doi:10.1006/jsvi.2000.3234, available online at http://www.idealibrary.com on

### VIBRATION ANALYSIS OF A ROTATING TIMOSHENKO

### BEAM

S. C. LIN

*Department of Power Mechanical Engineering, National Hu-wei Institute of ¹echnology, Hu-=ei,*
*632 >unlin, ¹aiwan, Republic of China*

AND K. M. HSIAO

*Department of Mechanical Engineering, National Chiao ¹ung ;niversity, Hsinchu, ¹aiwan,*
*Republic of China*

*(Received 15 April 1999, and in ,nal form 29 June 2000)*

The governing equations for linear vibration of a rotating Timoshenko beam are derived by the d&Alembert principle and the virtual work principle. In order to capture all inertia e!ect and coupling between extensional and #exural deformation, the consistent linearization of the fully geometrically non-linear beam theory is used. The e!ect of Coriolis force on the natural frequency of the rotating beam is considered. A method based on the power series solution is proposed to solve the natural frequency of the rotating Timoshenko beam. Numerical examples are studied to verify the accuracy of the proposed method and to investigate the e!ect of Coriolis force on the natural frequency of rotating beams with di!erent angular velocity, hub radius and slenderness ratio.

2001 Academic Press

1. INTRODUCTION

Rotating beams are often used as simple models for propellers, turbine blades, and satellite booms. The free vibration frequencies of rotating beams have been extensively studied [1}7]. Rotating beam di!ers from a non-rotating beam in having an additional centrifugal force and Coriolis e!ects on its dynamics. However, Coriolis e!ects were neglected in references [1, 3}6] and were considered in references [2, 7] for only Euler beam. In order to capture correctly all inertia e!ects and coupling between bending and stretching deformations of the beam, the equations of motion of beam might be derived by the fully geometrically non-linear beam theory [8]. However, in the conventional method [1, 3}6], the governing equations for the bending vibrations of rotating Timoshenko beam are not derived using consistent linearization of the fully geometrically non-linear beam theory. In references [1, 3}6], the beams are assumed to be linear elastic and inextensional. Thus, only bending vibrations are considered. However, in references [1, 3}6], the magnitudes of the steady state axial strain induced by the centrifugal force is not checked to verify the validity of their assumption of inextensional beam.

In this paper, exact governing equations for linear vibration of a rotating Timoshenko beam are derived based on the assumptions that the beam is linear elastic and the steady state axial strain is small. The e!ect of Coriolis force on the natural frequency of the rotating Timoshenko beam is considered. A method based on the power series solution to solve the

Figure 1. A rotating Timoshenko beam.

natural frequency of rotating Timoshenko beam is presented. The equations of motion for rotating Timoshenko beam are derived by the d'Alembert principle and the virtual work principle. In order to capture all inertia e!ect and coupling between extensional and #exural deformation, the consistent linearization [8, 9] of the fully geometrically non-linear beam theory is used in the derivation.

Numerical examples are studied to verify the accuracy of the proposed method and to investigate the e!ect of Coriolis force on the natural frequency of rotating beams with di!erent angular velocity, hub radius and slenderness ratio.

2. FORMULATION 2.1. DESCRIPTION OF PROBLEM

Consider a uniform Timoshenko beam rigidly mounted on the periphery of a rigid hub of
*radius R rotating about its axis "xed in space at a constant angular speed, as shown in*
Figure 1. The deformational displacements of the beam are de"ned in a rotating rectangular
Cartesian co-ordinate system which is rigidly tied to the hub. The origin of this co-ordinate
system is chosen to be the intersection of the centroid axes of the hub and the undeformed
*beam. The X-axis is chosen to coincide with the centroid axis of the undeformed beam, and*
*the X- and X-axis are chosen to be the principal directions of the beam cross-section at*
the undeformed state. In this paper, all vectors are referred to this co-ordinate system. The
angular velocity of the hub may be given by

*X"+0, X sin b, X cos b,,* (1)
where the symbol+ , denotes a column matrix, which is used throughout the paper; b, the
*angle between the hub axis and the X-axis, is the setting angle of the beam.*

Figure 2. Kinematics of deformed Timoshenko beam.

*Here it is assumed that the beam is only deformed in the X}X plane. As mentioned in*
reference [2], the #apwise and lagwise bending motions are coupled for setting angles other
thanb"03 and 903. Thus, onlyb"0 and 903 are considered in this study. Whenb"0 and
903, bending vibrations are #apwise and lagwise respectively. It is well known that the beam
sustains a steady state axial deformations (time-independent displacement) induced by
constant rotation [10]. In this study, the vibration (time-dependent displacement) of the
beam is measured from the position of the steady state axial deformation, and only
in"nitesimal free vibration is considered. Here the engineering strain and stress are used for
the measure of the strain and stress. It is assumed that the strains are small and the
stress}strain relationships are linear.

2.2. KINEMATICS OF TIMOSHENKO BEAM

*Let P (see Figure 2) be an arbitrary point in the beam element, and Q be the point*
*corresponding to P on the centroid axis. The position vector of point P in the undeformed*
and deformed con"gurations may be expressed as

*r"+R#x, y, z,,* (2)

r"*+R#x#uN(x, t)!z sin u, y, w(x, t)#z cos u,"rGeG,* (3)

*where t is the time, uQ(x) is the steady state axial deformations induced by the constant*
*rotation, u(x, t) and w(x, t) are the in"nitesimal displacements of point Q in the X and X*
directions, respectively, caused by the free vibration,*u"u(x, t) is the in*"nitesimal angle of
*rotation of the cross section passing through point Q about the negative X-axis, caused by*
*the free vibration, eG (i"1, 2, 3) denote the unit vectors associated with the XG-axis.*

For convenience, the engineering strains in the deformed beam are obtained from the
*corresponding Green strains in this study. If x, y and z in equation (2) are regarded as the*
Lagrange co-ordinates,e and e, the only non-zero components of the Green strains for
the Timoshenko beam, are given by [11]

*e"(rVrV!1),* (5)

*e"rVrX.* (6)

*From equations (3) and (4), rV and rX are expressed as*

*rV"(1#e)+cosh!iz cosu, 0, sinh!iz sinu,,* (7)

*rX"+!sinu, 0, cosu,,* (8)
e"**x*s*!*1"[(1#uN V)#wV]!1,* (9)
cosh"**(x#uN)*
**s* "
1
1#e*(1#uN V),* (10)
sinh"**w*
**s*"
1
1#e*wV,* (11)
i"**u*
**s*"
1
1#e*uV,* (12)

where*e is the unit extension of the centroid axis, h is angle measured from the X-axis to*
the tangent of the centroid axis. Making use of the assumption of small strain,e in equation
(9) may be approximated by

*e"uNV#(uNV#wV).* (13)

Substituting equations (7)}(12) into equations (5) and (6),e and e are given by

*e"+(1#e)[1#iz!2izcos(h!u)]!1,,* (14)

*e"(1#e)sin(h!u).* (15)

The engineering strain corresponding toe and e is given by [11]

e"(1#2e)!1, (16)

c"sin\

2eFigure 3. Free body of a portion of deformed beam.

Note thate and c in equations (16) and (17) are exact expression of engineering strain for
the Timoshenko beam. The exact expression ofe and c in equations (16) and (17) are quite
complicated. However, from the assumption of small strains, the approximationsh+sin h,
cos(*h!u)+1 and 1#e+1 may be used in the expression of e and c. Substituting*
equations (11), (12), (14) and (15) into equations (16) and (17), and using the
above-mentioned approximations,e and c may be approximated by

*e"e!zuV,* (18)

*c"wV!u.* (19)

2.3. EQUATIONS OF MOTION

The equations of motion for rotating Timoshenko beam are derived by the d'Alembert principle and the virtual work principle. The consistent linearization of the fully geometrically non-linear beam theory is used in the derivation.

Figure 3 shows a portion of the deformed centerline of the beam. Here the generalized
*displacements are chosen to be uN , w, and u de"ned in equation (3). The corresponding*
*generalized forces are F, F, and M, the forces in X, X directions, and moment about*
*negative X-axis. FH, FH, and MH ( j"1, 2) in Figure 3 denote the values of F, F, and M at*
*sections j.*

For linear elastic material, the virtual work principle may be written as

*d=CVR"d=GLR,* (20)
*d=CVR"(FduN#Fdw#Mdu)",* (21)
*d=GLR"E*

*4*

*deRe d<#aQ*

*4*

*dcRc d<#o*

*4*rKdr d<, (22) where

*d=CVR and d=GLR are the virtual work of the external forces and the internal stresses,*respectively, ( )" is the value of ( ) in section 2 minus the value of ( ) in section 1, de is the variation ofe given in equation (18), dc is the variation of c given in equation (19), dr is the

variation of r given in equation (3), and rK"d*r/dt. In this article, the symbol (*)) denotes
*di!erentiation with respect to time t. E is Young's modulus, G is shear modulus,aQ is the*
shear correction factor,o is the density, < is the volume of the undeformed beam between
*sections 1 and 2. The di!erential volume d< may be expressed as d<"dA dx, where dA is*
the di!erential cross-sectional area of the beam.

Equation (21) may be equal to [12]

*d=CVR"*

*dx(Mdu#FduN#Fdw)dx*"

*(MVdu#MduV#FVduN#FduNV#FVdw#FdwV)dx.*(23)

The exact expression of *d=GLR may be very complicated. However, due to the*
*assumption of in"nitesimal vibration, the quantities u, w, and* *u de*"ned in equations
*(3) and (4), and their derivatives with respect to x and t are all in"nitesimal quantities.*
For linear vibration analysis only the terms up to the "rst order of in"nitesimal quantities
are required. In order to retain all terms up to the "rst order of in"nitesimal
quantities in *d=GLR, all terms up to the "rst order of in"nitesimal quantities are retained*
for de, e, dc, c, dr, and rK in equation (21). Note that the steady state axial deformations

*uQ(x) in equation (4) and its derivatives with respect to x are small "nite quantities,*

not in"nitesimal quantities, and are all retained as zeroth order terms of in"nitesimal quantities.

From equations (3), (13), (18), and (19),dr, de, and dc may be approximated by

*dr"+duN!zdu, 0, dw!zudu,,* (24)

*de"(1#uNV)duNV#wVdwV!zduV,* (25)

*dc"dwV#du.* (26)

The second time derivative of r in equation (3) may be expressed as

*rK"rKGeG#2rRGeG#rGeKG,* (27)

e* G"X;eG, eKG"X;eG,* (28)

where**X is given in equation (1).**

From equations (1), (3), (27), and (28), rK in equation (27) may be approximated by

rK"

*uK!zuK #2wR X sin b#X[!(R#x#uN)#zu]*
2*X(uR!zuR)cos b!yX cos b#X(z#w) sin b cos b*

*wK#2X(!uR#zuR)sin b#yX sinb cos b!X(z#w) sin b*

. (29)

Substituting equations (18), (19), (24)}(26) and (29) into equation (22), using

*z dA"0,*

order of in"nitesimal quantities, one may obtain
*d=GLR"E*

*(AuN VduNV#Au1VwVdwV#IuVduV)dx*#

*aQGA*

*[(wV!u)dwV!(wV!u)du] dx*#

*oA*

*[uK#2wR X sin b!(R#x#uN)X]duN dx*

#*oI*

[*u*K !*uX cos b]du dx*

#*oA*

*[wK!2uR X sin b!wX sinb]dw dx,* (30)
*where I"*

*z dA is the moment of inertia of the cross-section.*

Substituting equations (23) and (30) into equation (20), and equating the terms on both
sides of equation (20) corresponding to the same generalized virtual displacements, one may
obtain
*FV"oA[uK#2wRX sinb!(R#x#uN)X],* (31)
*FV"oA(wK!2uRX sinb!wXsinb),* (32)
*MV"oI(uK!uXcosb)!aQGA(wV!u),* (33)
*M"EIuV,* (34)
*F"EAuNV,* (35)
*F"EAuQVwV#aQGA(wV!u).* (36)

Equations (31)}(33) are equations of motion and equations (34)}(36) are constitutive equations.

Substituting equations (34)}(36) into equations (31)}(33), one may obtain

*AEu _{N VV"oA[uNG#2wRXsinb!(R#x#uN)X],}* (37)

*AE(uQVwV)V#aQGA(wVV!uV)"oA(wK!2uRX sinb!wXsinb),* (38)

*EIuVV"oI(uK!uXcosb)!aQGA(wV!u),* (39)
where the underlined terms in equations (37) and (38) are Coriolis force. Whenb"n/2 and

*X is constant, equation (39) is identical to that given in reference [8], which is obtained by*

a consistent linearization of a fully non-linear beam theory. The term*uX cos b in equation*
(39) is replaced by*uX in reference [4], where a linear beam theory is used. It can be seen*
that whenb"n/2, the governing equation for rotating Timoshenko beam obtained by the
linear beam theory is incorrect.

*The boundary conditions for a "xed end at x"0 and for a free end at x"¸ are given by*

*uQ(0)"u(0, t)"0, w(0, t)"0, u(0, t)"0,* (40)

*uQV(¸)"uV(¸, t)"0, wV(¸, t)!u(¸, t)"0, uV(¸, t)"0.* (41)
2.4. STEADY STATE AXIAL DEFORMATIONS

*For the steady state axial deformations, uN (x, t)"uQ(x), u(x, t)"w(x, t)"u(x, t)"0.*
Thus, equations (37)}(41) can be reduced to

*EuQVV"o(R#x#uQ)X,* (42)

*uQ(0)"0, uQV(¸)"0.* (43)
The solution of equation (42), which satis"es boundary conditions in equation (43) may
be given by
*uQ(x)"R*

*kx*¸ !1# ¸#

*Rk sin k*

*k cos k*sin

*kx*¸ !

*x,*

*k"X¸(o/E,*(44, 45)

*where k is a dimensionless rotation speed. When k1, u1(x) in equation (44) may be*approximated by

*uQ(x)"oXE*!

*x*6 !

*Rx*2 # ¸

*x*2 #

*R¸x*. (46)

The centrifugal force corresponding to equation (46) is identical to that for the inextensional beam [4]. In order to compare the results with those given in the literature,

*k is assumed to be much smaller than unity and equation (46) is used to calculate the*

centrifugal force in this study.

2.5. FREE VIBRATION

The vibration of the beam is measured from the position of the steady state axial deformation. From equations (4), (37)}(39), and (42), the governing equations for free vibration may be expressed as

;KK"o¸*E* [;G#2=*Q X sin b!X;],* (47)

*NK=K#N=KK#k(=KK!uK)"*o¸*E* (!2;*Q X sin b!=X sin b#=G),* (48)
*uKK"*o¸*E* (*u*K !*uX cos b)!k(=K!u),* (49)

where
*N"k(!0)5m!rm#r#0)5)* (50)
*NK"!k(m#r),* (51)
k"*aQGE* , *k"gk, g"(A¸/I,*
m"*x*
¸,
;"*u*
¸,
="*w*
¸, *r"*
*R*
¸, (52)

*and k is de"ned in equation (45). Note thatg is the slenderness ratio of the beam, and N is*
*the steady state axial strain. The maximum value of N occurs at the root of the beam and*
may be expressed as

*eK?V"k(r#0)5).* (53)

For linear elastic materials, it is reasonable to assume that*eK?V)10\.*
We shall seek a solution of equations (47)}49) in the form

U(*m, t)"(U0(m)#iU'(m))e SR* (54)

U(*m, t)"+;, =, u,, U0(m)"+;0, =0, u0,, U'(m)"+;', =', u',,* (55)
where i"(!1, and u is the natural frequency to be determined. Introducing equation
(54) into equations (47)}(49), we obtain

PAKK#QAK#RA"0, (56)

PBKK#QBK#SB"0, (57)

A"*+;0, =', u',, B"+;', =0, u0,,* (58)
P"
1 0 0
0 *N#*k 0
0 0 1
, Q"
0 0 0
0 * _{NK !k}*
0 k 0
,
R"

*a*

*d*0

*d*

*b*0 0 0

*c*, S"

*a*!

*d*0 !

*d*

*b*0 0 0

*c*, (59)

*where a"k#K, b"K#k sin b, c"!k#K#k cos b, d"2kK sin b, and K is*
a dimensionless natural frequency given by

*K"u¸(o/E.* (60)

*It can be seen from equations (56)}(59) that ;'";0, =0"!=', and u0"!u'. Thus,*
only equation (56) is solved in this study. In the next section, a power series method is
employed to obtain the natural frequencies and vibration modes for free vibration.

2.6. POWER SERIES SOLUTION

From equations (50), (51) and (59), one can express P and Q in equation (59) as

P"P#mP#mP, Q"Q#mQ, (61), (62)

where P, P, P, Q and Q are constant matrices.

From equations (61) and (62), it can be seen that equation (56) is a set of linear ordinary di!erential equations with variable coe$cients. The solution of equation (56) can be expressed as a power series in the independent variablem:

A(m)"

*LCLmL, CL"+CL, CL, CL,,*

(63), (64)

*where CGL (i"1, 2, 3) are undetermined coe$cients.*

Substituting equation (63) into equation (56) and equating coe$cients of like power ofm, we obtain the recurrence formula

*CL"ALCL\#BLCL\, n*2,*
*AL"n(n!1)*!1
*A A 0*
*A A 0*
0 0 * _{A}*
,

*!1*

_{BL"}*n*0 0 0 0

*0*

_{B B}*, (65)*

_{B 0}*where A"a, A"d, A"d/f, A"D[b!(n!2)k!(n!2)(n!3)k], A"c,*

*B"!(n!1)kr/ f, B"!k/f, B"k, in which f"k#k(r#0)5), a, b, c, and*
*d are de"ned in equation (59).*

From equation (65), it can be seen that only C and C are independent constants in
*equation (63), and CL can be rewritten as*

*CL"YLC#YLC, n*2,*
Y*L"ALYL\*
#
*BLYL\* , Y*L"ALYL\* #
*BLYL\* ,
Y"Y"I, Y"Y"0, (66)
where I and 0 are unit and zero matrices of order 3;3, respectively,

A(m)"

I#*LmLYL*C#mI#

*LmLYL*C " E(m)C#E(m)C. (67)

From the boundary conditions given in equations (40) and (41), and equations (52), (55), (58) and (67), one can obtain

C"0, (68)

where M" 0 0 0 0 0 1 0 0 0 ,

*and K(K) is a function of K given in equation (60).*

For a non-trivial C, the determinant of the 3;3 matrix K in equation (69) must be equal
*to zero. The values of K which make this determinant vanish are called eigenvalues of*
matrix K and give the natural frequencies of the rotating Timoshenko beam through
*equation (60). The bisection method is used here to "nd the eigenvalues. Let KG and*
X denote an eigenvalue and the corresponding eigenvector of equation (69). The eigenvector
X may be obtained by solving the following standard eigenvalue problem:

*[K(KG)#I]X"jX,* (70)

where I is a unit matrix of order 3;3. It can be seen that j"1 is an eigenvalue of equation (70). The eigenvector of equation (70) corresponding toj"1 is the required eigenvector of equation (69). Here an inverse power method is used to "nd the eigenvalue and eigenvector of equation (70).

Substituting equation (68) and C"X into equation (67), the mode shape corresponding
*to KG can be obtained.*

3. NUMERICAL EXAMPLES

To verify the accuracy of the present method and investigate the e!ect of the Coriolis force on the natural frequency of the rotating beam, several numerical examples are studied. Here cases with and without considering the Coriolis force, referred to as cases A and B, respectively, are considered, and the corresponding results are referred as to Present-A and Present-B respectively.

In order to compare present results with those reported in the literature, in which the
linear beam theory is used and the Coriolis e!ect is not considered, the dimensionless
natural frequency *K"gK"u¸(oA/EI and dimensionless rotational speed*
*a"gk"X¸(oA/EI are also employed here. While most analytical studies reported in*
the literature do not provide the experimental results, limited experimental measurements
on the fundamental frequency under several rotating speeds are given in reference [1]. The
experimental evidence did back up the predicted analytical results.

With the consideration of the Coriolis force, except *b"0 or k"0, the axial and*
lateral vibrations are coupled in the vibration modes. However, for convenience, the
*dimensionless natural frequencies are divided into KG (KG) and K?G (K?G), where KG (KG) and K?G*
(*K?G) denote the ith dimensionless natural frequencies of lateral and axial vibration,*
*respectively, at k"0.*

The dimensionless natural frequencies of the rotating beams with dimensionless variables

*r"3,gk"10, and k"0)32693 are listed in Table 1. As expected, for b"03 the results of*

Present-A, Present-B and those reported in the literature are in close agreement. For b"903, the results of Present-B and those reported in the literature are in close agreement, but the discrepancy between the results of Present-A and Present-B are remarked. It is interesting to note that the governing equations of the Present-B for lateral vibration

TABLE1

*Dimensionless frequencies for rotating ¹imoshenko beam (a"gk"10, r"3, k"0)32693)*

g"10 g"20
b _{K}_{K}_{K}_{K}_{K}_{K}_{K}* _{K}*
03 A 22)938 44)781 66)287 71)967 23)491 55)984 96)913 143)71
B 22)938 44)791 66)287 71)967 23)491 55)984 96)913 143)71
C 23)050 45)598 67)716 73)076 23)524 56)105 97)188 144)49
D 22)938 44)781 66)287 71)967 23)491 55)984 96)913 143)71
E 23)037 45)428 66)854 72)313 23)514 56)072 97)011 143)82
903 A 8)500 29)152 49)372 74)141 16)491 37)751 91)041 140)98
B 20)853 44)957 66)677 71)985 21)298 55)240 96)594 143)50
C 20)867 45)115 67)520 72)756 21)313 55)284 96)747 144)21
D 20)753 44)315 66)109 71)620 21)277 55)162 96)473 143)43
E 20)850 44)955 66)668 71)982 21)302 55)250 96)570 143)53

*Note: A: Present-A; B: Present-B; C: reference [3]; D: reference [4]; E: reference [1].*

(equations (38) and 39)) are identical to those given in reference [4] forb"0. Thus, the
Present-B results are the same as those given in reference [4] forb"0. However, equation
(39) of the Present-B for lateral vibration is di!erent from that given in reference [4] for
b"903 as mentioned in section 2.3. Thus, the results are di!erent forb"903. Note that the
*maximum steady state axial strain (see equation (53)) for the rotating beam at k"0)5 and*
1 are 0)875 and 3)5 respectively. Thus, all results shown in Table 1 are only for academic
interest and may be meaningless in practice.

Table 2 presents the "rst and second dimensionless natural frequencies for in-plane
vibration (b"903) of a slender rotating beam. The slenderness ratio of the beam considered
isg"10. Thus, the beam may be regarded as the Euler}Bernoulli beam. The maximum
*steady state axial strain (see equation (53)) for the rotating beam corresponding to r"1 and*

*k"0)05 is 3)75*;10\. It is seen that the results of Present-A, Present-B and those reported

in the literature are in close agreement. It indicates that when the steady state axial strain is small, the e!ect of the Coriolis force on the natural frequencies of the rotating beam may be negligible.

*Figure 4 shows the "rst four dimensionless natural frequencies, KG (i"1, 2, 3) and K?, for*
the rotating beam at di!erent angular velocities. The setting angle b"903 and
*dimensionless variables r"0)1,*g"70, and k"0)32693 are used for this example. This
example was also studied by Yoo and Shin [7]. In reference [7], the beam was considered to
be Euler beam. The results of the present study and reference [7] are essentially in
*agreement. As can be seen that as k increases, K? increases for Present-A but decreases for*
*Present-B. The K and K? curves cross at k"0)34. The mode shapes corresponding to KG*
*and K? are shown in Figure 5 for k"0)3 and 0)4. As can be seen that at k"0)3 and 0)4, the*
*corresponding mode shapes of K and K? are similar. It is noted that u has appreciable*
*change. This result may be explained as follows. Even mode shapes of w' corresponding to*

*K? for k"0)3 and 0)4 look similar, however, the slopes of mode shapes of w' have*

*appreciable change. The values of wV and u should be very close, because the shear strain is*
small in this study. Thus, the corresponding *u' has appreciable change as well. This*
*observation veri"es that K and K? curves cross rather than veer at k"0)34. In reference*
*[7], it is stated that K and K? curves veer rather than cross at k"0)34. This statement*
*might be incorrect. It is also interesting to note that K and K? curves veer rather than*

TABLE2

*Dimensionless frequencies for rotating ¹imoshenko beam (*b"903,g"10, k"0)32693)

*r"0* *r"1*
*k (10*\) _{K (10\)}_{K (10\)}_{K (10\)}* _{K (10\)}*
0 Present-A 3)516 22)033 3)516 22)033
Present-B 3)516 22)033 3)516 22)033
Reference [3] 3)516 22)036 3)516 22)036
2 Present-A 3)622 22)525 4)400 23)279
Present-B 3)622 22)525 4)400 23)279
Reference [3] 3)622 22)528 4)401 23)282
5 Present-A 4)074 23)949 7)411 28)922
Present-B 4)074 24)949 7)411 28)922
Reference [3] 4)074 24)952 7)412 28)926
10 Present-A 5)048 32)118 13)257 43)224
Present-B 5)049 32)118 13)258 43)225
Reference [3] 5)050 32)123 13)261 43)237
20 Present-A 6)772 51)349 25)278 76)585
Present-B 6)774 51)351 25)286 76)588
Reference [3] 6)794 51)372 25)318 76)659
50 Present-A 10)416 116)148 61)463 181)780
Present-B 10)437 116)175 61)584 181)824
Reference [3] 10)899 116)200 61)641 181)936

Figure 4. Variation of natural frequency with rotational speed (b"903, r"0)1;g"70; k"0)32693): **, Present-A; - - - -, Present-B.

cross. This phenomenon might be explained as follows. When the rotation speed increases, the Coriolis force has stronger e!ect on the natural frequency of the lower lateral vibration mode, and the natural frequency of lower lateral vibration mode increases slower than that of the higher lateral vibration mode due to the centrifugal force. Thus, the net increase rate

Figure 5. Variation of mode shape (b"90*3, r"0)1;*g"70; k"0)32693):䊉*, ;0;*䊏*, =';*䊐,*u'.*
*of k is slower than that of k with the increase of rotation speed. In order to compare the*
*results with those given in reference [7], the frequency curves are presented for k"0}1)8.*
*However, the maximum steady state axial strain at k"0)2 is 0)024. Thus, as k'0)2, the*
results shown in Figure 4 might be meaningless.

*Tables 3 and 4 present dimensionless natural frequencies KG (i"1}4) and K?G (i"1, 2) at*
*dimensionless rotation speeds k"0, 0)05, and 0)1 for the rotating beams with di!erent*
slenderness ratios. The maximum steady state axial strain (see equation (53)) for the rotating
*beam with r"0 and 1 at k"0)1 are 0)005 and 0)015 respectively. Because the Coriolis force*
vanishes atb"0*3 and k"0, the results of cases A and B are identical for*b"0*3 and k"0.*
Thus, only the results of case A are presented in Tables 3 and 4 orb"0*3 and k"0. It is*
observed that forb"90*3, the values of K?G obtained by excluding the Coriolis force are*
always lower than those by including the Coriolis force. As can be seen from Tables 1, 3 and
*4, with the consideration of the Coriolis force, the values of KG (i"1}4) are reduced for*
*g*20. However, for g"10, only the values of KG (i"1}3) are reduced, but the value of K*
is increased. From equations (56) and (59), it seems that the e!ect of the Coriolis forces on
*the natural frequencies of lateral vibrations is a!ected by the distribution of the X*
component of the vibrations modes, which may be a!ected by slenderness ratiog. However,
the discrepancy between the results obtained by excluding and including the Coriolis force
is insigni"cant for all cases presented in Tables 3 and 4. It seems that the e!ect of the
Coriolis force on the natural frequencies of vibrating beams might be negligible.

TABLE3

*Dimensionless frequencies for rotating beam (k"0)32693, r"0)*

b g *k* Case _{K}_{K}_{K}_{K}*K?* *K?*
03 10 0 A 0)3231 1)4531 3)1671 4)8228 1)5708 4)7124
0)05 A 0)3272 1)4575 3)1729 4)8295 1)5700 4)7121
0)1 A 0)3391 1)4705 3)1902 4)8492 1)5676 4)7113
20 0 A 0)1718 0)9570 2)3376 3)9620 1)5708 4)7124
0)05 A 0)1800 0)9646 2)3460 3)9720 1)5700 4)7121
0)1 A 0)2025 0)9870 2)3711 4)0019 1)5676 4)7113
50 0 A 0)0701 0)4296 1)1640 2)1836 1)5708 4)7123
0)05 A 0)0886 0)4476 1)1823 2)2031 1)5700 4)7120
0)1 A 0)1285 0)4979 1)2354 2)2604 1)5676 4)7112
903 10 0 A 0)3231 1)4531 3)1671 4)8228 1)5708 4)7124
0)05 A 0)3230 1)4549 3)1722 4)8295 1)5748 4)7129
B 0)3236 1)4569 3)1726 4)8294 1)5700 4)7121
0)1 A 0)3226 1)4604 3)1876 4)8492 1)5867 4)7144
B 0)3251 1)4681 3)1892 4)8488 1)5676 4)7113
20 0 A 0)1718 0)9570 2)3376 3)9620 1)5708 4)7124
0)05 A 0)1728 0)9630 2)3453 3)9716 1)5733 4)7132
B 0)1731 0)9634 2)3456 3)9718 1)5700 4)7121
0)1 A 0)1753 0)9808 2)3684 4)0001 1)5809 4)7156
B 0)1766 0)9825 2)3694 4)0009 1)5675 4)7113
50 0 A 0)0701 0)4296 1)1640 2)1836 1)5708 4)7123
0)05 A 0)0731 0)4474 1)1812 2)2025 1)5732 4)7131
B 0)0732 0)4489 1)1813 2)2026 1)5700 4)7120
0)1 A 0)0803 0)4874 1)2310 2)2580 1)5804 4)7155
B 0)0809 0)4880 1)2316 2)2584 1)5676 4)7112
TABLE4

*Dimensionless frequencies for rotating beam (k"0)32693, r"1)*

b g *k* Case _{K}_{K}_{K}_{K}*K?* *K?*
03 10 0 A 0)3231 1)4531 3)1671 4)8228 1)5708 4)7124
0)05 A 0)3327 1)4638 3)1817 4)8395 1)5700 4)7121
0)1 A 0)3600 1)4953 3)2247 4)8887 1)5676 4)7113
20 0 A 0)1718 0)9570 2)3376 3)9620 1)5708 4)7124
0)05 A 0)1904 0)9749 2)3581 3)9868 1)5700 4)7121
0)1 A 0)2371 1)0266 2)4185 4)0600 1)5676 4)7113
50 0 A 0)0701 0)4296 1)1640 2)1836 1)5708 4)7123
0)05 A 0)1083 0)4707 1)2075 2)2308 1)5700 4)7120
0)1 A 0)1782 0)5760 1)3280 2)3653 1)5676 4)7112
903 10 0 A 0)3231 1)4531 3)1671 4)8228 1)5708 4)7124
0)05 A 0)3286 1)4611 3)1810 4)8395 1)5749 4)7129
B 0)3292 1)4632 3)1814 4)8394 1)5700 4)7121
0)1 A 0)3442 1)4837 3)2221 4)8883 1)5883 4)7148
B 0)3468 1)4929 3)2224 4)8882 1)5676 4)7113
20 0 A 0)1718 0)9570 2)3376 3)9620 1)5708 4)7124
0)05 A 0)1835 0)9733 2)3574 3)9863 1)5733 4)7132
B 0)1838 0)9737 2)3577 3)9854 1)5700 4)7121
0)1 A 0)2139 1)0205 2)4158 4)0582 1)5810 4)7157
B 0)2155 1)0225 2)4168 4)0590 1)5676 4)7113
50 0 A 0)0701 0)4296 1)1640 2)1836 1)5708 4)7123
0)05 A 0)0960 0)4680 1)2064 2)2302 1)5732 4)7131
B 0)0961 0)4681 1)2065 2)2303 1)5700 4)7120
0)1 A 0)1465 0)5668 1)3239 2)3630 1)5805 4)7155
B 0)1476 0)5675 1)3245 2)3634 1)5676 4)7112

Figure 6. Mode shape of a rotating beam (b"903, r"0;g"10; k"0)32693):䊉* _{, ;0;}*䊏

*䊐,*

_{, =';}*u' **, b"03;*} } },b"903.

Figure 6 Continued.

*As a "nal result, Figures 6 and 7 present the mode shapes corresponding to KG (i"1}4)*
*and K?G (i"1, 2) of case A at k"0 and 0)1 for g"10 and 50 given in Table 3 to illustrate the*
coupling between the axial and lateral vibrations of rotating beams. The mode shapes for
b"0*3 and 903 are identical at k"0. Thus, only the mode shapes for*b"03 are presented at

*k"0. Note that when the mode shapes for*b"03 and 903 are virtually indistinguishable at

*k"0)1, only the mode shapes for*b"03 are presented in Figures 6 and 7. As expected, for
b"03, the axial and lateral vibrations of rotating beams are not coupled. The coupling
between the axial and lateral vibrations of rotating beams can be observed from the mode
*shapes corresponding to KG and K?G (i"1, 2) at k"0)1 for b"903.*

4. CONCLUSIONS

In this paper, the correct governing equations for the linear vibration of a rotating uniform Timoshenko beam are derived based on the assumptions that the beam is linear elastic and the steady state axial strain is small. The e!ect of Coriolis force on the natural frequency of the rotating Timoshenko beam is investigated. The vibration of the beam is measured from the position of the steady-state axial deformation, and only in"nitesimal free vibration is considered. The equations of motion for rotating Timoshenko beam are derived by the d'Alembert principle and the virtual work principle. It is seen that even for linear vibration of a rotating Timoshenko beam, the exact governing equations might be derived by the consistent linearization of the fully geometrically non-linear beam theory. A method based on the power series solution is proposed to solve the natural frequency of rotating Timoshenko beam.

Figure 7. Mode shape of a rotating beam (b"903, r"0;g"50; k"0)32693):䊉* _{, ;0;}*䊏

*䊐,*

_{, =';}*u'; **, b"03;*} } },b"903.

Figure 7 Continued.

The results of numerical examples show that the e!ect of the Coriolis force on the natural frequencies of the rotating Timoshenko beam may be negligible when the beam is linear elastic and the steady state axial strain is small. It is suggested that the value of the maximum steady state axial strain should be checked to ensure meaningful results.

Finally, it may be emphasized that, although the proposed formulation and numerical procedure are applied to the uniform rotating cantilever beams here, the method described here can be easily extended to non-uniform rotating beams with discontinuities, as well as with other end conditions.

REFERENCES

1. J. T. S. WANG, O. MAHRENHOLTZand J. BOHM *1976 Solid Mechanics Archives 1, 341}365.*
Extended Galerkin's method for rotating beam vibrations using Legendre polynomials.
2. K. B. SUBRAHMANYAMand K. R. V. KAZA*1986 ASME Journal of <ibration, Acoustics, Stress, and*

*Reliability in Design 108, 140}149. Vibration and buckling of rotating, pretwisted preconed beams*
including Coriolis e!ects.

3. T. YOKOYAMA*1988 International Journal of Mechanical Sciences 30, 743}755. Free vibration*
characteristics of rotating Timoshenko beam.

4. S. Y. LEE and S. M. LIN *1994 ASME Journal of Applied Mechanics 61, 949}955. Bending*
vibrations of rotating nonuniform Timoshenko beams with an elastically restrained root.
5. H. DU, M. K. LIMand K. M. LIEW*1994 Journal of Sound and <ibration 175, 505}523. A power*

series solution for vibration of a rotating Timoshenko beam.

6. V. T. NAGARAJ*1996 Journal of Aircraft 33, 637}639. Approximate formula for the frequencies of*
a rotating Timoshenko beam.

7. H. H. YOOand S. H. SHIN*1988 Journal of Sound and <ibration 212, 807}828. Vibration analysis of*
rotating cantilever beams.

8. J. C. SIMO and L. VU-QUOC*1987 Journal of Sound and <ibration 119, 487}508. The role of*
non-linear theories in transient dynamic analysis of #exible structures.

9. K. M. HSIAO*1992 AIAA Journal 30, 797}804. Corotational total Lagrangian formulation for*
three-dimensional beam element.

10. P. W. LIKINS*1973 AIAA Journal 11, 1251}1258. Mathematical modeling of spinning elastic*
bodies for model analysis.

11. T. J. CHUNG*1988 Continuum Mechanics. Englewood Cli!s, NJ: Prentice Hall.*

12. D. J. MALVERN*1969 Introduction to the Mechanics of the Continuous Medium. Englewood Cli!s,*
NJ: Prentice-Hall.