Shock and Vibration

Volume 2018, Article ID 1056397, 14 pages

https://doi.org/10.1155/2018/1056397

## Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

School of Civil Engineering and Architecture, Xi’an University of Technology, Xi’an 710048, China

Correspondence should be addressed to Zhongmin Wang; nc.ude.tuax@mgnohzgnaw

Received 26 October 2017; Accepted 29 January 2018; Published 28 March 2018

Academic Editor: Matteo Filippi

Copyright © 2018 Zhongmin Wang and Rongrong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

#### 1. Introduction

The dynamical problem of rotating uniform and nonuniform solid beam is widely used in many practical engineering, such as helicopter rotor blades and wind turbine blades. Also, the dynamics of rotating tapered hollow beams is of practical significant, for example, rotating tank gun barrel (hollow circular cross-section). As pointed out in [1], in dynamical analysis, a rotating beam differs from a nonrotating beam because it also possesses centrifugal stiffness and Coriolis effects that influence its dynamical characteristics. Besides the above effects, there are some complicated factors, including the secondary coupling deformation term, coupling effect, and the variable coefficient differential equation. Therefore, the methodologies and solutions for rotating nonuniform beam turn out to be cumbersome.

The dynamic analysis of rotating uniform beams has been the subject of many articles and received much attention. Yoo and Shin [2] investigated the effect of centrifugal force for rotating uniform cantilever beams and used a modal formulation to obtain the natural frequencies and mode shape. Tsai et al. [3] proposed the corotational finite element method combined with floating frame method to derive differential equation of motion for the rotating inclined Euler uniform beams at constant angular speed and investigated the steady-state deformation and the natural frequencies of infinitesimal free vibration. Vinod Kumar and Ganguli [4] used the static part of the homogeneous differential equation of violin strings to obtain new shape functions for the finite element analysis of rotating Timoshenko beams. Aksencer and Aydogdu [5] studied flapwise vibration of rotating composite beams, which are used in different beam theories, including Euler-Bernoulli, Timoshenko, and Reddy beam theories, and obtained some results for different orthotropy ratios, rotation speed, hub ratio, length to thickness ratio of the rotating composite beam, and different boundary conditions. Li et al. [6] developed a new dynamic model of a planar rotating hub-beam system, where the beam is of an Euler-Bernoulli type and the deformation of the beam is described by the slope angle and stretch strain of the centroid line of the beam. They obtained four corresponding spatially discretized models, that is, ESA, FOSA, SOSA, and SSOSA model, and calculated natural frequencies and mode shapes of the system with the chordwise bending and stretching coupling effect. J. W. Lee and J. Y. Lee [7] investigated the effects of cracks on the natural frequencies of a rotating Bernoulli-Euler beam using a new numerical method in which these effects can be computed simply using the transfer matrix method.

In recent years, more studies related to transverse vibration of rotating nonuniform beams can be found in the following papers. Gunda and Ganguli [8] developed new interpolating functions which satisfy the static part of the homogenous governing differential equation for rotating uniform and tapered beams and imposed as a constraint equation in the derivation of the shape functions. Cheng et al. [9] investigated vibration characteristics of cracked rotating tapered beam by p-version finite element method and analyzed the effects of crack location, crack size, rotating speed, and hub radius on vibration characteristics of the beam. Bulut [10] considered out-of-rotation plane bending vibrations of rotating composite beam with periodically varying speed and further examined the effect of taper ratio on dynamic stability of this parametrically system. Banerjee and Jackson [11] addressed the free vibration problem of a rotating tapered Rayleigh beam by developing its dynamic stiffness matrix. In their analysis, the effects of centrifugal stiffening, an outboard force, an arbitrary hub radius, and importantly, the rotary inertia (Rayleigh beam) are included. Sarkar and Ganguli [12] proposed an inverse problem approach for dynamics of the rotating nonuniform Euler-Bernoulli beam and showed that there exists a certain class of rotating Euler-Bernoulli beam, having cantilever and pinned-free boundary conditions, which has a closed-form polynomial solution to its governing differential equation. At the same year, they also studied the free vibration of a nonhomogeneous rotating Timoshenko beam, having uniform cross-section, using an inverse problem approach, for both cantilever and pinned-free boundary conditions [13]. Tang et al. [14] studied free vibration of rotating tapered cantilever beams with rotary inertia using the integral equation method and analyzed the effects of the rotary inertia, angular speed, taper ratio, and hub radius. Li and Zhang [15] developed a new rigid-flexible coupled dynamic model to study dynamics of rotating axially functionally graded (FG) tapered beams by using the B-spline method (BSM) and observed some new interesting phenomena of frequency veering and mode shift in a rotating axially FG tapered beam when the B-S coupling effect is included. Huo and Wang [16] derived the nonlinear dynamic equations of a rotating, double-tapered, cantilever Timoshenko beam and analyzed the effect of angular speed, hub radius, slenderness ratio, and the height and width taper ratios on the natural frequencies of the rotating Timoshenko beam when the rotation beam is in a steady state, in which the extensional deformation of the beam is considered. Panchore et al. [17, 18] investigated free vibration problem of a rotating Euler-Bernoulli beam and a rotating Timoshenko beam using meshless local Petrov-Galerkin method and introduced a locking-free shape function formulation with an improved radial basis function interpolation. Ghafarian and Ariaei [19] presented a new procedure for determining natural frequencies and mode shapes of a system of elastically connected multiple rotating tapered beams through a differential transform method, which obey the Timoshenko beam theory, and discussed the effects of the rotational speed, hub radius, taper ratios, rotary inertia, shear deformation, slenderness ratio, and elastic layer stiffness coefficients on the natural frequencies. Ghafari and Rezaeepazhand [20] presented free vibration analysis of rotating composite beams with arbitrary cross-section using dimensional reduction method. Adair and Jaeger [1] used the computational approach of AMDM to analyze the free vibration of nonuniform Euler-Bernoulli beams under various boundary conditions, rotation speeds, and hub radii and simultaneously obtained the natural frequencies and corresponding closed-form series solution of the mode shape. Panchore and Ganguli [21] studied the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Other researchers also investigated the relevant second-order coupling term that represents longitudinal shrinking of the rotating beam caused by the transverse displacement. Li et al. [22] introduced a dynamic model of a rotating hub-functionally graded material beam system with the dynamic stiffening effect. In their work, the dynamic stiffening effect of the rotating hub-FGM beam system is captured by a second-order coupling term. Zhao and Wu [23] established the coupling equations of motion of a rotating three-dimensional cantilever beam to study the effects of Coriolis term and steady-state axial deformation on coupling vibration, which considered the longitudinal shrinkage caused by flapwise and chordwise bending displacement. At present, a large amount of articles relating to free vibration of rotating functionally graded plates or disk can be found (see, for instance, [24–26]).

In the above referenced articles, the model of rotating uniform beam and nonuniform beam have been considered, especially for rotating tapered beam, which has rectangular cross-section with linearly varying width and constant height, with linearly varying height and constant width, and with linearly varying width and height. However, to the best of the authors’ knowledge, no research work related to the dynamics of a rotating beam with varying hollow circular cross-section (or rotating tapered hollow beam) has been yet presented. The dynamical of the system is of practical significant because rotating tapered hollow beams are widely used as structural components in the engineering field.

In this paper, the investigation proceeds as follows. First the geometry parameters of a rotating tapered cantilever beam with hollow circular cross-section are described, and the governing differential equation of motion for transverse free vibration of a rotating tapered Rayleigh beam is derived using Hamilton variational principle. Next, for harmonic oscillation, the differential equation with variable coefficients is solved using the differential quadrature method, and an eigen-equation of the system for dimensionless parameters is formulated in explicit algebraic form. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

#### 2. Parameters of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

Figure 1 shows the schematic diagram of a rotating tapered cantilever beam with hollow circular cross-section, which has length* L*, elastic modulus , and density* ρ *and is fixed at point of a rigid hub with radius . The hub is rotating in the horizontal plane around point with a rotating angular speed . A fixed (inertial) planar coordinate system

*OXY*through the fixed point and a floating coordinate system

*oxy*that is tangent to the attachment point of the beam to the hub are prescribed, respectively. The latter relative to the former rotates with a rotation angle of large range motion.