Applied Mathematics and Mechanics (English Edition) ›› 2013, Vol. 34 ›› Issue (8): 1001-1016.doi:

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Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

Zhan-fang LIU, Shi-jun YAN, Zhi FU   

  1. Department of Engineering Mechanics, College of Resource and Environmental Sciences, Chongqing University, Chongqing 400030, P. R. China
  • Online:2013-07-15 Published:2013-07-15
  • Contact: Zhan-fang LIU

Abstract: The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kinematics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their increments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motiondeformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.

Key words: stabilization, exact controllability, limit problems, semilinear, subcritical, partial differential equations, Cauchy-Ventcel, generalized linearly elastic model, rotational deformation, dynamics, motion-deformation coupled model, finite element method

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