Journal of Sound and <ibration (2001) 242(1), 103}124doi:10.1006/jsvi.2000.3362, available online at http://www.idealibrary.com on
FINITE ELEMENT VIBRATION ANALYSIS OF ROTATINGTIMOSHENKO BEAMS
S. S. RAO
Department of Mechanical Engineering, ;niversity of Miami, Coral Gables, F¸ 33124-0624, ;.S.A.
AND
R. S. GUPTA
Department of Mechanical Engineering, Punjab Engineering College, Chandigarh 11, India
(Received 12 January 2000, and in ,nal form 13 September 2000)
The sti!ness and mass matrices of a rotating twisted and tapered beam element arederived. The angle of twist, breadth and depth are assumed to vary linearly along the lengthof beam. The e!ects of shear deformation and rotary inertia are also considered in derivingthe elemental matrices. The "rst four natural frequencies and mode shapes inbending}bending mode are calculated for cantilever beams. The e!ects of twist, o!set, speedof rotation and variation of depth and breadth taper ratios are studied.
( 2001 Academic Press
1. INTRODUCTION
Generally, turbomachine blades are idealized as tapered cantilever beams. In order to re"nethe analysis the e!ects of pre-twist and rotation are also included. As the blades are short insome of the designs and may vibrate in higher-frequency ranges, the e!ects of sheardeformation and rotary inertia may be of considerable magnitude. Various investigators inthe "eld of turbine-blade vibrations have solved the di!erential equations of motion, bytaking into account one or more of the above-mentioned e!ects.
The analysis of tapered beams has been made by many investigators using di!erenttechniques. Rao [1] used the Galerkin method, Housner and Keightley [2] applied theMyklestad procedure, Rao and Carnegie [3] used the "nite di!erences approach, Martin[4] adopted a perturbation technique and Mabie and Rogers [5] solved the di!erentialequations using Bessel functions to "nd the natural frequencies of vibration of taperedcantilever beams.
In analysing the pre-twisted beams, various approaches have been used by variousinvestigators. Mendelson and Gendler [6] used the station functions, Rosard [7] appliedthe Myklestad method, Di Prima and Handleman [8] adopted the Rayleigh}Ritzprocedure, Carnegie and Thomas [9] used the "nite di!erence techniques and Rao [10]considered the Galerkin method for the analysis of pre-twisted beams. The bendingvibration of a twisted rotating beam was considered by Targo! [11].
The rotation e!ect in beam analysis has also been considered using di!erent formulationand solution procedures. Targo! applied the matrix method, Yntema [12] applied theRayleigh}Ritz procedure, Isakson and Eisley [13] used the Rayleigh}Southwell procedure,Banerjee and Rao [14] applied the Galerkin procedure, Carnegie et al. [15] applied the"nite di!erence scheme, Krupka and Baumanis [16] used the Myklestad method and Rao
0022-460X/01/160103#22 $35.00/0 ( 2001 Academic Press
104 S. S. RAO AND R. S. GUPTA
and Carnegie [17] used the extended Holzer method to "nd the vibration characteristics ofrotating cantilever beams.
The "nite element technique has also been applied by many investigators, mostly for thevibration analysis of beams of uniform cross-section. All these investigations di!er from oneanother in the nodal degrees of freedom taken for deriving the elemental sti!ness and massmatrices. McCalley [18] derived the consistent mass and sti!ness matrices by selecting thetotal de#ection and total slope as nodal co-ordinates. Kapur [19] took bending de#ection,shear de#ection, bending slope and shear slope as nodal degrees of freedom and derived theelemental matrices of beams with linearly varying inertia. Carnegie et al. [20] analyzeduniform beams by taking few internal nodes in it. Nickel and Sector [21] used totalde#ection, total slope and bending slope of the two nodes and the bending slope at themid-point of the beam as the degrees of freedom to derive the elemental sti!ness and themass matrices of order seven. Thomas and Abbas [22] analyzed uniform Timoshenkobeams by taking total de#ection, total slope, bending slope and the derivative of thebending slope as nodal degrees of freedom.
The "nite element analysis of vibrations of twisted blades based on beam theory wasconsidered by Sisto and Chang [23]. Sabuncu and Thomas [24] studied the vibrationcharacteristics of pre-twisted aerofoil cross-section blade packets under rotating conditions.An improved two-node Timoshenko beam "nite element was derived by Friedman andKosmatka [25]. The vibration of Timoshenko beams with discontinuities in cross-sectionwas investigated by Farghaly and Gadelab [26, 27]. Corn et al. [28] derived "nite elementmodels through Guyan condensation method for the transverse vibration of short beams.Gupta and Rao [29] considered the "nite element analysis of tapered and twistedTimoshenko beams.
In this work, the "nite element technique is applied to "nd the natural frequenciesand mode shapes of beams in the bending}bending mode of vibration by takinginto account the taper, the pre-twist and the rotation simultaneously. The coupling thatexists between the #exural and torsional vibration is not considered. The taper and theangle of twist are assumed to vary linearly along the length of the beam. The elementsti!ness and mass matrices are derived and the e!ects of o!set, rotation, pre-twist, depthand breadth taper ratios and rotary inertia and shear deformation on the naturalfrequencies are studied. Various special cases of beam vibration can be obtained from thegeneral equations derived.
2. ELEMENT STIFFNESS AND MASS MATRICES
2.1. DISPLACEMENT MODEL
Figure 1(a) shows a doubly tapered, twisted beam element of length l with the nodes as1 and 2. The breadth, depth and the twist of the element are assumed to be linearly varyingalong its length. The breadth and depth at the two nodal points are shown as b
1, h
1and
b2, h
2respectively. The pre-twist at the two nodes is denoted by h
1and h
2. Figure 1(b)
shows the nodal degrees of freedom of the element where bending de#ection, bending slope,shear de#ections and shear slope in the two planes are taken as the nodal degrees offreedom. Figure 1(c) shows the angle of twist h at any section z. The beam is assumed torotate about the x}x-axis at a speed of X rad/s.
The total de#ections of the element in the y and x directions at a distance z from node 1,w(z) and v(z), are taken as
w (z)"wb(z)#w
s(z), v(z)"v
b(z)#v
s(z), (1)
Figure 1. (a) An element of tapered and twisted beam, (b) degrees of freedom of an element, (c) angle of twist h,(d) rotation of tapered beam.
ROTATING TIMOSHENKO BEAMS 105
where wb(z) and v
b(z) are the de#ections due to bending in the yz and xz planes respectively,
and ws(z) and v
s(z) are the de#ections due to shear in the corresponding planes.
The displacement models for wb(z), w
s(z), v
b(z) and v
s(z) are assumed to be polynomials of
third degree. They are similar in nature except for the nodal constants. These expressionsare given by
wb(z)"
u1
l3(2z3!3lz2#l3)#
u2
l3(3lz2!2z3)!
u3
l2(z3!2lz2#l2z)!
u4
l2(z3!lz2),
ws(z)"
u5
l3(2z3!3lz2!l3)#
u6
l3(3lz2!2z3)!
u7
l2(z3!2lz2#l2z)!
u8
l2(z3!lz2),
106 S. S. RAO AND R. S. GUPTA
vb(z)"
u9
l3(2z3!3lz2!l3)#
u10l3
(3lz2!2z3)!u11l2
(z3!2lz2#l2z)
(2)
!
u12l2
(z3!lz2),
vs(z)"
u13l3
(2z3!3lz2!l3)#u14l3
(3lz2!2z3)!u15l2
(z3!2lz2#l2z)
!
u16l2
(z3!lz2),
where u1, u
2, u
3and u
4represent the bending degrees of freedom and u
5, u
6, u
7and u
8are
the shear degrees of freedom in the yz plane; u9, u
10, u
11and u
12represent the bending
degrees of freedom and u13
, u14
, u15
and u16
shear degrees of freedom in the xz plane.
2.2. ELEMENT STIFFNESS MATRIX
The total strain energy ; of a beam of length l, due to bending and shear deformationincluding rotary inertia and rotation e!ects is given by
;"Pl
0CG
EIxx
2 AL2w
bLz2 B
2#EI
xy
L2wb
Lz2
L2vb
Lz2#
EIyy
2 AL2v
bLz2 B
2
H#
kAG
2 GALw
sLz B
2#A
Lvs
LzB2
HD dz#1
2 Pl
0
P(z) ALw
bLz
#
Lws
Lz B2dz (3)
#
1
2 Pl
0
P (z) ALv
bLz
#
Lvs
LzB2dz!P
l
0
pw(z) (w
b#w
s) dz!P
l
0
pv(z) (v
b#v
s) dz.
where
P (z)"PL`e
e`ze`z
mX2m dm+oAX2
2g[(¸#e)2!(e#z
e#z)2]
(4)
"
oAX2
g CAe¸#
1
2¸2!ez
e!
1
2z2eB!(e#z
e)z!
1
2z2D ,
pw(z)"
oAX2
g(w
b#w
s), p
v(z)"
oAX2
g(v
b#v
s), (5, 6)
where e is the o!set and zeis the distance of the "rst node of the element from the root of the
beam as shown in Figure 1(d), and P (z) is the axial force acting at section z.
ROTATING TIMOSHENKO BEAMS 107
As the cross-section of the element changes with z and as the element is twisted, thecross-sectional area A, and the moments of inertia I
xx, I
yyand I
xywill be functions of z:
A(z)"b (z)h(z)"Gb1#(b2!b
1)z
lH Gh1#(h2!h
1)z
lH(7)
"
1
l2(c
1z2#c
2lz#c
3l2),
where
c1"(b
2!b
1) (h
2!h
1), c
2"b
1(h
2!h
1)#h
1(b
2!b
1), c
3"b
1h1, (8)
Ixx
(z)"Ix{x{
cos2 h#Iy{y{
sin2 h, Iyy
(z)"Iy{y{
cos2 h#Ix{x{
sin2 h,
Ixy
(z)"(Ix{x{
!Iy{y{
)sin 2h
2, (9)
where x@x@ and y@y@ are the axes inclined at an angle h, the angle of twist, at any point in theelement, to the original axes xx and yy as shown in Figure 1(c). The value of I
x{y{"0 and the
values of Ix{x{
and Iy{y{
can be computed as
Ix{x{
(z)"b(z)h3 (z)
12"
1
12l4[a
1z4#a
2lz3#a
3l2z2#a
4l3z#a
5l4], (10)
where
a1"(b
2!b
1) (h
2!h
1)3, a
2"b
1(h
2!h
1)3#3(b
2!b
1) (h
2!h
1)2h
1,
a3"3Mb
1h1(h
2!h
1)3#(b
2!b
1) (h
2!h
1)h2
1, (11)
a4"3b
1h21(h
2!h
1)#(b
2!b
1)h3
1, a
5"b
1h31,
Iy{y{
(z)"h(z)b3 (z)
12"
1
12l4[d
1z4#d
2lz3#d
3l2z2#d
4l3z#d
5l4], (12)
where
d1"(h
2!h
1) (b
2!b
1)3, d
2"h
1(b
2!b
1)3#3(h
2!h
1) (b
2!b
1)2b
1,
d3"3Mh
1b1(b
2!b
1)3#(h
2!h
1) (b
2!b
1)b2
1, (13)
d4"3h
1b21(b
2!b
1)#(h
2!h
1)b3
1, d
5"h
1b31.
By substituting the expressions of wb, w
s, v
b, v
s, A, I
xx, I
xyand I
yyfrom equations (2), (7)
and (9) into equation (3), the strain energy ; can be expressed as
;"12uT[K]u, (14)
108 S. S. RAO AND R. S. GUPTA
where u is the vector of nodal displacements u1, u
2,2, u
16, and [K] is the elemental
sti!ness matrix of order 16. Denoting the integrals
Pl
0
EIxxA
L2wb
Lz2 B2
dz"[u1
u2
u3
u4]T[AK][u
1u2
u3
u4], (15)
Pl
0
EIyyA
L2vb
Lz2B2
dz"[u9
u10
u11
u12
]T[BK][u9
u10
u11
u12
], (16)
Pl
0
kAGALw
sLz B
2dz"[u
5u6
u7
u8]T[CK][u
5u6
u7
u8], (17)
Pl
0
EIxyA
L2wb
Lz2 B AL2v
bLz2B dz"[u
1u2
u3
u4]T[DK][u
9u10
u11
u12
], (18)
Pl
0
P (z)ALw
bLz B
2dz"[u
1u2
u3
u4]T[EK][u
1u2
u3
u4] (19)
and
Pl
0
2oAX2
g(w2
b) dz"[u
1u2
u3
u4]T[FK][u
1u2
u3
u4], (20)
the element sti!ness matrix can be expressed as
[K]"
[AK]#[EK]![FK] [EK]![FK] [DK] [0]
[EK]![FK] [CK]#[EK]![FK] [0] [0]
[DK] [0] [BK]#[EK]![FK] [EK]![FK]
[0] [0] [EK]![FK] [CK]#[EK]![FK]
,
(21)
where [AK], [BK], [CK], [DK], MEK] and [FK] are symmetric matrices of order 4 andtheir elements are formulated in Appendix B. [0] is a null matrix of order 4.
2.3. ELEMENT MASS MATRIX
The kinetic energy of the element ¹ including the e!ects of shear deformation and rotaryinertia is given by
¹"Pl
0CoA
2g ALw
bLt
#
Lws
Lt B2#
oA
2g ALv
bLt
#
Lvs
Lt B2#
oIyy
2g AL2v
bLz LtB
(22)
#
oIxy
g AL2w
bLz LtB A
L2vb
Lz LtB#oI
xx2g A
L2wb
Lz LtB2
D dz.
ROTATING TIMOSHENKO BEAMS 109
By de"ning
Pl
0
oA
g ALw
bLt B
2dz"[uR
1uR2
uR3
uR4]T[AM][uR
1uR2
uR3
uR4], (23)
Pl
0
oIxx
g AL2w
bLz LtB
2dz"[uR
1uR2
uR3
uR4]T[BM][uR
1uR2
uR3
uR4], (24)
Pl
0
oIyy
g AL2v
bLz LtB
2dz"[uR
9uR10
uR11
uR12
]T[CM][uR9
uR10
uR11
uR12
], (25)
and
Pl
0
oIxy
g AL2w
bLz LtB A
L2vb
Lz LtB dz"[uR1
uR2
uR3
uR4]T[DM][uR
9uR10
uR11
uR12
], (26)
where uRl
denotes the time derivative of the nodal displacement ui, i"1, 2,2 , 16, the
kinetic energy of the element can be expressed as
¹"12u5 [M]u5 , (27)
where [M] is the mass matrix given by
[M]"
16]16
[AM]#[BM] [AM] [DM] [0]
[AM] [AM] [AM] [0]
[DM] [AM] [AM]#[CM] [AM]
[0] [0] [AM] [AM]
(28)
and [AM], [BM] [CM] and [DM] are symmetric matrices of order 4 whose elements arede"ned in Appendix A.
2.4. BOUNDARY CONDITIONS
The following boundary conditions are to be applied depending on the type of endconditions.
Free endLw
sLz
"0,Lv
sLz
"0,L2v
bLz2
"0,L2w
bLz2
"0. (29)
Clamped end ws"0, w
b"0, v
s"0, v
b"0,
Lwb
Lz"0,
Lvb
Lz"0. (30)
Hinged end ws"0, w
b"0, v
s"0, v
b"0.
L2vb
Lz2"0,
L2wb
Lz2"0. (31)
TABLE 1
Natural frequencies (Hz)
No ofelements
First mode Second mode Third mode Fourth mode
1 304.3 1191)66 2327)07 4552)232 296)22 1161)70 1779)50 4106)833 295)03 1155)97 1746)55 3747)044 294)85 1154)94 1741)39 3697)055 294)78 1154)67 1739)99 3689)826 294)78 1154)58 1739)40 3683)987 294)78 1154)54 1739)25 3683)918 294)78 1154)50 1739)10 3683)85
Data: length of beam"0)1524 m, breadth at root"0)0254 m, depth at root"0)0046 m, depth taperratio"2)29, breadth taper ratio"2)56, twist angle"453, shear coe$cient"0)833, E"2)07]1011 N/m2,G"E/2)6, o!set"0, mass density"800 kg/m3.
Figure 2. E!ect of depth taper ratio and shear deformation on frequency ratio of rotating beam:**, withoutshear deformation; } } } }, with shear deformation.
110 S. S. RAO AND R. S. GUPTA
ROTATING TIMOSHENKO BEAMS 111
3. SPECIAL CASES
The various special cases of the beam vibration problem can be solved by applying one ormore of the following four conditions:
(a) For non-rotating beams: X"0 which results in [EK]"[FK]"[0]. (32)
(b) For uniform beams: by setting b2"b
1and h
2"h
1, one obtains
c1"c
2"0 and c
3"b
1h1, a
1"a
2"a
3"a
4"0 and a
5"b
1h31, (33)
d1"d
2"d
3"d
4"0 and d
5"h
1b31.
Figure 3. E!ect of breadth taper ratio and shear deformation on frequency ratio of rotating beam: **, withshear deformation; } } } }, without shear deformation.
112 S. S. RAO AND R. S. GUPTA
(c) For neglecting the e!ect of shear deformation: ws"v
s"0 so that equations (1) and (2)
become
w (z)"wb(z) and v (z)"v
b(z). (34)
Due to this, the order of [K] and [M] matrices reduces from 16 to 8.(d) For beams without pre-twist: in this case, there will be no coupling between the moment
of inertia terms and one obtains
Ixx"I
x{x{, I
yy"I
y{y{, I
xy"0. (35)
For vibration in the yz plane, vb"v
s"0; for vibration in the xz plane, w
b"w
s"0.
This condition further reduces the size of matrices [K] and [M] by half. Thus, for thegeneral case (with shear deformation) the matrices will be of order 8 and, if coupled withcondition (c) these will be of order 4.
If the conditions (a), (b) and (d) are applied one gets the following expressions for [K] and[M] for non-rotating uniform beams without pre-twist but with a consideration of rotaryinertia and shear deformation e!ects (for vibration in the y}z plane):
[K]"EI
xxl3
12 !12 !6l !6l 0 0 0 0
12 6l 6l 0 0 0 0
4l2 2l2 0 0 0 0
4l2 0 0 0 0
36J !36J !3lJ !3lJ
Symmetric 36J 3lJ 3lJ
4l2J !l2J
4l2J
, (36)
[M]"oAI
420g
156#36PM 54!36PM 22l!3lPM 13l!3lPM 156 54 22l 13l
156#36PM !13l#3lPM 22l#3lPM 54 156 !13l 22l
4l2#4l2PM !3l2!l2PM 22l !13l 4l2 !3l2
4l2!4l2PM 13l 22l !3l2 4l2
156 54 22l 13l
156 !13l 22l
4l2 !3l2
4l2
, (37)
where J"kGb1h1l2/(30EI
xx) , and PM "14I
xx/(b
1h1l2),
ROTATING TIMOSHENKO BEAMS 113
Equations (36) and (37) further reduce to the following well-known equations if the e!ectsof shear deformation and rotary inertia are neglected:
[K]"EI
xxl3
12 !12 !6l !6l
12 6l 6l
Symmetric 4l2 2l2
4l2
, (38)
[M]"ob
1h1
420g
156 54 !22l 13l
156 !13l 22l
Symmetric 4l2 !3l
4l2
. (39)
4. NUMERICAL RESULTS
The element sti!ness and mass matrices developed are used for the dynamic analysis ofcantilever beams. By using the standard procedures of structural analysis, the eigenvalue
Figure 4. E!ect of rotation and twist on "rst and second natural frequencies.
114 S. S. RAO AND R. S. GUPTA
problem can be stated as
([K3]!u2[M
3])U
3"0, (40)
where [K3] and [M
3] denote the sti!ness and mass matrices of the structure, respectively,
U3
indicates nodal displacement vector of the structure, and u is the natural frequency ofvibration.
A study of the convergence properties of the general element developed has been madeand the results are given in Table 1. It is seen that the results converge well even with onlyfour elements. The e!ects of shear deformation and depth taper ratio on the naturalfrequencies of a rotating twisted beam are shown in Figure 2 for a beam of length 0)254 m,o!set zero, depth at root 0)00865 m, breadth at root 0)0173 m, twist 453, rotation 100 r.p.s.
Figure 5. E!ect of rotation and twist on third and fourth natural frequencies: **, third mode; } } }, fourthmode.
ROTATING TIMOSHENKO BEAMS 115
and breadth taper ratio 3. The material properties of the beam are taken the same as thosegiven in Table 1. The e!ect of shear deformation is found to reduce the frequencies at highermodes while at lower modes the results are nearly una!ected. There is an increase in thefrequencies of vibration with an increase in the depth taper ratio in the "rst, second andfourth modes while a decrease has been observed in the case of third mode (vibration ina perpendicular plane). Figure 3 shows the variation of natural frequencies with breadthtaper ratio. In this case the data are the same as in the case of Figure 2.
In Figures 4 and 5, the variation of frequency ratio with rotation and pre-twist isstudied. It can be seen that the frequency ratio changes slightly with the rotationbut appreciably with the twist. At higher modes (in Figure 5) the e!ect of twist can be seento be more pronounced. It is also observed that the frequency ratio increases withan increase in the twist in the case of "rst and third modes while it decreases with an increaseof the twist in the case of second and fourth modes of vibration. In Figure 6, the e!ect ofo!set is studied for a twisted blade having 603 twist with the other data the same as that ofFigure 4. It is observed that an increase in o!set changes the frequency ratio more at highervalues of rotation. The frequency ratio has been found to increase with an increase in theo!set.
Figure 6. E!ect of o!set and rotation on frequency ratio. O!set 1: e"0 m; o!set 2: e"0.0254 m; o!set 3:e"0.0508 m; o!set 4: e"0.0762 m.
116 S. S. RAO AND R. S. GUPTA
5. CONCLUSION
The mass and sti!ness matrices of a thick rotating beam element with taper and twist aredeveloped for the eigenvalue analysis of rotating, doubly tapered and twisted Timoshenkobeams. The element has been found to give reasonably accurate results even with four "niteelements. The e!ects of breadth and depth taper ratios, twist angle, shear deformation, o!setand rotation on the natural frequencies of vibration of cantilever beams are found. Theconsideration of shear deformation is found to reduce the values of the higher naturalfrequencies of vibration of the beam. An increase in the breadth and depth taper ratios isfound to increase the "rst two modes of vibration. The frequency ratio is found to changeonly slightly with rotation but appreciably with the twist of the beam.
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11. W. P. TARGOFF 1957 Proceedings of third Midwestern Conference on Solid Mechanics, 177. Thebending vibrations of a twisted rotating beam.
12. R. T. YNTEMA 1955 NACA ¹N 3459. Simpli"ed procedures and charts for the rapid estimation ofbending frequencies of rotating beams.
13. G. ISAKSON and G. J. EISLEY 1964 NASA No. NSG-27-59. Natural frequencies in coupled bendingand torsion of twisted rotating and nonrotating blades.
14. S. BANERJEE and J. S. RAO American Society of Mechanical Engineers Paper No. 76-GT.43.Coupled bending-torsion vibrations of rotating blades.
15. W. CARNEGIE, C. STIRLING and J. FLEMING 1966 Paper No. 32. Applied Mechanics Convention,Cambridge, MA. Vibration characteristics of turbine blading under rotation-results of an initialinvestigation and details of a high-speed test installation.
16. R. M. KRUPKA and A. M. BAUMANIS 1969 Journal of Engineering for Industry American Society ofMechanical Engineers 91, 1017. Bending bending mode of a rotating tapered twistedturbomachine blade including rotary inertia and shear de#ection.
17. J. S. RAO and W. CARNEGIE 1973 International Journal of Mechanical Engineering Education 1,37}47. Numerical procedure for the determination of the frequencies and mode shapes of lateralvibration of blades allowing for the e!ects of pretwist and rotation.
18. R. MCCALLEY 1963 Report No. DIG SA, 63}73, General Electric Company, Schenectady, New>ork. Rotary inertia correction for mass matrices.
19. K. K. KAPUR 1966 Journal of Acoustical Society of America 40, 1058}1063. Vibrations ofa Timoshenko beam, using "nite element approach.
ROTATING TIMOSHENKO BEAMS 117
20. W. CARNEGIE, J. THOMAS and E. DOCUMAKI 1969 Aeronautical Quarterly 20, 321}332. Animproved method of matrix displacement analysis in vibration problems.
21. R. NICKEL and G. SECOR 1972 International Journal of Numerical Methods in Engineering 5,243}253. Convergence of consistently derived Timoshenko beam "nite elements.
22. J. THOMAS and B. A. H. ABBAS 1975 Journal of Sound and <ibration 41, 291}299. Finite elementmodel for dynamic analysis of Timoshenko beam.
23. F. SISTO and A. T. CHANG 1984 American Institute of Aeronautics and Astronautics Journal 22,1646}1651. A "nite element for vibration analysis of twisted blades based on beam theory.
24. M. SABUNCU and J. THOMAS 1992 American Institute of Aeronautics and Astronautics Journal 30,241}250. Vibration characteristics of pretwisted aerofoil cross-section blade packets underrotating conditions.
25. Z. FRIEDMAN and J. B. KOSMATKA 1993 Computers and Structures 47, 473}481. An improvedtwo-node Timoshenko beam "nite element.
26. S. H. FARGHALY Journal of Sound and <ibration 174, 591}605. Vibration and stability analysis ofTimoshenko beams with discontinuities in cross-section.
27. S. H. FARGHALY and R. M. GADELRAB 1995 Journal of Sound and <ibration 187, 886}896. Freevibration of a stepped composite Timoshenko cantilever beam.
28. S. CORN, N. BUHADDI and J. PIRANDA 1997 Journal of Sound and <ibration 201, 353}363.Transverse vibrations of short beams: "nite element models obtained by a condensation method.
29. R. S. GUPTA and S. S. RAO 1978 Journal of Sound and <ibration 56, 187}200. Finite elementeigenvalue analysis of tapered and twisted Timoshenko beams.
APPENDIX A: EXPRESSIONS FOR STRAIN AND KINETIC ENERGIES
A.1. EXPRESSION FOR STRAIN ENERGY (;)
A.1.1. Strain energy due to bending
If the bending de#ections in yz and xz planes of a beam are wband v
b, respectively, the
axial strain and stress induced due to wband v
bare given by e
xxdue to w
b"!y (L2w
b/Lz2);
exx
due to vb"!xL2v
b/Lz2 and p
xxdue to w
b"!EyL2w
b/Lz2; p
xxdue to v
b"
!ExL2vb/Lz2 .
The strain energy stored in the beam due to bending is given by: ; due to bending
"
1
2 PV
exx
pxx
d<
"
1
2 PV
(exx
due to wb#e
xxdue to v
b) (p
xxdue to w
b#p
xxdue to v
b) d<
"
E
2 Pl
0
dz PAAy
L2wb
Lz2#x
L2vb
Lz2B2dA
"
E
2 Pl
0
dzCAL2w
bLz2 B
2
PA
y2 dA#AL2v
bLz2B
2
PA
x2 dA#2L2w
bLz2
L2vb
Lz2 PA
xy dAD
"Pl
0CEI
xx2 A
L2wb
Lz2 B2#EI
xy
L2wb
Lz2
L2vb
Lz2#
EIyy
2 AL2v
bLz2B
2
D dz, (A.1)
where < is the volume, l is the length and A is the cross-sectional area of the beam.
118 S. S. RAO AND R. S. GUPTA
A.1.2. Strain energy due to shearing
Let Fx
and Fy
be the shear forces that produce the shear de#ections dvsand dw
sin an
element of length dz respectively. Then the strain energy of the beam due to shearing isgiven by
;due to shearing
"
1
2 Pl
0AFx
dvs
dz#F
y
dws
dz B dz.
By substituting AGk dvs/dz and AGk dw
s/dz for F
xand F
y, respectively, one obtains
;due to shearing
"Pl
0
kAG
2 CAdw
sdz B
2#A
dvs
dzB2
Ddz. (A.2)
A.1.3. Strain energy due to rotation
The rotation of a beam induces an axial force P in the beam due to centrifugal action. Ifthe beam is bending in the yz plane (Figure A1), the change in the horizontal projection ofan element of length ds is given by
ds!dz"G(dz)2#ALw
LzdzB
2
H1@2
!dz+1
2 ALw
LzB2dz.
Since the axial force P acts against the changes in the horizontal projection, the work doneby P is given by
;due to P andw
"!
1
2 Pl
0
P(z) Adw
dzB2dz. (A.3)
The work done by the transverse distributed force pw(z) can be written as
;due to pw
"Pl
0
pw(z)w dz. (A.4)
Figure A.1. An element of the beam in equilibrium.
ROTATING TIMOSHENKO BEAMS 119
The expressions corresponding to the bending of the beam in the xz plane can be obtainedsimilarly as
;due to P and v
"
1
2 Pl
0
P(z) Adv
dzB2
dz, ;due to pv
"Pl
0
pv(z)v dz. (A.5, 6)
The total strain energy of the beam can be obtained as given in equation (3) by combiningequations (A.1)}(A.6).
A.2. EXPRESSION FOR KINETIC ENERGY (¹)
Consider a small element of area dA and length dz at a point in the cross-section havingco-ordinates (x, y) with respect to x- and y-axes. The kinetic energy of this element is givenby
o2g
[(wR 2#vR 2)#(y/Qx#x/Q
y)2] dz dA,
where /xand /
ydenote the bending slopes, Lv
b/Lz and Lw
b/Lz, respectively, and a dot over
a symbol represents derivative with respect to time. Integrating this equation over the beamcross-section, the kinetic energy of an element of length dz can be obtained as
d¹"
o2g
[A(wR 2#vR 2)#(Ixx
/Q 2y#2I
xy/Qx/Q
y#I
yy/Q 2x)] dz.
The kinetic energy of the entire beam (¹ ) can be expressed as
¹"Pl
0CoA
2g(wR 2#vR 2)#
o2g
(Ixx
/Q 2y#2I
xy/Q
x/Qy#I
yy/Q 2
x)] dz, (A.7)
which can be seen to be the same as in equation (22).
APPENDIX B: EXPRESSIONS FOR [AK], MBK],2 , [DM]
The following notation is used for convenience:
;i"P
l
0
zi~1 dz, ¸i"li~1, i"1, 2,2, n, (B.1, 2)
<i"P
l
0
zi~1 cos2C(h2!h1)z
l#h
1D dz, i"1, 2,2 , n, (B.3)
Si"P
l
0
zi~1 sin 2C(h2!h1)z
l#h
1D dz, i"1, 2,2 , n, (B.4)
where h1
and h2
denote the values of pre-twist at nodes 1 and 2, respectively, of the element.As the nature of w
b, w
s, v
band v
sis the same except for their positions in the sti!ness and
mass matrices, one can use wN to denote any one of the quantities wb, w
s, v
bor v
sand in
TABLE B1
<alues of Hi,j
, Ri, j,k
, Qi,j,k
Ri, j,k
Qi, j,k
i j Hi,j
k"1 k"2 k"3 k"1 k"2 k"3 k"4 k"5
1 1 0 144)0 !144)0 36)0 36)0 !72)0 36)0 0)0 0)01 2 0 !144)0 144)0 !36)0 !36)0 72)0 !36)0 0)0 0)01 3 1 !72)0 84)0 !24)0 !18)0 42)0 !30)0 6)0 0)01 4 1 !72)0 60)0 !12)0 !18)0 30)0 !12)0 0)0 0)02 2 0 144)0 !144)0 36)0 36)0 !72)0 36)0 0)0 0)02 3 1 72)0 !84)0 24)0 18)0 !42)0 30)0 !6)0 0)02 4 1 72)0 !60)0 12)0 18)0 !30)0 12)0 0)0 0)03 3 2 36)0 !48)0 16)0 9)0 !24)0 22)0 !8)0 1)03 4 2 36)0 !36)0 8)0 9)0 !18)0 11)0 !2)0 0)04 4 2 36)0 !24)0 4)0 9)0 !12)0 4)0 0)0 0)0
120 S. S. RAO AND R. S. GUPTA
a similar manner the set (uN1, uN
2, uN
3, uN
4) can be used to represent any one of the sets
(u1, u
2, u
3, u
4), (u
5, u
6, u
7, u
8), (u
9, u
10, u
11, u
12) or (u
13, u
14, u
15, u
16). Thus,
wN (z)"uN1
l3(2z3!3lz2#l2)!
uN3
l2(z3!2lz2#l2z)#
uN2
l3(3lz2!2z3)!
uN4
l2(z3!lz2), (B.5)
dwNdz
"
uN1
l3(6z2!6lz)!
uN3
l2(3z2!4lz#l2)#
uN2
l3(6lz!6z2)!
uN4
l2(3z2!2lz), (B.6)
d2wNdz2
"
uN1
l3(12z!6l)!
uN3
l2(6z!4l)#
uN2
l3(6l!12z)!
uN4
l2(6z!2l). (B.7)
By letting Pi,j,k
(i"1,2, 4; j"1,2, 4; k"1,2, 7) denote the coe$cient of zk~1 l7~k forthe uN
iuNj
term in the expression of wN 2, Qi, j,k
(i"1,2, 4; j"1,2, 4; k"1,2, 5), thecoe$cient of zk~1 l5~k for the uN
iuNjterm in the expression of (dwN /dz)2, R
i, j,k(i"1,2, 4;
j"1,2, 4; k"1,2, 3), the coe$cient of zk~1 l3~k for the uNiuNjterm in the expression of
(d2wN /dz2)2, Hi,j
(i"1,2, 4; j"1,2, 4), the index coe$cient of l to account for thedi!erence in the index of l due to multiplication of rotational degrees of freedom uN
1and
uN2
and the displacement degrees of freedom uN3
and uN4, the values of P
i, j,k, Q
i, j,k, R
i, j,kand
Hi,j
can be obtained as shown in Tables B1 and B2.
B.1. EVALUATION OF [BK]
As the procedure for the derivation of [AK], [BK],2, [DM] is the same, the expressionfor [BK] is derived here as an illustration:
[u9
u10
u11
u12
]T[BK][u9
u10
u11
u12
]"Pl
0
EIyyA
L2vb
Lz2B2dz
(B.8)
"Pl
0
EIyy A
L2wNLz2B
2dz,
TABLE B2
<alues of Pi, j,k
Pi, j,k
i j k"1 k"2 k"3 k"4 k"5 k"6 k"7
1 1 4)0 !12)0 9)0 4)0 !6)0 0)0 1)01 2 !4)0 12)0 !9)0 !2)0 3)0 0)0 0)01 3 !2)0 7)0 !8)0 2)0 2)0 !1)0 0)01 4 !2)0 5)0 !3)0 !1)0 1)0 0)0 0)02 2 4)0 !12)0 9)0 0)0 0)0 0)0 0)02 3 2)0 !7)0 8)0 !3)0 0)0 0)0 0)02 4 2)0 !5)0 3)0 0)0 0)0 0)0 0)03 3 1)0 !4)0 6)0 !4)0 1)0 0)0 0)03 4 1)0 !3)0 3)0 !1)0 0)0 0)0 0)04 4 1)0 !2)0 1)0 0)0 0)0 0)0 0)0
ROTATING TIMOSHENKO BEAMS 121
where
wN "vb
and GuN1
uN2
uN3
uN4H"G
u9
u10
u11
u12H .
Substituting the values of Iyy
and wN into equation (B.8),
Pl
0
EIyy A
L2wNLz2B
2dz"P
l
0
ECIx{x{#(Iy{y{
!Ix{x{
) cos2G(h2!h1)z
l#h
1HD(B.9)
CuN1
l3(12z!6l)!
uN3
l2(6z!4l)#
uN2
l3(6l!12z)!
uN4
l2(6z!2l)D
2dz
with
BK1,1
"coe$cient of uN1uN1"coe$cient of u
9u9
"
E
12l10lHi,j P
l
0CT(a1z4#a
2lz3#a
3l2z2#a
4l3z#a
5l4)
#M(d1z4#d
2lz3#d
3l2z2#d
4l3z#d
5l4)
!(a1z4#a
2lz3#a
3l2z2#a
4l3z#a
5l4)N cos2G(h2
!h1)z
l#h
1HU]MR
1,1,1z2#R
1,1,2lz#R
1,1,3l2ND dz
122 S. S. RAO AND R. S. GUPTA
"
E
12l10¸
(H1,1`1)
5+i/1
ai[R
1,1,1;
8~i#¸
i`1R
1,1,2;
7~i#¸
i`2R
1,1,3<6~i
]
(di!a
i) [¸
iR
1,1,1<8~i
#¸i`1
R1,1,2<7~i
#¸i`2
R1,1,3<6~i
] (B.10)
"
E
12l10¸
(H1,1`1)
5+i/1
3+j/1
Mai¸i`j~1
;9~i~j
R1,1, j
N#(di!a
i)M¸
i`j~1<9~i~j
R1,1,j
N .
This relation can be generalized as
BKI,J
"
E
12l10¸
(H1,J`1)
5+i/1
3+j/1
Mai¸
(i`j~1);
(9~i~j)R
I,J,jN
#(di!a
i)M¸
(i`j~1)<(9~i~j )
RI,J,j
N, I"1,2, 4, J"I ,2, 4,
(B.11)
"
E
12l10
5+i/1
3+j/1
[Mai¸(i`j`HI,J)
;(9~i~j)
RI,J,j
N
#(di!a
i)M¸
(i`j`HI,J)<(9~i~j)
RI,J,j
N], I"1,2, 4, J"1,2, 4.
Similarly,
AKI,J
"
E
12l10
5+i/1
3+j/1
[Mdi¸
(i`j`HI,J);
(9~i~j)R
I,J,jN
(B.12)
#(ai!d
i)M¸
(i`j`HI,J)<(9~i~j)
RI,J,j
N], I"1,2, 4, J"1,2, 4,
CKI,J
"
kG
l8
5+i/1
3+j/1
[ci¸(i`j`HI,J)
;(9~i~j)
QI,J,j
], I"1,2, 4, J"1,2, 4, (B.13)
DKI,J
"
E
12l10
5+i/1
3+j/1
[(ai!d
i)¸
(i`j`HI,J)S(9~i~j)
RI,J,j
], I"1,2, 4, J"1,2, 4,
(B.14)
AMI,J
"
ogl8
3+i/1
7+j/1
[ci¸
(i`j`HI,J);
(11~i~j)PI,J,j
], I"1,2, 4, J"1,2, 4, (B.15)
BMI,J
"
o12gl10
5+i/1
5+j/1
[Mdi¸(i`j`HI,J)
;(11~i~j)
QI,J,j
N
(B.16)
#(ai!d
i)M¸
(i`j`HI,J)<(11~i~j)
QI,J,j
N], I"1,2, 4, J"1,2, 4,
CMI,J
"
o12gl10
5+i/1
5+j/1
[Mai¸(i`j`HI,J)
;(11~i~j)
QI,J,j
N
(B.17)
#(di!a
i)M¸
(i`j`HI,J)<(11~i~j)
QI,J,j
N], I"1,2, 4, J"1,2, 4,
ROTATING TIMOSHENKO BEAMS 123
DMI,J
"
o12gl10
5+i/1
5+j/1
[(ai!d
i)¸
(i`j`HI,J)S(11~i~j)
QI,J,j
],
(B.18)I"1,2, 4, J"1,2, 4.
B.2. EVALUATION OF [EK] AND [FK]
[u1
u2
u3
u4]T[EK] [u
1u2
u3
u4]"P
l
0
P(z) ALw
bLz B
2dz
"Pl
0
oAX2
g CAe¸#
1
2¸2!ez
e!z2
eB!(e#ze)z!
1
2z2D A
Lwb
Lz B2dz
(B.19)
"Pl
0
oAX2
g Ae¸#
1
2¸2!ez
e!z2
eB ALw
bLz B
2dz
!Pl
0
oAX2
g(e#z
e) z A
Lwb
Lz B2dz!P
l
0
1
2
oAX2
gz2 A
Lwb
Lz B2dz.
Thus,
EKI,J
"
oX2
gl8 Ae¸#
1
2¸2!ez
e!z2
eB3+i/1
5+j/1
[Ci¸(i`j`HI,J)
;(9~i~j)
QI,J, j
]
!
oX2
gl8(e#z
e)
3+i/1
5+j/1
[Ci¸
(i`j`HI,J);
(10~i~j)Q
I,J,j]
(B.20)
!
oX2
2gl8
3+i/1
5+j/1
[Ci¸(i`j`HI,J)
;(11~i~j)
QI,J,j
], I"1,2, 4, J"1,2, 4.
Similarly,
FKI,J
"
2oX2
gl8
3+i/1
7+j/1
[Ci¸(i`j`HI,J)
;(11~i~j)
PI,J,j
], I"1,2, 4, J"I ,2, 4. (B.21)
APPENDIX C: NOMENCLATURE
A area of cross-sectionb breadth of beame o!setE Young's modulusg acceleration due to gravityG shear modulush depth of beamIxx
, Iyy
, Ixy
moment of inertia of beam cross-section about xx-, yy- and xy-axis[K] element sti!ness matrixl length of an element¸ length of total beam[M] element mass matrixt time parameteru nodal degrees of freedom
124 S. S. RAO AND R. S. GUPTA
; strain energyv displacement in xz planew displacement in yz planex, y co-ordinate axesz co-ordinate axis and length parameterze
distance of the "rst node of the element from the root of the beamfrequency ratio ratio of modal frequency to frequency of fundamental mode of uniform beam with
the same root cross-section and without shear deformation e!ectsa depth taper ratio"h
1/h
2b breadth taper ratio"b1/b
2h angle of twisto weight densityk shear coe$cientX rotational speed of the beam (rad/s)
Subscriptsb bendings shear
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