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On the extensional and flexural vibrations of rotating bars
ON THE EXTENSIONAL AND FLEXURAL VIBRATIONS OF ROTATING BARS G. L. ANDERSON Be&t Weapons Laboratory, Watervliet Arsenal, Watervliet, New York 12189, U.S.A. (Receiced 4 April
1975)
Abstract-The non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated. Under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration, linearized equations of motion for the longitudinal and flexural deformations of a rotating bar carrying a tip mass are derived. Numerical computations for the natural frequencies of the lowest three modes of free vibration reveal that the values of the extensional frequencies increase monotonically, contrary to previously published results, as the angular velocity of rotation increases. 1. INTRODUCTION
The longitudinal static displacement and vibration of a slender bar rotating about a vertical axis with a constant angular velocity R have been investigated[l-31. In these studies, two observations were reported: (i) rotation of the bar tends to lower the natural frequencies of longitudinal vibration as compared to the natural frequencies of the nonrotating bar, and (ii) there exists a critical angular velocity at which the longitudinal displacement becomes arbitrarily large. The latter phenomenon was called “static resonance” in [l] and “static instability” or “divergence” in [2-31. In the related problem of flexural oscillations of rotating bars, several authors[4-8) have found that the rotation of the bar tends to increase the natural frequencies of flexural vibration relative to those for the non-rotating bar. The effects of hub radius and tip mass on the bending frequencies of rotating bars have been studied [S-lo] and [l l-121, respectively, where it was observed that, for small hub radius relative to the length of the beam, the bending frequency is nearly a linear function of hub radius, whereas the bending frequencies in the various natural modes may be increased or decreased as compared to those of the non-rotating bar by the presence of a tip mass. In the present investigation, the author reconsiders certain aspects of the boundary value problems described in references [l-12], deriving the non-linear equations for the flexural and extensional motions of slender rotating, elastic bars. When these non-linear equations are separated into the so-called initial stress problem and the deformed motion problem and linearized in the manner of [13-15-J, it is found that the boundary value problem discussed in reference [l] has not been formulated completely and that the assertion that the natural frequencies of longitudinal vibrations are lowered by the effect of whirling is incorrect. In addition, it appears that the use of the terminology “static instability” and “divergence” in references [2-31 is a misinterpretation of the actual physical situation. Furthermore, the present analysis reveals that in [l-12] the terms in the partial differential equations of both flexural and extensional motions of rotating bars which account for the initial stress field in the bar due to the centrifugal force effect are accurate only for relatively low angular velocities of rotation. This situation has arisen since the effect of the initial longitudinal displacement, arising because of the rotation of the system, in the initial stress problem is generally disregarded, either inadvertently or possibly as a first approximation. 2. THE
NON-LINEAR
FORMULATION
Consider a long, slender bar of length G whose axis lies along the xi-axis and whose left end is displaced a horizontal distance cl from the vertical y3-axis, as shown in Fig. 1. It is assumed that the y1yzy3-coordinate frame remains fixed in space, whereas the horizontal xi-axis rotates about the vertical y,-axis with constant angular velocity Q. 223
G.L.
ANDERSON
L-*,, y, d’
@
x,
=o
*I
Fig. I. The coordinate
Fig. 2. A deformed
element
=P
systems.
of the bar and the reactive
forces and moments.
Let the line element AB shown in Fig. 2 represent a generic (infinitesimal) deformed element of the axis of the bar. Shown also in Fig. -- 2 are the reactive forces and moments acting on the ends of this element; specifically, N, Q, and Ei denote the longitudinal force, the shear force, and the bending moment, respectively, acting at a point on the axis of the bar. The angle 8 designates the angle of deviation of the forces from ti,; undeformed state. One can easily show that, in terms of ii and iT;,the longitudinal and transverse deflections of the bar, respectively, the sine and cosine of the angle 0 can be determined from (2.1) Next, let the acceleration vector of a generic point in the bar, as measured relative to the origin C’ of the inertial coordinate frame _Y~ y2 y3, be denoted by i=
nlel+n2e2+a3e3
(; = dv,idr)
(2.2)
where t denotes time and the a& i = 1,2,3, are the components of the acceleration vector relative to the basis vectors of the rotating xi x2 x3-coordinate frame. Then, to a degree of approximation satisfactory for present purposes and in view of equation (2.2). one may suppose that the inertia terms associated with the motion of the bar can be expressed as ; (m;) = pA;;As = pA(a, el + ~1~e2 + n3 e3)As
(2.3)
where m denotes the mass of the element, p the density, A the cross sectional area of the bar, and As the element of arc length measured along the axis of the bar. Summing the forces acting in the horizontal and vertical directions, as depicted in Fig. 2 and using equation (2.3), one obtains the following equations of motion: (R cos 0). i -(& sin t3),1 = pAai(ds/dxi)
(2.4)
(fl sin 8)
(2.5)
I +(Q
cos 0). 1 =
pAn,(ds/dx,)
where the notation (. . .), 1 = 2(. . .)/Sx, is used. In addition, summing the moments of force acting on the element, one finds ~,~+m[~,,cos8-(1+ii,,)sin8]-Q[~.,sine+(l+~,,)cos8] = rotatory inertia term
(2.6)
On the extensional
and flexural vibrations
of rotating
225
bars
However, if the effect of rotatory inertia is neglected and if cognizance is taken of the expressions appearing in equation (2.1) for sin tI and cos 8, then equation (2.6) becomes simply a, 1- Q(ds/dx,)’ = 0
(2.7)
Equations (2.4), (2.Q and (2.7) may now be compared with the corresponding results obtained by Eringen[l6] and Woodall[l7]. Specific expressions for ai and a3, which appear on the right sides of equations (2.4) and (2.5), must now be found. The position vector Yfrom the origin c” to a generic point x1 in the bar is r = (cl +x1 + ii)e, + iCe3
Thus, upon differentiation, it follows that i= fe1+Q(cl+xi+fi)e2+~+e3 and
(2.5)
i: = [i--R2(cl +x1 +17)]ei +2Rfie2-tGe3 where the well known formula of elementary mechanics e=Rxe with R = Re3, has been used. Consequently, reveals that CIi= ; - R’(Ci + xi + ii),
a comparison
of equations (2.2) and (2.5)
02 = 2nt,
a3 =
”
1v
(2.9)
Therefore, substitution of equation (2.9) into equations (2.4-2.5) yieldsf (N
COS~),~-(Q
sine),
1 = pA[ii-R2(~l+.~l
+ii)](ds/dxJ
(2.10)
(fl sine), I +(Q cos ~9).1 = pA$ds/d.u,)
(2.11)
Hence, the non-linear differential equations of motion of the (2.7), (2.10-2.11). Finally, to complete this system of partial boundary conditions must now be appended. Hereinafter let it be supposed that the bar is clamped at its tip mass ,Dat its free right end. Consequently, the boundary end are
beam are given in equations differential equations, the left end and that it carries a conditions at the clamped
fi = ii: = \T,1 = 0
(2.12)
whereas at the free end one has, upon balancing forces and moments, flc0se-Qsin8+~[~-QZ(c,+e+ii)] flsinQ+Qcose+$=
3.
EQUATIONS
FOR
THE
UNDISTURBED
= 0 0 a=0
(2.13)
EQUILIBRIUM
Bolotin[l3], p. 43, discusses the motion and stability of a given form of equilibrium which is often called the rrndismrbed form of equilibrium. Of importance, furthermore, in the present investigation are the disturbed forms of motion close to the undisturbed form of equilibrium. It is also stated on p. 35 of this same reference that in this situation it is permissible to consider displacements and angles of rotation as finite and to suppose that changes in areas and linear dimensions of the bar are negligibly small. Thus, to this degree of approximation, one may write ds/dx, 5 1, so that equation (2.1) may be replaced by c0se2 i+ii,, sine 2: E,, (3.1) 7 It is imminent in the theory developed herein that the bar is much stiKer in the horizontal plane than in the transverse so that the deflection and the Coriolis effect in the horizontal plane can be neglected.
G. L. ANDERSON
216
Consequently, the equations of motion (2.7), (2.10-2.11) simplify to [~(l+u,l)].l-(QIV,,),I
(3.2)
= pA[LR’(ci+si+L1)]
(m~,,,,,+[Q(l+ii,,)],, J&Q
= pA$
(3.3)
= 0
(3.4)
Also, the boundary conditions at the free end of the bar as given in equation (2.13) become rn(l+a,,,-Ql?,,+c([U’-R*(C1+&+ii)] KJ+Q(l+li,)+/G
= 0 = 0 R=O 1
(3.5)
at ?cl = t. The accuracy of equations (3.2-3.5) is not less than the accuracy of the corresponding linear theories for the extensional and flexural deformations of bars. Consider the rotating elastic bar under consideration here to be in a state of undisturbed equilibrium characterized by deflections uO(_yi),M’~(-Y~), longitudinal and shear forces .V,,(.Y,), Qo(xi), and a bending moment M,(x,). Therefore, in view of equations (3.2-3.4). these quantities must satisfy the following differential equations :
[~o~~+~~.~~l,~-~Qo~.~,~).~+~~~~~~~+.~~+~~ 0 (3.6) (~owo,~~~+[Qo(l+~o,,)l,, =0 (3.7) MO,,-Qo=O (3.8) In addition, according to equations (2.12) and (3.5), the accompanying boundary conditions are ug = Wg= We.1 = 0 at _yi =0 (3.9) and No(l+uo,1)-Qo~vo,1-~tR2(~1+L+t~O)= 0 (3.10) No~~~o,, +Qo(l +uo.d = 0 at _yi = /
MO = 0 4.
EQUATIONS
FOR
THE
DISTURBED
MOTION
Under the assumption that the bar undergoes small deviations from the configuration of the undisturbed equilibrium, which was determined in general terms in Section 3, the changes in these deviations with time will now be investigated. The components of the characteristics of the perturbed motion will be denoted hereinafter by u, W, N, Q. and M. Furthermore, it is assumed that “(.Xi,1) = WO(Xi)fW(Xi, t), W1, r) = uo(xJ+u(% r), mcx,, r) = No(x,)+N(x,, R(xi,t)=
r),
Q(% r) =
Qoh)+Qh, r),
(4.1)
Mo(x,)+M(x,,t)
Substituting equation (4.1) into equations (3.2-3.4) and taking into account equations (3.6-3.8), one obtains the following differential equations for the disturbed motion:
[~(1+~o,~+~~.~)+~,~,,],,-[Q(wo,,+~~,~)+Qo~~,~ = p4ti'-R24(4.2) (4.3) [~(~~o,~+~~)+~ow.~]+[Q(~+~o,~+~~,~)+Qo~,,]. = PA@ M,,-Q = 0
(4.4)
An analogous treatment of the boundary conditions in equations (2.12), (3.5), taking also into account the conditions in equation (3.10), yields u=w=w
.1 --0
at
)cl = 0
(4.5)
and Nou,~+N(1+uo,~+u,~)-Qo~v,,-Q(~c~o,~+~~,l)+~~(ii-~2~~)= 0 NOW,~+N(~v~,~+~v,,)+Q~~I,~+ Q(l+no, I+u,l)+p"= 0
M=O
at _~i= t
(4.6)
On the extensional
and flexurai vibrations
of rotating
bars
227
Equations (4.2-4.6), which are non-linear-and hence rather complicated, can be linearized and consequently simplified while still retaining a level of accuracy which is nonetheless satisfactory for a large class of small amplitude vibration problems which arise in certain applied engineering analyses of the extensional and flexural vibrations of rotating slender bars. Therefore, if the disturbances are small, then the non-linear terms Nu, t, Qw, i, NW,i, and Qu,i can be neglected in the equations of motion and the boundary conditions in equations (4.2-4.4) and (4.6), respectively, the results being
[N(1+uo,t)+Not~.t],t-[Q~~o.,+Qo~~,t],t
=pA(ii-Vu)
(4.7)
= p~\ij
(4.8)
[N~~,,t+No~:t],t+[Q(l+~~~,t)+Q~t~.t],t M, i - Q = 0
(4.9)
and, at the free end xi = t, Nou,l+N(l+uo,l)-Qo~~,,-Qw,,,+jl(ii-~’~~) = 0 ~V,,v,,+N~cg.~+Q~~~,,+Q(l+~~~,~)+j~ii=o M=O 1
(4.10)
One additional simplification may often be made. In many engineering problems, the difference between the undisturbed state and the initial undeformed state is comparatively small, and, therefore, in the solution of small amplitude vibration problems, the geometry in the undisturbed state of equilibrium is usually taken to be the same as that in the undeformed state. Furthermore, Bolotin[ 131 argues that with no significant loss of accuracy in, at least, a rather broad class of problems, it is permissible to neglect all gradients of initial displacements, namely, u,,i and We,,, in the so-called “initial stress problem” in equations (3.6-3.10) as well as in the deformed motion problem in equations (4.7-4.10). Doing this, one finds No,, +pAQ2(c1+x1 +uo) = 0 Qo,
l
=
’
MO,, = 0
(4.11)
and 110= It’0= “(‘&I= 0 No-j&2(~1+~+c~o)
at xi = 0 at .x1 = /
= 0, Q. = MO = 0
(4.12)
for the simplified equations for the undisturbed state of equilibrium, and N,~+(NoL~,~),~-QO~V,~~
Q.1 +(N~tvt),t
= pA(ii-R2u)
+Qou,tt
(4.13) (4.14)
= pAii
(4.15)
M,,--Q = 0 for 0 < x1 c k and u=1v=\v
*I -0 -
at .~i = 0
(4.16)
and N+Nou,l-Qo’y,+~c(ii-R’rr)=
0
Q+NoM: i+Qoat+joK=O
at xi = 1
(4.17)
M=O
for the disturbed state of motion. To the accuracy of terms linear with respect to uo, wo, u, and W, the constitutive equations which must accompany equations (4.11-4.17) are No = EAcI~,~, MO = -EZw,.,,,
N = EA+, M = -EI\v,~~
where I denotes the moment of inertia of the cross section of the bar.
(4.18)
5.
From equations
THE
(4.11-4.12),
INITIAL
STRESS
it is immediately
PROBLEM
evident
that
Q&Y,) = M~(.Ul) = 0 for all s1 in the interval
0 < .yl < k. In view of equations E&J. 11+ pR’uo = - pR’(q
The solution
ofthe elementary
boundary
H~(.Y~)= -(c,
+sl)+
(4.11-4.12)
+ Xl),
uo(O) = EAL~o.l(~)-@2’[cl
(5.1)
0 < Xl < t
+/+LLJ(r)]
value in equation c,
and (4.18). one also has (5.2)
= 0
(5.2) can easily be shown to be sin(t0,.y,!i)
cos(ox//)+b
(5.3)
where b = i (!J
1 +to< sin(0+rO’< [
tc) = R// (E;‘p)l 2?
cos(jJ
(5.4)
cos(ti--rr(!0sinc9
1
5 = c*./.
r = &pA/
co, <, and r being dimensionless
the hub radius, found that
parameters associated with the angular velocity of rotation, and the tip mass to beam mass ratio. From equations (4.18) and (5.31, it is &(x,)
= - EA [ 1 + U< sin(OJsl//)]
+ EA(wb;O
cos(tur,:‘/)
(5.5)
Thisexpression togetherwith those inequation(5.1) will be needed later in the investigation of the deformed motion problem embodied in equations (4.13-4.18). It is necessary to consider equations (5.3) and (5.5) in some detail. Of particular interest is the fact that the displacement u,,(s~) and the longitudinal force M,(.Y,) become arbitrarily large as the denominator in the expression for b given in equation (5.4) tends to zero. This denominator vanishes whenever the dimensionless angular velocity parameter OJ satisfies the transcendental equation cot 01 = PC?) (5.6) In the event that r = 0, i.e. in the absence of a tip mass, equation (5.6) becomes simply cos w = 0, which has the zeros LI), = (2~ - 1)x/2, tl = 1.2, 3. . These values of the angular velocity parameter correspond to the so-called “static resonance” case discussed in [l], and they were erroneously termed the divergence angular velocities in [2-31. The roots of equation (5.6) may not be associated with a loss of stability, at least not within the framework of the concept of stability as discussed by Bolotin[ 131, because at such angular velocities the undisturbed state of equilibrium has not even been attained yet. Since the development and linearization presented herein have been based upon small displacements, attention throughout the remainder of this study will be centered upon the range of angular velocities 0 d o < CL)*,where co* denotes the lowest zero of equation (5.6). A plot of the variation of o* with the tip mass parameter r appears in Fig. 3, from which it is evident that o* decreases monotonically from co* = ~~2 to w* = 0 as r increases from zero and becomes arbitrarily large. It is clear, then, that the theory outlined above loses its validity if the angular velocity R approaches the value R*, where 1 2 **=y
$
0 If R << R*, equation (5.5) can be replaced with a rather simple approximation. that, say 0 d r < 1 and o cc CL)* < ~‘2. In this event, one can write cos t!J = 1 -&!J’ + c (OJ), etc. to a good degree of accuracy. Consequently, after some algebraic can be approximated by &(.U,) =
manipulation,
~IR2(C1+/)+pAR2[C1(/-_S~)+t(/‘-Sf)]+C’(~J3)
Suppose
sinw = u-((w’) it can be verified
that equation
(5.5) (5.7)
On the extensional
0.2
and flexural vibrations
0.4
0.6
08
I.0 0.6
of rotating
0.6 0.4
r
Fig 3. Variation
6.
THE
bars
0.2
0
I/r
of w* with the tip mass parameter
DEFORMED
229
MOTION
r.
PROBLEM
By virtue ofequation (5. l), the flexural and extensional equations for the deformed motion problem become uncoupled. With the aid of equation (4.18), equations (4.13-4.17) may now be expressed as EAYII+(Nol!l),l+pAR’u EAu,I+Nou,,+~((i~-nzu)
= phi, ti = 0 =0
O
(6.1)
at xi = 0 at xi={ I
(6.2)
O
(6.3)
for the extensional motion and Ellc,l,l,-(Not~,l),l+pAii
= 0,
VV= \(: 1 = 0 \V,ll = El\v,i,,-NOC~,i-~iC = 0
at .Y,= 0 (6.4) at .~i = e 1 for the flexural motion, where the form of N,(_K,)is given in equation (5.5) for 0 < Q < R* or, for relatively low rotational speeds, in equation (5.7). If the form of &(x1) given in equation (5.7) is inserted into equations (6.1-6.2), the result is [EA + PR2(C, + tjju, 11+pAR2{[cl(/-xl)+i_(~2-.~:)]~,1),I+pAR2u u(0, t) = 0
[EA+,LlR’(c*+lq]zfJ(!, t)+p[ii(P, t)-n2u(t,
t,] = 0
= phi
1
(6.5)
(6.6)
In the case ofvanishing tip mass and hub radius, i.e. ,U= cl = 0, equations (6.5-6.6) reduce to EAI~,~~+~~AR’[(/‘--.~:)~,~],~
+pAQ’u
u(0, t) = u*,(k, t) = 0
= pAii
(6.7) (6.8)
A comparison of equations (6.7-6.8) with the corresponding boundary value problem discussed in reference [l] reveals that Bhuta and Jones failed to include the initial stress term +pAR2 [(C’ -xf)u, J 1 in their investigation. This term is of the same order of magnitude as the other two terms on the left side of equation (6.7), and hence it cannot be
G. L. ANDERSON
230
discarded. Indeed, the neglect of this term led Bhuta and Jones to conclude that in the extensional motion of a rotating bar whirling lowers the natural frequencies of vibration. That precisely the opposite conclusion is true will be demonstrated in Section 8. Similarly, substitution of equation (5.7) into equations (6.3-6.4) leads to EIw,i,,, -YQ2(c1 +W*,,
-pAR’([~~(/-x,)+~(e’-x:)]w,~~~+pAi~ \v = W’, 1= 0
=
w.11
at x, = 0 at x1 = / 1
Efw,,,,-~R2(cl+e)~~,,-~~=o
Various forms ofequations
= 0
(6.9) (6.10)
(6.9-6.10) have been derived and discussed in [5], [9-121, and
[151. 7. APPLICATION
OF
THE
RITZ
METHOD
If next &(x1) as given in equation (5.5) is substituted into equations (6.1-6.4) and if the changes of independent variables xi = lx and t = Pr(p/E)‘!’ are made, one finds (Pu’)’ + 02u = ii,
O
(7.1)
u(O,s)= P(l)u’(l,T)+r[ii(l,r)-w%(l,r)]
= 0
(7.2)
where now U’= &@x, ti = an/&, etc. and P(x) = p cos ox -
w
l+w
O
,t’ = \V’= 0 w” = W“’+ (&() w’_ r* = O
(7.3)
at x=0 atx=l
(7.4)
where u = AL’JI,
&(x1)
= 1 -t w< sin wx - j? COsox
for the flexural motion. Assuming now that U(X,5) = 4(x) cos AT,
W(X,5) = I++) cos A?
it follows directly from equations (7.1-7.4) that 4(x) and $(x) are solutions of the eigenvalue problems (Pqfl’)‘+(w2+%2)#
4(O) = P(1)#(1)-r(i.2-to2)&1)
= 0
O
= 0
(7.5) 1
and *““+a(&$‘)‘-i2*
= 0
O
i//(O)= I//‘(O)= 0 $“(l) = 1(/“‘(1)+aF&(l)$‘(1)+rl.21(/(1) = 0
(7.6) 1
It is a straightforward matter to verify that the boundary value problems in equations (7.5-7.6) can be generated from the variational principles (7.7) and 6
[(@02 -a&($‘)2
- i.‘G2] dx - ri’$‘(l)
1= 0
(7.8)
On the extensional
and Aexural vibrations
of rotating
bars
231
These variational principles will now be used in conjunction with Ritz’ method for the purpose of determining the natural frequencies of vibration for the first few longitudinal and flexural modes of motion. Following the Ritz procedure, let it be assumed that C#J(X) and Ii/(x) can be approximated by 4(x) - “ii &4”(X),
$(x) z “ii 4$“(X)
(7.9)
where N is a positive integer, the coefficients n, are unknown, and the coordinate functions 4.(x), $Jx) are selected to be the sets of polynomials ci, = (n + l)(cos w + r<02) - rw2(cos 0 - i-0 sin 0)
(7.10)
Pn = n(cos w + rtw’) - ~0~(cos 0 - r-0 sin 0) and l)“(X) = x”+1(~n+/3”s+i’“X2) c(, = (n+2)(n+3)[(n+l)(n+2)-&(l)] /?n= -2(n+
(7.11)
l)(n+3)[n(n+2)-r&(l)]
7” = (n+ l)(n+2)[n(n+
l)-r&(l)]
which satisfy the boundary conditions A(0) = P(l)dXI)-~~244) = 0 IC/“(O) = rl/A(O) = l&(l) = I1/~‘(1)+c&(l)rl/~(l) = 0
(7.12)
respectively. Not all the boundary conditions appearing in equation (7.12) are identical to the corresponding boundary conditions in equations (7.5-7.6). It may be recalled, however, that, in the application of the Ritz method, the coordinate functions &(x) and G,,(x) need satisfy only the geometric boundary conditions 4(O) = @(O)= $‘(O) = 0 and not necessarily the natural boundary conditions P(l)@(l)-r(L2+o’)4(1) $“(l) = l+V”(1)+c&(l)iJ/‘(l)+ri2$(1)
= 0 = 0
Substitution of equations (7.9) into the apposite variational principles in equations (7.7-7.8) leads to the homogeneous system of algebraic equations i
a,(Ati
2.’+ Bnk) = 0,
n,k=l,2
,..., N
where ‘%k
=
0’
h(bk
dx+
s &k
=
@,(1)6,(l) I
P&, 4; dx
w2Ad s 0
for the extensional motion problem and An.+=
’ $n$kd-~+rtin(l)‘bk(l) s 0
for the flexural motion problem. Therefore, because, in view of equations (7.10-7.1 l), the values of the coefficients A,k and Bnkare known, the natural frequencies of vibration (&}, n = 1,2,3,. . . ) N, for the extensional and flexural modes of a bar rotating at a uniform rate can be computed from the equation det(A,ki.2 + &) = 0
(7.13)
232
L.
G.
8.
ANDERSON
NUMERICAL
RESULTS
Equation (7.13) was solved numerically for the frequencies E.,, i.z,. . . for combinations of the following values of the tip mass parameter r and the hub radius parameter <: r, < = 0, l/5, I/.5. The value chosen for z was r = (4:‘3) x 106, which is typical of some helicopter blades that are currently in use. Numerical calculations were performed with N = 6,7,8, and it appeared that for N = 8 and for the range of the angular velocity
/
,.I
1.4
-
1.3
-
I.2
-
4 /
/.’
r=2/5
1.0
I
0
I
0.1
I
1
0.2
0.3
I
I
0.4
0.5
0.6
w
Fig. .l, Variation
of i, for the lowest extensional and -.~
mode with w I-< = 2 j).
5 = 0, ___
< = l.!j.
w
Fig. 5. Variation
of Lz for the second
extensional mode with w I--and -.< = 2: j),
2 = 0. ---
5 = II j,
parameter, o, considered, the numerical procedure had essentially converged to four digit accuracy in the lowest three modes of oscillation. Attention here is confined to the domain 0 Q w < 0.7 because of the considerations described in Section 5. The variations of the lowest three natural frequencies i.I, j.2, ;.3 of the extensional modes ofvibration of a rotating bar are plotted in Figs. 4-6, respectively, for several combinations of the tip mass parameter r and the hub radius parameter 5. Firstly, these plots reveal that for given values of I and < the values of the natural frequencies of the lowest three extensional modes increase monotonically as the angular velocity increases. This result
On the extensional
and flexural vibrations
of rotating
bars
233
9
6.5 -
gL 0
I 0.3
0.2
0.1
I 0.5
0.4
0.6
w
Fig. 6. Variation
of i., for the third
extensional and -‘-
mode with w (< = 2/5).
2 = 0, ---
g = 115,
confutes the assertion in [I] that whirling tends to lower the natural frequencies of vibration as compared to those of the non-rotating bar. Secondly, it is evident that for a fixed value of r the frequencies increase as the hub radius parameter 5 is increased, whereas, if the value of 5 is held fixed, the frequencies decrease as the tip mass parameter r is increased, provided that the angular velocity parameter w is sufficiently small. However, one should note that the curves shown in Fig. 6 for iW3obtained with I = l/5 and I = 2/5 for a given value of 5 intersect one another when w becomes sufficiently large. In other words, for a given value of 5 and for o sufficiently large, the natural frequencies may increase with increasing tip mass. The variation of the lowest three natural frequencies of the flexural modes of oscillation of a rotating bar are plotted in Figs. 7-9 again for various combinations of the parameter r and 5. These curves are very similar in appearance to those presented in Figs. 4-6, and the conclusions drawn regarding the extensional modes of oscillation are therefore applicable to the flexural modes also.
1 I
5.5
5.0
4.5
4.0 h 3.5
3.0
2.5
0
0.5
1.0
1.5
2.0
2.5
3.0
wx103
Fig. 7. Variation
of i., for the lowest
flexural and -.-
mode with 5 = 215).
UJ (---
< = 0, ---
< = 115,
G.
L.
ALDERSO~
25
23
22
2 I
20
x2 19
18
17
0
0.5
1.0
I.5
wx
Fig. 8. Variation
of il
for the second
Aexural and
64
_._
2.0
2.5
3.0
I03
mode ,'=
5 = 0, ___
with o (-
<=
l;j,
2.j)
./.’
-
./ /’ /’
63
,,=,:A
53
52
0
J 0.5
1.0 w
Fig. 9. Variation
of ;.3 for the third flexud -.-
1.5
2.0
2.5
30
x103 mode with w C-----~~~o,---~c
1 j,and
< = 2;s).
Finally, a few remarks regarding the range of applicability of equation (5.7), which in various forms has been used in references [4-131 in place of the more complete expression for &,(x1) as given in equation (5.5), are in order. In the Appendix, equations (6.9-6.10) are solved approximately by the Ritz method, and the natural frequencies were determined numerically for the case r = < = l/5, for the sake of example. These values, labeled j.ip (n = 1,2,3 and “ap” for “approximate”) are displayed in Table 1 and are compared with the corresponding values i.‘,” (“e.u” for “exact”) obtained from equation (7.13) over the
On the extensional
and flexural vibrations
of rotating
bars
235
domain 0 < w ,< 0.7. At least for the three modes considered, the percentage difference between Lip and nGXis virtually independent of the mode number n for a given angular velocity, however as the value of w increases the percentage difference also increases. Thus, for the combination of parameters r = 5 = l/5, the relatively simple expression for &,(x1) in equation (5.7) is sufficiently accurate for computing the natural frequencies of the lowest three flexural modes whenever w < 0.1, but, if o > 0.1, then the somewhat more complicated expression for &,(x1) given in equation (5.5) should be used. Table
of ,1:p and jex .” 7n =
1. Comparison
L2.3, for several values of the angular velocity parameter
w,
for r = 5 = l/5. %
W
Zyp
i;
difference
i”9
0.1 02 0.3 @4 @5 0.6 0.7
131.4 262.9 394.7 526.0 6577 789.0 9206
131.8 266.1 405.3 5524 711.1 887.1 1088
0.3 1.2 2.6 4.8 7.5 11.1 15.4
359.6 724.2 1087 1452 1815 2179 2542
% d@erence
J.‘i’
.?y
0.3 1.1 2.8 4.7 7.5 11.0 15.2
3606 732.5 1118 1523 1962 2447 2999
636.3 1283 1929 2574 3218 3863 4507
r 5’ 638.2 1298 1981 2701 3479 4340 5322
% difference 02 1.2 2.6 4.7 7.5 11.0 15.3
REFERENCES 1. P. G. Bhuta and J. P. Jones, On axial vibrations of a whirling bar, J. acousr. Sot. Am. 35, 217-221 (1963). 2. E. J. Brunelle, Stress redistribution and instability of rotating beams and disks, A.I.A.A. Jl9, 758-759 (1971). 3. E. J. Brunelle, The super flywheel: a second look, J. Engng Math. Tech. 95, 63-65 (1973), and Discussion, 95, 195-196 (1973). 4. H. Lo and J. L. Renbarger, Bending vibrations of a rotating beam, First U.S. natn. Cong. appl. Mech. 75-79 (1951). 5. W. E. Boyce, R. C. DiPrima and G. H. Handleman, Vibrations of rotating beams of constant section, Second U.S. natn. Cong. appl. Mech. 165-173 (1954). 6. M. J. Schilhansl, Bending frequency of a rotating cantilever beam, J. appl. Mech. 25, 28-30 (1958). 7. N. Rubinstein and J. T. Stadter, Bounds to bending frequencies of a rotating beam, J. Franklin Insr. 294. 217-229 (1972). 8. D. Pnueli, Natural bending frequency comparable to rotational frequency in rotating cantilever beam. J. appl. Mech. 39,602-604 (1972). (1956). 9. W. E. Boyce, Effect of hub radius on the vibrations of a uniform bar, J. oppl. Mech. 23.287-290 10. H. Lo, J. E. Goldberg and J. H. Bogdanoff, Effect of small hub radius change on bending frequencies of a rotating beam, J. appl. Mech. 27, 548-550 (1960). W. Boyce and H. Cohen, Vibrations of a uniform rotating beam with tip mass, Third t’.S. 11. G. Handleman, mtn. Coy. appl. Mech. 175-180 (1958). Vibrations of rotating beams with tip mass, Z. angew. Marh. Phys. 12, 12. W. E. Boyce and G. Handleman, 369-392 (1961). 13. V. V. Bolotin, Nonconservative Problems in the Theory 01 Elastic Stability. Macmillan, New York (1963). on the dynamic behavior of large flexible bodies in orbit, A.I.A.A. Jl 5, 460-469 14. H. Ashley, Observations (1967). (1968). 15. R. D. Mime, Some remarks on the dynamics of deformable bodies, A.I.A.A. Jl6,556-558 16. A. C. Eringen, On the non-linear vibration of elastic bars, Q. appl. Math. 9, 361-369 (1952). 17. S. R. Woodall, On the large amplitude oscillations of a thin elastic beam, Inc. J. Non-linear Mech. 1, 217-238 (1966). APPENDIX Proceeding
as in Section
7, one can show that equations
(6.9-6.10)
become
JI”“-rKo*(l+~)~~-K0*(fiSJ/‘)l--,is~
= 0
$(O) = tj’(O) = (1/“(l) = t/“(l)-rzo2(1+~)ljl’(l)+r12~(1)
= 0
O
where 1s(x) = C(l-s)+f(l-x2) This boundary
value problem
s Equation boundary
can be obtained
from the foIlowing
variational
principle:
rKw’(1 +~)(1(/‘)2+~~2~(J1’)2--j.L~z]dx-r~.2[~(1)]2
(7.9) will again be assumed, conditions
however
now the coordinate
functions
IL.(x) will be required
+.(O) = p,(O) = $:( 1) = +:“( 1) - rzw’( 1 + t)&( 1) = 0
(A.1) to satisfy the
(A.2)
G. L.
236
The set of polynomial
coordinate
,A\IDEKSOh
functions ti”(.Y) = Y-l IZ.+/i..Y+:‘,Y2)
where 1” = oI+2)(n+3)[ol+ /r.=
.’,n = (n+ I)(n+:!)[n(nc satisfies all the boundary equation
conditions
(7.13) still obtains,
I)(~I+2)+rX’P(l
+<,I
-2(~1+1)(t1+3)[n(n+2~+r20~(1+~)]
in equation
I)+rX92(I+,)]
(A.?). Furthermore.
It is easy to show from equation
tA.1) that
where now
R&sum&On formule Ies kquatiolls non lintaires du mouvrment d’une barre mince tournant a\ec une vitesse angulnire constante autourd‘un axe transversal. Sous I’hypoth2se qu’un petit mouvement perturb& se produise autour d’une configuration d’iquilibre ayant une contrainte initiale, on etablit des kquations linearistes du mouvement pour les dXormations longitudinales et en Hexion d’une barre en rotation portnnt une masse ti son extrbmitC. Des calculs numiriques pour Its friquences naturelles dcs trois modes de vibration libre les plus has rt2lL;lent que lorsque la \.ites,e angulaire augmente les caleurs des friquences d’extension augmentent de facon monotone. contrairement aux r&sultats prCctdemment publiis.
Zusammenfassung-Die nichtlinearen Bewegungsgleichungen eines schlanken Stabes. der mit konstanter Winkelgeschuindigkeit urn eine Querachse rotiert. werden formuliert. Unter der Annahme. dass sich eine kleine Stdrungsbewegun g der korgespannten Gleichgeuichtsanordnun: tiberlagert. werden linearisierte Bewegungsgleichungen tir die LLngs- und Biege\ erformungen eines rotierenden Stabes mit einer Endmasse hergeleitet. Numerische Berechnungen der Eigenfrequenzen der drei niedregsten freien Eigenschwingungsforem zeigen, dass im Gegensatz zu veriii?entlichten Ergebnissen die Werte der Ausdehnungsfrequenzen monotonisch zunehmen. wenn die Winkelgeschwindigkeit der Drehung zunimmt.
1975)
Abstract-The non-linear equations of motion of a slender bar rotating at constant angular velocity about a transverse axis are formulated. Under the assumption that a small perturbed motion occurs about an initially stressed equilibrium configuration, linearized equations of motion for the longitudinal and flexural deformations of a rotating bar carrying a tip mass are derived. Numerical computations for the natural frequencies of the lowest three modes of free vibration reveal that the values of the extensional frequencies increase monotonically, contrary to previously published results, as the angular velocity of rotation increases. 1. INTRODUCTION
The longitudinal static displacement and vibration of a slender bar rotating about a vertical axis with a constant angular velocity R have been investigated[l-31. In these studies, two observations were reported: (i) rotation of the bar tends to lower the natural frequencies of longitudinal vibration as compared to the natural frequencies of the nonrotating bar, and (ii) there exists a critical angular velocity at which the longitudinal displacement becomes arbitrarily large. The latter phenomenon was called “static resonance” in [l] and “static instability” or “divergence” in [2-31. In the related problem of flexural oscillations of rotating bars, several authors[4-8) have found that the rotation of the bar tends to increase the natural frequencies of flexural vibration relative to those for the non-rotating bar. The effects of hub radius and tip mass on the bending frequencies of rotating bars have been studied [S-lo] and [l l-121, respectively, where it was observed that, for small hub radius relative to the length of the beam, the bending frequency is nearly a linear function of hub radius, whereas the bending frequencies in the various natural modes may be increased or decreased as compared to those of the non-rotating bar by the presence of a tip mass. In the present investigation, the author reconsiders certain aspects of the boundary value problems described in references [l-12], deriving the non-linear equations for the flexural and extensional motions of slender rotating, elastic bars. When these non-linear equations are separated into the so-called initial stress problem and the deformed motion problem and linearized in the manner of [13-15-J, it is found that the boundary value problem discussed in reference [l] has not been formulated completely and that the assertion that the natural frequencies of longitudinal vibrations are lowered by the effect of whirling is incorrect. In addition, it appears that the use of the terminology “static instability” and “divergence” in references [2-31 is a misinterpretation of the actual physical situation. Furthermore, the present analysis reveals that in [l-12] the terms in the partial differential equations of both flexural and extensional motions of rotating bars which account for the initial stress field in the bar due to the centrifugal force effect are accurate only for relatively low angular velocities of rotation. This situation has arisen since the effect of the initial longitudinal displacement, arising because of the rotation of the system, in the initial stress problem is generally disregarded, either inadvertently or possibly as a first approximation. 2. THE
NON-LINEAR
FORMULATION
Consider a long, slender bar of length G whose axis lies along the xi-axis and whose left end is displaced a horizontal distance cl from the vertical y3-axis, as shown in Fig. 1. It is assumed that the y1yzy3-coordinate frame remains fixed in space, whereas the horizontal xi-axis rotates about the vertical y,-axis with constant angular velocity Q. 223
G.L.
ANDERSON
L-*,, y, d’
@
x,
=o
*I
Fig. I. The coordinate
Fig. 2. A deformed
element
=P
systems.
of the bar and the reactive
forces and moments.
Let the line element AB shown in Fig. 2 represent a generic (infinitesimal) deformed element of the axis of the bar. Shown also in Fig. -- 2 are the reactive forces and moments acting on the ends of this element; specifically, N, Q, and Ei denote the longitudinal force, the shear force, and the bending moment, respectively, acting at a point on the axis of the bar. The angle 8 designates the angle of deviation of the forces from ti,; undeformed state. One can easily show that, in terms of ii and iT;,the longitudinal and transverse deflections of the bar, respectively, the sine and cosine of the angle 0 can be determined from (2.1) Next, let the acceleration vector of a generic point in the bar, as measured relative to the origin C’ of the inertial coordinate frame _Y~ y2 y3, be denoted by i=
nlel+n2e2+a3e3
(; = dv,idr)
(2.2)
where t denotes time and the a& i = 1,2,3, are the components of the acceleration vector relative to the basis vectors of the rotating xi x2 x3-coordinate frame. Then, to a degree of approximation satisfactory for present purposes and in view of equation (2.2). one may suppose that the inertia terms associated with the motion of the bar can be expressed as ; (m;) = pA;;As = pA(a, el + ~1~e2 + n3 e3)As
(2.3)
where m denotes the mass of the element, p the density, A the cross sectional area of the bar, and As the element of arc length measured along the axis of the bar. Summing the forces acting in the horizontal and vertical directions, as depicted in Fig. 2 and using equation (2.3), one obtains the following equations of motion: (R cos 0). i -(& sin t3),1 = pAai(ds/dxi)
(2.4)
(fl sin 8)
(2.5)
I +(Q
cos 0). 1 =
pAn,(ds/dx,)
where the notation (. . .), 1 = 2(. . .)/Sx, is used. In addition, summing the moments of force acting on the element, one finds ~,~+m[~,,cos8-(1+ii,,)sin8]-Q[~.,sine+(l+~,,)cos8] = rotatory inertia term
(2.6)
On the extensional
and flexural vibrations
of rotating
225
bars
However, if the effect of rotatory inertia is neglected and if cognizance is taken of the expressions appearing in equation (2.1) for sin tI and cos 8, then equation (2.6) becomes simply a, 1- Q(ds/dx,)’ = 0
(2.7)
Equations (2.4), (2.Q and (2.7) may now be compared with the corresponding results obtained by Eringen[l6] and Woodall[l7]. Specific expressions for ai and a3, which appear on the right sides of equations (2.4) and (2.5), must now be found. The position vector Yfrom the origin c” to a generic point x1 in the bar is r = (cl +x1 + ii)e, + iCe3
Thus, upon differentiation, it follows that i= fe1+Q(cl+xi+fi)e2+~+e3 and
(2.5)
i: = [i--R2(cl +x1 +17)]ei +2Rfie2-tGe3 where the well known formula of elementary mechanics e=Rxe with R = Re3, has been used. Consequently, reveals that CIi= ; - R’(Ci + xi + ii),
a comparison
of equations (2.2) and (2.5)
02 = 2nt,
a3 =
”
1v
(2.9)
Therefore, substitution of equation (2.9) into equations (2.4-2.5) yieldsf (N
COS~),~-(Q
sine),
1 = pA[ii-R2(~l+.~l
+ii)](ds/dxJ
(2.10)
(fl sine), I +(Q cos ~9).1 = pA$ds/d.u,)
(2.11)
Hence, the non-linear differential equations of motion of the (2.7), (2.10-2.11). Finally, to complete this system of partial boundary conditions must now be appended. Hereinafter let it be supposed that the bar is clamped at its tip mass ,Dat its free right end. Consequently, the boundary end are
beam are given in equations differential equations, the left end and that it carries a conditions at the clamped
fi = ii: = \T,1 = 0
(2.12)
whereas at the free end one has, upon balancing forces and moments, flc0se-Qsin8+~[~-QZ(c,+e+ii)] flsinQ+Qcose+$=
3.
EQUATIONS
FOR
THE
UNDISTURBED
= 0 0 a=0
(2.13)
EQUILIBRIUM
Bolotin[l3], p. 43, discusses the motion and stability of a given form of equilibrium which is often called the rrndismrbed form of equilibrium. Of importance, furthermore, in the present investigation are the disturbed forms of motion close to the undisturbed form of equilibrium. It is also stated on p. 35 of this same reference that in this situation it is permissible to consider displacements and angles of rotation as finite and to suppose that changes in areas and linear dimensions of the bar are negligibly small. Thus, to this degree of approximation, one may write ds/dx, 5 1, so that equation (2.1) may be replaced by c0se2 i+ii,, sine 2: E,, (3.1) 7 It is imminent in the theory developed herein that the bar is much stiKer in the horizontal plane than in the transverse so that the deflection and the Coriolis effect in the horizontal plane can be neglected.
G. L. ANDERSON
216
Consequently, the equations of motion (2.7), (2.10-2.11) simplify to [~(l+u,l)].l-(QIV,,),I
(3.2)
= pA[LR’(ci+si+L1)]
(m~,,,,,+[Q(l+ii,,)],, J&Q
= pA$
(3.3)
= 0
(3.4)
Also, the boundary conditions at the free end of the bar as given in equation (2.13) become rn(l+a,,,-Ql?,,+c([U’-R*(C1+&+ii)] KJ+Q(l+li,)+/G
= 0 = 0 R=O 1
(3.5)
at ?cl = t. The accuracy of equations (3.2-3.5) is not less than the accuracy of the corresponding linear theories for the extensional and flexural deformations of bars. Consider the rotating elastic bar under consideration here to be in a state of undisturbed equilibrium characterized by deflections uO(_yi),M’~(-Y~), longitudinal and shear forces .V,,(.Y,), Qo(xi), and a bending moment M,(x,). Therefore, in view of equations (3.2-3.4). these quantities must satisfy the following differential equations :
[~o~~+~~.~~l,~-~Qo~.~,~).~+~~~~~~~+.~~+~~ 0 (3.6) (~owo,~~~+[Qo(l+~o,,)l,, =0 (3.7) MO,,-Qo=O (3.8) In addition, according to equations (2.12) and (3.5), the accompanying boundary conditions are ug = Wg= We.1 = 0 at _yi =0 (3.9) and No(l+uo,1)-Qo~vo,1-~tR2(~1+L+t~O)= 0 (3.10) No~~~o,, +Qo(l +uo.d = 0 at _yi = /
MO = 0 4.
EQUATIONS
FOR
THE
DISTURBED
MOTION
Under the assumption that the bar undergoes small deviations from the configuration of the undisturbed equilibrium, which was determined in general terms in Section 3, the changes in these deviations with time will now be investigated. The components of the characteristics of the perturbed motion will be denoted hereinafter by u, W, N, Q. and M. Furthermore, it is assumed that “(.Xi,1) = WO(Xi)fW(Xi, t), W1, r) = uo(xJ+u(% r), mcx,, r) = No(x,)+N(x,, R(xi,t)=
r),
Q(% r) =
Qoh)+Qh, r),
(4.1)
Mo(x,)+M(x,,t)
Substituting equation (4.1) into equations (3.2-3.4) and taking into account equations (3.6-3.8), one obtains the following differential equations for the disturbed motion:
[~(1+~o,~+~~.~)+~,~,,],,-[Q(wo,,+~~,~)+Qo~~,~ = p4ti'-R24(4.2) (4.3) [~(~~o,~+~~)+~ow.~]+[Q(~+~o,~+~~,~)+Qo~,,]. = PA@ M,,-Q = 0
(4.4)
An analogous treatment of the boundary conditions in equations (2.12), (3.5), taking also into account the conditions in equation (3.10), yields u=w=w
.1 --0
at
)cl = 0
(4.5)
and Nou,~+N(1+uo,~+u,~)-Qo~v,,-Q(~c~o,~+~~,l)+~~(ii-~2~~)= 0 NOW,~+N(~v~,~+~v,,)+Q~~I,~+ Q(l+no, I+u,l)+p"= 0
M=O
at _~i= t
(4.6)
On the extensional
and flexurai vibrations
of rotating
bars
227
Equations (4.2-4.6), which are non-linear-and hence rather complicated, can be linearized and consequently simplified while still retaining a level of accuracy which is nonetheless satisfactory for a large class of small amplitude vibration problems which arise in certain applied engineering analyses of the extensional and flexural vibrations of rotating slender bars. Therefore, if the disturbances are small, then the non-linear terms Nu, t, Qw, i, NW,i, and Qu,i can be neglected in the equations of motion and the boundary conditions in equations (4.2-4.4) and (4.6), respectively, the results being
[N(1+uo,t)+Not~.t],t-[Q~~o.,+Qo~~,t],t
=pA(ii-Vu)
(4.7)
= p~\ij
(4.8)
[N~~,,t+No~:t],t+[Q(l+~~~,t)+Q~t~.t],t M, i - Q = 0
(4.9)
and, at the free end xi = t, Nou,l+N(l+uo,l)-Qo~~,,-Qw,,,+jl(ii-~’~~) = 0 ~V,,v,,+N~cg.~+Q~~~,,+Q(l+~~~,~)+j~ii=o M=O 1
(4.10)
One additional simplification may often be made. In many engineering problems, the difference between the undisturbed state and the initial undeformed state is comparatively small, and, therefore, in the solution of small amplitude vibration problems, the geometry in the undisturbed state of equilibrium is usually taken to be the same as that in the undeformed state. Furthermore, Bolotin[ 131 argues that with no significant loss of accuracy in, at least, a rather broad class of problems, it is permissible to neglect all gradients of initial displacements, namely, u,,i and We,,, in the so-called “initial stress problem” in equations (3.6-3.10) as well as in the deformed motion problem in equations (4.7-4.10). Doing this, one finds No,, +pAQ2(c1+x1 +uo) = 0 Qo,
l
=
’
MO,, = 0
(4.11)
and 110= It’0= “(‘&I= 0 No-j&2(~1+~+c~o)
at xi = 0 at .x1 = /
= 0, Q. = MO = 0
(4.12)
for the simplified equations for the undisturbed state of equilibrium, and N,~+(NoL~,~),~-QO~V,~~
Q.1 +(N~tvt),t
= pA(ii-R2u)
+Qou,tt
(4.13) (4.14)
= pAii
(4.15)
M,,--Q = 0 for 0 < x1 c k and u=1v=\v
*I -0 -
at .~i = 0
(4.16)
and N+Nou,l-Qo’y,+~c(ii-R’rr)=
0
Q+NoM: i+Qoat+joK=O
at xi = 1
(4.17)
M=O
for the disturbed state of motion. To the accuracy of terms linear with respect to uo, wo, u, and W, the constitutive equations which must accompany equations (4.11-4.17) are No = EAcI~,~, MO = -EZw,.,,,
N = EA+, M = -EI\v,~~
where I denotes the moment of inertia of the cross section of the bar.
(4.18)
5.
From equations
THE
(4.11-4.12),
INITIAL
STRESS
it is immediately
PROBLEM
evident
that
Q&Y,) = M~(.Ul) = 0 for all s1 in the interval
0 < .yl < k. In view of equations E&J. 11+ pR’uo = - pR’(q
The solution
ofthe elementary
boundary
H~(.Y~)= -(c,
+sl)+
(4.11-4.12)
+ Xl),
uo(O) = EAL~o.l(~)-@2’[cl
(5.1)
0 < Xl < t
+/+LLJ(r)]
value in equation c,
and (4.18). one also has (5.2)
= 0
(5.2) can easily be shown to be sin(t0,.y,!i)
cos(ox//)+b
(5.3)
where b = i (!J
1 +to< sin(0+rO’< [
tc) = R// (E;‘p)l 2?
cos(jJ
(5.4)
cos(ti--rr(!0sinc9
1
5 = c*./.
r = &pA/
co, <, and r being dimensionless
the hub radius, found that
parameters associated with the angular velocity of rotation, and the tip mass to beam mass ratio. From equations (4.18) and (5.31, it is &(x,)
= - EA [ 1 + U< sin(OJsl//)]
+ EA(wb;O
cos(tur,:‘/)
(5.5)
Thisexpression togetherwith those inequation(5.1) will be needed later in the investigation of the deformed motion problem embodied in equations (4.13-4.18). It is necessary to consider equations (5.3) and (5.5) in some detail. Of particular interest is the fact that the displacement u,,(s~) and the longitudinal force M,(.Y,) become arbitrarily large as the denominator in the expression for b given in equation (5.4) tends to zero. This denominator vanishes whenever the dimensionless angular velocity parameter OJ satisfies the transcendental equation cot 01 = PC?) (5.6) In the event that r = 0, i.e. in the absence of a tip mass, equation (5.6) becomes simply cos w = 0, which has the zeros LI), = (2~ - 1)x/2, tl = 1.2, 3. . These values of the angular velocity parameter correspond to the so-called “static resonance” case discussed in [l], and they were erroneously termed the divergence angular velocities in [2-31. The roots of equation (5.6) may not be associated with a loss of stability, at least not within the framework of the concept of stability as discussed by Bolotin[ 131, because at such angular velocities the undisturbed state of equilibrium has not even been attained yet. Since the development and linearization presented herein have been based upon small displacements, attention throughout the remainder of this study will be centered upon the range of angular velocities 0 d o < CL)*,where co* denotes the lowest zero of equation (5.6). A plot of the variation of o* with the tip mass parameter r appears in Fig. 3, from which it is evident that o* decreases monotonically from co* = ~~2 to w* = 0 as r increases from zero and becomes arbitrarily large. It is clear, then, that the theory outlined above loses its validity if the angular velocity R approaches the value R*, where 1 2 **=y
$
0 If R << R*, equation (5.5) can be replaced with a rather simple approximation. that, say 0 d r < 1 and o cc CL)* < ~‘2. In this event, one can write cos t!J = 1 -&!J’ + c (OJ), etc. to a good degree of accuracy. Consequently, after some algebraic can be approximated by &(.U,) =
manipulation,
~IR2(C1+/)+pAR2[C1(/-_S~)+t(/‘-Sf)]+C’(~J3)
Suppose
sinw = u-((w’) it can be verified
that equation
(5.5) (5.7)
On the extensional
0.2
and flexural vibrations
0.4
0.6
08
I.0 0.6
of rotating
0.6 0.4
r
Fig 3. Variation
6.
THE
bars
0.2
0
I/r
of w* with the tip mass parameter
DEFORMED
229
MOTION
r.
PROBLEM
By virtue ofequation (5. l), the flexural and extensional equations for the deformed motion problem become uncoupled. With the aid of equation (4.18), equations (4.13-4.17) may now be expressed as EAYII+(Nol!l),l+pAR’u EAu,I+Nou,,+~((i~-nzu)
= phi, ti = 0 =0
O
(6.1)
at xi = 0 at xi={ I
(6.2)
O
(6.3)
for the extensional motion and Ellc,l,l,-(Not~,l),l+pAii
= 0,
VV= \(: 1 = 0 \V,ll = El\v,i,,-NOC~,i-~iC = 0
at .Y,= 0 (6.4) at .~i = e 1 for the flexural motion, where the form of N,(_K,)is given in equation (5.5) for 0 < Q < R* or, for relatively low rotational speeds, in equation (5.7). If the form of &(x1) given in equation (5.7) is inserted into equations (6.1-6.2), the result is [EA + PR2(C, + tjju, 11+pAR2{[cl(/-xl)+i_(~2-.~:)]~,1),I+pAR2u u(0, t) = 0
[EA+,LlR’(c*+lq]zfJ(!, t)+p[ii(P, t)-n2u(t,
t,] = 0
= phi
1
(6.5)
(6.6)
In the case ofvanishing tip mass and hub radius, i.e. ,U= cl = 0, equations (6.5-6.6) reduce to EAI~,~~+~~AR’[(/‘--.~:)~,~],~
+pAQ’u
u(0, t) = u*,(k, t) = 0
= pAii
(6.7) (6.8)
A comparison of equations (6.7-6.8) with the corresponding boundary value problem discussed in reference [l] reveals that Bhuta and Jones failed to include the initial stress term +pAR2 [(C’ -xf)u, J 1 in their investigation. This term is of the same order of magnitude as the other two terms on the left side of equation (6.7), and hence it cannot be
G. L. ANDERSON
230
discarded. Indeed, the neglect of this term led Bhuta and Jones to conclude that in the extensional motion of a rotating bar whirling lowers the natural frequencies of vibration. That precisely the opposite conclusion is true will be demonstrated in Section 8. Similarly, substitution of equation (5.7) into equations (6.3-6.4) leads to EIw,i,,, -YQ2(c1 +W*,,
-pAR’([~~(/-x,)+~(e’-x:)]w,~~~+pAi~ \v = W’, 1= 0
=
w.11
at x, = 0 at x1 = / 1
Efw,,,,-~R2(cl+e)~~,,-~~=o
Various forms ofequations
= 0
(6.9) (6.10)
(6.9-6.10) have been derived and discussed in [5], [9-121, and
[151. 7. APPLICATION
OF
THE
RITZ
METHOD
If next &(x1) as given in equation (5.5) is substituted into equations (6.1-6.4) and if the changes of independent variables xi = lx and t = Pr(p/E)‘!’ are made, one finds (Pu’)’ + 02u = ii,
O
(7.1)
u(O,s)= P(l)u’(l,T)+r[ii(l,r)-w%(l,r)]
= 0
(7.2)
where now U’= &@x, ti = an/&, etc. and P(x) = p cos ox -
w
l+w
O
,t’ = \V’= 0 w” = W“’+ (&() w’_ r* = O
(7.3)
at x=0 atx=l
(7.4)
where u = AL’JI,
&(x1)
= 1 -t w< sin wx - j? COsox
for the flexural motion. Assuming now that U(X,5) = 4(x) cos AT,
W(X,5) = I++) cos A?
it follows directly from equations (7.1-7.4) that 4(x) and $(x) are solutions of the eigenvalue problems (Pqfl’)‘+(w2+%2)#
4(O) = P(1)#(1)-r(i.2-to2)&1)
= 0
O
= 0
(7.5) 1
and *““+a(&$‘)‘-i2*
= 0
O
i//(O)= I//‘(O)= 0 $“(l) = 1(/“‘(1)+aF&(l)$‘(1)+rl.21(/(1) = 0
(7.6) 1
It is a straightforward matter to verify that the boundary value problems in equations (7.5-7.6) can be generated from the variational principles (7.7) and 6
[(@02 -a&($‘)2
- i.‘G2] dx - ri’$‘(l)
1= 0
(7.8)
On the extensional
and Aexural vibrations
of rotating
bars
231
These variational principles will now be used in conjunction with Ritz’ method for the purpose of determining the natural frequencies of vibration for the first few longitudinal and flexural modes of motion. Following the Ritz procedure, let it be assumed that C#J(X) and Ii/(x) can be approximated by 4(x) - “ii &4”(X),
$(x) z “ii 4$“(X)
(7.9)
where N is a positive integer, the coefficients n, are unknown, and the coordinate functions 4.(x), $Jx) are selected to be the sets of polynomials ci, = (n + l)(cos w + r<02) - rw2(cos 0 - i-0 sin 0)
(7.10)
Pn = n(cos w + rtw’) - ~0~(cos 0 - r-0 sin 0) and l)“(X) = x”+1(~n+/3”s+i’“X2) c(, = (n+2)(n+3)[(n+l)(n+2)-&(l)] /?n= -2(n+
(7.11)
l)(n+3)[n(n+2)-r&(l)]
7” = (n+ l)(n+2)[n(n+
l)-r&(l)]
which satisfy the boundary conditions A(0) = P(l)dXI)-~~244) = 0 IC/“(O) = rl/A(O) = l&(l) = I1/~‘(1)+c&(l)rl/~(l) = 0
(7.12)
respectively. Not all the boundary conditions appearing in equation (7.12) are identical to the corresponding boundary conditions in equations (7.5-7.6). It may be recalled, however, that, in the application of the Ritz method, the coordinate functions &(x) and G,,(x) need satisfy only the geometric boundary conditions 4(O) = @(O)= $‘(O) = 0 and not necessarily the natural boundary conditions P(l)@(l)-r(L2+o’)4(1) $“(l) = l+V”(1)+c&(l)iJ/‘(l)+ri2$(1)
= 0 = 0
Substitution of equations (7.9) into the apposite variational principles in equations (7.7-7.8) leads to the homogeneous system of algebraic equations i
a,(Ati
2.’+ Bnk) = 0,
n,k=l,2
,..., N
where ‘%k
=
0’
h(bk
dx+
s &k
=
@,(1)6,(l) I
P&, 4; dx
w2Ad s 0
for the extensional motion problem and An.+=
’ $n$kd-~+rtin(l)‘bk(l) s 0
for the flexural motion problem. Therefore, because, in view of equations (7.10-7.1 l), the values of the coefficients A,k and Bnkare known, the natural frequencies of vibration (&}, n = 1,2,3,. . . ) N, for the extensional and flexural modes of a bar rotating at a uniform rate can be computed from the equation det(A,ki.2 + &) = 0
(7.13)
232
L.
G.
8.
ANDERSON
NUMERICAL
RESULTS
Equation (7.13) was solved numerically for the frequencies E.,, i.z,. . . for combinations of the following values of the tip mass parameter r and the hub radius parameter <: r, < = 0, l/5, I/.5. The value chosen for z was r = (4:‘3) x 106, which is typical of some helicopter blades that are currently in use. Numerical calculations were performed with N = 6,7,8, and it appeared that for N = 8 and for the range of the angular velocity
/
,.I
1.4
-
1.3
-
I.2
-
4 /
/.’
r=2/5
1.0
I
0
I
0.1
I
1
0.2
0.3
I
I
0.4
0.5
0.6
w
Fig. .l, Variation
of i, for the lowest extensional and -.~
mode with w I-< = 2 j).
5 = 0, ___
< = l.!j.
w
Fig. 5. Variation
of Lz for the second
extensional mode with w I--and -.< = 2: j),
2 = 0. ---
5 = II j,
parameter, o, considered, the numerical procedure had essentially converged to four digit accuracy in the lowest three modes of oscillation. Attention here is confined to the domain 0 Q w < 0.7 because of the considerations described in Section 5. The variations of the lowest three natural frequencies i.I, j.2, ;.3 of the extensional modes ofvibration of a rotating bar are plotted in Figs. 4-6, respectively, for several combinations of the tip mass parameter r and the hub radius parameter 5. Firstly, these plots reveal that for given values of I and < the values of the natural frequencies of the lowest three extensional modes increase monotonically as the angular velocity increases. This result
On the extensional
and flexural vibrations
of rotating
bars
233
9
6.5 -
gL 0
I 0.3
0.2
0.1
I 0.5
0.4
0.6
w
Fig. 6. Variation
of i., for the third
extensional and -‘-
mode with w (< = 2/5).
2 = 0, ---
g = 115,
confutes the assertion in [I] that whirling tends to lower the natural frequencies of vibration as compared to those of the non-rotating bar. Secondly, it is evident that for a fixed value of r the frequencies increase as the hub radius parameter 5 is increased, whereas, if the value of 5 is held fixed, the frequencies decrease as the tip mass parameter r is increased, provided that the angular velocity parameter w is sufficiently small. However, one should note that the curves shown in Fig. 6 for iW3obtained with I = l/5 and I = 2/5 for a given value of 5 intersect one another when w becomes sufficiently large. In other words, for a given value of 5 and for o sufficiently large, the natural frequencies may increase with increasing tip mass. The variation of the lowest three natural frequencies of the flexural modes of oscillation of a rotating bar are plotted in Figs. 7-9 again for various combinations of the parameter r and 5. These curves are very similar in appearance to those presented in Figs. 4-6, and the conclusions drawn regarding the extensional modes of oscillation are therefore applicable to the flexural modes also.
1 I
5.5
5.0
4.5
4.0 h 3.5
3.0
2.5
0
0.5
1.0
1.5
2.0
2.5
3.0
wx103
Fig. 7. Variation
of i., for the lowest
flexural and -.-
mode with 5 = 215).
UJ (---
< = 0, ---
< = 115,
G.
L.
ALDERSO~
25
23
22
2 I
20
x2 19
18
17
0
0.5
1.0
I.5
wx
Fig. 8. Variation
of il
for the second
Aexural and
64
_._
2.0
2.5
3.0
I03
mode ,'=
5 = 0, ___
with o (-
<=
l;j,
2.j)
./.’
-
./ /’ /’
63
,,=,:A
53
52
0
J 0.5
1.0 w
Fig. 9. Variation
of ;.3 for the third flexud -.-
1.5
2.0
2.5
30
x103 mode with w C-----~~~o,---~c
1 j,and
< = 2;s).
Finally, a few remarks regarding the range of applicability of equation (5.7), which in various forms has been used in references [4-131 in place of the more complete expression for &,(x1) as given in equation (5.5), are in order. In the Appendix, equations (6.9-6.10) are solved approximately by the Ritz method, and the natural frequencies were determined numerically for the case r = < = l/5, for the sake of example. These values, labeled j.ip (n = 1,2,3 and “ap” for “approximate”) are displayed in Table 1 and are compared with the corresponding values i.‘,” (“e.u” for “exact”) obtained from equation (7.13) over the
On the extensional
and flexural vibrations
of rotating
bars
235
domain 0 < w ,< 0.7. At least for the three modes considered, the percentage difference between Lip and nGXis virtually independent of the mode number n for a given angular velocity, however as the value of w increases the percentage difference also increases. Thus, for the combination of parameters r = 5 = l/5, the relatively simple expression for &,(x1) in equation (5.7) is sufficiently accurate for computing the natural frequencies of the lowest three flexural modes whenever w < 0.1, but, if o > 0.1, then the somewhat more complicated expression for &,(x1) given in equation (5.5) should be used. Table
of ,1:p and jex .” 7n =
1. Comparison
L2.3, for several values of the angular velocity parameter
w,
for r = 5 = l/5. %
W
Zyp
i;
difference
i”9
0.1 02 0.3 @4 @5 0.6 0.7
131.4 262.9 394.7 526.0 6577 789.0 9206
131.8 266.1 405.3 5524 711.1 887.1 1088
0.3 1.2 2.6 4.8 7.5 11.1 15.4
359.6 724.2 1087 1452 1815 2179 2542
% d@erence
J.‘i’
.?y
0.3 1.1 2.8 4.7 7.5 11.0 15.2
3606 732.5 1118 1523 1962 2447 2999
636.3 1283 1929 2574 3218 3863 4507
r 5’ 638.2 1298 1981 2701 3479 4340 5322
% difference 02 1.2 2.6 4.7 7.5 11.0 15.3
REFERENCES 1. P. G. Bhuta and J. P. Jones, On axial vibrations of a whirling bar, J. acousr. Sot. Am. 35, 217-221 (1963). 2. E. J. Brunelle, Stress redistribution and instability of rotating beams and disks, A.I.A.A. Jl9, 758-759 (1971). 3. E. J. Brunelle, The super flywheel: a second look, J. Engng Math. Tech. 95, 63-65 (1973), and Discussion, 95, 195-196 (1973). 4. H. Lo and J. L. Renbarger, Bending vibrations of a rotating beam, First U.S. natn. Cong. appl. Mech. 75-79 (1951). 5. W. E. Boyce, R. C. DiPrima and G. H. Handleman, Vibrations of rotating beams of constant section, Second U.S. natn. Cong. appl. Mech. 165-173 (1954). 6. M. J. Schilhansl, Bending frequency of a rotating cantilever beam, J. appl. Mech. 25, 28-30 (1958). 7. N. Rubinstein and J. T. Stadter, Bounds to bending frequencies of a rotating beam, J. Franklin Insr. 294. 217-229 (1972). 8. D. Pnueli, Natural bending frequency comparable to rotational frequency in rotating cantilever beam. J. appl. Mech. 39,602-604 (1972). (1956). 9. W. E. Boyce, Effect of hub radius on the vibrations of a uniform bar, J. oppl. Mech. 23.287-290 10. H. Lo, J. E. Goldberg and J. H. Bogdanoff, Effect of small hub radius change on bending frequencies of a rotating beam, J. appl. Mech. 27, 548-550 (1960). W. Boyce and H. Cohen, Vibrations of a uniform rotating beam with tip mass, Third t’.S. 11. G. Handleman, mtn. Coy. appl. Mech. 175-180 (1958). Vibrations of rotating beams with tip mass, Z. angew. Marh. Phys. 12, 12. W. E. Boyce and G. Handleman, 369-392 (1961). 13. V. V. Bolotin, Nonconservative Problems in the Theory 01 Elastic Stability. Macmillan, New York (1963). on the dynamic behavior of large flexible bodies in orbit, A.I.A.A. Jl 5, 460-469 14. H. Ashley, Observations (1967). (1968). 15. R. D. Mime, Some remarks on the dynamics of deformable bodies, A.I.A.A. Jl6,556-558 16. A. C. Eringen, On the non-linear vibration of elastic bars, Q. appl. Math. 9, 361-369 (1952). 17. S. R. Woodall, On the large amplitude oscillations of a thin elastic beam, Inc. J. Non-linear Mech. 1, 217-238 (1966). APPENDIX Proceeding
as in Section
7, one can show that equations
(6.9-6.10)
become
JI”“-rKo*(l+~)~~-K0*(fiSJ/‘)l--,is~
= 0
$(O) = tj’(O) = (1/“(l) = t/“(l)-rzo2(1+~)ljl’(l)+r12~(1)
= 0
O
where 1s(x) = C(l-s)+f(l-x2) This boundary
value problem
s Equation boundary
can be obtained
from the foIlowing
variational
principle:
rKw’(1 +~)(1(/‘)2+~~2~(J1’)2--j.L~z]dx-r~.2[~(1)]2
(7.9) will again be assumed, conditions
however
now the coordinate
functions
IL.(x) will be required
+.(O) = p,(O) = $:( 1) = +:“( 1) - rzw’( 1 + t)&( 1) = 0
(A.1) to satisfy the
(A.2)
G. L.
236
The set of polynomial
coordinate
,A\IDEKSOh
functions ti”(.Y) = Y-l IZ.+/i..Y+:‘,Y2)
where 1” = oI+2)(n+3)[ol+ /r.=
.’,n = (n+ I)(n+:!)[n(nc satisfies all the boundary equation
conditions
(7.13) still obtains,
I)(~I+2)+rX’P(l
+<,I
-2(~1+1)(t1+3)[n(n+2~+r20~(1+~)]
in equation
I)+rX92(I+,)]
(A.?). Furthermore.
It is easy to show from equation
tA.1) that
where now
R&sum&On formule Ies kquatiolls non lintaires du mouvrment d’une barre mince tournant a\ec une vitesse angulnire constante autourd‘un axe transversal. Sous I’hypoth2se qu’un petit mouvement perturb& se produise autour d’une configuration d’iquilibre ayant une contrainte initiale, on etablit des kquations linearistes du mouvement pour les dXormations longitudinales et en Hexion d’une barre en rotation portnnt une masse ti son extrbmitC. Des calculs numiriques pour Its friquences naturelles dcs trois modes de vibration libre les plus has rt2lL;lent que lorsque la \.ites,e angulaire augmente les caleurs des friquences d’extension augmentent de facon monotone. contrairement aux r&sultats prCctdemment publiis.
Zusammenfassung-Die nichtlinearen Bewegungsgleichungen eines schlanken Stabes. der mit konstanter Winkelgeschuindigkeit urn eine Querachse rotiert. werden formuliert. Unter der Annahme. dass sich eine kleine Stdrungsbewegun g der korgespannten Gleichgeuichtsanordnun: tiberlagert. werden linearisierte Bewegungsgleichungen tir die LLngs- und Biege\ erformungen eines rotierenden Stabes mit einer Endmasse hergeleitet. Numerische Berechnungen der Eigenfrequenzen der drei niedregsten freien Eigenschwingungsforem zeigen, dass im Gegensatz zu veriii?entlichten Ergebnissen die Werte der Ausdehnungsfrequenzen monotonisch zunehmen. wenn die Winkelgeschwindigkeit der Drehung zunimmt.