Modal characteristics of a rotating flexible beam with a concentrated mass based on the absolute nodal coordinate formulation

Abstract

A new dynamic model of a rotating flexible beam with a concentrated mass located in arbitrary position is derived based on the absolute nodal coordinate formulation, and its modal characteristics are investigated in this paper. To consider the concentrated mass at an arbitrary location of the beam, a Dirac's delta function is used to express the mass per unit length of the beam. Based on the proposed dynamic model, the frequency analysis is performed. The nonlinear equation is transformed into the linear one via employing the linear perturbation analysis method. The stiffness matrix of static equilibrium of the system under the deformed condition is obtained, in which the effect of coupling between the longitudinal deformation and transversal deformation is included. This means even if only the chordwise bending equation is solved, the longitudinal vibration effect can be still considered. As we know, once the longitudinal deformation is large, it will significantly affect the chordwise bending vibration. So the proposed model in this paper is more accurate than the traditional dynamic models which are usually lack of the coupling terms between the longitudinal deformation and transversal deformation. In fact, the traditional dynamic models for the chordwise vibration analysis in the existing literature are usually linear due to neglecting the coupling terms, and consequently, they are only suitable for the modal characteristic analysis of a beam under small deformations. In order to get some general conclusions of the natural frequencies and mode shapes, the equation which governs the chordwise bending vibration of the rotating beam is transformed into a dimensionless form. The dynamic model presented in this paper is nonlinear and can be conveniently used to analyze the modal characteristics of a rotating flexible beam with large deformations. To demonstrate the power of the new dynamic model presented in this paper, the dynamic simulations involving the comparisons between the different frequencies obtained using the model proposed in this paper and the models in the existing literature and the investigating in frequency veering and mode shift phenomena are given. The simulation results show that the angular velocity of the flexible beam will give rise to the phenomena of the natural frequency loci veering and the associated mode shift which is verified in the previous studies. In addition, the phenomena of the natural frequency loci veering rather than crossing can 123 X. Zhang et al.be observed due to the changing of the magnitude of the concentrated mass or of the location of the concentrated mass which are found for the first time. Furthermore, there is an interesting phenomenon that the natural frequency loci will veer more than once due to different types of mode coupling between the bending and stretching vibrations of the rotating beam. At the same time, the mode shift phenomenon will occur correspondingly. Additionally, the characteristics of the vibration nodes are also investigated in this paper.

On the Existence of Time Delay for Rotating Beam with Proportional–Derivative Controller

“…The effect of rotation speed on the nonlinear vibrations of the beam and particularly on the hardening/softening behavior of its resonances and the occurrence of high amplitude jump phenomena were investigated. A new dynamic model of a rotating flexible beam with a condensed mass positioned in an arbitrary location, based on the absolute nodal coordinate formulation, was investigated by Zhang et al [2]. They found that both the magnitude and the direction of the condensed mass impact the normal frequencies and the mode shapes.…”

Position analysis and nonlinear phenomena of flexible manipulator with generic payload mounted on a moving base

“…Saad et al 11 presented the mathematical model of an Euler-Bernoulli link using the Lagrange equations by taking into account the foreshortening effect and incorporating a payload having mass, inertia, and offset. Zhang et al 12 studied the natural frequencies of a flexible Euler-Bernoulli rotating beam with a concentrated mass at an arbitrary position. The effect of variation of the mass, rotating frequency, and mass position on the natural frequencies has been discussed thoroughly.…”

“…Zhang et al. 12 studied the natural frequencies of a flexible Euler–Bernoulli rotating beam with a concentrated mass at an arbitrary position. The effect of variation of the mass, rotating frequency, and mass position on the natural frequencies has been discussed thoroughly.…”

Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components

“…Rotating structures have seen a great variety of applications to industrial products, such as a turbo engine, a helicopter and a spinning solar sail [1][2][3][4]. Previous studies have shown that the dynamic characteristics of a rotating structure are more complicated than non-rotating ones because of the centrifugal stiffening effect [1,3,[5][6][7]. For example, the eigenfrequency loci veering, eigenfrequency loci crossing and mode switching phenomena have been observed with an increase of the rotating speed.…”