FreeVibrationAnalysisofaRotatingCompositeShaftUsing the p ...downloads.hindawi.com/journals/ijrm/2008/752062.pdfDepartment of Mechanical Engineering, Faculty of Sciences Engineering,

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2008, Article ID 752062, 10 pagesdoi:10.1155/2008/752062

Research ArticleFree Vibration Analysis of a Rotating Composite Shaft Usingthe p-Version of the Finite Element Method

A. Boukhalfa, A. Hadjoui, and S. M. Hamza Cherif

Department of Mechanical Engineering, Faculty of Sciences Engineering, Abou Bakr Belkaid University, Tlemcen 13000, Algeria

Correspondence should be addressed to A. Boukhalfa, [email protected]

Received 19 November 2007; Accepted 24 May 2008

Recommended by Agnes Muszynska

This paper is concerned with the dynamic behavior of the rotating composite shaft on rigid bearings. A p-version, hierarchicalfinite element is employed to define the model. A theoretical study allows the establishment of the kinetic energy and the strainenergy of the shaft, necessary to the result of the equations of motion. In this model the transverse shear deformation, rotaryinertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated.A hierarchical beam finite element with six degrees of freedom per node is developed and used to find the natural frequenciesof a rotating composite shaft. A program is elaborate for the calculation of the eigenfrequencies and critical speeds of a rotatingcomposite shaft. To verify the present model, the critical speeds of composite shaft systems are compared with those available inthe literature. The efficiency and accuracy of the methods employed are discussed.

Copyright © 2008 A. Boukhalfa et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

The application of composite shafts has come a long wayfrom early low-speed automotive driveshafts to helicoptertail rotors operating above the second critical speed. Withoperation at supercritical speeds, a substantial amount ofpayoffs and net system weight reductions are possible. Atthe same time, the rotordynamic aspects assume moreimportance, and detailed analysis is required. There are sometechnological problems associated with implementation,such as joints with bearings, affixing of lumped masses,couplings, provision of external damping, and so forth. Thesolutions proposed are just adequate, but require substantialrefinements, which might explain some of the differingexperiences of various authors.

Zinberg and Symonds [1] described a boron/epoxy com-posite tail rotor driveshaft for a helicopter. The critical speedswere determined using equivalent modulus beam theory(EMBT), assuming the shaft to be a thin-walled circular tubesimply supported at the ends. Shear deformation was nottaken into account. The shaft critical speed was determinedby extrapolation of the unbalance response curve which wasobtained in the subcritical region.

Dos Reis et al. [2] published analytical investigations onthin-walled layered composite cylindrical tubes. In part III

of the series of publications, the beam element was extendedto formulate the problem of a rotor supported on generaleight coefficient bearings. Results were obtained for shaftconfiguration of Zinberg and Symmonds. The authors haveshown that bending-stretching coupling and shear-normalcoupling effects change with stacking sequence, and alterthe frequency values. Gupta and Singh [3] studied the effectof shear-normal coupling on rotor natural frequencies andmodal damping. Kim and Bert [4] have formulated theproblem of determination of critical speeds of a compositeshaft including the effects of bending-twisting coupling. Theshaft was modelled as a Bresse-Timoshenko beam. The shaftgyroscopics have also been included. The results comparewell with Zinberg’s rotor [1]. In another study, Bert andKim [5] have analysed the dynamic instability of a compositedrive shaft subjected to fluctuating torque and/or rotationalspeed by using various thin shell theories. The rotationaleffects include centrifugal and Coriolis forces. Dynamicinstability regions for a long span simply supported shaft arepresented.

M.-Y. Chang et al. [6] published the vibration behavioursof the rotating composite shafts. In the model, the transverseshear deformation, rotary inertia, and gyroscopic effects,as well as the coupling effect due to the lamination ofcomposite layers have been incorporated. The model based

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2 International Journal of Rotating Machinery

on a first-order shear deformable beam theory (continuum-based Timoshenko beam theory).

C.-Y. Chang et al. [7] published the vibration analysisof rotating composite shafts containing randomly orientedreinforcements. The Mori-Tanaka mean-field theory isadopted here to account for the interaction at the finiteconcentrations of reinforcements in the composite material.

Additional recent work on composite shafts dealing withboth the theoretical and experimental aspects was reportedby Singh [8], Gupta and Singh [3], and Singh and Gupta[9, 10]. Rotordynamic formulation based on equivalentmodulus beam theory was developed for a compositerotor with a number of lumped masses and supported ongeneral eight coefficient bearings. A layerwise beam theorywas derived by Gupta and Singh [3] from an availableshell theory, with a layerwise displacement field, and wasthen extended to solve a general composite rotordynamicproblem. The conventional rotor dynamic parameters aswell as critical speeds, natural frequencies, damping factors,unbalance response, and threshold of stability were analysedin detail and results from the formulations based on thetwo theories, namely, the equivalent modulus beam theory(EMBT) and layerwise beam theory (LBT) were compared[9, 10]. The experimental rotordynamic studies carried bySingh and Gupta [11, 12] were conducted on two filamentwound carbon/epoxy shafts with constant winding angles(±45◦ and ±60◦). Progressive balancing had to be carriedout to enable the shaft to traverse through the first criticalspeed. Inspite of the very different shaft configurations used,the authors’ have shown that bending-stretching couplingand shear-normal coupling effects change with stackingsequence, and alter the frequency values.

Some practical aspects such as effect of shaft discangular misalignment, interaction between shaft bow, whichis common in composite shafts and rotor unbalance, and anunsuccessful operation of a composite rotor with an externaldamper were discussed and reported by Singh and Gupta[11]. The Bode and cascade plots were generated and orbitalanalysis at various operating speeds was performed. Theexperimental critical speeds showed good correlation withthe theoretical prediction.

This paper deals with the p-version, hierarchical finiteelement method applied to free vibration analysis of rotatingcomposite shafts. The hierarchical concept for finite elementshape functions has been investigated during the past 25years. Babuska et al. [13] established a theoretical basis forp-elements, where the mesh keeps unchanged and the poly-nomial degree of the shape functions is increased; however,in the standard h-version of the finite element method themesh is refined to achieve convergence and the polynomialdegree of the shape functions remains unchanged. Sincethen, standard forms of the hierarchical shape functions havebeen represented in the literature elsewhere; see for instance[14, 15].

Meirovitch and Baruh [16] and Zhu [17] have shownthat the hierarchical finite element method yields a betteraccuracy than the h-version for eigenvalues problems. Thehierarchical shape functions used by Bardell [18] are basedon integrated Legendre orthogonal polynomials; the sym-

z

x

y

�eθ

�er�k�i �j

o

rθ

Figure 1: The cylindrical coordinate system.

bolic computing is used to calculate the mass and stiffnessmatrices of beams and plates. Côté and Charron [19] givethe selection of p-version shape functions for plate vibrationanalysis.

In the presented composite shaft model, the Timoshenkotheory will be adopted. It is the purpose of the present workto study dynamic characteristics such as natural frequencies,whirling frequencies, and the critical speeds of the rotatingcomposite shaft. In the model, the transverse shear defor-mation, rotary inertia, and gyroscopic effects, as well as thecoupling effect due to the lamination of composite layershave been incorporated. To determine the rotating shaftsystem’s responses, the hierarchical finite element methodwith trigonometric shape functions [20, 21] is used here toapproximate the governing equations by a system of ordinarydifferential equations.

2. EQUATIONS OF MOTION

2.1. Kinetic and strain energy expressions

The shaft is modelled as a Timoshenko beam, that is, first-order shear deformation theory with rotary inertia andgyroscopic effect is used. The shaft rotates at constant speedabout its longitudinal axis. Due to the presence of fibersoriented than axially or circumferentially, coupling is madebetween bending and twisting. The shaft has a uniform,circular cross-section.

The following displacement field of a rotating shaft isassumed by choosing the coordinate axis x to coincide withthe shaft axis:

U(x, y, z, t) = U0(x, t) + zβx(x, t)− yβy(x, t),V(x, y, z, t) = V0(x, t)− zφ(x, t),W(x, y, z, t) =W0(x, t) + yφ(x, t),

(1)

where U , V , and W are the flexural displacements of anypoint on the cross-section of the shaft in the x, y, and zdirections (see Figure 1), the variables U0, V0, and W0 arethe flexural displacements of the shaft’s axis, while βx and βyare the rotation angles of the cross-section, about the y andz axis, respectively. The φ is the angular displacement of thecross-section due to the torsion deformation of the shaft.

A. Boukhalfa et al. 3

z

y

R0R1Rk

r

θ

e

Figure 2: k-layers of composite shaft.

3 (T’)

2 (T)

1 (L)

Transversal directions

Longitudinal direction

Figure 3: A typical composite lamina and its principal axes.

The strain components in the cylindrical coordinatesystem (as shown in Figures 1-2) can be written in terms ofthe displacement variables defined earlier as

εxx = ∂U0∂x

+ r sin θ∂βx∂x

− r cos θ ∂βy∂x

,

εrr = εθθ = εrθ = 0,εxθ = εθx

= 12

(βy sin θ+ βx cos θ−sin θ ∂V0

∂x+ cos θ

∂W0∂x

+ r∂φ

∂x

),

εxr = εrx

= 12

(βx sin θ − βy cos θ − sin θ ∂W0

∂x+ cos θ

∂V0∂x

).

(2)

Let us consider a composite shaft consists of k layered(Figure 2) of fiber inclusion reinforced laminate. The con-stitutive relations for each layer are described by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σxx

σθθ

σrr

τrθ

τxr

τxθ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C′11 C′12 C

′13 0 0 C

′16

C′12 C′22 C

′23 0 0 C

′26

C′13 C′23 C

′33 0 0 C

′36

0 0 0 C′44 C′45 0

0 0 0 C′45 C′55 0

C′16 C′26 C

′36 0 0 C

′66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εxx

εθθ

εrr

γrθ

γxr

γxθ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (3)

, 3

2

1�er

η�eθ

xx

Figure 4: The definitions of the principal coordinate axes on anarbitrary layer of the composite.

whereC′i j are the effective elastic constants, they are related tolamination angle η (as shown in Figures 3-4) and the elasticconstants of principal axes.

The stress-strain relations of the nth layer expressed inthe cylindrical coordinate system (Figure 5) can be expressedas

σxx = C′11nεxx + ksC′16nγxθ ,τxθ = τθx = ksC′16nεxx + ksC′66nγxθ ,τxr = τrx = ksC′55nγxr ,

(4)

where ks is the transverse shear correction factor.The formula of the strain energy is

Ed = 12∫V

(σxxεxx + 2τxrεxr + 2τxθεxθ

)dV. (5)

The various components of strain energy come from theshaft:

Ed = 12A11∫ L

0

(∂U0∂x

)2dx

+12B11

[∫ L0

(∂βx∂x

)2dx +

∫ L0

(∂βy∂x

)2dx]

+12ksB66

∫ L0

(∂φ

∂x

)2dx +

12ksA16

×[

2∫ L

0

∂φ

∂x

∂U0∂x

dx +∫ L

0βy∂βx∂x

dx −∫ L

0βx∂βy∂x

dx

−∫ L

0

∂V0∂x

∂βx∂x

dx −∫ L

0

∂W0∂x

∂βy∂x

dx]

+12ks(A55 + A66

)

×[∫ L

0

(∂V0∂x

)2dx +

∫ L0

(∂W0∂x

)2dx +

∫ L0β2xdx

+∫ L

0β2ydx + 2

∫ L0βx∂W0∂x

dx − 2∫ L

0βy∂V0∂x

dx]

,

(6)

where Aij and Bij are given in the appendix.


τxz

τxy

σxx

τyz

τyx

σyy

�k

�i�j

(a)

τxr

τxθ

σxx

τθxτθr

σθθ

�er�eθ

�i(b)

Figure 5: The stress components: (a) in the coordinate axes (x, y, z), (b) in the coordinate axes (x , r , θ).

L

1 2

ξ = 0

ξ = x/L

ξ = 1x, ξ

Figure 6: Beam element with two nodes.

0

200

400

600

800

1000

1200

1400

1600

1800

ωfr

equ

ency

(Hz)

7 8 9 10 11 12

Number of hierarchical terms p

ω1ω2ω3

ω4ω5

Figure 7: Convergence of the frequency ω for the 5 bending modesof the simply-supported (S-S) shaft as a function of the number ofhierarchical terms p.

The kinetic energy of the rotating composite shaft,including the effects of translatory and rotary inertia, can be

0

200

400

600

800

1000

1200

1400

1600

1800

ωfr

equ

ency

(Hz)

7 8 9 10 11 12

Number of hierarchical terms p

ω1ω2ω3

ω4ω5

Figure 8: Convergence of the frequency ω for the 5 bending modesof the clamped-clamped (C-C) shaft as a function of the number ofhierarchical terms p.

written as

Ec = 12∫ L

0

[Im(U̇20 + V̇

20 + Ẇ

20

)+ Id

(β̇2x + β̇

2y

)− 2ΩIpβxβ̇y+ 2ΩIpφ̇ + Ipφ̇2 +Ω2Ip +Ω2Id

(β2x + β

2y

)]dx,

(7)

where Ω is the rotating speed of the shaft which is assumedconstant, L is the length of the shaft, the 2ΩIpβxβ̇y termaccounts for the gyroscopic effect, and Id(β̇2x + β̇

2y) represent

the rotary inertia effect. The mass moments of inertia Im,the diametrical mass moments of inertia Id, and polar massmoment of inertia Ip of rotating shaft per unit length aredefined in the appendix. As the Ω2Id(β2x + β

2y) term is far

smaller than Ω2Ip, it will neglected in further analysis.


0

1000

2000

3000

4000

5000

6000

7000

8000

9000

ωfr

equ

ency

(rad

/s)

Ω rotating speed (rpm)

0 1000 2000 3000 4000 5000 6000 7000 8000

1B1F2B2F3B

3F4B4F5B5F

Figure 9: The Campbell diagram of a composite shaft ((F) forwardmodes, (B) backward modes).

400

500

600

700

800

900

1000

1100

1200

1300

ωfr

equ

ency

(rad

/s)

0 1 2 3 4 5 6 7 8 9 10×104Ω rotating speed (rpm)

1B (S-S)1F (S-S)1B (C-C)1F (C-C)

1B (C-S)1F (C-S)1B (C-F)1F (C-F)

Figure 10: The first backward (1B) and forward (1F) bendingmode of a composite shaft for different boundary conditions anddifferent rotating speeds (S: simply supported; C: clamped; F: free).

2.2. Hierarchical beam element formulation

The spinning flexible beam is descretised into one hierarchi-cal finite element is shown in Figure 6. The element’s nodald.o.f. at each node are U0, V0, W0, βx, βy , and φ. The localand nondimensional coordinates are related by

ξ = x/L, (0 ≤ ξ ≤ 1). (8)

Table 1: Properties of composite materials [22].

Boron-epoxy Graphite-epoxy

E11 (GPa) 211.0 139.0

E22 (GPa) 24.1 11.0

G12 (GPa) 6.9 6.05

G23 (GPa) 6.9 3.78

v12 0.36 0.313

ρ (kg/m3) 1967.0 1578.0

The vector displacement formed by the variables U0, V0, W0,βx, βy , and φ can be written as

U0 =pU∑m=1

xm(t)· fm(ξ),

V0 =pV∑m=1

ym(t)· fm(ξ),

W0 =pW∑m=1

zm(t)· fm(ξ),

βx =pβx∑m=1

βxm(t)· fm(ξ),

βy =pβy∑m=1

βym(t)· fm(ξ),

φ =pφ∑m=1

φm(t)· fm(ξ).

(9)

And it can be expressed as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

U0

V0

W0

βx

βy

φ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

= [N] {q}, (10)

where [N] is the matrix of the shape functions, given by

[N] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[NU]

0 0 0 0 0

0[NV]

0 0 0 0

0 0[NW

]0 0 0

0 0 0[Nβx]

0 0

0 0 0 0[Nβy

]0

0 0 0 0 0[Nφ]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (11)

[NU ,V ,V ,βx ,βy ,φ

] = [ f1 f2 · · · fpU ,pV , pW ,pβx ,pβy ,pφ], (12)

where pU , pV , pW , pβx , pβy , and pφ are the numbers ofhierarchical terms of displacements (are the numbers ofshape functions of displacements).

In this work, pU = pV = pW = pβx = pβy = pφ = p.


Table 2: The critical speed of the boron-epoxy composite shaft.

L = 2.47 m, D = 12.69 cm, e = 1.321 mm, 10 layers of equal thickness (90◦, 45◦, −45◦, 0◦6, 90◦), ks = 0.503

Theory or methodFirst critical

speed (rpm)

Zinberg and Symonds [1]Measured experimentally 6000

EMBT 5780

dos Reis et al. [2]Bernoulli-Euler beam theory with stiffness

4942determined by shell finite elements

Kim and Bert [4] Sanders shell theory 5872

Bert [23] Donnell shallow shell theory 6399

Bert and Kim [22] Bernoulli-Euler beam theory 5919

Bresse-Timoshenko beam theory 5788

Gupta and Singh [3] EMBT 5747

LBT 5620

M.-Y. Chang et al. [6] Continuum-based Timoshenko beam theory 5762

PresentTimoshenko beam theory by the p-version

5760of the finite element method

The vector of generalised coordinates given by

{q} = {qU , qV , qW , qβx , qβy , qφ}T , (13)where

{qU} = {x1, x2, x3, . . . , xpU}T exp( jωt),

{qV} = {y1, y2, y3, . . . , ypV }T exp( jω t),

{qW} = { z1, z2, z3, . . . , ypW }T exp( jω t),

{qβx} = {βx1 ,βx2 ,βx3 , . . . ,βxPβx

}Texp( jωt),

{qβy} = {βy1 ,βy2 ,βy3 , . . . ,βyPβy

}Texp( jωt),

{qφ} = {φ1,φ2,φ3, . . . ,φpφ}T exp( jωt).

(14)

The group of the shape functions used in this study isgiven by

f1 = 1− ξ,f2 = ξ,

fr+2 = sin(δrξ),

δr = rπ; r = 1, 2, 3, . . . .

(15)

The functions ( f1, f2) are those of the finite elementmethod necessary to describe the nodal displacements of theelement, whereas the trigonometric functions fr+2 contributeonly to the internal field of displacement and do not affectnodal displacements. The most attractive particularity of thetrigonometric functions is that they offer great numericalstability. The shaft is modeled by only one element calledhierarchical finite element.

By applying the Euler-Lagrange equations, the equationsof motion of free vibration of spinning flexible shaft can beobtained:

[M]{q̈}

+ [G]{q̇}

+ [K]{q} = {0}. (16)The system obtained is linear of which equations are

coupled once by the gyroscopic effect, represented by thematrix [G].

[M] and [K] are the conventional hierarchical finiteelement mass and stiffness matrix, [G] is the gyroscopicmatrix (the different matrices of the system of equation aregiven in the appendix).

3. RESULTS

A program based on the formulation proposed was devel-oped for the resolution of (16).

3.1. Convergence

First, the material properties for boron-epoxy are listed inTable 1, and the geometric parameters are L = 2.47 m, D =12.69 cm, e = 1.321 mm, 10 layers of equal thickness (90◦,45◦, −45◦, 0◦6, 90◦). The shear correction factor ks = 0.503and the rotating speed Ω = 0.

The results of the five bending modes for variousboundary conditions of the composite shaft as a function ofthe number of hierarchical terms are shown in Figures 7 and8. Figures clearly show that rapid convergence from above tothe exact values occur as the number of hierarchical terms isincreased.

3.2. Numerical examples and discussions

In all the following examples, p = 10.In the first example, the properties of the boron-epoxy

composite shaft are given by Table 1. The results obtained


Table 3: The critical speed of the graphite-epoxy composite shaft.

L = 2.47 m, D = 12.69 cm, e = 1.321 mm, 10 layers of equal thickness (90◦, 45◦, −45◦, 0◦6, 90◦), ks = 0.503

Theory or methodFirst critical

speed (rpm)

Bert and Kim [22]

Sanders shell theory 5349

Donnell shallow shell theory 5805

Bernoulli-Euler beam theory 5302

Bresse-Timoshenko beam theory 5113

M.-Y. Chang et al. [6] Continuum-based Timoshenko beam theory 5197

PresentTimoshenko beam theory by the p-version

5200of the finite element method

Table 4: The critical speed (rpm) of the graphite-epoxy composite shaft for various lengths to mean diameter ratio.

Theory or methodL/D

2 5 10 15 20 25 30 35

Sanders shell 112400 41680 16450 8585 5183 3441 2440 1816

Bert and Kim [22] Bernoulli-Euler 329600 76820 20210 9072 5121 3283 2282 1677

Bresse-Timoshenko 176300 54830 17880 8543 4945 3209 2246 1658

M.-Y. Chang et al. [6]Continuum-based Timoshenko

181996 55706 17929 8527 4925 3192 2233 1648beam theory

Present

Timoshenko beam theory by184667 56196 18005 8549 4934 3198 2236 1650the p-version of the finite

element method

using the present model are shown in Table 2 together withthose of referenced papers. As can be seen from the table, ourresults are close to those predicted by other beam theories.Since in the studied example the wall of the shaft is relativelythin, models based on shell theories [4] are expected toyield more accurate results. In the present example, thecritical speed measured from the experiment however isstill underestimated by using Sander shell theory, whileoverestimated by Donnell shallow shell theory. When thematerial of the shaft is changed to the graphite-epoxy givenin Table 1 with other conditions left unchanged, the criticalspeed obtained from the present model is shown in Table 3.In this case, the result from the present model is compatibleto that of the Bresse-Timoshenko beam theory of M.-Y.Chang et al. [6].

Next, comparisons are made with those of [6, 22]for different length to mean diameter ratios L/D. Theshafts being analysed are made of the graphite-epoxymaterial given in Table 1 and all have the same lamination[90◦/45◦/45◦/0◦6/90◦]. The mean diameter and the wallthickness of the shaft remain the same as the previousexamples. The shear correction factor being used is again0.503. The results are listed in Table 4. Further results beingcompared are for generalized orthotropic composite tubeof different lamination angles η. The results are shown inTable 5.

In our work, the shaft is modelled by only one elementwith two nodes, but in the model of [6] the shaft is modelledby 20 finite elements of equal length (h-version). The rapid

convergence while taking only one element and a reducednumber of shape functions shows the advantage of themethod used.

From the thin-walled shaft systems studied above, thepresent shaft model yields result in all cases close to thoseof the model of Bert and Kim [22] based on the Bresse-Timoshenko theory. One should stress here that the presentmodel is not only applicable to the thin-walled compositeshafts as studied above, but also to the thick-walled shafts aswell as to the solid ones.

In the following example, the frequencies responsesof a graphite-epoxy composite shaft system are analysed.The material properties are those listed in Table 1. Thelamination scheme of the shaft remains the same as previousexamples, while its geometric properties, the Campbelldiagram containing the frequencies of the first five pairs ofbending whirling modes of the above composite system isshown in Figure 9. The intersection point of the line (Ω =ω) with the whirling frequency curves indicate the speedat which the shaft will vibrate violently (i.e., the criticalspeed). The first 10 eigenvalues correspond to 5 forward(F) and 5 backward (B) whirling bending modes of theshaft.

The gyroscopic effect causes a coupling of orthogonaldisplacements to the axis of rotation, and by consequenceseparates the frequencies in two branches: backward pre-cession mode and forward precession mode. In all cases,the backward modes increase with increasing rotating speedhowever the forward modes decrease.


Table 5: The critical speed (rpm) of the graphite-epoxy composite shaft for various lamination angles.

Theory or methodLamination angle η (◦)

0 15 30 45 60 75 90

Sanders shell 5527 4365 3308 2386 2120 2020 1997

Bert and Kim [22] Bernoulli-Euler 6425 5393 4269 3171 2292 1885 1813

Bresse-Timoshenko 6072 5209 4197 3143 2278 1874 1803

M.-Y. Chang et al. [6]Continuum-based Timoshenko

6072 5331 4206 3124 2284 1890 1816beam theory

Timoshenko beam theory by the6094 5359 4222 3129 2284 1890 1816Present p-version of the finite element

method

Figure 10 shows the variation of the bending funda-mental frequency ω as a function of the rotating speed Ω(Campbell diagram) for different boundary conditions anddifferent rotating speeds.

The gyroscopic effect inherent to rotating structuresinduces a precession motion. The backward precessionmodes (1B) increase with increasing the rotating speed,however the forward precession (1F) modes decrease.

4. CONCLUSION

This paper has presented the free vibration analysis ofspinning composite shaft using the p-version, hierarchicalfinite element method with trigonometric shape functions.Results obtained using the method has been evaluatedagainst those available in the literature and the agreement hasbeen found to be good. The main conclusions have emergedfrom this work, these are itemised below.

(1) Monotonic and uniform convergence is found tooccur as the number of hierarchical functions isincreased.

(2) The dynamic characteristics of rotating compositeshaft are influenced significantly by varying thefiber orientation, the rotating speed, and boundaryconditions.

(3) The gyroscopic effect causes a coupling of orthogonaldisplacements to the axis of rotation, and by con-sequence separates the frequencies in two branches:backward and forward precession modes. In all cases,the backward modes increase with increasing rotatingspeed however the forward modes decrease.

APPENDIX

The terms Aij , Bij of (6) and Im, Id, Ip of (7) are given as fol-lows:

A11 = πk∑

n=1C′11n

(R2n − R2n−1

),

A55 = π2k∑

n=1C′55n

(R2n − R2n−1

),

A66 = π2k∑

n=1C′66n

(R2n − R2n−1

),

A16 = 2π3k∑

n=1C′16n

(R3n − R3n−1

),

B11 = π4k∑

n=1C′11n

(R4n − R4n−1

),

B66 = π2k∑

n=1C′66n

(R4n − R4n−1

),

Im = πk∑

n=1ρn(R2n − R2n−1

),

Id = π4k∑


),

Ip = π2k∑


),

(A.1)

where k is the number of the layer, Rn−1 is the nth layer innerradius of the composite shaft, and Rn is the nth layer outerof the composite shaft. L is the length of the composite shaftand ρn is the density of the nth layer of the composite shaft.

Whereas the various matrices of (16) are expressed asfollows:

[M] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[MU

]0 0 0 0 0

0[MV

]0 0 0 0

0 0[MW

]0 0 0

0 0 0[Mβx

]0 0

0 0 0 0[Mβy

]0

0 0 0 0 0[Mφ]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

[K] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

[KU]

0 0 0 0[K1]

0[KV]

0[K2] [

K3]

0

0 0[KW

] [K4] [

K5]

0

0[K2]T [

K4]T [

Kβx] [

K6]

0

0[K3]T [

K5]T [

K6]T [

Kβy]

0[K1]T

0 0 0 0[Kφ]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,


[G] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0[G1]

0

0 0 0 −[G1]T 0 00 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

[MU

] = ImL∫ 1

0

[NU]T[

NU]dξ,

[MV

] = ImL∫ 1

0

[NV]T[

NV]dξ,

[MW

] = ImL∫ 1

0

[NW

]T[NW

]dξ,

[Mβx

] = IdL∫ 1

0

[Nβx]T[

Nβx]dξ,

[Mβy

] = IdL∫ 1

0

[Nβy

]T[Nβy

]dξ,

[Mφ] = IpL

∫ 10

[Nφ]T[

Nφ]dξ,

[KU] = 1

LA11

∫ 10

[N ′U]T[

N ′U]dξ,

[KV] = 1

Lks(A55 + A66

)∫ 10

[N ′V]T[

N ′V]dξ,

[KW

] = 1Lks(A55 + A66

)∫ 10

[N ′W

]T[N ′W

]dξ,

[K1] = 1

LksA16

∫ 10

[N ′φ]T[

N ′U]dξ,

[K2] = − 1

2LksA16

∫ 10

[N ′V]T[

N ′βx]dξ,

[K3] = −ks(A55 + A66)

∫ 10

[Nβy

]T[N ′V]dξ,

[K4] = ks(A55 + A66)

∫ 10

[Nβx]T[

N ′W]dξ,

[K5] = − 1

2LksA16

∫ 10

[N ′W

]T[N ′βy

]dξ,

[K6] =

[12ksA16

∫ 10

[Nβy

]T[N ′βx]dξ]

−[

12ksA16

∫ 10

[Nβx]T[

N ′βy]dξ]

,

[Kβx] =

[1LB11

∫ 10

[N ′βx]T[

N ′βx]dξ]

+[Lks(A55 + A66

)∫ 10

[Nβx]T[

Nβx]dξ]

,

[Kβy] =

[1LB11

∫ 10

[N ′βy

]T[N ′βy

]dξ]

+[Lks(A55 + A66

)∫ 10

[Nβy

]T[Nβy

]dξ]

,

[Kφ] = 1

LB66

∫ 10

[N ′φ]T[

N ′φ]dξ,

[G1] = ΩIpL

∫ 10

[Nβx]T[

Nβy]dξ,

(A.2)

where [N ′i ] = ∂[Ni]/∂ξ, with (i = U ,V ,W ,βx,βy ,φ).The terms of the matrices are a function of the integrals,

Jαβmn =

∫ 10 f

αm(ξ) f

βn (ξ)dξ, (m,n) indicate the number of the

shape functions used, where (α,β) is the order of derivation.

NOMENCLATURE

U(x, y, z): Displacement in x directionV(x, y, z): Displacement in y directionW(x, y, z): Displacement in z directionβx: Rotation angles of the cross-section on y axisβy : Rotation angles of the cross-section on z axisφ: Angular displacement of the cross-section

due to the torsion deformation of the shaft(x, y, z): Cartesian coordinates

(�i , �j , �k): Axes of the Cartesian coordinates(x , r , θ): Cylindrical coordinates

(�i , �er , �eθ): Axes of the cylindrical coordinates(1, 2, 3): Principal axes of a layer of laminateC′i j : Elastic constantsE: Young modulusG: Shear modulusv: Poisson coefficientρ: Masse densityks: Shear correction factorL: Length of the shaftD: Mean radius of the shafte: Wall thickness of the shaftk: Number of the layer of the composite shaftRn−1: The nth layer inner radius of the composite

shaftRn: The nth layer outer radius of the composite

shaftR0: Inner radius of the composite shaftRk: Outer radius of the composite shaftη: Lamination angleθ: Circumferential coordinateξ: Local and nondimensional coordinatesεi j : Strain tensorσi j : Stress tensorω: Frequency, eigenvalueΩ: Rotating speedf (ξ): Shape functions[N]: Matrix of the shape functionsp: Number of the shape functions or number of

hierarchical termst: Time


Ec: Kinetic energyEd: Strain energy{qi}: Generalised coordinates, with

(i = U ,V ,W ,βx,βy ,φ)[M]: Masse matrix[K]: Stiffness matrix[G]: Gyroscopic matrix.

REFERENCES

[1] H. Zinberg and M. F. Symonds, “The development of anadvanced composite tail rotor driveshaft,” in Proceedings ofthe 26th Annual Forum of the American Helicopter Society,Washington, DC, USA, June 1970.

[2] H. L. M. dos Reis, R. B. Goldman, and P. H. Verstrate,“Thin-walled laminated composite cylindrical tubes—part III:critical speed analysis,” Journal of Composites Technology andResearch, vol. 9, no. 2, pp. 58–62, 1987.

[3] K. Gupta and S. E. Singh, “Dynamics of composite rotors,” inProceedings of the Indo-US Symposium on Emerging Trends inVibration and Noise Engineering, pp. 59–70, New Delhi, India,March 1996.

[4] C. D. Kim and C. W. Bert, “Critical speed analysis of laminatedcomposite, hollow drive shafts,” Composites Engineering, vol. 3,pp. 633–643, 1993.

[5] C. W. Bert and C. D. Kim, “Dynamic instability of composite-material drive shaft subjected to fluctuating torque and/orrotational speed,” Dynamics and Stability of Systems, vol. 10,no. 2, pp. 125–147, 1995.

[6] M.-Y. Chang, J.-K. Chen, and C.-Y. Chang, “A simple spinninglaminated composite shaft model,” International Journal ofSolids and Structures, vol. 41, no. 3-4, pp. 637–662, 2004.

[7] C.-Y. Chang, M.-Y. Chang, and J. H. Huang, “Vibrationanalysis of rotating composite shafts containing randomlyoriented reinforcements,” Composite Structures, vol. 63, no. 1,pp. 21–32, 2004.

[8] S. P. Singh, Some studies on dynamics of composite shafts, Ph.D.thesis, Mechanical Engineering Department, IIT, Delhi, India,1992.

[9] S. P. Singh and K. Gupta, “Dynamic analysis of compositerotors,” in Proceedings of the 5th International Symposium onRotating Machinery (ISROMAC ’94), pp. 204–219, Kaanapali,Hawaii, USA, May 1994.

[10] S. P. Singh and K. Gupta, “Dynamic analysis of compositerotors,” International Journal of Rotating Machinery, vol. 2, no.3, pp. 179–186, 1996.

[11] S. E. Singh and K. Gupta, “Experimental studies on compositeshafts,” in Proceedings of the International Conference onAdvances in Mechanical Engineering, pp. 1205–1221, Banga-lore, India, December 1995.

[12] S. P. Singh and K. Gupta, “Composite shaft rotordynamicanalysis using a layerwise theory,” Journal of Sound andVibration, vol. 191, no. 5, pp. 739–756, 1996.

[13] I. Babuska, B. A. Szabó, and I. N. Katz, “The p-version of thefinite element method,” SIAM Journal on Numerical Analysis,vol. 18, no. 3, pp. 515–545, 1981.

[14] B. A. Szabó and G. J. Sahrmann, “Hierarchic plate andshell models based on p-extension,” International Journal forNumerical Methods in Engineering, vol. 26, no. 8, pp. 1855–1881, 1988.

[15] B. A. Szabó and I. Babuska, Finite Element Analysis, John Wiley& Sons, New York, NY, USA, 1991.

[16] L. Meirovitch and H. Baruh, “On the inclusion principle forthe hierarchical finite element method,” International Journalfor Numerical Methods in Engineering, vol. 19, no. 2, pp. 281–291, 1983.

[17] D. C. Zhu, “Development of hierarchical finite elementmethod at BIAA,” in Proceedings of the International Confer-ence on Computational Mechanics, pp. 123–128, Tokyo, Japan,May 1986.

[18] N. S. Bardell, “The application of symbolic computing to thehierarchical finite element method,” International Journal forNumerical Methods in Engineering, vol. 28, no. 5, pp. 1181–1204, 1989.

[19] A. Côté and F. Charron, “On the selection of p-versionshape functions for plate vibration problems,” Computers &Structures, vol. 79, no. 1, pp. 119–130, 2001.

[20] A. Houmat, “A sector fourier p-element applied to freevibration analysis of sectorial plates,” Journal of Sound andVibration, vol. 243, no. 2, pp. 269–282, 2001.

[21] S. M. Hamza Cherif, “Free vibration analysis of rotatingflexible beams by using the fourier p- version of the finiteelement,” International Journal of Computational Methods, vol.2, no. 2, pp. 255–269, 2005.

[22] C. W. Bert and C.-D. Kim, “Whirling of composite-materialdriveshafts including bending-twisting coupling and trans-verse shear deformation,” Journal of Vibration and Acoustics,vol. 117, no. 1, pp. 17–21, 1995.

[23] C. W. Bert, “The effect of bending-twisting coupling on thecritical speed of a driveshafts,” in Proceedings of the 6th Japan-US Conference on Composites Materials, pp. 29–36, Technomic,Orlando, Fla, USA, June 1992.

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