Moving Force-Induced Vibration of a Rotating Beam with Elastic Boundary Conditions

Abstract

In this paper, an analytical technique, the so-called Fourier Spectral method (FSM), is extended to the vibration analysis of a rotating Rayleigh beam considering the gyroscopic effect. The model presented can have arbitrary boundary conditions specified in terms of elastic constraints in the translations and rotations or even in terms of attached lumped masses and inertias. Each displacement function is universally expressed as a linear combination of a standard Fourier cosine series and several supplementary functions introduced to ensure and accelerate the convergence of the series expansion. Lagrange's equation is established for all the unknown Fourier coefficients viewed as a set of independent generalized coordinates. A numerical model is constructed for the rotating beam. First, a numerical example considering simply supported boundary conditions at both ends is calculated and the results are compared with those of a published paper to show the accuracy and convergence of the proposed model. Then, the method is applied to one real work piece structure with elastically supported boundary conditions updated from the modal experiment results including both the frequencies and mode shapes using the method of least squares. Several numerical examples of the updated model are studied to show the effects of some parameters on the dynamic characteristics of the work piece subjected to moving loads at different constant velocities.

A Variable-Length Beam Element Incorporating the Effect of Spinning

“…During the turning process of a slender shaft (Lv et al, 2015;Han et al, 2012), complex interactions between the rotating workpiece and the translating cutter may cause excessive vibrations that not only reduce machining accuracy and lead to poor surface quality of the workpiece but also impose restrictions on production efficiency. To study the dynamic characteristics of the shaft, it could be divided into several VLBSEs using the intersection of the shaft and the cutter as one of the nodes.…”

“…The VLBSE can be used in the dynamic analyses of (i) the drill stem of a planetary soil-drilling sampler in a deep-space exploration mission (Tian et al, 2012;Deng et al, 2014;Yang et al, 2014), (ii) the slender workpiece of a cylinder shaft in a turning process (Lv et al, 2015;Han et al, 2012), and (iii) slender rods in linear motion elements (Katz, 2001;Huang and Wang, 2014).…”

Simulation of the Vibratory Behavior of Slender Shafts Subject to Transverse Loads Moving in the Axial Direction

“…Moreover, the deflection-dependent loading produces timedependent coefficients in the governing equations and the possibility of parametric resonance and instability when time dependence is periodic. e vibration in the lathing process has been specifically investigated considering a moving transverse force in [7] and a more realistic threedirectional force on the rotating workpiece modelled as a Rayleigh or Timoshenko beam in [8][9][10], assuming the clamped-hinged supports and including as in [11] the e ect of support elasticity.…”

An improved variable-length beam element with a torsion effect based on the absolute nodal coordinate formulation

“…In the following, the novel element is termed an improved variable-length beam-shaft element (IVLBSE) considering the introduction of both bending and torsion effects, and its improvement compared to Zhao's variable-length element with torsion. This proposed element can be used in multiple types of dynamic problems of flexible multi-body systems, such as the dynamic analysis of drill stems in the drilling process, [33][34][35] slender workpieces of cylinder shafts in the turning process 36,37 and slender rods in linear motion elements. 38,39 This paper introduces variables to describe the element motion, the generalized coordinates of the element and corresponding shape functions firstly, then the axial strain, bending curvature and change rate of the rotational angle of the cross section, which describe the deformation of the element.…”

“…In the following, the novel element is termed an improved variable-length beam-shaft element (IVLBSE) considering the introduction of both bending and torsion effects, and its improvement compared to Zhao’s variable-length element with torsion. This proposed element can be used in multiple types of dynamic problems of flexible multi-body systems, such as the dynamic analysis of drill stems in the drilling process, 33–35 slender workpieces of cylinder shafts in the turning process 36,37 and slender rods in linear motion elements. 38,39…”