-
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
mailto:[email protected]
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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.
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4 International Journal of Rotating Machinery
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.
-
A. Boukhalfa et al. 5
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.
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6 International Journal of Rotating Machinery
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
-
A. Boukhalfa et al. 7
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.
-
8 International Journal of Rotating Machinery
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∑
n=1ρn(R4n − R4n−1
),
Ip = π2k∑
n=1ρn(R4n − R4n−1
),
(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]
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
,
-
A. Boukhalfa et al. 9
[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
-
10 International Journal of Rotating Machinery
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.
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