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A note on the stability of a rod subjected to compression by centrifugal force
Journal of Sound and Vibration (1976) 46( 1), 105-l 11
A NOTE
ON THE STABILITY COMPRESSION
OF A ROD
BY CENTRIFUGAL
SUBJECTED
TO
FORCE
H. I. WEBER Departamento
Engenharia Mecrinica, Universidade Estadual de Campinas, Brasil
(Received 8 July 1975, and in revisedform
24 November 1975)
A flexible bar with discrete masses, clamped at the inner side of a rotating ring, is investigated. The rod is initially radial and the ring has rotational motion about its radial symmetry axis. The vibration of the masses is studied for small deformations of the rod in the plane of rotation and out of that plane. The critical velocity for stability due to the centrifugal force and its dependence on different parameters, as well as the influence of the Coriolis force on the motion, are investigated.
1. INTRODUCTION
The stability investigation of a structural element undergoing steady rotation may be critical in the case of a slender element subjected to high rotational speed. Some parts of chemical equipment present a rotating cross-section, as indicated in Figure l(a), for which this analysis is important. In a first approximation for the solution of this problem, one needs the solution for the stability of a rotating rod carrying some discrete masses.
(0)
Figure 1. Physical models.
In this work is presented the analysis of motion of a simplified massless bar with one or two concentrated masses. The stability is investigated for the motion in the plane of rotation and out of that plane. The limits of stability are determined and the influence of the Coriolis force is shown. 105
106
H. I. WEBER
The results are compared with those of the work of Mostaghel and Tadjbakhsh [l] who investigated the buckling of a rod under similar conditions. The approximations done for these calculations [I] raise some doubts [2] concerning their precise relevance to physical problems.
2. MATHEMATICAL
MODELS
The type of system to be studied is represented in Figures I(b) and l(c). It is a rod fixed by one end at an external ring and pointing initially in the radial direction. One considers the ring as rigid. The rod is massless with flexural rigidity EZ, discrete massesm or Mare considered at one or more points, and the whole system is rotating about a perpendicular axis passing through the center of the ring. The angular velocity, o, is considered constant. The mathematical model is obtained by Newton’s law (the mass is hypothetically cut from the rod and its motion is described) and from the equilibrium of the massless rod. 2.1.
SYSTEM WITH ONE MASS, MOTION
IN THE PLANE OF ROTATION
The motion of the rod represented in Figure l(b) will be restrained to the plane of rotation by a convenient form of the cross-section. In Figure 2 both problems that must be solved to obtain the mathematical model are represented. Two reference frames are considered: X, Y, Z, inertial, with origin at the center of the ring; and x, y, z, with origin at the point where the rod is built into the ring, maintaining the x-axis radial.
Figure
2. System with one mass, vibration
in the plane of rotation.
A mass is fixed at the end of the rod with length 1. Small deformations are considered so that its motion can be described by x(t) = I = const., and y(t) = q(t). The motion of the mass in a rotating plane under the action of forces (Figure 2) is then given by -2mwd
= F2 - mo2 R,
mfj-mw2q=-F,.
(1)
On the other hand the elastic curve of a beam subjected to forces Fl and F2 (Figure 2), with only the bending deformation considered, is given, for small curvature, by d2y/dx2 = -(l/EI)[F,y The rod is clamped at x = 0. The solution situated is therefore
+ F,(Z-x)
+ F2q].
for the displacement
F2
’
0,
rl =
V’,IF2)[~
F2
<
0,
q =
(F,/F,) [m
tan
C-0
tanh (m
(2)
at the end where the mass is
-
4, I) - 11.
The system of equations (1) and (3) is the mathematical model for this problem. convenient to introduce following parameters for the non-dimensionalization : vTmFm=@,
c2 = m13/EI,
T = (l/c) 1,
6=CcW,
Y = RII,
(3) It is very
ii = V/l.
(4)
107
STABILITY UNDER CENTRIFUGAL FORCE
The mathematical model, with derivatives in respect to 7, is obtained as (v- l)O-2f’>O, ij” + [a’/(tan a/a - 1) - 6’1 f = 0, ‘I”+ [a*/( 1 - tanh a/a) - G’] ij = 0,
(y -
or
1) w - 2ij’< 0,
(5a) (5b)
with a* = I(7 - 1) W*- 2Zj’].
(5c)
2.2. SYSTEMWITH ONE MASS, MOTION ORTHOGONAL TO THE PLANE OF ROTATION The acceleration on the mass is different in the case represented in Figure 3. Considering x=I,y=O,z=[thenonehas F2 = mw*(R - I), ml = -F,. (6)
Figure 3. System with one mass, vibration orthogonal to the plane of rotation.
The equation for the rod is analogous to equation (2) and one obtains, after non-dimensionalizing, y>l, or c” + [a’/(tan a/a - l)] 5 = 0, (7a) 5” + [a*/( 1 - tanh a/a)] [ = 0, Y
-2Mwrj, + Mo*(R - 2L) = F,,
Mij2 - Mw* q2 = -(F3 - F,),
-2Moij,
+ Mo*(R
-L)
= F4 - F2.
Figure 4. System with two masses, vibration in the plane of rotation.
(8)
108
H. I. WEBER
The elastic curve of the rods is described by the solution of d’y,/dx: + (F,IEZ)y, = (IIEZ)]F,(L -xl) + F~.Y,LI, d2y/dx2 + (F,/EZ)y = (I/EZ)[F,(L - x) + FbyL + fil,
(9)
where the moment J? = FI L + F2 y,,. The relations between the displacements of the masses and of the ends of the half rods are qI = y,, + y, and q2 = y,. As boundary conditions one uses the clamping of the outer rod, y(x = 0) = 0, (dy/dx)l,,, = 0, and also the continuity condition between the rods, yI(x, = 0) = 0, (dy,/dx,)l,,=, = (dy/dx)l,_,. The application of these conditions to the solutions of equations (9) for the displacements at the ends of each of the rods results in the following system :
(10)
Mm = Plm where
with F = (F/EZ)L and f = y/L. Matrices [A] and [B] are functions only of x1 = ,mL and a2 = -L. Three different possibilities must be considered independently, resulting in different coefficients in equations (10). Where F2 > 0, F4 > 0
then
m = 0, n = 0, su, = sin a,, ca, = cos ff,, su2 = sin az, ccLz= cos uz,
F2 < 0, F4 > 0
then
m = 1, n = 0, sal = sinha,, ccz, = coshu,, suz = sina,,
cc!*= cos c?z, F, < 0, F4 < 0
then
m=l,n=l,sa,=sinhcc,,ca,=coshu,,sa,=sinha,, cciz = cash uz,
one has a 11= [ScL&I + (-l)“/a:ls%
- (-l)mca,/%,
a I* = [(-l)“ca&
+ sc4-t, - (-l)“/a:] sar,
a21 = 1 - CCL~,
a22 = SC(~/CL, - cct2,
b,, =-(-1)“cI:(sc(*/a2)sa,
+ a,cc(,,
/I,, = (-l)ma: ca2 - (-1)” cc:,
b,, =-(-l)“C(~ScY~SrX*,
b,, = (-l)“a:ca,.
Using the same non-dimensionalizing parameters defined in equation (4), and introducing & = qI/2L and jj2 = v,JL, then one may calculate cr: = (1/16)12G2(y - 1) - 4G7;/,
a; = (1/16)jW2(4y - 3) - 40& - 2Wq;/,
(11)
and separate the different cases according to the signs of the arguments of aI and ~1~. Equations (8), with [Z] denoting the identity matrix, can be written as
(12) By substituting equation (10) in this equation it can be transformed to
[ml” + WI - w2mrl> = Kc where [C] = [T,][A]-‘[B][T,]
(13)
and
P’11=16[lif p], ,,I=[;
-j
are matrices due to the differences in normalization and due to the fact that one wishes later to compare discrete and continuous results.
STABILITY UNDER CENTRIFUGAL FORCE
Equation (13) is the mathematical model of the motion discrete masses that were defined. The same equation can of n masses if one conveniently adapts the matrices. The equivalent model for the motion in the orthogonal comparing equation (6) with equation (l), or (7) with (5)
m5>” + [WC>= w.
109
in the plane of rotation of the two also be used for a generalized case plane can be obtained directly by and is given by (14)
3. STABILITY The stability considered is of the oscillating type, in the sense of Lyapunov. For the determination of the limits of stability one proceeds to a linearization of the problem. That corresponds to supposing O’f(y) 9 ~$((r’) in expressions (5~) and (11). According to this CI becomes constant and one concludes that the equations to be used are only functions of y. From sections 2.1 and 2.2, if y > 1 (R > I) the solution is expression (5a) and if y < 1 it is expression (5b). For the case of section 2.3, one has three distinct intervals, cri> y > 1, 1 > y > 0.75, 0.75 > y > 0, determined by the signs of the forces on the rods (traction or compression) and not by the coincidence of the masses with the center of rotation. From consideration of equation (13) stability exists if ([Cl - Wz[Z])is positive definite, or in the orthogonal motion from equation (14) if [C] is positive definite. Remember that [C] = [C(a,)] and cli = or,(y,W). Therefore one obtains intervals for LY where the motion is stable and which lead to critical values for 0 at which instability begins. For the systems described in sections 2.1 and 2.2, the investigation is simple to do. One gets stability in the plane of rotation, from equations (5) for y > 1 when tancc < yc~and tancr > a and for y < 1 when tanhcc > ycl. When the motion is in the orthogonal plane one has stability, from equation (7) for y > 1 when tana > c(, i.e., 0 < LY< rc/2 (first interval) and for y < 1 always, and in this case the result coincides with that obtained for the critical buckling in equation (6) to this load : load. This can be seen by equating I;; mo2(R -I) = rr2E1/412. (15) Then one has from equation (4) that a = X/Z for this first limit of stability. In Figure 5 the curves for 0 critical, the first limit of stability, are represented as functions of y = R/I.The curve “1” is obtained for the case of section 2.1, motion in the plane of rotation, and curve “2” for that of section 2.2, motion orthogonal to that plane. In the latter case c[= 71/2is equivalent, due to equation (5c), to ID’= 7c/4(y- 1).
Y
mass in plane; curve “2”-one mass out of plane; Figure 5. The first limit of stability: curve “I”-one curve “3’‘-two masses in plane; curve “4’-two masses out of plane; curve “5”-continuous rod.
110
H. I. WEBER
Comparing both curves one notes that for a certain y the maximum admissible value for the angular velocity before instability is less in the case of motion in the plane of rotation. This is due to the component of centrifugal force normal to the rod. Both curves approach each other for large values of y, when this effect loses its relative importance. This means, in general, that a rod with circular cross-section will first be unstable in the plane of rotation. The same effect is also responsible for instability when the whole rod is under traction: i.e., y < 1 for the case of section 2.1 and y < 0.75 for that of section 2.3. In Figure 5 there are also represented the curves “3” and “4” corresponding to the case of section 2.3 for stability in plane and out of plane, respectively. Note that curves “1” and “3” cut at y = 0.5 and that curve “4” is asymptotic to y = O-75. Curve “5” is calculated in reference [1] in the represented interval and shows the approximation obtained by the discrete models compared to the continuous case result. There are some other interesting things to point out. In the case of section 2.1, when y = 1, i.e., the mass is in the center of rotation, the critical angular velocity is equal to the natural frequency of the system : cocri, = 1/(3EI/m13).
(16)
Curve “2” has no values for y < 1 because when the mass is beyond the center of rotation (I > R) there is only traction in the rod (see equation (6)) and there is no transversal force in this case. One can generalize this conclusion. For a rod with length 1 with a certain number of masses there is no longer instability in the orthogonal plane if the forces on all masses are of traction. 4
F; -Z 5
4
. . . . . . . . . . ~-._-_--E to F2”-f&-2
Z 6-F.
Z 6 -5
Figure 6. Compressive force distribution.
In Figure 6, diminishing R, all forces become less maintaining F2 < F4 < Fe < **a< Fz. (positive means compression) till F2 becomes zero. Then there is no more instability possible. In the case of section 2.3, F4 = Mo”2(2R - 3L) and generally Fz, = Mw’(nR
-L
2 i),
where m = Mn, I= Ln.
i=l
There is compression
for nR > (x:=, i)L or y > (l/2)(1
(17)
+ l/n).
Some limit values are then, for n = 1, y = 1; for n = 2, y = O-750; for n = 3, y = O-666; for n co, y = O-5. The last result is the same obtained in the direct analysis of the continuous rodf 11. Therefore, analogous to the case of section 2.2, one can expect a curve that in each case is asymptotic to the corresponding y = yLimit axis on one end and approaches the curve for the stability in the plane on the other end.
4. ANALYSIS
OF MOTION,
SYSTEM WITH ONE MASS
To characterize the influence of the non-linear term of expression (5~) one proceeds by an analysis of motion in the plane of rotation. The best way to realize this investigation is in the phase plane. It is possible to transform equation (5) into MI= v,
v’ = - [a2/(tan u/a - 1) - G2] 24,
with c? = (y - 1) W - 26~. A corresponding
expression
is obtained
fory>
1,
for y < 1. The equation
(1%
of
STABILITY UNDER CENTRIFUGAL FORCE
111
Figure 7. Phase plane for a systemwith one mass,vibration in the plane of rotation, y = 1. the trajectory of equations (18) is given by
s ”
242= u; - 2
vdv
“o [u2/(tan a/a
- 1) - 0’1’
where a = a(v).
(19)
The results are shown in Figure 7. The motion in the first interval of stability is represented and the influence of the angular velocity is investigated. The distortion is of a considerable amount only in the neighborhood of the limit of stability. This effect of asymmetry to the u-axis is wholly due to the non-linear term, the Coriolis force. 5. CONCLUSIONS The problem of the stability of a system comprising a rod with some discrete masses is studied when it is rotating about an orthogonal axis. The stability in the plane of rotation and orthogonal to that plane is considered. It is shown that instability is first reached in the plane of rotation and only for very large values of R/l are the limits almost the same in both cases. It is verified also that only the motion in the orthogonal plane has a minimum value of R/I and then instability is no longer possible. The influence of the Coriolis force on the motion is remarkable only in the neighborhood of the stability limits. REFERENCES 1.
N. MOSTAGHELand I. TADJBAKHSH1973 International Journal of Mechanical Science 15,429-434. Buckling of rotating rods and plates. 2. F. G. RAMMERSTORFER 1974 International Journal of Mechanical Science 16,515-517. Comments on buckling of rotating rods and plates.
A NOTE
ON THE STABILITY COMPRESSION
OF A ROD
BY CENTRIFUGAL
SUBJECTED
TO
FORCE
H. I. WEBER Departamento
Engenharia Mecrinica, Universidade Estadual de Campinas, Brasil
(Received 8 July 1975, and in revisedform
24 November 1975)
A flexible bar with discrete masses, clamped at the inner side of a rotating ring, is investigated. The rod is initially radial and the ring has rotational motion about its radial symmetry axis. The vibration of the masses is studied for small deformations of the rod in the plane of rotation and out of that plane. The critical velocity for stability due to the centrifugal force and its dependence on different parameters, as well as the influence of the Coriolis force on the motion, are investigated.
1. INTRODUCTION
The stability investigation of a structural element undergoing steady rotation may be critical in the case of a slender element subjected to high rotational speed. Some parts of chemical equipment present a rotating cross-section, as indicated in Figure l(a), for which this analysis is important. In a first approximation for the solution of this problem, one needs the solution for the stability of a rotating rod carrying some discrete masses.
(0)
Figure 1. Physical models.
In this work is presented the analysis of motion of a simplified massless bar with one or two concentrated masses. The stability is investigated for the motion in the plane of rotation and out of that plane. The limits of stability are determined and the influence of the Coriolis force is shown. 105
106
H. I. WEBER
The results are compared with those of the work of Mostaghel and Tadjbakhsh [l] who investigated the buckling of a rod under similar conditions. The approximations done for these calculations [I] raise some doubts [2] concerning their precise relevance to physical problems.
2. MATHEMATICAL
MODELS
The type of system to be studied is represented in Figures I(b) and l(c). It is a rod fixed by one end at an external ring and pointing initially in the radial direction. One considers the ring as rigid. The rod is massless with flexural rigidity EZ, discrete massesm or Mare considered at one or more points, and the whole system is rotating about a perpendicular axis passing through the center of the ring. The angular velocity, o, is considered constant. The mathematical model is obtained by Newton’s law (the mass is hypothetically cut from the rod and its motion is described) and from the equilibrium of the massless rod. 2.1.
SYSTEM WITH ONE MASS, MOTION
IN THE PLANE OF ROTATION
The motion of the rod represented in Figure l(b) will be restrained to the plane of rotation by a convenient form of the cross-section. In Figure 2 both problems that must be solved to obtain the mathematical model are represented. Two reference frames are considered: X, Y, Z, inertial, with origin at the center of the ring; and x, y, z, with origin at the point where the rod is built into the ring, maintaining the x-axis radial.
Figure
2. System with one mass, vibration
in the plane of rotation.
A mass is fixed at the end of the rod with length 1. Small deformations are considered so that its motion can be described by x(t) = I = const., and y(t) = q(t). The motion of the mass in a rotating plane under the action of forces (Figure 2) is then given by -2mwd
= F2 - mo2 R,
mfj-mw2q=-F,.
(1)
On the other hand the elastic curve of a beam subjected to forces Fl and F2 (Figure 2), with only the bending deformation considered, is given, for small curvature, by d2y/dx2 = -(l/EI)[F,y The rod is clamped at x = 0. The solution situated is therefore
+ F,(Z-x)
+ F2q].
for the displacement
F2
’
0,
rl =
V’,IF2)[~
F2
<
0,
q =
(F,/F,) [m
tan
C-0
tanh (m
(2)
at the end where the mass is
-
4, I) - 11.
The system of equations (1) and (3) is the mathematical model for this problem. convenient to introduce following parameters for the non-dimensionalization : vTmFm=@,
c2 = m13/EI,
T = (l/c) 1,
6=CcW,
Y = RII,
(3) It is very
ii = V/l.
(4)
107
STABILITY UNDER CENTRIFUGAL FORCE
The mathematical model, with derivatives in respect to 7, is obtained as (v- l)O-2f’>O, ij” + [a’/(tan a/a - 1) - 6’1 f = 0, ‘I”+ [a*/( 1 - tanh a/a) - G’] ij = 0,
(y -
or
1) w - 2ij’< 0,
(5a) (5b)
with a* = I(7 - 1) W*- 2Zj’].
(5c)
2.2. SYSTEMWITH ONE MASS, MOTION ORTHOGONAL TO THE PLANE OF ROTATION The acceleration on the mass is different in the case represented in Figure 3. Considering x=I,y=O,z=[thenonehas F2 = mw*(R - I), ml = -F,. (6)
Figure 3. System with one mass, vibration orthogonal to the plane of rotation.
The equation for the rod is analogous to equation (2) and one obtains, after non-dimensionalizing, y>l, or c” + [a’/(tan a/a - l)] 5 = 0, (7a) 5” + [a*/( 1 - tanh a/a)] [ = 0, Y
-2Mwrj, + Mo*(R - 2L) = F,,
Mij2 - Mw* q2 = -(F3 - F,),
-2Moij,
+ Mo*(R
-L)
= F4 - F2.
Figure 4. System with two masses, vibration in the plane of rotation.
(8)
108
H. I. WEBER
The elastic curve of the rods is described by the solution of d’y,/dx: + (F,IEZ)y, = (IIEZ)]F,(L -xl) + F~.Y,LI, d2y/dx2 + (F,/EZ)y = (I/EZ)[F,(L - x) + FbyL + fil,
(9)
where the moment J? = FI L + F2 y,,. The relations between the displacements of the masses and of the ends of the half rods are qI = y,, + y, and q2 = y,. As boundary conditions one uses the clamping of the outer rod, y(x = 0) = 0, (dy/dx)l,,, = 0, and also the continuity condition between the rods, yI(x, = 0) = 0, (dy,/dx,)l,,=, = (dy/dx)l,_,. The application of these conditions to the solutions of equations (9) for the displacements at the ends of each of the rods results in the following system :
(10)
Mm = Plm where
with F = (F/EZ)L and f = y/L. Matrices [A] and [B] are functions only of x1 = ,mL and a2 = -L. Three different possibilities must be considered independently, resulting in different coefficients in equations (10). Where F2 > 0, F4 > 0
then
m = 0, n = 0, su, = sin a,, ca, = cos ff,, su2 = sin az, ccLz= cos uz,
F2 < 0, F4 > 0
then
m = 1, n = 0, sal = sinha,, ccz, = coshu,, suz = sina,,
cc!*= cos c?z, F, < 0, F4 < 0
then
m=l,n=l,sa,=sinhcc,,ca,=coshu,,sa,=sinha,, cciz = cash uz,
one has a 11= [ScL&I + (-l)“/a:ls%
- (-l)mca,/%,
a I* = [(-l)“ca&
+ sc4-t, - (-l)“/a:] sar,
a21 = 1 - CCL~,
a22 = SC(~/CL, - cct2,
b,, =-(-1)“cI:(sc(*/a2)sa,
+ a,cc(,,
/I,, = (-l)ma: ca2 - (-1)” cc:,
b,, =-(-l)“C(~ScY~SrX*,
b,, = (-l)“a:ca,.
Using the same non-dimensionalizing parameters defined in equation (4), and introducing & = qI/2L and jj2 = v,JL, then one may calculate cr: = (1/16)12G2(y - 1) - 4G7;/,
a; = (1/16)jW2(4y - 3) - 40& - 2Wq;/,
(11)
and separate the different cases according to the signs of the arguments of aI and ~1~. Equations (8), with [Z] denoting the identity matrix, can be written as
(12) By substituting equation (10) in this equation it can be transformed to
[ml” + WI - w2mrl> = Kc where [C] = [T,][A]-‘[B][T,]
(13)
and
P’11=16[lif p], ,,I=[;
-j
are matrices due to the differences in normalization and due to the fact that one wishes later to compare discrete and continuous results.
STABILITY UNDER CENTRIFUGAL FORCE
Equation (13) is the mathematical model of the motion discrete masses that were defined. The same equation can of n masses if one conveniently adapts the matrices. The equivalent model for the motion in the orthogonal comparing equation (6) with equation (l), or (7) with (5)
m5>” + [WC>= w.
109
in the plane of rotation of the two also be used for a generalized case plane can be obtained directly by and is given by (14)
3. STABILITY The stability considered is of the oscillating type, in the sense of Lyapunov. For the determination of the limits of stability one proceeds to a linearization of the problem. That corresponds to supposing O’f(y) 9 ~$((r’) in expressions (5~) and (11). According to this CI becomes constant and one concludes that the equations to be used are only functions of y. From sections 2.1 and 2.2, if y > 1 (R > I) the solution is expression (5a) and if y < 1 it is expression (5b). For the case of section 2.3, one has three distinct intervals, cri> y > 1, 1 > y > 0.75, 0.75 > y > 0, determined by the signs of the forces on the rods (traction or compression) and not by the coincidence of the masses with the center of rotation. From consideration of equation (13) stability exists if ([Cl - Wz[Z])is positive definite, or in the orthogonal motion from equation (14) if [C] is positive definite. Remember that [C] = [C(a,)] and cli = or,(y,W). Therefore one obtains intervals for LY where the motion is stable and which lead to critical values for 0 at which instability begins. For the systems described in sections 2.1 and 2.2, the investigation is simple to do. One gets stability in the plane of rotation, from equations (5) for y > 1 when tancc < yc~and tancr > a and for y < 1 when tanhcc > ycl. When the motion is in the orthogonal plane one has stability, from equation (7) for y > 1 when tana > c(, i.e., 0 < LY< rc/2 (first interval) and for y < 1 always, and in this case the result coincides with that obtained for the critical buckling in equation (6) to this load : load. This can be seen by equating I;; mo2(R -I) = rr2E1/412. (15) Then one has from equation (4) that a = X/Z for this first limit of stability. In Figure 5 the curves for 0 critical, the first limit of stability, are represented as functions of y = R/I.The curve “1” is obtained for the case of section 2.1, motion in the plane of rotation, and curve “2” for that of section 2.2, motion orthogonal to that plane. In the latter case c[= 71/2is equivalent, due to equation (5c), to ID’= 7c/4(y- 1).
Y
mass in plane; curve “2”-one mass out of plane; Figure 5. The first limit of stability: curve “I”-one curve “3’‘-two masses in plane; curve “4’-two masses out of plane; curve “5”-continuous rod.
110
H. I. WEBER
Comparing both curves one notes that for a certain y the maximum admissible value for the angular velocity before instability is less in the case of motion in the plane of rotation. This is due to the component of centrifugal force normal to the rod. Both curves approach each other for large values of y, when this effect loses its relative importance. This means, in general, that a rod with circular cross-section will first be unstable in the plane of rotation. The same effect is also responsible for instability when the whole rod is under traction: i.e., y < 1 for the case of section 2.1 and y < 0.75 for that of section 2.3. In Figure 5 there are also represented the curves “3” and “4” corresponding to the case of section 2.3 for stability in plane and out of plane, respectively. Note that curves “1” and “3” cut at y = 0.5 and that curve “4” is asymptotic to y = O-75. Curve “5” is calculated in reference [1] in the represented interval and shows the approximation obtained by the discrete models compared to the continuous case result. There are some other interesting things to point out. In the case of section 2.1, when y = 1, i.e., the mass is in the center of rotation, the critical angular velocity is equal to the natural frequency of the system : cocri, = 1/(3EI/m13).
(16)
Curve “2” has no values for y < 1 because when the mass is beyond the center of rotation (I > R) there is only traction in the rod (see equation (6)) and there is no transversal force in this case. One can generalize this conclusion. For a rod with length 1 with a certain number of masses there is no longer instability in the orthogonal plane if the forces on all masses are of traction. 4
F; -Z 5
4
. . . . . . . . . . ~-._-_--E to F2”-f&-2
Z 6-F.
Z 6 -5
Figure 6. Compressive force distribution.
In Figure 6, diminishing R, all forces become less maintaining F2 < F4 < Fe < **a< Fz. (positive means compression) till F2 becomes zero. Then there is no more instability possible. In the case of section 2.3, F4 = Mo”2(2R - 3L) and generally Fz, = Mw’(nR
-L
2 i),
where m = Mn, I= Ln.
i=l
There is compression
for nR > (x:=, i)L or y > (l/2)(1
(17)
+ l/n).
Some limit values are then, for n = 1, y = 1; for n = 2, y = O-750; for n = 3, y = O-666; for n co, y = O-5. The last result is the same obtained in the direct analysis of the continuous rodf 11. Therefore, analogous to the case of section 2.2, one can expect a curve that in each case is asymptotic to the corresponding y = yLimit axis on one end and approaches the curve for the stability in the plane on the other end.
4. ANALYSIS
OF MOTION,
SYSTEM WITH ONE MASS
To characterize the influence of the non-linear term of expression (5~) one proceeds by an analysis of motion in the plane of rotation. The best way to realize this investigation is in the phase plane. It is possible to transform equation (5) into MI= v,
v’ = - [a2/(tan u/a - 1) - G2] 24,
with c? = (y - 1) W - 26~. A corresponding
expression
is obtained
fory>
1,
for y < 1. The equation
(1%
of
STABILITY UNDER CENTRIFUGAL FORCE
111
Figure 7. Phase plane for a systemwith one mass,vibration in the plane of rotation, y = 1. the trajectory of equations (18) is given by
s ”
242= u; - 2
vdv
“o [u2/(tan a/a
- 1) - 0’1’
where a = a(v).
(19)
The results are shown in Figure 7. The motion in the first interval of stability is represented and the influence of the angular velocity is investigated. The distortion is of a considerable amount only in the neighborhood of the limit of stability. This effect of asymmetry to the u-axis is wholly due to the non-linear term, the Coriolis force. 5. CONCLUSIONS The problem of the stability of a system comprising a rod with some discrete masses is studied when it is rotating about an orthogonal axis. The stability in the plane of rotation and orthogonal to that plane is considered. It is shown that instability is first reached in the plane of rotation and only for very large values of R/l are the limits almost the same in both cases. It is verified also that only the motion in the orthogonal plane has a minimum value of R/I and then instability is no longer possible. The influence of the Coriolis force on the motion is remarkable only in the neighborhood of the stability limits. REFERENCES 1.
N. MOSTAGHELand I. TADJBAKHSH1973 International Journal of Mechanical Science 15,429-434. Buckling of rotating rods and plates. 2. F. G. RAMMERSTORFER 1974 International Journal of Mechanical Science 16,515-517. Comments on buckling of rotating rods and plates.