ON COMPARISON OF 3D ISOGEOMETRIC TIMOSHENKO AND BERNOULLI BEAM FORMULATIONS

Application of isogeometric analysis (IGA) for curved beams is very convenient for its ability of exact representation of curved geometries. Several beam formulation has been presented since the introduction of IGA. In this paper, two different beam formulations are presented: Bernoulli beam formulation of A. M. Bauer et al. [1], and Timoshenko beam element introduced by G. Zhang et al. [2]. Both beam elements are implemented and their performance is documented on the fully threedimensionalexample of helicoidal spring.

A New Three-Dimensional Moving Timoshenko Beam Element for Moving Load Problem Analysis

A new three-dimensional moving Timoshenko beam element is developed for dynamic analysis of a moving load problem with a very long beam structure. The beam has small deformations and rotations, and bending, shear, and torsional deformations of the beam are considered. Since the dynamic responses of the beam are concentrated on a small region around the moving load and most of the long beam is at rest, owing to the damping effect, the beam is truncated with a finite length. A control volume that is attached to the moving load is introduced, which encloses the truncated beam, and a reference coordinate system is established on the left end of the truncated beam. The arbitrary Lagrangian–Euler method is used to describe the relationship of the position of a particle on the beam between the reference coordinate system and the global coordinate system. The truncated beam is spatially discretized using the current beam elements. Governing equations of a moving element are derived using Lagrange’s equations. While the whole beam needs to be discretized in the finite element method or modeled in the modal superposition method (MSM), only the truncated beam is discretized in the current formulation, which greatly reduces degrees-of-freedom and increases the efficiency. Furthermore, the efficiency of the present beam element is independent of the moving load speed, and the critical or supercritical speed range of the moving load can be analyzed through the present method. After the validation of the current formulation, a dynamic analysis of three-dimensional train–track interaction with a non-ballasted track is conducted. Results are in excellent agreement with those from the commercial software simpack where the MSM is used, and the calculation time of the current formulation is one-third of that of simpack. The current beam element is accurate and more efficient than the MSM for moving load problems of long three-dimensional beams. The derivation of the current beam element is straightforward, and the beam element can be easily extended for various other moving load problems.

Dynamic Analysis and Optimization on Assembly Parameters of Rod Fastening Rotor System With Manufacturing Tolerances

Rod fastening rotor (RFR), as a typical rotor structure of gas turbine which is different from the integral rotor, is comprised of a set of discs clamped together by a central tie rod or several tie rods on the pitch circle diameters. In process of machining, tolerances of the disc are inevitable, of which the parallelism error and mass imbalance are focused on in this paper. Firstly, the complex bending of RFR by accumulation of parallelism errors of discs is derived through the coordinate transmission. Then the static analysis of RFR is performed to obtain the additional pressure by the effect of unbalanced forces, which is related to the assembly angles and rotating speed, on contact surfaces using a linear hypothesis, based on which the distribution of contact pressure considering the original pre-tightening force is obtained. Then the Bifractal-Regular theory is adopted to acquire the micro-topography of the contact interface and derive the contact stiffness related to normal contact pressure, fractal upper length limit and regular shape of the contact interfaces. After that, the zero thickness element is introduced to obtain the equivalent stiffness matrices of the contact surface. In addition, the circumferential uniformly distributed rods are modeled as a spring element which provides additional bending stiffness for the RFR. Based on the analysis above, the dynamic model of the RFR-bearing system containing 10 discs is established using the Timoshenko beam element where the continuous part of the shaft is modeled by Timoshenko beam element considering shear effect. Finally, the multi-optimization is carried out on the vibration response by the coupled effects of both initial bending and mass imbalance of the RFR-bearing system through which the optimal assembly angles are obtained. The results show a good performance in decreasing vibration as well as bending of the RFR system.