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

Authors: Zhongmin Wang; Rongrong Li
Journal: Shock and Vibration
 (2018)

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## 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.


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## Body

## 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

Figure1 shows the schematic diagram of a rotating tapered cantilever beam with hollow circular cross-section, which has lengthL, elastic modulus E, and densityρand is fixed at point o of a rigid hub with radius a. The hub is rotating in the horizontal plane around point O with a rotating angular speed ω~. A fixed (inertial) planar coordinate systemOXYthrough the fixed point O and a floating coordinate systemoxy that is tangent to the attachment point of the beam to the hub are prescribed, respectively. The latter (oxy) relative to the former (OXY) rotates with a rotation angle θ of large range motion.Figure 1
Schematic diagram of a rotating tapered cantilever beam with hollow circular cross-section.The rotating beam with varying hollow circular cross-section is considered, whose outer diameter varies linearly and the inner diameter keeps unchanged along its longitudinalx-axis, as shown in Figure 2. The beam has the root radius R1 (atx = 0) and the end radius R2 (atx =L), the wall thickness ex versus the coordinate x, and the radius Rpx at the middle line of the wall thickness for any cross-section. A local coordinate system with a normal direction n and tangential direction s at the central line of hollow circular cross-section is adopted.Figure 2
Geometry of rotating tapered cantilever beam with hollow circular cross-section and coordinate system.
(a) (b)The average radiusRpx and the wall thickness ex can be expressed, respectively, as follows:(1)Rpx=R121-1-λxL+d4,ex=R11-1-λxL-d2=2Rpx-d,where λ=R2/R1 is called the taper ratio of cross-section. It is also stipulated that the section size of the beam decreases and increases linearly from the root to the end, that is, λ∈d/2R1,∞, in which d/2R1 is denoted by β (called the ratio of inner radius to the root radius). There are two particular cases: one is a uniform beam when R2=R1, that is, λ=1, and the other is a particular varying cross-section beam when R2=d/2, that is, λmin=d/2R1.The area of any cross-section and its moment of inertia with respect to axisz can be expressed, respectively, as (2)Apx=πRp+e22-πd22=πR121-1-λxL2-β2=A1Aλx,Ix=π64D4x-d4=π642ex+d4-d4=πRp3xex1+e2x4Rp2x=π642R141-1-λxL4-β4=I1Iλx,where A1=πR12 and I1=π/4R14 are the area and the moment of inertia with respect to axis z of the root cross-section of the beam, respectively, A2=π/4d2, Iλx and Aλx are given by(3)Iλx=1-1-λxL4-β4,Aλx=1-1-λxL2-β2.Any hollow circular cross-section of the beam is shown as Figure3. The vertical coordinate of any pointM can be expressed, respectively, as(4)y=ys+nsin⁡α=ys-ndzsds.Figure 3
Any hollow circular cross-section of the beam and infinitesimal arc length.
(a) (b)


## 3. Differential Equation of Motion

### 3.1. The Description of the Deformation Field

Figure4 shows that ra is a radius vector of the original pointo of floating coordinate systemoxy with respect to the pointO of the inertial coordinate systemOXY, xp is a radius vector of any point P0, which is on the axis of the beam before deformation, the pointP is the positions of the point P0 after deformation, and up is a displacement vector of the point P0.Figure 4
Schematic diagram of the deformation field.The vector of pointP0 relative to original pointO of inertial coordinate systemOXY can be expressed as(5)rp=ra+Aθxp+up,where(6)ra=acos⁡θasin⁡θT,xp=x0T,Aθ=cos⁡θ-sin⁡θsin⁡θcos⁡θis a direction-cosine matrix of the floating base relative to the inertial base,up=upxupy=u+uc+ubwT=u-12∫0xw′2dx-yw′win which u and w are the axial displacement and the transverse bending deflection, respectively; ub=-yw′=-w′ys-n(dzs/ds) is axial displacement of any point caused by transverse bending, in which the prime denotes spatial derivatives with respect to x; uc=-1/2∫0x(w′)2dx is a second-order coupling term that represents longitudinal shrinking of the rotating tapered beam with hollow circular cross-section beam caused by the transverse displacement w. It includes the coupling effect between the axial displacement and transverse displacement of rotating tapered hollow beam.Taking the derivative with respect to time for (4), the velocity vector of the point P0 at the inertial coordinate system can be obtained(7)r˙p=r˙a+θ˙Aθxp+up+Aθu˙p=-θ˙a+x+uc+ubsin⁡θ-θ˙wcos⁡θ+u˙c+u˙bcos⁡θ-w˙sin⁡θθ˙a+x+uc+ubcos⁡θ-θ˙wsin⁡θ+u˙c+u˙bsin⁡θ+w˙cos⁡θ,where θ˙=0-θ˙θ˙0 is a antisymmetric matrix relating to angular speed θ˙, in which the over dot denotes derivative with respect to time t.


### 3.2. Differential Equation of Motion

#### 3.2.1. Kinetic Energy of System

Kinetic energy of system consists of two parts: one is the kinetic energy of the hub and the other is the kinetic energy of the beam with hollow circular cross-section; namely,(8)T=TH+TP.The kinetic energy of the hub is given by(9)TH=12JHθ˙2,where JH is rotary inertia of the hub with respect to central axis.Neglecting the axis displacement of the beam with hollow circular cross-section, the kinetic energy of the per unit length of the beam can be expressed as(10)T~p=∫02πRp∫-e/2e/212ρr˙pTr˙pdnds=12mcw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc+12ρI1Iλxθ˙2w,x2+ρI1Iλxw˙,x2,where mc is the effective mass of the per unit length of the beam; it can be expressed as(11)mc=∫02πRpxρexds=∫02πRpxρexds=ρ·2πRpxex=ρApx=ρA1Aλx.The kinetic energy of the beam with hollow circular cross-section can be rewritten as(12)TP=∫0LT~pdx=12∫0Lmcu˙c-θ˙w2+w˙+θ˙a+x+uc2dx=12∫0LρA1Aλxw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc2dx+12∫0LρI1Iλxθ˙2w,x2+w˙,x2dx.Thus, substituting (9) and (12) into (8), the total kinetic energy of the system can be expressed as(13)T=12JHθ˙2+12∫0LρA1Aλxw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc2dx+12∫0LρI1Iλxθ˙2w,x2+w˙,x2dx.


#### 3.2.2. Strain Energy of System

Neglecting the deformation energy caused by shear deformation, the strain energy of rotating beam with hollow circular cross-section is written as(14)Vε=∫V12σxεxdV=∫V12Eεx2dV,where E is the elastic modulus of material; σx and εx represent the normal stress and normal strain in x direction, respectively.In (6), ignoring the axial displacement and nonlinear term, the normal strain can be got by the relationship between strain and displacement: namely,(15)εx=∂upx∂x=-ys-ndzsdsw″.According to Figure3, a geometrical relationship is given by(16)dzsds=-sin⁡α=-sin⁡sRp.Thus, the strain energy of the beam with hollow circular cross-section can be rewritten as(17)Vε=12∫0L∫-e/2e/2∫02πεx2Rpx+ndαdsdn=12∫0L∫-e/2e/2∫02πRpE-ys-ndzsdsw″2Rpx+nRpxdsdndx=12∫0LEIxw″2dx=12EI1∫0LIλxw″2dx.


#### 3.2.3. Derivation of Differential Equation of Motion

In this paper, Hamilton variational principle for elastic system is used to derive the differential equation of motion. The basis form of Hamilton variational principle can be showed as(18)δ∫t1t2T-Vεdt=0.Substituting (13) and (17) into (18) and implementing a lot of variational operation and integration by parts, a variational expression is given by (19)∫0L-EI1Iλxw″″-2EI1Iλ′xw′′′-EI1Iλ′′xw″+ρA1Ax-w¨+θ˙2w-θ¨a+x+uc-2θ˙u˙c+ρI′xθ˙2w,x+ρIxθ˙2w,xx-ρI′xw¨,x-ρIxw¨,xxdx+∂∂xw′∫xLBx,tdx=0,where Bx,t=ρA1Aλx-u¨c+θ˙2a+x+uc+2w˙θ˙+θ¨w.Because second-order coupling deformation termuc  is a second-order small quantity, we can neglect some nonlinear terms and time-varying coupling terms in (19) to simplify the equation appropriately. Thus, the differential equation of motion of the rotating beam with hollow circular cross-section can be derived(20)-EI1Iλxw″″-2EI1Iλ′xw′′′-EI1Iλ′′xw″+ρA1Aλx-w¨+θ˙2w-θ¨a+x-θ˙2w′a+x-2w˙θ˙w′-θ¨ww′+2θ˙w′w˙′ρA1L-x+1-λ23L2L3-x3-1-λLL2-x2-2θ˙w′w˙′ρA2L-x+θ¨w′2ρA1L-x+1-λ23L2L3-x3-1-λLL2-x2-θ¨w′2ρA2L-x+ρA1w′′2w˙θ˙+θ¨w+θ˙aL-x+1-λ23L2L3-x3-1-λLL2-x2+θ˙2L2-x22+1-λ24L2L4-x4-21-λ3LL3-x3-ρA2w,xxθ˙22L2-x2+2w˙θ˙+θ¨w+θ˙2aL-x+ρI′xθ˙2w′+ρIxθ˙2w′′-ρI′xw¨′-ρIxw¨″=0.Taking uniform rotation into consideration, that is,θ¨=0, θ˙=ω0, a differential equation of motion of the beam can be expressed as(21)EI1Iλxw′′′′+2EI1Iλ′xw′′′+EI1Iλ′′xw′′+ρA1Aλxw¨-ρI1Iλxw¨″-ρI1Iλ′xw¨′+ρI1Iλ′xω02w′-ρA1ω02aγ1+γ2w′′+ρI1Iλxω02w′′+ρA1β2ω02aL-x+L2-x22w′′+ρA1Aλxω02a+xw′=0in which λ1=L-x+1-λ2L3-x3/3L2-1-λL2-x2/L,  λ2=(L2-x2)/2+1-λ2L4-x4/4L2-21-λ/3LL3-x3.The boundary conditions of the cantilever beam are as follows:(22)x=0:w=0,w′=0,x=L:w″=0,EddxIxw,xx-ρIxw¨,x+ρIxω02w,x=0.


### 3.3. Dimensionless Method of the Equation

For simplicity, the following dimensionless quantities are introduced:ξ=x/L, w-=w/L, τ=t/L2EI1/ρA1, ω=ω0L2ρA1/EI1 (called dimensionless angular speed of the hub), r0=a/L (called ratio of hub radius to beam length), and r=I1/A1L2 (called slenderness ratio).Dimensionless expression of (21) can be expressed as(23)∂4w-∂ξ4+2Iλ′ξIλξ∂3w-∂ξ3+Iλ′′ξIλξ∂2w-∂ξ2+AλξIλξ∂2w-∂τ2-Iλ′ξIλξr2∂3w-∂τ2∂ξ-r2∂4w-∂ξ2∂τ2+ω2r2∂2w-∂ξ2+Iλ′ξIλξω2r2∂w-∂ξ+AλξIλξω2r0+ξ∂w-∂ξ-ω2Iλξr0γ11+γ22∂2w-∂ξ2+β2ω2Iλξr01-ξ+1-ξ22∂2w-∂ξ2=0,where γ11=1-ξ-1-λ1-ξ2+1-λ2/31-ξ3,γ12=(1-ξ2)/2-21-λ/31-ξ3 + 1-λ2/41-ξ4.Let the solution of (23) be w-ξ,τ=Wξexp⁡Ωτ; a differential equation of mode shape can be written as(24)d4Wdξ4+2Iλ′ξIλξd3Wdξ3+Iλ′′ξIλξd2Wdξ2+Ω2AλξIλξW-Iλ′ξIλξr2Ω2dWdξ-r2Ω2d2Wdξ2+ω2r2d2Wdξ2+Iλ′ξω2r2IλξdWdξ+Aλξω2Iλξr0+ξdWdξ-ω2Iλξr0γ11+γ22d2Wdξ2+ω2β2Iλξr01-ξ+1-ξ22d2Wdξ2=0,where Ω is dimensionless natural frequency.The dimensionless forms of the boundary conditions (22) are rewritten as(25)W0=0,dW0dξ=0,d2W1dξ2=0,d3W1dξ3-r2Ω2-ω2dW1dξ=0.


## 3.1. The Description of the Deformation Field

Figure4 shows that ra is a radius vector of the original pointo of floating coordinate systemoxy with respect to the pointO of the inertial coordinate systemOXY, xp is a radius vector of any point P0, which is on the axis of the beam before deformation, the pointP is the positions of the point P0 after deformation, and up is a displacement vector of the point P0.Figure 4
Schematic diagram of the deformation field.The vector of pointP0 relative to original pointO of inertial coordinate systemOXY can be expressed as(5)rp=ra+Aθxp+up,where(6)ra=acos⁡θasin⁡θT,xp=x0T,Aθ=cos⁡θ-sin⁡θsin⁡θcos⁡θis a direction-cosine matrix of the floating base relative to the inertial base,up=upxupy=u+uc+ubwT=u-12∫0xw′2dx-yw′win which u and w are the axial displacement and the transverse bending deflection, respectively; ub=-yw′=-w′ys-n(dzs/ds) is axial displacement of any point caused by transverse bending, in which the prime denotes spatial derivatives with respect to x; uc=-1/2∫0x(w′)2dx is a second-order coupling term that represents longitudinal shrinking of the rotating tapered beam with hollow circular cross-section beam caused by the transverse displacement w. It includes the coupling effect between the axial displacement and transverse displacement of rotating tapered hollow beam.Taking the derivative with respect to time for (4), the velocity vector of the point P0 at the inertial coordinate system can be obtained(7)r˙p=r˙a+θ˙Aθxp+up+Aθu˙p=-θ˙a+x+uc+ubsin⁡θ-θ˙wcos⁡θ+u˙c+u˙bcos⁡θ-w˙sin⁡θθ˙a+x+uc+ubcos⁡θ-θ˙wsin⁡θ+u˙c+u˙bsin⁡θ+w˙cos⁡θ,where θ˙=0-θ˙θ˙0 is a antisymmetric matrix relating to angular speed θ˙, in which the over dot denotes derivative with respect to time t.


## 3.2. Differential Equation of Motion

### 3.2.1. Kinetic Energy of System

Kinetic energy of system consists of two parts: one is the kinetic energy of the hub and the other is the kinetic energy of the beam with hollow circular cross-section; namely,(8)T=TH+TP.The kinetic energy of the hub is given by(9)TH=12JHθ˙2,where JH is rotary inertia of the hub with respect to central axis.Neglecting the axis displacement of the beam with hollow circular cross-section, the kinetic energy of the per unit length of the beam can be expressed as(10)T~p=∫02πRp∫-e/2e/212ρr˙pTr˙pdnds=12mcw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc+12ρI1Iλxθ˙2w,x2+ρI1Iλxw˙,x2,where mc is the effective mass of the per unit length of the beam; it can be expressed as(11)mc=∫02πRpxρexds=∫02πRpxρexds=ρ·2πRpxex=ρApx=ρA1Aλx.The kinetic energy of the beam with hollow circular cross-section can be rewritten as(12)TP=∫0LT~pdx=12∫0Lmcu˙c-θ˙w2+w˙+θ˙a+x+uc2dx=12∫0LρA1Aλxw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc2dx+12∫0LρI1Iλxθ˙2w,x2+w˙,x2dx.Thus, substituting (9) and (12) into (8), the total kinetic energy of the system can be expressed as(13)T=12JHθ˙2+12∫0LρA1Aλxw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc2dx+12∫0LρI1Iλxθ˙2w,x2+w˙,x2dx.


### 3.2.2. Strain Energy of System

Neglecting the deformation energy caused by shear deformation, the strain energy of rotating beam with hollow circular cross-section is written as(14)Vε=∫V12σxεxdV=∫V12Eεx2dV,where E is the elastic modulus of material; σx and εx represent the normal stress and normal strain in x direction, respectively.In (6), ignoring the axial displacement and nonlinear term, the normal strain can be got by the relationship between strain and displacement: namely,(15)εx=∂upx∂x=-ys-ndzsdsw″.According to Figure3, a geometrical relationship is given by(16)dzsds=-sin⁡α=-sin⁡sRp.Thus, the strain energy of the beam with hollow circular cross-section can be rewritten as(17)Vε=12∫0L∫-e/2e/2∫02πεx2Rpx+ndαdsdn=12∫0L∫-e/2e/2∫02πRpE-ys-ndzsdsw″2Rpx+nRpxdsdndx=12∫0LEIxw″2dx=12EI1∫0LIλxw″2dx.


### 3.2.3. Derivation of Differential Equation of Motion

In this paper, Hamilton variational principle for elastic system is used to derive the differential equation of motion. The basis form of Hamilton variational principle can be showed as(18)δ∫t1t2T-Vεdt=0.Substituting (13) and (17) into (18) and implementing a lot of variational operation and integration by parts, a variational expression is given by (19)∫0L-EI1Iλxw″″-2EI1Iλ′xw′′′-EI1Iλ′′xw″+ρA1Ax-w¨+θ˙2w-θ¨a+x+uc-2θ˙u˙c+ρI′xθ˙2w,x+ρIxθ˙2w,xx-ρI′xw¨,x-ρIxw¨,xxdx+∂∂xw′∫xLBx,tdx=0,where Bx,t=ρA1Aλx-u¨c+θ˙2a+x+uc+2w˙θ˙+θ¨w.Because second-order coupling deformation termuc  is a second-order small quantity, we can neglect some nonlinear terms and time-varying coupling terms in (19) to simplify the equation appropriately. Thus, the differential equation of motion of the rotating beam with hollow circular cross-section can be derived(20)-EI1Iλxw″″-2EI1Iλ′xw′′′-EI1Iλ′′xw″+ρA1Aλx-w¨+θ˙2w-θ¨a+x-θ˙2w′a+x-2w˙θ˙w′-θ¨ww′+2θ˙w′w˙′ρA1L-x+1-λ23L2L3-x3-1-λLL2-x2-2θ˙w′w˙′ρA2L-x+θ¨w′2ρA1L-x+1-λ23L2L3-x3-1-λLL2-x2-θ¨w′2ρA2L-x+ρA1w′′2w˙θ˙+θ¨w+θ˙aL-x+1-λ23L2L3-x3-1-λLL2-x2+θ˙2L2-x22+1-λ24L2L4-x4-21-λ3LL3-x3-ρA2w,xxθ˙22L2-x2+2w˙θ˙+θ¨w+θ˙2aL-x+ρI′xθ˙2w′+ρIxθ˙2w′′-ρI′xw¨′-ρIxw¨″=0.Taking uniform rotation into consideration, that is,θ¨=0, θ˙=ω0, a differential equation of motion of the beam can be expressed as(21)EI1Iλxw′′′′+2EI1Iλ′xw′′′+EI1Iλ′′xw′′+ρA1Aλxw¨-ρI1Iλxw¨″-ρI1Iλ′xw¨′+ρI1Iλ′xω02w′-ρA1ω02aγ1+γ2w′′+ρI1Iλxω02w′′+ρA1β2ω02aL-x+L2-x22w′′+ρA1Aλxω02a+xw′=0in which λ1=L-x+1-λ2L3-x3/3L2-1-λL2-x2/L,  λ2=(L2-x2)/2+1-λ2L4-x4/4L2-21-λ/3LL3-x3.The boundary conditions of the cantilever beam are as follows:(22)x=0:w=0,w′=0,x=L:w″=0,EddxIxw,xx-ρIxw¨,x+ρIxω02w,x=0.


## 3.2.1. Kinetic Energy of System

Kinetic energy of system consists of two parts: one is the kinetic energy of the hub and the other is the kinetic energy of the beam with hollow circular cross-section; namely,(8)T=TH+TP.The kinetic energy of the hub is given by(9)TH=12JHθ˙2,where JH is rotary inertia of the hub with respect to central axis.Neglecting the axis displacement of the beam with hollow circular cross-section, the kinetic energy of the per unit length of the beam can be expressed as(10)T~p=∫02πRp∫-e/2e/212ρr˙pTr˙pdnds=12mcw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc+12ρI1Iλxθ˙2w,x2+ρI1Iλxw˙,x2,where mc is the effective mass of the per unit length of the beam; it can be expressed as(11)mc=∫02πRpxρexds=∫02πRpxρexds=ρ·2πRpxex=ρApx=ρA1Aλx.The kinetic energy of the beam with hollow circular cross-section can be rewritten as(12)TP=∫0LT~pdx=12∫0Lmcu˙c-θ˙w2+w˙+θ˙a+x+uc2dx=12∫0LρA1Aλxw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc2dx+12∫0LρI1Iλxθ˙2w,x2+w˙,x2dx.Thus, substituting (9) and (12) into (8), the total kinetic energy of the system can be expressed as(13)T=12JHθ˙2+12∫0LρA1Aλxw˙2+u˙c2+θ˙2w2+a+x+uc2+2θ˙-u˙cw+w˙a+x+uc2dx+12∫0LρI1Iλxθ˙2w,x2+w˙,x2dx.


## 3.2.2. Strain Energy of System

Neglecting the deformation energy caused by shear deformation, the strain energy of rotating beam with hollow circular cross-section is written as(14)Vε=∫V12σxεxdV=∫V12Eεx2dV,where E is the elastic modulus of material; σx and εx represent the normal stress and normal strain in x direction, respectively.In (6), ignoring the axial displacement and nonlinear term, the normal strain can be got by the relationship between strain and displacement: namely,(15)εx=∂upx∂x=-ys-ndzsdsw″.According to Figure3, a geometrical relationship is given by(16)dzsds=-sin⁡α=-sin⁡sRp.Thus, the strain energy of the beam with hollow circular cross-section can be rewritten as(17)Vε=12∫0L∫-e/2e/2∫02πεx2Rpx+ndαdsdn=12∫0L∫-e/2e/2∫02πRpE-ys-ndzsdsw″2Rpx+nRpxdsdndx=12∫0LEIxw″2dx=12EI1∫0LIλxw″2dx.


## 3.2.3. Derivation of Differential Equation of Motion

In this paper, Hamilton variational principle for elastic system is used to derive the differential equation of motion. The basis form of Hamilton variational principle can be showed as(18)δ∫t1t2T-Vεdt=0.Substituting (13) and (17) into (18) and implementing a lot of variational operation and integration by parts, a variational expression is given by (19)∫0L-EI1Iλxw″″-2EI1Iλ′xw′′′-EI1Iλ′′xw″+ρA1Ax-w¨+θ˙2w-θ¨a+x+uc-2θ˙u˙c+ρI′xθ˙2w,x+ρIxθ˙2w,xx-ρI′xw¨,x-ρIxw¨,xxdx+∂∂xw′∫xLBx,tdx=0,where Bx,t=ρA1Aλx-u¨c+θ˙2a+x+uc+2w˙θ˙+θ¨w.Because second-order coupling deformation termuc  is a second-order small quantity, we can neglect some nonlinear terms and time-varying coupling terms in (19) to simplify the equation appropriately. Thus, the differential equation of motion of the rotating beam with hollow circular cross-section can be derived(20)-EI1Iλxw″″-2EI1Iλ′xw′′′-EI1Iλ′′xw″+ρA1Aλx-w¨+θ˙2w-θ¨a+x-θ˙2w′a+x-2w˙θ˙w′-θ¨ww′+2θ˙w′w˙′ρA1L-x+1-λ23L2L3-x3-1-λLL2-x2-2θ˙w′w˙′ρA2L-x+θ¨w′2ρA1L-x+1-λ23L2L3-x3-1-λLL2-x2-θ¨w′2ρA2L-x+ρA1w′′2w˙θ˙+θ¨w+θ˙aL-x+1-λ23L2L3-x3-1-λLL2-x2+θ˙2L2-x22+1-λ24L2L4-x4-21-λ3LL3-x3-ρA2w,xxθ˙22L2-x2+2w˙θ˙+θ¨w+θ˙2aL-x+ρI′xθ˙2w′+ρIxθ˙2w′′-ρI′xw¨′-ρIxw¨″=0.Taking uniform rotation into consideration, that is,θ¨=0, θ˙=ω0, a differential equation of motion of the beam can be expressed as(21)EI1Iλxw′′′′+2EI1Iλ′xw′′′+EI1Iλ′′xw′′+ρA1Aλxw¨-ρI1Iλxw¨″-ρI1Iλ′xw¨′+ρI1Iλ′xω02w′-ρA1ω02aγ1+γ2w′′+ρI1Iλxω02w′′+ρA1β2ω02aL-x+L2-x22w′′+ρA1Aλxω02a+xw′=0in which λ1=L-x+1-λ2L3-x3/3L2-1-λL2-x2/L,  λ2=(L2-x2)/2+1-λ2L4-x4/4L2-21-λ/3LL3-x3.The boundary conditions of the cantilever beam are as follows:(22)x=0:w=0,w′=0,x=L:w″=0,EddxIxw,xx-ρIxw¨,x+ρIxω02w,x=0.


## 3.3. Dimensionless Method of the Equation

For simplicity, the following dimensionless quantities are introduced:ξ=x/L, w-=w/L, τ=t/L2EI1/ρA1, ω=ω0L2ρA1/EI1 (called dimensionless angular speed of the hub), r0=a/L (called ratio of hub radius to beam length), and r=I1/A1L2 (called slenderness ratio).Dimensionless expression of (21) can be expressed as(23)∂4w-∂ξ4+2Iλ′ξIλξ∂3w-∂ξ3+Iλ′′ξIλξ∂2w-∂ξ2+AλξIλξ∂2w-∂τ2-Iλ′ξIλξr2∂3w-∂τ2∂ξ-r2∂4w-∂ξ2∂τ2+ω2r2∂2w-∂ξ2+Iλ′ξIλξω2r2∂w-∂ξ+AλξIλξω2r0+ξ∂w-∂ξ-ω2Iλξr0γ11+γ22∂2w-∂ξ2+β2ω2Iλξr01-ξ+1-ξ22∂2w-∂ξ2=0,where γ11=1-ξ-1-λ1-ξ2+1-λ2/31-ξ3,γ12=(1-ξ2)/2-21-λ/31-ξ3 + 1-λ2/41-ξ4.Let the solution of (23) be w-ξ,τ=Wξexp⁡Ωτ; a differential equation of mode shape can be written as(24)d4Wdξ4+2Iλ′ξIλξd3Wdξ3+Iλ′′ξIλξd2Wdξ2+Ω2AλξIλξW-Iλ′ξIλξr2Ω2dWdξ-r2Ω2d2Wdξ2+ω2r2d2Wdξ2+Iλ′ξω2r2IλξdWdξ+Aλξω2Iλξr0+ξdWdξ-ω2Iλξr0γ11+γ22d2Wdξ2+ω2β2Iλξr01-ξ+1-ξ22d2Wdξ2=0,where Ω is dimensionless natural frequency.The dimensionless forms of the boundary conditions (22) are rewritten as(25)W0=0,dW0dξ=0,d2W1dξ2=0,d3W1dξ3-r2Ω2-ω2dW1dξ=0.


## 4. Differential Quadrature Method

In order to solve the differential equation with variable coefficients (24) and deal with the boundary conditions (25), the differential quadrature method (DQM) and the δ method are used, respectively. Selecting nonuniform nodes, the node coordinates are as follows [27, 28]:(26)X1=0,X2=δ,XN-1=1-δ,XN=1,Xi=121-cos⁡i-2N-3πi=3,4,…,N-2,where N is the numbers of nodes and δ is small parameter.According to the DQM procedures, (24) can be discretized as follows:(27)
∑
j
=
1
N
A
i
j
4
W
j
+
2
I
λ
′
ξ
i
I
λ
ξ
i
∑
j
=
1
N
A
i
j
3
W
j
+
I
λ
′
′
ξ
i
I
λ
ξ
i
∑
j
=
1
N
A
i
j
2
W
j
+
Ω
2
A
λ
ξ
i
I
λ
ξ
i
W
i
-
I
λ
′
ξ
i
I
λ
ξ
i
r
2
Ω
2
∑
j
=
1
N
A
i
j
1
W
j
-
r
2
Ω
2
∑
j
=
1
N
A
i
j
2
W
j
+
r
2
ω
2
∑
j
=
1
N
A
i
j
2
W
j
+
I
λ
′
ξ
i
I
λ
ξ
i
r
2
ω
2
∑
j
=
1
N
A
i
j
1
W
j
+
ω
2
A
λ
ξ
i
I
λ
ξ
i
r
0
+
ξ
i
∑
j
=
1
N
A
i
j
1
W
j
-
ω
2
I
λ
ξ
i
r
0
γ
11
+
γ
22
-
β
2
r
0
1
-
ξ
i
-
β
2
1
-
ξ
i
2
2
∑
j
=
1
N
A
i
j
2
W
j
=
0
i
=
3,4
,
…
,
N
-
2
,where Aλξi=1-1-λξi2-β2,  Iλ(ξi)=1-(1-λ)ξi4-β4,dIλξi/dξ=-41-(1-λ)ξi3(1-λ), d2Iλξi/dξ2=121-1-λξi21-λ2.The boundary conditions (25) can also be discretized as follows:(28a)W1=0,(28b)∑j=1NA2j1Wj=0,(28c)∑j=1NAN-1j2Wj=0,(28d)∑j=1NANj3Wj-r2Ω2-ω2∑j=1NANj1Wj=0.Equations (27), (28a), (28b), (28c), and (28d) can be rewritten as a simpler matrix form(29)Ω2M+ΩC+KW=0,where M, C, and K are all N-2×N-2 square matrix; W=W3W4⋯Wn-2T is column matrix.Equation (29) denotes a generalized eigenvalue problem. Based on the linear algebra theory, the sufficient and necessary conditions of homogeneous linear algebraic equations which create the nonzero solution are that the determinant of coefficients equals zero; thus, one can arrive to the following generalized eigen-equation:(30)Ω2M+ΩC+K=0,where the square matrices M, C, and K involve some parameters, such as 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, each dimensionless natural frequencies, and dimensionless angular speed of the hub.


## 5. Numerical Results and Analyses

### 5.1. Rotating Tapered Cantilever Beam with Solid Circular Cross-Section

In this paragraph, settingβ=0 in (24), we can obtain the differential equation of motion for rotating tapered beam with solid circular cross-section, where taper ratio λ∈0,∞ is defined. It becomes evident that if the taper ratio λ=0 and λ=1, they are varying solid circular cross-section beam with zero radius at free end and entirely uniform solid circular cross-section, respectively. Prior to the presentation of our numerical results, let us first consider three particular cases to confirm the effectiveness of the present approach: a simple uniform nonrotating cantilever beam, a rotating tapered Euler-Bernoulli cantilever beam, and a rotating tapered Rayleigh cantilever beam, which is given by setting parameters ω=0, r0=0, r=0, and λ=1, parameters r0=0, r=0, and parameters r0=0, 1/r=30, respectively. In three cases, we calculate the first three-order dimensionless natural frequencies by selecting N=12 for different taper ratio of cross-section and dimensionless angular speed, and some numerical results are tabulated in Tables 1 and 2  (0≤λ≤1). From the two tables, we can see that the numerical results in the present coincide well with the existing ones [14]. These verify that the method presented in this paper is efficient and accurate. The first three dimensionless mode shapes are shown as Figure 5 for ω=10, λ=1, r0=0, β=0, and r=0.Table 1
Comparison of the first three dimensionless natural frequencies of rotating solid tapered Euler-Bernoulli beams atr0=0, β=0, r=0, 0≤λ≤1.
ω Ω λ = 1 λ = 0.75 λ = 0.5 λ = 0.25 λ = 0 Present Ref. [14] Present Ref. [14] Present Ref. [14] Present Ref. [14] Present 0 Ω 1 3.5164 3.5160 3.9570 3.9567 4.6257 4.6252 5.8293 5.8231 8.7445 Ω 2 22.0366 22.0345 20.8091 20.807 19.5514 19.5476 18.5183 18.480 20.1042 Ω 3 61.6960 61.6972 55.3500 55.3304 48.5953 48.5789 41.4611 41.321 36.5279  5 Ω 1 6.4499 6.4495 6.7732 6.7729 7.2905 7.2901 8.2653 8.2620 10.6634 Ω 2 25.4483 25.4461 24.0673 24.0660 22.6379 22.6360 21.4209 21.384 22.8601 Ω 3 65.2118 65.2050 58.6518 58.6364 51.7012 51.6918 44.4176 44.269 41.0619  10 Ω 1 11.2029 11.2023 11.4859 11.4856 11.9419 11.9415 12.7928 12.791 14.7477 Ω 2 33.6417 33.6404 31.8863 31.8895 30.0289 30.0299 28.3381 28.301 29.4213 Ω 3 74.6671 74.6493 67.5293 67.5316 60.0369 60.0399 52.2738 52.100 51.2478  15 Ω 1 16.1445 - 16.4165 - 16.8587 - 17.6745 - 19.4476 Ω 2 43.9323 - 41.7005 - 39.2756 - 36.9308 - 37.5106 Ω 3 87.9543 - 80.0153 - 71.7371 - 63.1548 - 59.9363Table 2
Comparison of the first three dimensionless natural frequencies of rotating solid tapered Rayleigh beams for differentλ(0≤λ≤1) at r0=0, β=0, 1/r=30.
ω Ω λ = 1 λ = 0.75 λ = 0.5 λ = 0.25 λ = 0 Present Ref. [14] Present Ref. [14] Present Ref. [14] Present Ref. [14] Present 0 Ω 1 3.5073 3.5070 3.9486 3.9483 4.6174 4.6168 5.8198 5.8136 8.7267 Ω 2 21.6497 21.6477 20.5498 20.5475 19.3883 19.3846 18.4214 18.3834 20.0372 Ω 3 59.2069 59.2073 53.7093 53.6911 47.6173 47.6021 40.9588 40.825 36.5169  5 Ω 1 6.4254 6.4251 6.7524 6.7521 7.2720 7.2717 8.2469 8.2436 10.6360 Ω 2 24.9759 24.9737 23.7510 23.7497 22.4394 22.4375 21.3029 21.266 22.7786 Ω 3 62.5464 663.5392 56.8916 56.8774 50.6483 50.6397 43.8720 43.730 41.0472  10 Ω 1 11.1604 11.1598 11.4494 11.4490 11.9087 11.9083 12.7584 12.767 14.6997 Ω 2 32.9667 32.9654 31.4395 31.4425 29.7515 29.7521 28.1722 28.136 29.3002 Ω 3 71.5142 71.4977 65.4444 65.4472 58.7850 58.7878 51.6155 51.450 50.5772  15 Ω 1 16.0851 - 16.3650 - 16.8105 - 17.6223 - 19.3744 Ω 2 43.0010 - 41.0951 - 38.9051 - 36.7064 - 37.3335 Ω 3 84.0806 - 77.4631 - 70.2081 - 62.3423 - 59.2553Figure 5
The first three dimensionless mode shapes forω=10, λ=1, r0=0, β=0, and r=0.Furthermore, in the case ofλ>1, for example, 1.25≤λ≤3.00, that is, the radius of beam cross-section at free end is more than one at the cantilevered end, the first three-order dimensionless natural frequencies of a rotating tapered Rayleigh cantilever solid beam for three dimensionless angular speeds are tabulated in Table 3. It can be seen in Table 3 that as a whole, the first three-order dimensionless natural frequencies increase with the increase of dimensionless angular speeds.Table 3
The first three dimensionless natural frequencies of rotating solid tapered Rayleigh beams for differentλ(1.25≤λ≤3) at r0=0, β=0, 1/r=30.
ω Ω λ = 1.25 λ = 1.50 λ = 1.75 λ = 2.00 λ = 2.25 λ = 2.5 λ = 2.75 λ = 3.00 0 Ω 1 3.1877 2.9421 2.7461 2.5882 2.4704 2.4129 2.4613 2.6858 Ω 2 22.6579 23.5718 24.3952 25.1281 25.7580 26.2480 26.5270 26.4787 Ω 3 64.2417 68.9100 73.2615 77.3249 81.1317 84.7333 88.2071 91.6568  5 Ω 1 6.1987 6.0331 5.9156 5.8549 5.8827 6.0577 6.4683 7.2429 Ω 2 26.0836 27.0745 27.9483 28.6698 29.2062 29.4743 29.3543 28.6666 Ω 3 67.7177 72.5014 76.9672 81.1744 85.1868 89.0823 92.9551 96.9133  10 Ω 1 10.9632 10.8352 10.7906 10.8785 11.1850 11.8383 13.0323 15.1307 Ω 2 34.3001 35.4363 36.3539 36.9976 37.2721 37.0347 36.0682 33.9720 Ω 3 77.0509 82.1465 86.9066 91.4446 95.8797 100.3326 104.9216 109.7575  15 Ω 1 15.9065 15.8297 15.9030 16.2187 16.9153 18.1923 20.3830 24.3670 Ω 2 44.5894 45.8474 46.7372 47.1832 47.0645 46.1947 44.2444 40.3254 Ω 3 90.1006 95.6133 100.7463 105.6461 110.4616 115.3326 120.3820 125.7134Figure6 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different ratios of hub radius to beam length r0=0,0.5,1 at λ=0.5, r=1/30. It can be found from Figure 6 that, with the increase of the ratios of hub radius to beam length, the first three-order dimensionless natural frequencies increase. Figure 7 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different slenderness ratios r=1/30,1/10 at λ=0.5, r0=0. It can be seen from Figure 7 that, for two different r=1/30,1/10, the increase of slenderness ratio has scarce influence on the first-order dimensionless natural frequencies; however, it has significant influence on the second- and the third-order dimensionless natural frequency. This shows that the increase of rotary inertia makes the system natural frequency decrease, and this conclusion is consistent with those given by Timoshenko beam. Figure 8 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different taper ratios of cross-section λ=0.25,0.5,0.75 at r0=0, r=1/30. It can be seen from Figure 8 that, with the increase of taper ratio, the first three-order dimensionless natural frequencies of the system decrease. It is noted that the increase of taper ratio has scarce influence on the first-order dimensionless natural frequency, it has slight influence on the second-order dimensionless natural frequency, and it has obviously an effect on the third-order dimensionless natural frequency. It should be pointed out that, as shown in Figures 6–8, in the case of the given ratios of hub radius to beam length, the slenderness ratio, and taper ratio of cross-section, the first three dimensionless natural frequencies of rotating tapered solid beam monotonically increase with dimensionless angular speed of the hub.Figure 6
Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for variousr0=0,0.5,1 at λ=0.5, r=1/30.Figure 7
Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for variousr=1/30, 1/10 at λ=0.5, r0=0.Figure 8
Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for variousλ=0.25, 0.5, 0.75 at r=1/30, r0=0.


### 5.2. Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

For a rotating tapered cantilever beam with hollow circular cross-section, its inner diameterd≠0, that is, β≠0, and the taper ratio λ∈[β,∞) is defined. When λ=β, the outer radius of beam at free end is the same as its inner radius. When λ=1, the beam is entirely uniform hollow circular cross-section. This section will mainly discuss the effect of 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 dimensionless natural frequencies of rotating tapered hollow beams.As a particular case of hollow circular cross-section, Table4 gives the variation of the first three dimensionless natural frequencies of rotating uniform thin-wall cross-section beams with dimensionless angular speed of the hub for three different slenderness ratios r=0,1/30,1/10 at r0=0, β=0.92, and λ=1. It should be pointed out that the first three dimensionless natural frequencies of nonrotating beam in Table 4 equal the first three dimensionless natural frequencies 3.5156, 22.0336, and 61.7010 [29] of cantilever beam multiplied by 1+β2. We can see that the numerical results in the present coincide well with the existing ones [29]. The first three dimensionless natural frequencies of the rotating uniform thin-wall cross-section beam decreased as the slenderness ratio was raised.Table 4
The first three order dimensionless natural frequencies of rotating circular thin-wall cross-section beams atr0=0, β=0.92, λ=1.
ω Ω r = 0 r = 1 / 30 r = 1 / 10 Present Ref. [29] 0 Ω 1 4.7781 4.7771 4.7555 4.5844 Ω 2 29.9438 29.9393 28.9935 23.6666 Ω 3 83.8339 83.8393 77.8914 54.3817  5 Ω 1 7.2326 - 7.1843 6.8112 Ω 2 32.5368 - 31.4682 25.4442 Ω 3 86.4579 - 80.2848 55.9058  10 Ω 1 11.7433 - 11.6598 10.9967 Ω 2 39.2955 - 37.9215 30.0678 Ω 3 93.8425 - 87.0105 60.1779  15 Ω 1 16.6095 - 16.4939 15.5592 Ω 2 48.4647 - 46.6776 36.2668 Ω 3 104.8728 - 97.0268 66.4979


#### 5.2.1. Effect of the Hub Dimensionless Angular Speed and Taper Ratio of Cross-Section

Table5 gives the variation of the first three dimensionless natural frequencies of rotating tapered hollow beams with dimensionless angular speed of the hub for two different slenderness ratios r=1/30,1/10 and three ratios of inner radius to the root radius β=0.2,0.4,0.6 at λ=0.75. With this table, it is obvious that the first three dimensionless natural frequencies increase with the ratio of inner radius to the root radius, except the second-order dimensionless natural frequencies at r=1/10, ω=15.Table 5
The first three order dimensionless natural frequencies of rotating tapered hollow beams for various ratio of inner radius to the root radiusβ(r0=0,λ=0.75).
ω Ω r = 1 / 30 r = 1 / 10 β = 0.2 β = 0.4 β = 0.6 β = 0.2 β = 0.4 β = 0.6 0 Ω 1 4.0743 4.4869 5.4689 4.0021 4.3928 5.3165 Ω 2 21.1137 22.8085 25.9958 19.1933 20.4539 22.8066 Ω 3 55.0759 59.0308 65.6330 45.1361 47.2503 50.6619  5 Ω 1 6.8283 7.0878 7.7613 6.6530 6.8756 7.4647 Ω 2 24.2176 25.6314 28.3193 21.8856 22.8463 24.6981 Ω 3 58.1504 61.7981 67.8694 47.5230 49.3294 52.2535  10 Ω 1 11.5023 11.6837 12.1584 11.1927 11.3104 11.6424 Ω 2 31.7436 32.6600 34.3608 28.4415 28.8297 29.6427 Ω 3 66.4593 69.3894 74.1474 53.9527 55.0169 56.7149  15 Ω 1 16.4109 16.5675 16.9725 15.9740 16.0415 16.2467 Ω 2 41.2695 41.7660 42.5327 36.7455 36.5930 36.3560 Ω 3 78.2143 80.3361 83.4935 62.9728 63.1570 63.3260Figure9 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for λ=0.5,0.75 and r=1/30,1/10 at λ=0.5, r0=0, β=0.2. With this figure, it is also understood that the first three-order dimensionless natural frequencies of rotating tapered hollow beam monotonically increase with dimensionless angular speed of the hub. Meanwhile, it is further noted in Figure 9(a) that, with the increase of the taper ratio of cross-section, the first-order dimensionless natural frequency of the system is reduced slightly. However, in Figures 9(b) and 9(c), with the increase of the taper ratio of cross-section, the values of the second- and third-order dimensionless natural frequencies of the system are increased.Figure 9
Variation of the first three dimensionless natural frequencies of rotating tapered hollow beam with dimensionless angular speed of the hub for different values ofλ=0.5, 0.75 and r=1/30, 1/10 at r0=0, β=0.2: (a) 1st mode; (b) 2nd mode; and (c) 3rd mode.
(a) (b) (c)In addition, it is also observed in Figure9 that the increase of slenderness ratio makes three dimensionless natural frequencies of the system decrease, contrasting Figures 9(a), 9(b), and 9(c), obviously, and the effect of the slenderness ratio on the second- and third-order dimensionless natural frequencies is relatively greater than the first-order dimensionless natural frequency.


#### 5.2.2. Effect of Ratios of Hub Radius to Beam Length and Slenderness Ratio

As it was expected in Table6, the change of ratios of hub radius to beam length has no effect on the natural frequencies of nonrotating tapered beam (ω=0). It is also observed in Table 6 that, for rotating tapered beam, the dimensionless natural frequencies of the system increase with ratios of hub radius to beam length.Table 6
The first three-order dimensionless natural frequencies of rotating tapered hollow beams for various ratios of hub radius to beam lengthr0(β=0.2).
ω Ω r = 1 / 10 r = 1 / 30 r 0 = 0.1 r 0 = 0.5 r 0 = 1 r 0 = 0.1 r 0 = 0.5 r 0 = 1 0 Ω 1 4.0021 4.0021 4.0021 4.0743 4.0743 4.0743 Ω 2 19.1933 19.1933 19.1933 21.1137 21.1137 21.1137 Ω 3 45.1361 45.1361 45.1361 55.0759 55.0759 55.0759  5 Ω 1 6.9391 7.9790 9.1094 7.1163 8.1648 9.3065 Ω 2 22.2340 23.5717 25.1345 24.6043 26.0908 27.8311 Ω 3 47.8600 49.1793 50.7682 58.5679 60.2042 62.1795  10 Ω 1 11.8553 14.1905 16.6439 12.1676 14.5191 16.9959 Ω 2 29.4809 29.4809 37.4245 32.8996 37.1390 41.7938 Ω 3 55.0961 59.0961 64.2301 67.8847 73.2615 79.3845  15 Ω 1 17.0078 20.6152 24.3667 17.4498 21.0838 24.8719 Ω 2 38.5013 44.7447 51.3531 43.2329 50.2622 57.7703 Ω 3 65.0693 72.6549 80.8113 80.8610 90.5387 101.1210Figure10 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for r0=0,0.5,1 at r=1/30, λ=0.5. With this figure, the first three-order dimensionless natural frequencies of the system increase with dimensionless angular speed of the hub and ratios of hub radius to beam length.Figure 10
The first three-order dimensionless natural frequencies with dimensionless angular speed of the hub for differentr0=0,0.5,1 at β=0.3.Figure11 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different taper ratios of cross-section λ=0.5, 0.75 at ω=5, r=1/30, β=0.3. Figure 12 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different slenderness ratios r=1/30,1/10 at ω=5, λ=0.5, β=0.3. It can be seen in Figures 11 and 12 that, for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of the system are almost linearly increased. Meanwhile, it is noted that the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.Figure 11
The first three-order dimensionless natural frequencies with ratios of hub radius to beam length forλ=0.5, 0.75 at ω=5, r=1/30, and β=0.3.Figure 12
The first three-order dimensionless natural frequencies with ratios of hub radius to beam length forr=1/30, 1/10 at ω=5, λ=0.5, and β=0.3.


## 5.1. Rotating Tapered Cantilever Beam with Solid Circular Cross-Section

In this paragraph, settingβ=0 in (24), we can obtain the differential equation of motion for rotating tapered beam with solid circular cross-section, where taper ratio λ∈0,∞ is defined. It becomes evident that if the taper ratio λ=0 and λ=1, they are varying solid circular cross-section beam with zero radius at free end and entirely uniform solid circular cross-section, respectively. Prior to the presentation of our numerical results, let us first consider three particular cases to confirm the effectiveness of the present approach: a simple uniform nonrotating cantilever beam, a rotating tapered Euler-Bernoulli cantilever beam, and a rotating tapered Rayleigh cantilever beam, which is given by setting parameters ω=0, r0=0, r=0, and λ=1, parameters r0=0, r=0, and parameters r0=0, 1/r=30, respectively. In three cases, we calculate the first three-order dimensionless natural frequencies by selecting N=12 for different taper ratio of cross-section and dimensionless angular speed, and some numerical results are tabulated in Tables 1 and 2  (0≤λ≤1). From the two tables, we can see that the numerical results in the present coincide well with the existing ones [14]. These verify that the method presented in this paper is efficient and accurate. The first three dimensionless mode shapes are shown as Figure 5 for ω=10, λ=1, r0=0, β=0, and r=0.Table 1
Comparison of the first three dimensionless natural frequencies of rotating solid tapered Euler-Bernoulli beams atr0=0, β=0, r=0, 0≤λ≤1.
ω Ω λ = 1 λ = 0.75 λ = 0.5 λ = 0.25 λ = 0 Present Ref. [14] Present Ref. [14] Present Ref. [14] Present Ref. [14] Present 0 Ω 1 3.5164 3.5160 3.9570 3.9567 4.6257 4.6252 5.8293 5.8231 8.7445 Ω 2 22.0366 22.0345 20.8091 20.807 19.5514 19.5476 18.5183 18.480 20.1042 Ω 3 61.6960 61.6972 55.3500 55.3304 48.5953 48.5789 41.4611 41.321 36.5279  5 Ω 1 6.4499 6.4495 6.7732 6.7729 7.2905 7.2901 8.2653 8.2620 10.6634 Ω 2 25.4483 25.4461 24.0673 24.0660 22.6379 22.6360 21.4209 21.384 22.8601 Ω 3 65.2118 65.2050 58.6518 58.6364 51.7012 51.6918 44.4176 44.269 41.0619  10 Ω 1 11.2029 11.2023 11.4859 11.4856 11.9419 11.9415 12.7928 12.791 14.7477 Ω 2 33.6417 33.6404 31.8863 31.8895 30.0289 30.0299 28.3381 28.301 29.4213 Ω 3 74.6671 74.6493 67.5293 67.5316 60.0369 60.0399 52.2738 52.100 51.2478  15 Ω 1 16.1445 - 16.4165 - 16.8587 - 17.6745 - 19.4476 Ω 2 43.9323 - 41.7005 - 39.2756 - 36.9308 - 37.5106 Ω 3 87.9543 - 80.0153 - 71.7371 - 63.1548 - 59.9363Table 2
Comparison of the first three dimensionless natural frequencies of rotating solid tapered Rayleigh beams for differentλ(0≤λ≤1) at r0=0, β=0, 1/r=30.
ω Ω λ = 1 λ = 0.75 λ = 0.5 λ = 0.25 λ = 0 Present Ref. [14] Present Ref. [14] Present Ref. [14] Present Ref. [14] Present 0 Ω 1 3.5073 3.5070 3.9486 3.9483 4.6174 4.6168 5.8198 5.8136 8.7267 Ω 2 21.6497 21.6477 20.5498 20.5475 19.3883 19.3846 18.4214 18.3834 20.0372 Ω 3 59.2069 59.2073 53.7093 53.6911 47.6173 47.6021 40.9588 40.825 36.5169  5 Ω 1 6.4254 6.4251 6.7524 6.7521 7.2720 7.2717 8.2469 8.2436 10.6360 Ω 2 24.9759 24.9737 23.7510 23.7497 22.4394 22.4375 21.3029 21.266 22.7786 Ω 3 62.5464 663.5392 56.8916 56.8774 50.6483 50.6397 43.8720 43.730 41.0472  10 Ω 1 11.1604 11.1598 11.4494 11.4490 11.9087 11.9083 12.7584 12.767 14.6997 Ω 2 32.9667 32.9654 31.4395 31.4425 29.7515 29.7521 28.1722 28.136 29.3002 Ω 3 71.5142 71.4977 65.4444 65.4472 58.7850 58.7878 51.6155 51.450 50.5772  15 Ω 1 16.0851 - 16.3650 - 16.8105 - 17.6223 - 19.3744 Ω 2 43.0010 - 41.0951 - 38.9051 - 36.7064 - 37.3335 Ω 3 84.0806 - 77.4631 - 70.2081 - 62.3423 - 59.2553Figure 5
The first three dimensionless mode shapes forω=10, λ=1, r0=0, β=0, and r=0.Furthermore, in the case ofλ>1, for example, 1.25≤λ≤3.00, that is, the radius of beam cross-section at free end is more than one at the cantilevered end, the first three-order dimensionless natural frequencies of a rotating tapered Rayleigh cantilever solid beam for three dimensionless angular speeds are tabulated in Table 3. It can be seen in Table 3 that as a whole, the first three-order dimensionless natural frequencies increase with the increase of dimensionless angular speeds.Table 3
The first three dimensionless natural frequencies of rotating solid tapered Rayleigh beams for differentλ(1.25≤λ≤3) at r0=0, β=0, 1/r=30.
ω Ω λ = 1.25 λ = 1.50 λ = 1.75 λ = 2.00 λ = 2.25 λ = 2.5 λ = 2.75 λ = 3.00 0 Ω 1 3.1877 2.9421 2.7461 2.5882 2.4704 2.4129 2.4613 2.6858 Ω 2 22.6579 23.5718 24.3952 25.1281 25.7580 26.2480 26.5270 26.4787 Ω 3 64.2417 68.9100 73.2615 77.3249 81.1317 84.7333 88.2071 91.6568  5 Ω 1 6.1987 6.0331 5.9156 5.8549 5.8827 6.0577 6.4683 7.2429 Ω 2 26.0836 27.0745 27.9483 28.6698 29.2062 29.4743 29.3543 28.6666 Ω 3 67.7177 72.5014 76.9672 81.1744 85.1868 89.0823 92.9551 96.9133  10 Ω 1 10.9632 10.8352 10.7906 10.8785 11.1850 11.8383 13.0323 15.1307 Ω 2 34.3001 35.4363 36.3539 36.9976 37.2721 37.0347 36.0682 33.9720 Ω 3 77.0509 82.1465 86.9066 91.4446 95.8797 100.3326 104.9216 109.7575  15 Ω 1 15.9065 15.8297 15.9030 16.2187 16.9153 18.1923 20.3830 24.3670 Ω 2 44.5894 45.8474 46.7372 47.1832 47.0645 46.1947 44.2444 40.3254 Ω 3 90.1006 95.6133 100.7463 105.6461 110.4616 115.3326 120.3820 125.7134Figure6 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different ratios of hub radius to beam length r0=0,0.5,1 at λ=0.5, r=1/30. It can be found from Figure 6 that, with the increase of the ratios of hub radius to beam length, the first three-order dimensionless natural frequencies increase. Figure 7 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different slenderness ratios r=1/30,1/10 at λ=0.5, r0=0. It can be seen from Figure 7 that, for two different r=1/30,1/10, the increase of slenderness ratio has scarce influence on the first-order dimensionless natural frequencies; however, it has significant influence on the second- and the third-order dimensionless natural frequency. This shows that the increase of rotary inertia makes the system natural frequency decrease, and this conclusion is consistent with those given by Timoshenko beam. Figure 8 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different taper ratios of cross-section λ=0.25,0.5,0.75 at r0=0, r=1/30. It can be seen from Figure 8 that, with the increase of taper ratio, the first three-order dimensionless natural frequencies of the system decrease. It is noted that the increase of taper ratio has scarce influence on the first-order dimensionless natural frequency, it has slight influence on the second-order dimensionless natural frequency, and it has obviously an effect on the third-order dimensionless natural frequency. It should be pointed out that, as shown in Figures 6–8, in the case of the given ratios of hub radius to beam length, the slenderness ratio, and taper ratio of cross-section, the first three dimensionless natural frequencies of rotating tapered solid beam monotonically increase with dimensionless angular speed of the hub.Figure 6
Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for variousr0=0,0.5,1 at λ=0.5, r=1/30.Figure 7
Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for variousr=1/30, 1/10 at λ=0.5, r0=0.Figure 8
Variation of the first three dimensionless natural frequencies of rotating solid tapered beams with dimensionless angular speed of the hub for variousλ=0.25, 0.5, 0.75 at r=1/30, r0=0.


## 5.2. Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

For a rotating tapered cantilever beam with hollow circular cross-section, its inner diameterd≠0, that is, β≠0, and the taper ratio λ∈[β,∞) is defined. When λ=β, the outer radius of beam at free end is the same as its inner radius. When λ=1, the beam is entirely uniform hollow circular cross-section. This section will mainly discuss the effect of 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 dimensionless natural frequencies of rotating tapered hollow beams.As a particular case of hollow circular cross-section, Table4 gives the variation of the first three dimensionless natural frequencies of rotating uniform thin-wall cross-section beams with dimensionless angular speed of the hub for three different slenderness ratios r=0,1/30,1/10 at r0=0, β=0.92, and λ=1. It should be pointed out that the first three dimensionless natural frequencies of nonrotating beam in Table 4 equal the first three dimensionless natural frequencies 3.5156, 22.0336, and 61.7010 [29] of cantilever beam multiplied by 1+β2. We can see that the numerical results in the present coincide well with the existing ones [29]. The first three dimensionless natural frequencies of the rotating uniform thin-wall cross-section beam decreased as the slenderness ratio was raised.Table 4
The first three order dimensionless natural frequencies of rotating circular thin-wall cross-section beams atr0=0, β=0.92, λ=1.
ω Ω r = 0 r = 1 / 30 r = 1 / 10 Present Ref. [29] 0 Ω 1 4.7781 4.7771 4.7555 4.5844 Ω 2 29.9438 29.9393 28.9935 23.6666 Ω 3 83.8339 83.8393 77.8914 54.3817  5 Ω 1 7.2326 - 7.1843 6.8112 Ω 2 32.5368 - 31.4682 25.4442 Ω 3 86.4579 - 80.2848 55.9058  10 Ω 1 11.7433 - 11.6598 10.9967 Ω 2 39.2955 - 37.9215 30.0678 Ω 3 93.8425 - 87.0105 60.1779  15 Ω 1 16.6095 - 16.4939 15.5592 Ω 2 48.4647 - 46.6776 36.2668 Ω 3 104.8728 - 97.0268 66.4979


### 5.2.1. Effect of the Hub Dimensionless Angular Speed and Taper Ratio of Cross-Section

Table5 gives the variation of the first three dimensionless natural frequencies of rotating tapered hollow beams with dimensionless angular speed of the hub for two different slenderness ratios r=1/30,1/10 and three ratios of inner radius to the root radius β=0.2,0.4,0.6 at λ=0.75. With this table, it is obvious that the first three dimensionless natural frequencies increase with the ratio of inner radius to the root radius, except the second-order dimensionless natural frequencies at r=1/10, ω=15.Table 5
The first three order dimensionless natural frequencies of rotating tapered hollow beams for various ratio of inner radius to the root radiusβ(r0=0,λ=0.75).
ω Ω r = 1 / 30 r = 1 / 10 β = 0.2 β = 0.4 β = 0.6 β = 0.2 β = 0.4 β = 0.6 0 Ω 1 4.0743 4.4869 5.4689 4.0021 4.3928 5.3165 Ω 2 21.1137 22.8085 25.9958 19.1933 20.4539 22.8066 Ω 3 55.0759 59.0308 65.6330 45.1361 47.2503 50.6619  5 Ω 1 6.8283 7.0878 7.7613 6.6530 6.8756 7.4647 Ω 2 24.2176 25.6314 28.3193 21.8856 22.8463 24.6981 Ω 3 58.1504 61.7981 67.8694 47.5230 49.3294 52.2535  10 Ω 1 11.5023 11.6837 12.1584 11.1927 11.3104 11.6424 Ω 2 31.7436 32.6600 34.3608 28.4415 28.8297 29.6427 Ω 3 66.4593 69.3894 74.1474 53.9527 55.0169 56.7149  15 Ω 1 16.4109 16.5675 16.9725 15.9740 16.0415 16.2467 Ω 2 41.2695 41.7660 42.5327 36.7455 36.5930 36.3560 Ω 3 78.2143 80.3361 83.4935 62.9728 63.1570 63.3260Figure9 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for λ=0.5,0.75 and r=1/30,1/10 at λ=0.5, r0=0, β=0.2. With this figure, it is also understood that the first three-order dimensionless natural frequencies of rotating tapered hollow beam monotonically increase with dimensionless angular speed of the hub. Meanwhile, it is further noted in Figure 9(a) that, with the increase of the taper ratio of cross-section, the first-order dimensionless natural frequency of the system is reduced slightly. However, in Figures 9(b) and 9(c), with the increase of the taper ratio of cross-section, the values of the second- and third-order dimensionless natural frequencies of the system are increased.Figure 9
Variation of the first three dimensionless natural frequencies of rotating tapered hollow beam with dimensionless angular speed of the hub for different values ofλ=0.5, 0.75 and r=1/30, 1/10 at r0=0, β=0.2: (a) 1st mode; (b) 2nd mode; and (c) 3rd mode.
(a) (b) (c)In addition, it is also observed in Figure9 that the increase of slenderness ratio makes three dimensionless natural frequencies of the system decrease, contrasting Figures 9(a), 9(b), and 9(c), obviously, and the effect of the slenderness ratio on the second- and third-order dimensionless natural frequencies is relatively greater than the first-order dimensionless natural frequency.


### 5.2.2. Effect of Ratios of Hub Radius to Beam Length and Slenderness Ratio

As it was expected in Table6, the change of ratios of hub radius to beam length has no effect on the natural frequencies of nonrotating tapered beam (ω=0). It is also observed in Table 6 that, for rotating tapered beam, the dimensionless natural frequencies of the system increase with ratios of hub radius to beam length.Table 6
The first three-order dimensionless natural frequencies of rotating tapered hollow beams for various ratios of hub radius to beam lengthr0(β=0.2).
ω Ω r = 1 / 10 r = 1 / 30 r 0 = 0.1 r 0 = 0.5 r 0 = 1 r 0 = 0.1 r 0 = 0.5 r 0 = 1 0 Ω 1 4.0021 4.0021 4.0021 4.0743 4.0743 4.0743 Ω 2 19.1933 19.1933 19.1933 21.1137 21.1137 21.1137 Ω 3 45.1361 45.1361 45.1361 55.0759 55.0759 55.0759  5 Ω 1 6.9391 7.9790 9.1094 7.1163 8.1648 9.3065 Ω 2 22.2340 23.5717 25.1345 24.6043 26.0908 27.8311 Ω 3 47.8600 49.1793 50.7682 58.5679 60.2042 62.1795  10 Ω 1 11.8553 14.1905 16.6439 12.1676 14.5191 16.9959 Ω 2 29.4809 29.4809 37.4245 32.8996 37.1390 41.7938 Ω 3 55.0961 59.0961 64.2301 67.8847 73.2615 79.3845  15 Ω 1 17.0078 20.6152 24.3667 17.4498 21.0838 24.8719 Ω 2 38.5013 44.7447 51.3531 43.2329 50.2622 57.7703 Ω 3 65.0693 72.6549 80.8113 80.8610 90.5387 101.1210Figure10 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for r0=0,0.5,1 at r=1/30, λ=0.5. With this figure, the first three-order dimensionless natural frequencies of the system increase with dimensionless angular speed of the hub and ratios of hub radius to beam length.Figure 10
The first three-order dimensionless natural frequencies with dimensionless angular speed of the hub for differentr0=0,0.5,1 at β=0.3.Figure11 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different taper ratios of cross-section λ=0.5, 0.75 at ω=5, r=1/30, β=0.3. Figure 12 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different slenderness ratios r=1/30,1/10 at ω=5, λ=0.5, β=0.3. It can be seen in Figures 11 and 12 that, for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of the system are almost linearly increased. Meanwhile, it is noted that the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.Figure 11
The first three-order dimensionless natural frequencies with ratios of hub radius to beam length forλ=0.5, 0.75 at ω=5, r=1/30, and β=0.3.Figure 12
The first three-order dimensionless natural frequencies with ratios of hub radius to beam length forr=1/30, 1/10 at ω=5, λ=0.5, and β=0.3.


## 5.2.1. Effect of the Hub Dimensionless Angular Speed and Taper Ratio of Cross-Section

Table5 gives the variation of the first three dimensionless natural frequencies of rotating tapered hollow beams with dimensionless angular speed of the hub for two different slenderness ratios r=1/30,1/10 and three ratios of inner radius to the root radius β=0.2,0.4,0.6 at λ=0.75. With this table, it is obvious that the first three dimensionless natural frequencies increase with the ratio of inner radius to the root radius, except the second-order dimensionless natural frequencies at r=1/10, ω=15.Table 5
The first three order dimensionless natural frequencies of rotating tapered hollow beams for various ratio of inner radius to the root radiusβ(r0=0,λ=0.75).
ω Ω r = 1 / 30 r = 1 / 10 β = 0.2 β = 0.4 β = 0.6 β = 0.2 β = 0.4 β = 0.6 0 Ω 1 4.0743 4.4869 5.4689 4.0021 4.3928 5.3165 Ω 2 21.1137 22.8085 25.9958 19.1933 20.4539 22.8066 Ω 3 55.0759 59.0308 65.6330 45.1361 47.2503 50.6619  5 Ω 1 6.8283 7.0878 7.7613 6.6530 6.8756 7.4647 Ω 2 24.2176 25.6314 28.3193 21.8856 22.8463 24.6981 Ω 3 58.1504 61.7981 67.8694 47.5230 49.3294 52.2535  10 Ω 1 11.5023 11.6837 12.1584 11.1927 11.3104 11.6424 Ω 2 31.7436 32.6600 34.3608 28.4415 28.8297 29.6427 Ω 3 66.4593 69.3894 74.1474 53.9527 55.0169 56.7149  15 Ω 1 16.4109 16.5675 16.9725 15.9740 16.0415 16.2467 Ω 2 41.2695 41.7660 42.5327 36.7455 36.5930 36.3560 Ω 3 78.2143 80.3361 83.4935 62.9728 63.1570 63.3260Figure9 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for λ=0.5,0.75 and r=1/30,1/10 at λ=0.5, r0=0, β=0.2. With this figure, it is also understood that the first three-order dimensionless natural frequencies of rotating tapered hollow beam monotonically increase with dimensionless angular speed of the hub. Meanwhile, it is further noted in Figure 9(a) that, with the increase of the taper ratio of cross-section, the first-order dimensionless natural frequency of the system is reduced slightly. However, in Figures 9(b) and 9(c), with the increase of the taper ratio of cross-section, the values of the second- and third-order dimensionless natural frequencies of the system are increased.Figure 9
Variation of the first three dimensionless natural frequencies of rotating tapered hollow beam with dimensionless angular speed of the hub for different values ofλ=0.5, 0.75 and r=1/30, 1/10 at r0=0, β=0.2: (a) 1st mode; (b) 2nd mode; and (c) 3rd mode.
(a) (b) (c)In addition, it is also observed in Figure9 that the increase of slenderness ratio makes three dimensionless natural frequencies of the system decrease, contrasting Figures 9(a), 9(b), and 9(c), obviously, and the effect of the slenderness ratio on the second- and third-order dimensionless natural frequencies is relatively greater than the first-order dimensionless natural frequency.


## 5.2.2. Effect of Ratios of Hub Radius to Beam Length and Slenderness Ratio

As it was expected in Table6, the change of ratios of hub radius to beam length has no effect on the natural frequencies of nonrotating tapered beam (ω=0). It is also observed in Table 6 that, for rotating tapered beam, the dimensionless natural frequencies of the system increase with ratios of hub radius to beam length.Table 6
The first three-order dimensionless natural frequencies of rotating tapered hollow beams for various ratios of hub radius to beam lengthr0(β=0.2).
ω Ω r = 1 / 10 r = 1 / 30 r 0 = 0.1 r 0 = 0.5 r 0 = 1 r 0 = 0.1 r 0 = 0.5 r 0 = 1 0 Ω 1 4.0021 4.0021 4.0021 4.0743 4.0743 4.0743 Ω 2 19.1933 19.1933 19.1933 21.1137 21.1137 21.1137 Ω 3 45.1361 45.1361 45.1361 55.0759 55.0759 55.0759  5 Ω 1 6.9391 7.9790 9.1094 7.1163 8.1648 9.3065 Ω 2 22.2340 23.5717 25.1345 24.6043 26.0908 27.8311 Ω 3 47.8600 49.1793 50.7682 58.5679 60.2042 62.1795  10 Ω 1 11.8553 14.1905 16.6439 12.1676 14.5191 16.9959 Ω 2 29.4809 29.4809 37.4245 32.8996 37.1390 41.7938 Ω 3 55.0961 59.0961 64.2301 67.8847 73.2615 79.3845  15 Ω 1 17.0078 20.6152 24.3667 17.4498 21.0838 24.8719 Ω 2 38.5013 44.7447 51.3531 43.2329 50.2622 57.7703 Ω 3 65.0693 72.6549 80.8113 80.8610 90.5387 101.1210Figure10 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for r0=0,0.5,1 at r=1/30, λ=0.5. With this figure, the first three-order dimensionless natural frequencies of the system increase with dimensionless angular speed of the hub and ratios of hub radius to beam length.Figure 10
The first three-order dimensionless natural frequencies with dimensionless angular speed of the hub for differentr0=0,0.5,1 at β=0.3.Figure11 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different taper ratios of cross-section λ=0.5, 0.75 at ω=5, r=1/30, β=0.3. Figure 12 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different slenderness ratios r=1/30,1/10 at ω=5, λ=0.5, β=0.3. It can be seen in Figures 11 and 12 that, for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of the system are almost linearly increased. Meanwhile, it is noted that the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.Figure 11
The first three-order dimensionless natural frequencies with ratios of hub radius to beam length forλ=0.5, 0.75 at ω=5, r=1/30, and β=0.3.Figure 12
The first three-order dimensionless natural frequencies with ratios of hub radius to beam length forr=1/30, 1/10 at ω=5, λ=0.5, and β=0.3.


## 6. Conclusion

In this paper, a new type of transverse vibration of a rotating tapered cantilever beam with linearly varying solid and hollow circular cross-section, that is, rotating tapered beam, was presented. The rotating beam, which is considered a tapered cantilever beam, is modeled by the Rayleigh beam theory. Considering 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. A differential quadrature method to solve the above-mentioned differential equation with variable coefficients was employed to simulate the dynamical behaviors of tapered rotating beams. Also, for two types of rotating tapered beams with solid and hollow circular cross-section, 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 depicted. The main results of this study are summarized as follows.When the rotating angular speed is constant, in the case of the given ratios of the hub radius to beam length, the slenderness ratio, and the taper ratio of cross-section, the first three-order dimensionless natural frequencies of rotating tapered solid and hollow beams monotonically ascend as the hub dimensionless angular speed increases. For a rotating tapered hollow beam at a constant angular speed, the first-order dimensionless natural frequency of the system is reduced slightly with the increase of the taper ratio of cross-section, the values of the second- and the third-order dimensionless natural frequencies of the system are increased; for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of system almost linearly increase, and the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.



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*Source: 1056397-2018-03-28.xml*