# Dynamic Analysis of a Tapered Composite Thin-Walled Rotating Shaft Using the Generalized Differential Quadrature Method

Authors: Weiyan Zhong; Feng Gao; Yongsheng Ren; Xiaoxiao Wu; Hongcan Ma
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A dynamic model of a tapered composite thin-walled rotating shaft is presented. In this model, the transverse shear deformation, rotary inertia, and gyroscopic effects have been incorporated. The equations of motion are derived based on a refined variational asymptotic method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the generalized differential quadrature method (GDQM). The validity of the dynamic model is proved by comparing the numerical results with those obtained in the literature and by using ANSYS. The effects of taper ratio, boundary conditions, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the natural frequencies and critical rotating speeds are investigated.


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## Body

## 1. Introduction

Composite materials have found numerous applications in many engineering fields, thanks to their outstanding engineering properties, such as high strength, high specific stiffness, light weight, and design ability. Specially, rotating composite shafts are widely used in the aerospace industry, automobile industry, and shipbuilding industry. Research on dynamic characteristics of rotating composite shaft is a focus of composite material structural dynamics. Chen and Peng [1, 2] studied the dynamic behavior of a rotating shaft subjected to axial periodic forces using the finite element method. Song et al. [3] analyzed the vibration and stability behavior of spinning circular cylindrical shafts modeled as thin-walled composite beams considering the effects of transverse shear, rotatory inertias, the axial compressive load, and various boundary conditions. Chang et al. [4] presented a simple spinning composite shaft model based on a first-order shear deformable beam theory. To determine the spinning shaft system’s responses, they used the numerical finite element method to approximate the governing equations by a system of ordinary differential equations. Chang et al. [5] analyzed the vibration behaviors of rotating composite shafts containing randomly oriented reinforcements. Sino et al. [6] developed a simplified homogenized beam theory (SHBT) to analysis the sensitivity of the frequencies and instability thresholds regarding shear effect, stacking order, and fiber orientation. Boukhalfa et al. [7] investigated free vibration analysis of rotating composite shafts on rigid bearings using the p-version hierarchical finite element method with trigonometric shape functions. Boukhalfa and Hadjoui [8] also presented the study of the vibratory behavior of rotating composite shafts. An hp-version of the finite element method was used to model the structure. Ren et al. [9] presented a dynamical model of a rotating composite shaft and studied dynamic characteristics of the shaft for simple support at the ends using the Galerkin method to discretize and solve the governing equations. Arab et al. [10] developed a finite element based on Equivalent Single Layer Theory (ESLT) to model the rotating composite shaft using the Timoshenko beam theory. Furthermore, Layerwise Shaft Theory (LST) is developed based on shaft finite element theory by Arab et al. [11]. They investigated the dynamic analysis of rotating laminated shafts including t the influences of stacking sequence, fiber orientation, and shear-normal coupling. Zhong et al. [12] conducted the free vibration analysis of a rotating composite thin-walled shaft using the generalized differential quadrature method.Several authors have focused their attention on the development of tapered shaft rotors. Bauchau [13] treated the optimization of the tapered wall thickness of hollow shafts using Rayleigh. A dynamic model was developed by Kim et al. [14–17] on tapered composite Timoshenko shafts which runs around its axis at a constant speed, which are used by the general Galerkin method. Na et al. [18] evaluated the vibration and stability of a cylindrical shaft modeled as a tapered thin-walled composite beam and adopted the extended Galerkin method to solve the eigenvalue problem. Ma et al. [19] studied the free vibration characteristics and stability of variable cross-section composite shaft for cantilever boundary condition and also used the Galerkin method to discretize and solve the governing equations. Rachid et al. [20] proposed a theoretical and numerical study on the behavior of a tapered shaft rotor made of composite materials by the classical version h and the version p of the finite element method.In this paper, a dynamic model for the analysis of a tapered composite thin-walled rotating shaft is presented. Based on a refined variational asymptotic method and Hamilton’s principle, the motion equations of the tapered composite rotating shaft are derived. To solve the motion equations of the shaft, the GDQM is carried out here. The natural frequencies and critical rotating speeds of the tapered composite thin-walled rotating shaft with different boundary conditions are analyzed. The validity of the model is proved by comparing the numerical results with those in the literature. The effects of taper ratio, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the natural frequencies, and critical rotating speeds of the tapered rotating shaft are then analyzed.


## 2. Model of Tapered Composite Thin-Walled Rotating Shaft

The model of a tapered composite thin-walled rotating shaft is shown in Figure1. The shaft rotates along its longitudinal x-axis with constant rate Ω. L, h, rR, and rT denote the length, the thickness, the radius of curvature of root middle surface, and the radius of curvature of tip middle surface of the shaft, respectively. As shown in Figure 1, (X,Y,Z) is the inertial coordinate system, (x,y,z) represents the rotating reference system, and (x,s,ξ) is a local coordinate system. Associated with the systems (X,Y,Z) and (x,y,z), they have the common origin O at the geometric center and the corresponding unit vectors are (I, J, K) and (i, j, k), respectively.Figure 1
Coordinate systems and geometry of a tapered composite thin-walled rotating shaft.The linear distribution along the tapered shaft of the radius of curvature of the mid-line cross-sectionsrζ varies according to the relationship [18]:(1)rζ=1−ζ1−σrR,where σ=rT/rR denotes the taper ratio and ζ=x/L (0≤ζ≤1) is the dimensionless cross-section coordinate.


## 3. Motion Equations

### 3.1. Strain Energy and Kinetic Energy of the Composite Rotating Shaft

The displacement function of the composite shaft based on the refined VAM thin-walled beam theory can be assumed as the following form [9]:(2)ux,y,z,t=Ux,t−ysψyx,t−zsψzx,t+gs,x,t,vx,y,z,t=Vx,t−zsϕx,t,wx,y,z,t=Wx,t+ysϕx,t,where Ux,t, Vx,t, and Wx,t denote the rigid body displacements along the x-, y-, and z-axis and ϕx,t, ψyx,t, and ψzx,t are the twist about x-axis and rotations about y- and z-axis, respectively.In which,ψyx,t and ψzx,t can be expressed as follows:(3)ψyx,t=V′x,t−2γzx,ψzx,t=W′x,t−2γyx,where γzx and γyx are the transverse shear strains in the planes xz and xy, respectively.From the classical VAM, the warping displacement functiongs,x,tis assumed as follows:(4)gs,x,t=Gsϕ′x,t+g1sU′x,t+g2sψy′x,t+g3sψz′x,t,where Gs,g1s,g2s, and g3s are associated with physical behavior for the torsion twist rate, the axial strain, and the bending curvatures, respectively.According to equations (2)–(4), the strains of the composite shaft are obtained by(5)εxx=U′x,t−ysψy′x,t−zsψz′x,t,2γxs=dgds+rnϕ′x,t+V′x,t−ψyx,tdyds+W′x,t−ψzx,tdzds,2γxξ=V′x,t−ψyx,tdzds+W′x,t−ψzx,tdyds,where rn denotes the normal projection of r which is the position vector of an arbitrary point on the cross section of the deformed shaft, and rn can be expressed as follows:(6)rn=ysdzsds−zsdysds.The position vectorr can be written as follows:(7)r=y+vi+z+wj+x+uk.From the above equation, the velocity of an arbitrary point can be given by(8)V=r˙=v˙−Ωz+wi+w˙+Ωy+vj+u˙k.The kinetic energy of the composite rotating shaftT can be written as follows:(9)T=12∫0L∬AρV⋅VdAdx.The strain energy of the composite shaftU can be expressed as follows:(10)U=12∫0L∬Aσxxεxx+τxsγxs+τxξγxξdAdx,where σxx, τxs, and τxξ represent the cross-section normal stress, in-plane shear stress, and transverse shear stress, respectively. εxx, γxs, and γxξare associated engineering strains.


### 3.2. Motion Equations

The governing equations of the composite rotating shaft can be derived based on Hamilton's principle, which is of the following form:(11)∫t0t1δU−δTdt=0.In the present study, a special ply-angle distribution referred to as circumferentially uniform stiffness (CUS) configuration [21] is considered. By employing Hamilton’s principle and considering the variable cross-section of the tapered shaft, the motion equations involving CUS configuration bending-transverse shear coupling can be written as follows:(12)−C35xψz″−C55xV″−ψy′+mcV¨−2ΩW˙−Ω2V=0,−C46xψy″−C66xW″−ψz′+mcW¨+2ΩV˙−Ω2W=0,−C44xψy″−C46xW″−ψz′−C35xψz′−C55xV′−ψy−Izψ¨y=0,−C33xψz″−C35xV″−ψy′−C46xψy′−C66xW′−ψz−Iyψ¨z=0,in which(13)C33x=∮ΓxA−B2Cz2ds+∮ΓxB/Czds2∮Γx1/Cds,C35x=−12∮ΓxBzdydsds,C44x=κ∮ΓxA−B2Cy2ds+∮ΓxB/Cyds2∮Γx1/Cds,C46x=−12κ∮ΓxBydzdsds,C55x=κ∮Γx14Cdyds2+Ddzds2ds,C66x=κ∮Γx14Cdzds2+Ddyds2ds,where(14)A=A11−A122A22,B=2A16−A12A26A22,C=4A66−A262A22,D=A44−A452A55,Aij=∑k=1NQ¯ijkzk−zk−1,i,j=1,2,6;i,j=4,5.ParametersA and B denote the reduced axial and coupling stiffness, parameters C and D are the reduced shear stiffness, and Aij is the local stretching stiffness,Q¯ijk is transformed stiffness of the kth layer. κ is shear factor of the cross-section which changes with the cross-section and material properties and is of the following form [14]:(15)κ=6Exx1−m¯41+m¯2Gxyvxy2m¯6+18m¯4−18m¯2−2−Exx7m¯6+27m¯4−27m¯2−7,where m¯ = r/R. r and R are the inner and outer radius, respectively. Exx, Gxy, and vxy are elastic modulus, shear modulus, and Poisson’s ratio:(16)Q¯=T−1QT−T,in which(17)T=m2n2002mnn2m200−2mn00m−n000nm0−mnmn00m2−n2,Q=Q11Q1200Q16Q21Q2200Q2600Q4400000Q5500000Q66,where T is transformation matrix, Q is the stiffness matrix, and m=cosα and n=sinα, α is the angle made by fiber direction with respect to the x-axis of the coordinate system (x,s,ξ). The stiffness arrays Qij are determined according to the material properties of the lamina:(18)Q11=E111−v12v21,Q12=Q21=v12E221−v12v21,Q22=E221−v12v21,Q44=κG23,Q55=κG13,Q66=G12.In addition, the mass termsmc, Iz, and Iy in (14) are expressed as(19)mc=∬AρdA,Iz=∬Aρy2dA,Iy=∬Aρz2dA.The motion equation (12) can be used for a tapered composite thin-walled rotating shaft with arbitrary boundary conditions. In this paper, three boundary conditions for the composite shaft are as follows [12]:(a)
Simply supported-simply supported (S-S):(20)V=0,W=0,My=0,Mz=0x=0,L.(b)
Clamped-clamped (C-C):(21)V=0,W=0,Ψy=0,Ψz=0x=0,L.(c)
Clamped-free (C-F):(22)V=0,W=0,Ψy=0,Ψz=0x=0,Qy=0,Qz=0,My=0,Mz=0x=L.


### 3.3. Approximate Solution Method

The GDQM is a global numerical approximate technique. The derivative of a sufficiently smooth functionfx,t at the jth discrete point can be approximated by weighted sums of the function values at all the discrete points. By applying the GDQM, the pth order derivative of fx,t is given by [22](23)∂pfx,t∂xpx=xj=∑k=1NGPCjkpfxk,t,j=1,2,…,NGP,where NGP is the number of total discrete grid points in thex direction and Cjkp is the corresponding weighting coefficient associated to the pth-order derivative which is obtained as below.For the first-order derivative, the weighting coefficients are given by(24)Cjk1=M1xjxj−xkM1xk,j≠k,j,k=1,2,…,NGP,Cjj1=−∑k=1k≠jNGPCjk1,j=1,2,…,NGP,where(25)M1xk=∏j=1j≠kNGPxk−xj.For the second and higher order derivatives, the weighting coefficients are acquired by using the following recurrence formulation:(26)Cjkp=pCjk1⋅Cjjp−1−Cjkp−1xj−xk,j≠k,j,k=1,2,…,NGP,p=2,3,…NGP,Cjjp=−∑k=1k≠jNGPCjkP,j=1,2,…,NGP,and the grid points are chosen as follows:(27)xj=L21−cosj−1NGP−1π,j=1,2,…,NGP.To find the approximate solution of the composite rotating shaft, bending deformation and bending angle are assumed as follows:(28)vx,t=Vxeiωt,wx,t=Wxeiωt,ψyx,t=Ψyxeiωt,ψzx,t=Ψzxeiωt,where Vx, Wx, Ψyx, and Ψzx are the indefinite functions of the x-direction, ω is the natural frequency, and i=−1.Substituting the displacement components (28) into the motion equations (12), a set of ordinary differential equations with variable coefficients toward the x-direction is derived:(29)L∗U∗=0,where U∗T=Vx,Wx,Ψyx,Ψzx is an unknown spatial function vector of mode shape, and L∗=Lij∗i,j=1,…,4 is a 4 × 4 matrix of U∗ and is defined as follows:(30)L∗=−ω2M1+iωG1+K1+KΩ,where(31)M1=mc0000mc0000−Iz0000−Iy,G1=0−2Ωmc002Ωmc00000000000,KΩ=−mcΩ20000−mcΩ20000000000,K1=−C55x∂2∂x20C55x∂∂x−C35x∂2∂x20−C66x∂2∂x2−C46x∂2∂x2C66x∂∂x−C55x∂∂x−C46x∂2∂x2−C44x∂2∂x2+C55xC46x−C35x∂∂x−C35x∂2∂x2−C66x∂∂xC35x−C46x∂∂x−C33x∂2∂x2+C66x.By imposing equations (23) to (29) and rearranging equation (29) according to the orders of derivatives, the approximate governing equations are obtained in the form of linear discrete algebraic equations:(32)L∗∗U∗∗x=xj=L4×15∗∗U15×1∗∗x=xj=0,j=1,2,…,NGP,where NGP is the total discrete grid points in the x-direction and U∗∗is written as follows:(33)U∗∗Tx=xj=Vxj,V1xj,V2xj,Wxj,V1xj,V2xj,Ψyxj,Ψy1xj,Ψy2xj,Ψzxj,Ψz1xj,Ψz2xj,where(34)Vpxj=∑k=1NGPCjkpVxk,t,Wpxj=∑k=1NGPCjkpWxk,t,Ψypxj=∑k=1NGPCjkpΨyxk,t,Ψzpxj=∑k=1NGPCjkpΨyxk,t,p=1,2.For given boundary conditions, imposing equation (32) on every discrete grid point, the discretized form of the boundary conditions is achieved.Therefore, the eigenvalue equation is obtained as follows:(35)Mω2+Giω+Kd=0,where M, G, and K are the 4NGP−2×4NGP−2numerical coefficient matrices. The dimensions of vector d is 4NGP−2 which can be expressed as follows:(36)dT=Vx2,…,VxNGP−1,Wx2,…,WxNGP−1,Ψyx2,…,ΨyxNGP−1,Ψzx2,…,ΨzxNGP−1.For a certain frequency, the eigenvalue equation (35) can be transformed into a standard form as follows [23]:(37)0I−K−G−I00Mωdωd=0,where I is the 4NGP−2×4NGP−2 identity matrix.The vibration characteristics of the tapered composite rotating shaft can be obtained by solving (37).


## 3.1. Strain Energy and Kinetic Energy of the Composite Rotating Shaft

The displacement function of the composite shaft based on the refined VAM thin-walled beam theory can be assumed as the following form [9]:(2)ux,y,z,t=Ux,t−ysψyx,t−zsψzx,t+gs,x,t,vx,y,z,t=Vx,t−zsϕx,t,wx,y,z,t=Wx,t+ysϕx,t,where Ux,t, Vx,t, and Wx,t denote the rigid body displacements along the x-, y-, and z-axis and ϕx,t, ψyx,t, and ψzx,t are the twist about x-axis and rotations about y- and z-axis, respectively.In which,ψyx,t and ψzx,t can be expressed as follows:(3)ψyx,t=V′x,t−2γzx,ψzx,t=W′x,t−2γyx,where γzx and γyx are the transverse shear strains in the planes xz and xy, respectively.From the classical VAM, the warping displacement functiongs,x,tis assumed as follows:(4)gs,x,t=Gsϕ′x,t+g1sU′x,t+g2sψy′x,t+g3sψz′x,t,where Gs,g1s,g2s, and g3s are associated with physical behavior for the torsion twist rate, the axial strain, and the bending curvatures, respectively.According to equations (2)–(4), the strains of the composite shaft are obtained by(5)εxx=U′x,t−ysψy′x,t−zsψz′x,t,2γxs=dgds+rnϕ′x,t+V′x,t−ψyx,tdyds+W′x,t−ψzx,tdzds,2γxξ=V′x,t−ψyx,tdzds+W′x,t−ψzx,tdyds,where rn denotes the normal projection of r which is the position vector of an arbitrary point on the cross section of the deformed shaft, and rn can be expressed as follows:(6)rn=ysdzsds−zsdysds.The position vectorr can be written as follows:(7)r=y+vi+z+wj+x+uk.From the above equation, the velocity of an arbitrary point can be given by(8)V=r˙=v˙−Ωz+wi+w˙+Ωy+vj+u˙k.The kinetic energy of the composite rotating shaftT can be written as follows:(9)T=12∫0L∬AρV⋅VdAdx.The strain energy of the composite shaftU can be expressed as follows:(10)U=12∫0L∬Aσxxεxx+τxsγxs+τxξγxξdAdx,where σxx, τxs, and τxξ represent the cross-section normal stress, in-plane shear stress, and transverse shear stress, respectively. εxx, γxs, and γxξare associated engineering strains.


## 3.2. Motion Equations

The governing equations of the composite rotating shaft can be derived based on Hamilton's principle, which is of the following form:(11)∫t0t1δU−δTdt=0.In the present study, a special ply-angle distribution referred to as circumferentially uniform stiffness (CUS) configuration [21] is considered. By employing Hamilton’s principle and considering the variable cross-section of the tapered shaft, the motion equations involving CUS configuration bending-transverse shear coupling can be written as follows:(12)−C35xψz″−C55xV″−ψy′+mcV¨−2ΩW˙−Ω2V=0,−C46xψy″−C66xW″−ψz′+mcW¨+2ΩV˙−Ω2W=0,−C44xψy″−C46xW″−ψz′−C35xψz′−C55xV′−ψy−Izψ¨y=0,−C33xψz″−C35xV″−ψy′−C46xψy′−C66xW′−ψz−Iyψ¨z=0,in which(13)C33x=∮ΓxA−B2Cz2ds+∮ΓxB/Czds2∮Γx1/Cds,C35x=−12∮ΓxBzdydsds,C44x=κ∮ΓxA−B2Cy2ds+∮ΓxB/Cyds2∮Γx1/Cds,C46x=−12κ∮ΓxBydzdsds,C55x=κ∮Γx14Cdyds2+Ddzds2ds,C66x=κ∮Γx14Cdzds2+Ddyds2ds,where(14)A=A11−A122A22,B=2A16−A12A26A22,C=4A66−A262A22,D=A44−A452A55,Aij=∑k=1NQ¯ijkzk−zk−1,i,j=1,2,6;i,j=4,5.ParametersA and B denote the reduced axial and coupling stiffness, parameters C and D are the reduced shear stiffness, and Aij is the local stretching stiffness,Q¯ijk is transformed stiffness of the kth layer. κ is shear factor of the cross-section which changes with the cross-section and material properties and is of the following form [14]:(15)κ=6Exx1−m¯41+m¯2Gxyvxy2m¯6+18m¯4−18m¯2−2−Exx7m¯6+27m¯4−27m¯2−7,where m¯ = r/R. r and R are the inner and outer radius, respectively. Exx, Gxy, and vxy are elastic modulus, shear modulus, and Poisson’s ratio:(16)Q¯=T−1QT−T,in which(17)T=m2n2002mnn2m200−2mn00m−n000nm0−mnmn00m2−n2,Q=Q11Q1200Q16Q21Q2200Q2600Q4400000Q5500000Q66,where T is transformation matrix, Q is the stiffness matrix, and m=cosα and n=sinα, α is the angle made by fiber direction with respect to the x-axis of the coordinate system (x,s,ξ). The stiffness arrays Qij are determined according to the material properties of the lamina:(18)Q11=E111−v12v21,Q12=Q21=v12E221−v12v21,Q22=E221−v12v21,Q44=κG23,Q55=κG13,Q66=G12.In addition, the mass termsmc, Iz, and Iy in (14) are expressed as(19)mc=∬AρdA,Iz=∬Aρy2dA,Iy=∬Aρz2dA.The motion equation (12) can be used for a tapered composite thin-walled rotating shaft with arbitrary boundary conditions. In this paper, three boundary conditions for the composite shaft are as follows [12]:(a)
Simply supported-simply supported (S-S):(20)V=0,W=0,My=0,Mz=0x=0,L.(b)
Clamped-clamped (C-C):(21)V=0,W=0,Ψy=0,Ψz=0x=0,L.(c)
Clamped-free (C-F):(22)V=0,W=0,Ψy=0,Ψz=0x=0,Qy=0,Qz=0,My=0,Mz=0x=L.


## 3.3. Approximate Solution Method

The GDQM is a global numerical approximate technique. The derivative of a sufficiently smooth functionfx,t at the jth discrete point can be approximated by weighted sums of the function values at all the discrete points. By applying the GDQM, the pth order derivative of fx,t is given by [22](23)∂pfx,t∂xpx=xj=∑k=1NGPCjkpfxk,t,j=1,2,…,NGP,where NGP is the number of total discrete grid points in thex direction and Cjkp is the corresponding weighting coefficient associated to the pth-order derivative which is obtained as below.For the first-order derivative, the weighting coefficients are given by(24)Cjk1=M1xjxj−xkM1xk,j≠k,j,k=1,2,…,NGP,Cjj1=−∑k=1k≠jNGPCjk1,j=1,2,…,NGP,where(25)M1xk=∏j=1j≠kNGPxk−xj.For the second and higher order derivatives, the weighting coefficients are acquired by using the following recurrence formulation:(26)Cjkp=pCjk1⋅Cjjp−1−Cjkp−1xj−xk,j≠k,j,k=1,2,…,NGP,p=2,3,…NGP,Cjjp=−∑k=1k≠jNGPCjkP,j=1,2,…,NGP,and the grid points are chosen as follows:(27)xj=L21−cosj−1NGP−1π,j=1,2,…,NGP.To find the approximate solution of the composite rotating shaft, bending deformation and bending angle are assumed as follows:(28)vx,t=Vxeiωt,wx,t=Wxeiωt,ψyx,t=Ψyxeiωt,ψzx,t=Ψzxeiωt,where Vx, Wx, Ψyx, and Ψzx are the indefinite functions of the x-direction, ω is the natural frequency, and i=−1.Substituting the displacement components (28) into the motion equations (12), a set of ordinary differential equations with variable coefficients toward the x-direction is derived:(29)L∗U∗=0,where U∗T=Vx,Wx,Ψyx,Ψzx is an unknown spatial function vector of mode shape, and L∗=Lij∗i,j=1,…,4 is a 4 × 4 matrix of U∗ and is defined as follows:(30)L∗=−ω2M1+iωG1+K1+KΩ,where(31)M1=mc0000mc0000−Iz0000−Iy,G1=0−2Ωmc002Ωmc00000000000,KΩ=−mcΩ20000−mcΩ20000000000,K1=−C55x∂2∂x20C55x∂∂x−C35x∂2∂x20−C66x∂2∂x2−C46x∂2∂x2C66x∂∂x−C55x∂∂x−C46x∂2∂x2−C44x∂2∂x2+C55xC46x−C35x∂∂x−C35x∂2∂x2−C66x∂∂xC35x−C46x∂∂x−C33x∂2∂x2+C66x.By imposing equations (23) to (29) and rearranging equation (29) according to the orders of derivatives, the approximate governing equations are obtained in the form of linear discrete algebraic equations:(32)L∗∗U∗∗x=xj=L4×15∗∗U15×1∗∗x=xj=0,j=1,2,…,NGP,where NGP is the total discrete grid points in the x-direction and U∗∗is written as follows:(33)U∗∗Tx=xj=Vxj,V1xj,V2xj,Wxj,V1xj,V2xj,Ψyxj,Ψy1xj,Ψy2xj,Ψzxj,Ψz1xj,Ψz2xj,where(34)Vpxj=∑k=1NGPCjkpVxk,t,Wpxj=∑k=1NGPCjkpWxk,t,Ψypxj=∑k=1NGPCjkpΨyxk,t,Ψzpxj=∑k=1NGPCjkpΨyxk,t,p=1,2.For given boundary conditions, imposing equation (32) on every discrete grid point, the discretized form of the boundary conditions is achieved.Therefore, the eigenvalue equation is obtained as follows:(35)Mω2+Giω+Kd=0,where M, G, and K are the 4NGP−2×4NGP−2numerical coefficient matrices. The dimensions of vector d is 4NGP−2 which can be expressed as follows:(36)dT=Vx2,…,VxNGP−1,Wx2,…,WxNGP−1,Ψyx2,…,ΨyxNGP−1,Ψzx2,…,ΨzxNGP−1.For a certain frequency, the eigenvalue equation (35) can be transformed into a standard form as follows [23]:(37)0I−K−G−I00Mωdωd=0,where I is the 4NGP−2×4NGP−2 identity matrix.The vibration characteristics of the tapered composite rotating shaft can be obtained by solving (37).


## 4. Numerical Results Analysis and Discussion

### 4.1. Convergence Analysis and Accuracy Verification

The numerical calculations are performed by considering the tapered shaft made of graphite-epoxy whose elastic characteristics are listed in Table1. The shaft has geometrical characteristics as rR = 0.0635 m, L = 2.47 m, and h = 1.321 mm. The stacking sequence of the shaft is±θ5.Table 1
Mechanical properties of the composite material [2].
ρkg/m3E11GPaE22GPaG12GPaG13GPaG23GPav1216101927.244.074.073.00.24In order to examine the influence of the number of grid pointsNGP on the accuracy of the results, a cylindrical thin-walled shaft with S-S boundary condition is first studied. The numerical results of natural frequencies are shown in Table 2 for an increasing number of grid points. It can be seen from Table 2 that no more than 12 grid points are required in order to obtain convergent results for the first three natural frequencies. So, for all results given in this paper, NGP = 12 unless otherwise noted.Table 2
Effect ofNGP on natural frequencies (Ω=0,θ=30°).
NGPω1ω2ω3ω4ω5ω6941.70135.27281.98516.74845.924117.671041.70135.03282.79484.27825.671259.901141.70135.01281.57485.74743.941231.971241.70135.02281.53481.22746.911063.691341.70135.02281.53480.95733.651070.461441.70135.02281.53481.15733.141038.211541.70135.02281.53481.17733.521037.62Figures2–3 show the variation of natural frequencies versus rotating speed for the cantilever composite shaft with different taper ratios from the present model with the ones from [18]. The numerical results are given according to the normalized natural frequencies and rotating speed rate which are defined by ω∗=ω/ω0 and Ω∗=Ω/ω0, where the normalizing factor ω0 = 138.85 rad/s corresponds to the fundamental frequency of the nonrotating shaft with θ = 0°.Figure 2
The natural frequency of a tapered composite thin-walled shaft versus rotating speed for different taper ratios (θ=60°).Figure 3
The natural frequency of a tapered composite thin-walled shaft versus rotating speed for different taper ratios (θ=90°).From Figures2 and 3, it clearly appears that when the rotating speed rate Ω∗ = 0 a single fundamental frequency is obtained. When the rotating speed rata Ω∗ > 0, a bifurcation of natural frequencies occurs due to the gyroscopic effect. Hence, the natural frequency curve splits into an upper one and a lower one. The upper curve goes up corresponding to the forward whirling (FW or F) mode, while the lower curve goes down corresponding to the backward whirling (BW or B) mode. The minimum rotating rate at which the natural frequency becomes zero valued is called the critical rotating speed, which will cause the dynamically unstable motion of the tapered composite rotating shaft. It also can obviously be seen that the present numerical results agree well with those presented in [18].In the absence of publications of vibration analysis of tapered composite thin-walled shafts with C-C and S-S boundary conditions, the formulation is verified by using the ANSYS 19.1. The FEA model of the tapered composite thin-walled rotating shaft withσ=0.6 was shown in Figure 4, and the critical settings in ANSYS are shown in Table 3.Figure 4
FEA model in ANSYS of the tapered thin-walled rotating shaft withσ=0.6.Table 3
Critical settings in ANSYS.
Element typeDisplacement constraintInertiaCoriolis inertia forceSHELL281Simply supported:Uy, UzAngular velocity: globalColoris: onClamped:Ux, Uy, Uz, Rotx,Roty, RotzReference frame: stationaryTables4 and 5 present the natural frequencies of the tapered composite thin-walled shafts obtained from both present model and ANSYS model. It can be seen from the tables that the numerical results obtained from the present model agree well with those obtained by using ANSYS.Table 4
The natural frequencies of the composite thin-walled shaft: C-C (θ=30°,Ω=0rpm).
Taper ratioMethodNatural frequencies (Hz)The first modeThe second modeThe third modeσ=0.2Present30.2483.05162.61ANSYS32.2288.19172.01σ=0.6Present46.72128.70252.16ANSYA48.20132.25257.76σ=1.0Present60.34165.89324.44ANSYS61.15167.47325.42Table 5
The natural frequencies of the composite thin-walled shaft: S-S (θ=30°,Ω=0rpm).
Taper ratioMethodNatural frequencies (Hz)The first modeThe second modeThe third modeσ=0.2Present21.1067.56140.57ANSYS20.8667.29143.85σ=0.6Present32.28104.57218.18ANSYS29.99102.43214.98σ=1.0Present41.70135.02281.53ANSYS39.62128.50266.86


### 4.2. Taper Ratio Effect

Figure5 reveals the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the taper ratio (Ω=0rpm,θ=30°). As shown in Figure 5, the natural frequencies increase with the taper ratio of the shaft with C-C and S-S boundary conditions. The first natural frequencies decrease with the taper ratio, while the second and third natural frequencies increase with the taper ratio of the shaft with C-F boundary condition. The tapered composite thin-walled shaft becomes cylindrical thin-walled when the taper ratio σ=1, which was previously investigated in Ref [12].Figure 5
The natural frequencies of the rotating shaft versus the taper for different boundary conditions (Ω=0rpm,θ=30°). (a) The first mode; (b) the second mode; (c) the third mode.
(a)(b)(c)Figure6 reveals the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the taper ratio (Ω=400rpm,θ=30°). Two natural frequencies occur for each mode because of the gyroscopic effect. The higher value is called the forward mode, and the lower one represents the backward mode. The variation trend of natural frequencies is the same as in Figure 5.Figure 6
The natural frequencies of the rotating shaft versus the taper for different boundary conditions (Ω=400rpm,θ=30°). (a) The first mode; (b) the second mode; (c) the third mode.
(a)(b)(c)


### 4.3. Ply Angle Effect

Figure7 shows the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the ply angle (Ω=0rpm,σ=0.5). Figure 7(a) represents the first mode of the tapered shaft. As shown in the figure, the maximum values of the first natural frequencies occur at θ=80° for the shaft with C-C and S-S boundary conditions, while the first natural frequencies of the tapered shaft increase with the ply angle for C-F boundary conditions. It can be seen from Figures 7(b) and 7(c) that the maximum values of the second and third natural frequencies occur at θ=75° for the shaft with aforementioned boundary conditions. In addition, it can be seen from Figure 7 that the higher the mode, the lager the variation amplitude.Figure 7
The natural frequencies of the rotating shaft versus ply angle for different boundary conditions (Ω=0rpm,σ=0.5). (a) The first mode; (b) the second mode; (c) the third mode.
(a)(b)(c)Figure8 shows the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the ply angle (Ω=1000rpm,σ=0.5). Two natural frequencies occur for each mode because of the gyroscopic effect similar to Figure 6. It can be seen from Figures 8(a) and 8(b) that the change trend of the natural frequency curve is the same as Figure 7 for the tapered shaft with the ply angle for C-C and S-S boundary conditions. Figure 8(c) shows the change curves of the first three frequencies with increase of the ply angle for C-F boundary condition. It is worth noting that the first backward frequency decreases with increase of the ply angle until decreases to 0 when θ=52° (the critical speed of the tapered composite shaft is 1000 rpm at this ply angle) and then the frequency increases with the increase of the ply angle.Figure 8
The first three natural frequencies of the rotating shaft versus ply angle (Ω=1000rpm,σ=0.5). (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)


### 4.4. Stability Analysis

The Campbell diagrams for the first bending mode of the tapered composite thin-walled rotating shaft for different boundary conditions and different taper ratios are shown in Figure9 (θ =  60°). The results show that the critical rotating speeds increase with the taper ratio of the shaft with C-C and S-S boundary conditions, while the critical rotating speeds decrease with the taper ratio of the shaft with C-F boundary condition.Figure 9
Campbell diagrams of the tapered rotating shaft for different taper ratios withθ =  60°. (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)Figure10 shows the Campbell diagrams of the tapered composite thin-walled rotating shaft for different length to mean radius ratios (θ=75°,σ=0.6). The results show that the critical rotating speeds decrease as the ratio of length to mean radius increase.Figure 10
Campbell diagrams of the tapered rotating shaft for different length to mean radius ratios (θ=75°,σ=0.6). (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)The Campbell diagrams of the tapered composite thin-walled rotating shaft for different mean radius to thickness ratios are shown in Figure11 (θ=30°,σ=0.6). It can be seen from the figure that the critical rotating speeds increase as the ratio of mean radius to thickness increase.Figure 11
Campbell diagrams of the tapered rotating shaft for different mean radius to thickness ratios (θ=30°,σ=0.6). (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)


## 4.1. Convergence Analysis and Accuracy Verification

The numerical calculations are performed by considering the tapered shaft made of graphite-epoxy whose elastic characteristics are listed in Table1. The shaft has geometrical characteristics as rR = 0.0635 m, L = 2.47 m, and h = 1.321 mm. The stacking sequence of the shaft is±θ5.Table 1
Mechanical properties of the composite material [2].
ρkg/m3E11GPaE22GPaG12GPaG13GPaG23GPav1216101927.244.074.073.00.24In order to examine the influence of the number of grid pointsNGP on the accuracy of the results, a cylindrical thin-walled shaft with S-S boundary condition is first studied. The numerical results of natural frequencies are shown in Table 2 for an increasing number of grid points. It can be seen from Table 2 that no more than 12 grid points are required in order to obtain convergent results for the first three natural frequencies. So, for all results given in this paper, NGP = 12 unless otherwise noted.Table 2
Effect ofNGP on natural frequencies (Ω=0,θ=30°).
NGPω1ω2ω3ω4ω5ω6941.70135.27281.98516.74845.924117.671041.70135.03282.79484.27825.671259.901141.70135.01281.57485.74743.941231.971241.70135.02281.53481.22746.911063.691341.70135.02281.53480.95733.651070.461441.70135.02281.53481.15733.141038.211541.70135.02281.53481.17733.521037.62Figures2–3 show the variation of natural frequencies versus rotating speed for the cantilever composite shaft with different taper ratios from the present model with the ones from [18]. The numerical results are given according to the normalized natural frequencies and rotating speed rate which are defined by ω∗=ω/ω0 and Ω∗=Ω/ω0, where the normalizing factor ω0 = 138.85 rad/s corresponds to the fundamental frequency of the nonrotating shaft with θ = 0°.Figure 2
The natural frequency of a tapered composite thin-walled shaft versus rotating speed for different taper ratios (θ=60°).Figure 3
The natural frequency of a tapered composite thin-walled shaft versus rotating speed for different taper ratios (θ=90°).From Figures2 and 3, it clearly appears that when the rotating speed rate Ω∗ = 0 a single fundamental frequency is obtained. When the rotating speed rata Ω∗ > 0, a bifurcation of natural frequencies occurs due to the gyroscopic effect. Hence, the natural frequency curve splits into an upper one and a lower one. The upper curve goes up corresponding to the forward whirling (FW or F) mode, while the lower curve goes down corresponding to the backward whirling (BW or B) mode. The minimum rotating rate at which the natural frequency becomes zero valued is called the critical rotating speed, which will cause the dynamically unstable motion of the tapered composite rotating shaft. It also can obviously be seen that the present numerical results agree well with those presented in [18].In the absence of publications of vibration analysis of tapered composite thin-walled shafts with C-C and S-S boundary conditions, the formulation is verified by using the ANSYS 19.1. The FEA model of the tapered composite thin-walled rotating shaft withσ=0.6 was shown in Figure 4, and the critical settings in ANSYS are shown in Table 3.Figure 4
FEA model in ANSYS of the tapered thin-walled rotating shaft withσ=0.6.Table 3
Critical settings in ANSYS.
Element typeDisplacement constraintInertiaCoriolis inertia forceSHELL281Simply supported:Uy, UzAngular velocity: globalColoris: onClamped:Ux, Uy, Uz, Rotx,Roty, RotzReference frame: stationaryTables4 and 5 present the natural frequencies of the tapered composite thin-walled shafts obtained from both present model and ANSYS model. It can be seen from the tables that the numerical results obtained from the present model agree well with those obtained by using ANSYS.Table 4
The natural frequencies of the composite thin-walled shaft: C-C (θ=30°,Ω=0rpm).
Taper ratioMethodNatural frequencies (Hz)The first modeThe second modeThe third modeσ=0.2Present30.2483.05162.61ANSYS32.2288.19172.01σ=0.6Present46.72128.70252.16ANSYA48.20132.25257.76σ=1.0Present60.34165.89324.44ANSYS61.15167.47325.42Table 5
The natural frequencies of the composite thin-walled shaft: S-S (θ=30°,Ω=0rpm).
Taper ratioMethodNatural frequencies (Hz)The first modeThe second modeThe third modeσ=0.2Present21.1067.56140.57ANSYS20.8667.29143.85σ=0.6Present32.28104.57218.18ANSYS29.99102.43214.98σ=1.0Present41.70135.02281.53ANSYS39.62128.50266.86


## 4.2. Taper Ratio Effect

Figure5 reveals the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the taper ratio (Ω=0rpm,θ=30°). As shown in Figure 5, the natural frequencies increase with the taper ratio of the shaft with C-C and S-S boundary conditions. The first natural frequencies decrease with the taper ratio, while the second and third natural frequencies increase with the taper ratio of the shaft with C-F boundary condition. The tapered composite thin-walled shaft becomes cylindrical thin-walled when the taper ratio σ=1, which was previously investigated in Ref [12].Figure 5
The natural frequencies of the rotating shaft versus the taper for different boundary conditions (Ω=0rpm,θ=30°). (a) The first mode; (b) the second mode; (c) the third mode.
(a)(b)(c)Figure6 reveals the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the taper ratio (Ω=400rpm,θ=30°). Two natural frequencies occur for each mode because of the gyroscopic effect. The higher value is called the forward mode, and the lower one represents the backward mode. The variation trend of natural frequencies is the same as in Figure 5.Figure 6
The natural frequencies of the rotating shaft versus the taper for different boundary conditions (Ω=400rpm,θ=30°). (a) The first mode; (b) the second mode; (c) the third mode.
(a)(b)(c)


## 4.3. Ply Angle Effect

Figure7 shows the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the ply angle (Ω=0rpm,σ=0.5). Figure 7(a) represents the first mode of the tapered shaft. As shown in the figure, the maximum values of the first natural frequencies occur at θ=80° for the shaft with C-C and S-S boundary conditions, while the first natural frequencies of the tapered shaft increase with the ply angle for C-F boundary conditions. It can be seen from Figures 7(b) and 7(c) that the maximum values of the second and third natural frequencies occur at θ=75° for the shaft with aforementioned boundary conditions. In addition, it can be seen from Figure 7 that the higher the mode, the lager the variation amplitude.Figure 7
The natural frequencies of the rotating shaft versus ply angle for different boundary conditions (Ω=0rpm,σ=0.5). (a) The first mode; (b) the second mode; (c) the third mode.
(a)(b)(c)Figure8 shows the first three natural frequencies of the tapered composite thin-walled rotating shaft with different boundary conditions versus the ply angle (Ω=1000rpm,σ=0.5). Two natural frequencies occur for each mode because of the gyroscopic effect similar to Figure 6. It can be seen from Figures 8(a) and 8(b) that the change trend of the natural frequency curve is the same as Figure 7 for the tapered shaft with the ply angle for C-C and S-S boundary conditions. Figure 8(c) shows the change curves of the first three frequencies with increase of the ply angle for C-F boundary condition. It is worth noting that the first backward frequency decreases with increase of the ply angle until decreases to 0 when θ=52° (the critical speed of the tapered composite shaft is 1000 rpm at this ply angle) and then the frequency increases with the increase of the ply angle.Figure 8
The first three natural frequencies of the rotating shaft versus ply angle (Ω=1000rpm,σ=0.5). (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)


## 4.4. Stability Analysis

The Campbell diagrams for the first bending mode of the tapered composite thin-walled rotating shaft for different boundary conditions and different taper ratios are shown in Figure9 (θ =  60°). The results show that the critical rotating speeds increase with the taper ratio of the shaft with C-C and S-S boundary conditions, while the critical rotating speeds decrease with the taper ratio of the shaft with C-F boundary condition.Figure 9
Campbell diagrams of the tapered rotating shaft for different taper ratios withθ =  60°. (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)Figure10 shows the Campbell diagrams of the tapered composite thin-walled rotating shaft for different length to mean radius ratios (θ=75°,σ=0.6). The results show that the critical rotating speeds decrease as the ratio of length to mean radius increase.Figure 10
Campbell diagrams of the tapered rotating shaft for different length to mean radius ratios (θ=75°,σ=0.6). (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)The Campbell diagrams of the tapered composite thin-walled rotating shaft for different mean radius to thickness ratios are shown in Figure11 (θ=30°,σ=0.6). It can be seen from the figure that the critical rotating speeds increase as the ratio of mean radius to thickness increase.Figure 11
Campbell diagrams of the tapered rotating shaft for different mean radius to thickness ratios (θ=30°,σ=0.6). (a) C-C; (b) S-S; (c) C-F.
(a)(b)(c)


## 5. Conclusions

This paper studies the dynamical behavior of a tapered composite thin-walled rotating shaft. A dynamic model of the tapered composite shaft considering the transverse shear deformation, rotary inertia, and gyroscopic effects has been proposed. The motion equations of the tapered composite shaft are derived based on a refined variational asymptotic method and Hamilton’s principle. The GDQM is used to discretize and solve the motion governing equations. The model of the tapered composite thin-walled rotating shaft with C-F boundary condition is verified by comparing the numerical results with those obtained in the literature. In the absence of publications of vibration analysis of tapered composite thin-walled shafts with C-C and S-S boundary conditions, the validity of the model is verified by using the ANSYS. Discussions are made for the influences of the taper ratio, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the frequency characteristics of the tapered composite thin-walled rotating shaft. The results are helpful to improve the understanding about dynamic characteristics of the tapered composite thin-walled rotating shaft and to provide guidance for designing of the composite shaft.



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*Source: 1695430-2020-06-14.xml*