trajectory constraints are imposed on the coordinates of a flexible body reference that satisfies the mean-axis conditions, the centrifugal forces take the same form as in the case of rigid body dynamics. The resulting flexible body dynamic equations can be solved numerically in order to obtain the body coordinates and evaluate numerically the constraint and centrifugal forces. The results obtained for a flexible body negotiating a circular curve are compared with the results obtained for the rigid body in order to have a better understanding of the effect of the deformation on the centrifugal forces and the overall dynamics of the body.

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## Figures

Fig. 1

Planar rigid body negotiating a curve

Fig. 2

Beam deformation in the CRP case: (a) BFM, (b) SSM, (c) FFM. (–◻–) E=2.0685×109Pa; (–○–) E=2.0685×108Pa; (–Δ–) E=2.0685×107Pa.

Fig. 3

Beam deformation in the CMP case: (a) BFM, (b) SSM, (c) FFM. (–◻–) E=2.0685×109Pa; (–○–) E=2.0685×108Pa; (–Δ–) E=2.0685×107Pa.

Fig. 4

Constraint forces due to irregularity in the CMP case. (——) ms˙2/r; (–––) FFM; (…) BFM.

Fig. 5

Difference in the constraint force between FFM and BFM in the CMP case

Fig. 6

Effect of the forward velocity on the constraint force in the CMP case (r=100 m). (–◻–) FFM; (–○–) BFM; (–Δ–) SSM.

Fig. 7

Effect of the curve radius on the constraint force in the CMP case (s˙=25 mph). (–◻–) FFM; (–○–) BFM; (–Δ–) SSM.

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