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Critical speeds for a flexible spinning disk
Int. J. Mech. Sci. Vol. 29, No. 8, pp. 525-531, 1987
0020-7403/87 $ 3 . 0 0 + . 0 0 © 1987 Pergamon Journals Ltd.
Printed in Great Britain.
CRITICAL SPEEDS FOR A FLEXIBLE SPINNING
DISK
GEORGE G. ADAMS Department of Mechanical Engineering, Northeastern University, Boston, MA 02115, U.S.A. (Received 26 September 1986; and in revised form 5 April 1987)
Abstraet--A flexible uniform elastic disk is clamped at its inner radius and rotates at constant speed. It is shown that certain critical speeds exist at which the spinning disk is unable to support arbitrary spatially fixed transverse loads. These critical speeds are in the range of rotational speeds relevant in certain floppy disk magnetic recording applications. It is then shown that the inclusion of an elastic foundation parameter into the governing equations can raise the critical speeds of rotation. In the case of the floppy disk this corresponds to the use of a baseplate located in close proximity to the spinning disk. The results presented are the effect of the foundation stiffness parameter and clamping radius ratio on the critical speeds. Also included, for the magnetic recording application, is the effect of the baseplate position on the critical speeds. NOTATION a apni
b Cvi
d D E h K P P
Q F
R S t w Z
13, 7 0 # v
P ~7r 170 , ~7~
O)
f~ V2
inner radius of disk coefficients of series solution outer radius of disk constants of integration disk-to-baseplate separation at inner radius flexural rigidity of disk elastic (Young's) modulus of disk thickness of disk foundation stiffness Fourier mode number pressure from air film net through-flow radial co-ordinate Fourier component of the transverse displacement root of the indicial equation time transverse displacement of the disk disk-to-baseplate spacing profile coefficients of ordinary differential equations disk parameter dimensionless radial co-ordinate angular co-ordinate (body-fixed) viscosity of air Poisson's ratio of the disk mass density of the disk normal stress in the radial direction due to disk rotation normal stress in the circumferential direction due to disk rotation angular co-ordinate (space-fixed) angular velocity of the disk dimensionless angular velocity of the disk Laplacian operator
INTRODUCTION
The study of the transverse deflection of spinning disks has drawn attention for many years. It
is both an important theoretical problem as well as one having engineering relevance due to such applications as the circular saw, turbine rotors and floppy disks. The first analysis of the spinning disk was by Lamb and SouthweU [1] who determined the natural frequencies and mode shapes for a solid circular disk. They included results for a very flexible disk (negligible bending stiffness), a slowly rotating disk (negligible membrane forces) and an approximate solution for the case when both bending stiffness and membrane forces MS
29/8
-
A
525
526
GEORGE G. ADAMS
are important. In [2] Southwell generalized the results of [1] by obtaining solutions for a disk which is clamped at its inner radius and free at its outer edge. A partial clamping condition is used in [2] in which the disk is constrained in the transverse, but not in the radial direction. Eversman and Dodson [3] then solved the problem of the spinning disk which is clamped at its inner edge by using a series solution. A different method is used by Barasch and Chen [4] in order to solve a similar problem. In [5], Iwan and Moeller use an approximate method in order to estimate the stability characteristics of a rotating solid disk subject to a concentrated transverse load system. In [6] Mote obtained an approximate solution for the case of a variable thickness disk subject to arbitrary in-plane loading. Other studies, such as Simmonds [7], Johnson [8] and Eversman [9] deal with the spinning disk using membrane theory. Benson and Bogy [10] studied the spinning disk problem with the objective of determining the transverse deflection due to spatially stationary loads. They showed that if the effect of bending stiffness was not included, then the eigenvalue problem for the determination of the natural frequencies of vibration was singular. Hence the disk is unable to support an arbitrary transverse pressure. They thus concluded that the effect of bending stiffness, no matter how small, still had to be included in the analysis. Analyses which treat the coupled system of a rotating flexible disk with a read/write head are applicable to magnetic recording and are given by Greenberg [l 1]~ Adams [12, 13] and White [14]. In the present analysis it is shown that even with the effect of bending stiffness included, there exist certain critical speeds at which the spinning disk is incapable of supporting arbitrary spatially fixed loads. This is shown to occur at speeds relevant in the operation of floppy disks. The analysis is generalized to include the effect of an elastic foundation which is shown to raise the critical speeds of rotation. In magnetic recording the placement of a baseplate can be treated as an elastic foundation [12] and raises the lowest critical speed above the usual operating speed. A N A L Y S I S OF T H E S P I N N I N G D I S K
The transverse deflection of the spinning disk (Fig. 1) is governed by the partial differential equation [1]
1 t~ I[dw]+tro(r~r 2~2w r fir L a'(r)r ~r ] 902
-
D V 4w h
K
- ~ w
= p
g2w , t~t2
(1)
where the in-plane membrane stresses of rotation corresponding to a spinning disk clamped in the radial direction at the inner radius and free at the outer radius are given by [l 5]
pro "3
trr(r)=~-
ao(r) =
(
+v)(b2-r 2)
l_ - V ?a2 /r2
1+3+
v
(1 + v) (a 2 + 7b 2) - (1 + 3v)r 2 - (1 - v)Ta2b2/r 2 ,
7 = [(3+v)b 2 - ( 1 +v)aa]/[(1 +v)b 2 + ( 1 - v ) a 2 ] ,
FIG. 1. Flexible disk of thickness h rotating with constant angular velocity to.
(21
Critical speeds for a flexible spinning disk
527
and w(r, O) is the transverse deflection in body fixed co-ordinates, p is the mass density, D is the plate flexural rigidity (Eha/12(1 -v2)), v is Poisson's ratio, K is the elastic foundation stiffness, co is the angular velocity and h is the thickness. The appropriate boundary conditions for the spinning disk are that it is clamped (zero deflection and slope) at the inside radius and free (zero bending moment and effective shear) at the outside radius. These conditions become [16]
Ow w(a, O) = O, ~r(a, O) = O, a2w ( l O w 1 O2w'~l ~-r2+Vkr 0r + ~- ~----0-T)J,=b = 0,
(3)
The partial differential equation of motion (1) and boundary conditions (3) are obtained under the assumption that the in-plane membrane forces of rotation are unaffected by the transverse motion of the disk [1]. Thus the membrane stresses (2) act as restoring forces which are treated using the classical theory of a membrane-plate. Through the use of the co-ordinate transformation 49 = 0 + o~t
(4)
equation (1) is rewritten as
lt3[-Ow] ;Srl ,(r)r
+a4,(r)
_~O2wD ~349z
K
t3Zw h V4W-~w=po)2ac~,
(5)
in which a steady solution [time invariant in space fixed co-ordinates (r, 49)] has been assumed. Now the angular co-ordinate is eliminated by expanding the unknown transverse deflection as an exponential Fourier series using
1 12" w(r, 49) = L R*(r) e~p*, R*(r) = ~n.]o w(r, 49)e-ipod49
(6)
p=0
resulting in an infinite set of ordinary differential equations which is then written in dimensionless form ~4 R . . . . . . .
n,,, ..p -r z~ 3 ~9 -+- (fl, + f12~2 +/33~4)~ 2 R~ + (f14 + f15 ~2 +/36~4)~R,p
+ (/37 +f18~ 2 "k-/39~4)Rp = 0,
p = 0, 1. . . .
oo,
(7)
where
= r/b,
R(() = R*(r)/b,
fll = - - ( l + 2 p 2)-f~2(1-v)ya2/b 2 = --f14, f12 = - ~ 2 [ 3 + v - ( l - v ) y a 2 / b f13 = ~'~2 (3 + v) = f 1 6 / 3 , f17 f18 = D 2 (1 + v)p 2 (y + a2/b2),
2] =fls, = p 2 ( p 2 _ 4 ) - ~,~2 (1 -
v)~p2a2/b2,
(8)
139 = - f~23(3 + v)p 2 + Kb4/D,
~2 = poj2b,,h/8D and prime (') denotes differentiation with respect to radial position ((). Each of the equations (7) can be solved using a series expansion given by Rpi(()= ~ Ap,i("+s,,
p=0,1 ....
o0,
i = 1,2,3,4.
(9)
n=O
The indicial equation, which determines the values of s and the recursion relations can be obtained by substituting (9) into (7) and equating like coefficients of r ~+". For n = 0, this results in the indicial equation s 4 -- 4s 3 + (5 + fll )S2 __ 2(1 + fl~ )s + f17 = O,
(10)
528
GEORGE G. ADAMS
whose four roots (s~, i = 1, 2, 3, 4) can be easily determined. The use of the n = 1 equation does not result in a solution which is independent of (10). The recursion relations for each root are Ap2, = -
(l&s~ + fls)Apo,/{s~[s,
+ 2)E(s, + 1)/ + fl, ] + f17 },
(11)
from n = 2, and [ (n + si) (n + si - 1 )2 (n + si - 2) + [31 (n + si) in + si - 2) + fl~ ] Ap,i
+ [ f l 2 ( n + s , - 2 ) 2+flS]Ap. 2 . + [ f l 3 i n + s , - 4 ) ( n + s , - 2 ) + f l g ] A p , - 4 , = O , n ~4,
p = O , l . . . . oo, i = 1,2,3,4.
Hence in a given Fourier mode (p) the deflection is described by a linear combination of the four functions defined by (9~(11)
Rp(~)=CplRpl(()+Cp2Rp2(()+Cp3Rp3(~)+Cp4Rp4(~),
p=0,1 ....
~.
(12)
The constants given in (12) can be determined by applying the boundary conditions R p i a / b ) = 0,
R'p(a/b) = O,
Rp(1) + vR'p(1) - v p 2 R p ( 1 ) = 0, Rp'(1)+R~(l)-[l+p2(2-v)]R'p(1)+(3-v)p2Rp(1)=0,
(13) p=O,
1. . . .
~,
which was obtained by applying (3) and (6). By using (12) and (13), a set of four linear homogenous algebraic equations results, for each value of p, which has non-trivial solutions for certain values of the dimensionless angular velocity (f~). Thus the vanishing of the determinant determines the critical speeds at which the spinning disk is unable to support arbitrary spatially stationary load distributions.
D E T E R M I N A T I O N OF T H E F O U N D A T I O N S T I F F N E S S
In some instances (e.g. floppy disks) the foundation stiffness ( K ) is not a constant but depends upon the speed of rotation. In certain applications the disk rotates in close proximity to a stationary baseplate [ 12]. Because of the viscosity of air, the spinning disk causes an air flow out of the small disk-to-baseplate air gap thus producing a negative pressure which causes the disk to deform downward toward the fixed baseplate (Fig. 2). As described in [12], this produces a reference configuration from which the total deflection can be measured. Also as described in [12], deflection away from the reference configuration cause pressure changes which can be estimated by P = -Kw, ,<
_
[ 'dx ~-
where/~ (Fig. 2). rotation of K, an
frdx fbdx ] / fbdx,
-- 3a X23 L XZ4
3aXZ43.XZ3J/J. XZ3
(14)
is the viscosity of air, Q is the net throughflow and z = z ( x ) is the spacing profile Both Q and z are calculated as described in [12] and depend upon the speed of (~o)of the disk as well as the separation distance (d). Thus by using the average value estimate of the effect of the baseplate location on the critical speed can be obtained.
z
L,a ,t.
r
b
f I
~7FIG. 2. Floppy disk rotating in close proximity to a stationary baseplate.
Critical speeds for a flexiblespinning disk
529
RESULTS AND DISCUSSION The eigenvalue problem described by (13) was solved for each Fourier mode (p). In each instance the lowest critical speed corresponded to zero nodal circles. It is noted that the Fourier mode number (p) corresponds to the number of spatial nodal diameters. In Figs 3-5 are shown the effect of the foundation stiffness parameter on the six lowest critical speeds for different values of the clamping radius ratio (a/b). The value of a/b in Fig. 4 corresponds to that used for floppy disks 1-14]. As expected an increase in the foundation stiffness always produces an increase in the critical speed, but the effect is much more pronounced for the lower Fourier mode numbers. This is because higher values of p correspond to shorter deflection wavelengths and hence proportionally higher strain energy in bending compared with the foundation energy. In Fig. 6 is shown the effect of clamping radius ratio (a/b) on the critical speed. As a/b increases the critical speed at first decreases before eventually increasing. This initial decrease may appear to be counter-intuitive because the bending stiffness of the disk increases monotonically with a/b. However the membrane stiffness decreases monotonically with a/b and it is this effect which causes the initial decline of the critical speed value. It is noted that there are no critical speeds for the p = 0 or p = 1 modes (Figs 3-6). The p = 0 mode would 4.0
....
3.5
n 3.0
2.5
2.0
1.5
0.0
200.0
400.0
600.0
Kb41D
FIG. 3. Dimensionlesscritical speed (f~)vs dimensionlessfoundation stiffness (Kb*/D)with a/b = O, for various modes p.
4.0
3,5
3.0 N 2.5
2.0
=
0.0
200.0
400.0
600.0
Kb41D
FIG. 4. Dimensionless critical speed (Q) vs dimensionless foundation stiffness (Kb4/D) with a/b = 0.1904, for various modes p.
530
GEORGEG. ADAMS
3.5 f13'0~ 2.5
~
r
2.0~
0.0
200.0
400.0
600.0
Kb4/D
FIG. 5. Dimensionless critical speed (f~) vs dimensionless foundation stiffness (Kb'~/D) with a/b = 0.4, for various modes p.
4.0~_~
3.o ~
7
~
I
J v~
-]
/
n2"5 ~
1.5<~ p 2 0.0
0.2 0.4 0.6 0.8 a/b
FIG. 6. Dimensionless critical speed vs clamping radius ratio (a/b) with Kb4/D = 0, for various modes p.
not be expected to exist because the corresponding deflection pattern would not alter the kinetic energy of the system but would increase the strain energy. The p = 1 mode does not exist because the effect of disk rotation is to stiffen the disk with increasing speed to such an extent that a critical speed does not exist for p = 1. If the stiffening effect was not included in the analysis then a critical speed would occur. This is consistent with the results of [5] in which an approximate solution for a complete disk (a/b = 0) is presented. In Fig. 7 is shown the effect of the baseplate position upon the critical speeds in an application relevant to magnetic recording. The dimensions used are those given in 1-14-1in which ~ = 62.8 rads -1. As the baseplate-to-disk distance (d) decreases the foundation stiffness coefficient increases, thus raising the critical speed. Note, however, that as d decreases the minimum critical speed occurs at progressively higher values ofp. This is once again due to higher values of p being less sensitive to increases in the foundation stiffness. In 1-5] an estimate of the critical speed is made for the special case when the foundation stiffness and the clamping radius are both equal to zero. This approximation was said to be a lower bound and, in fact, is about 5 % less than the exact solution obtained here.
Critical speeds for a flexible spinning disk
531
80.0
70,0 w 60.0
3
50,0
40.0
0.0
0.5
1.0
1.5
2,0
2.5
Fl~. 7. Critical speed oJ (rads -1) vs disk-to-baseplate separation distance (d, in mm) with
a = 12.7 ram, b = 66.7 ram, h = 0.076 ram, E = 3.45 GPa, p = 1400 k g m -3, v = 0.3.
CONCLUSIONS
The critical speeds for a spinning elastic disk, which includes the effect of an elastic foundation, have been determined. These critical speeds were found to be in the range of rotational speeds relevant in certain floppy disk magnetic recording applications. In all cases the lowest critical speed in a given circumferential mode corresponded to zero nodal cricles. Furthermore no critical speeds were found for less than two nodal diameters due to the stiffening effect of increasing speed. REFERENCES 1. H. LAMB and R. V. SOUTHWELL, The vibrations of a spinning disk. Proc. R. Soc. Lond. A99, 272-280 (1921). 2. R. V. SOUTHWELL,On the free transverse vibrations of a uniform circular disc clamped at its centre; and on the effects of rotation. Proc. R. Soc. Lond. AI01, 133-i53 (1922). 3. W. EVERSMANand R. O. DODSON, Free vibration of a centrally clamped spinning circular disk. AIAA J. 7, 2010--2012 (1969). 4. S. BARASCHand Y. CHEW, On the vibration of a rotating disk. ASME J. appl. Mech. 39, 1143-1144 (1972). 5. W. D. IWANand T. L. MOELLER, The stability of a spinning elastic disk with a transverse load system. ASME J. appl. Mech. 43, 485-490 (1976). 6. C. D. MOTE, Free vibration of initially stressed circular disks. ASME J. Engng Ind. 89, 258-265 (1965). 7. J. G. SIMMONDS, The transverse vibrations of a flat spinning membrane. J. aeronaut. Sci. 29, 16-18 (1962). 8. M.W. JOHNSON, On the dynamics of shallow elastic membranes. Proceedings of the Symposium on Thin Elastic Shells (Edited by W. KOOTER), pp. 281-300. North-Holland Publishing Co., Amsterdam (1960). 9. W. EVERSMAN, Transverse vibrations of a clamped spinning membrane. AIAA J. 6, 1395-1397 (1968). 10. R. C. BENSONand D. B. BOGY, Deflection of a very flexible spinning disk due to a stationary transverse load. ASME J. appl. Mech. 45, 636--642 (1978). 11. H. J. GREENBERG, Flexible disk--reed/write interface. IEEE Trans. Magnetics 14, 336-338 (1978). 12. G. G. ADAMS, Analysis of the flexible disk/heed interface. ASME J. lubric. Technol. 102, 86-90 (1980). 13. G. G. ADAMS, Procedures for the study of the flexible-disk to head interface. IBM J. Res. Dev. 24, 512-517 (1980). 14. J. W. WHITE, On the design of low flying heeds for floppy disk magnetic recording. Tribotogy and Mechanics of Magnetic Storage Systems, ASLE Special Publication SP-16, Vol. 1, pp. 126-131 (1984). 15. A. C. U GURALand S. K. F ENsTER• Ad•anced Stren•th and Applied Elasticity. E•sevier-N•rth H•••and• New Y•rk (1975). 16. J. W. S. RAYLEIGH, The Theory of Sound, Vol. 1. Dover Publications, New York (1945).
0020-7403/87 $ 3 . 0 0 + . 0 0 © 1987 Pergamon Journals Ltd.
Printed in Great Britain.
CRITICAL SPEEDS FOR A FLEXIBLE SPINNING
DISK
GEORGE G. ADAMS Department of Mechanical Engineering, Northeastern University, Boston, MA 02115, U.S.A. (Received 26 September 1986; and in revised form 5 April 1987)
Abstraet--A flexible uniform elastic disk is clamped at its inner radius and rotates at constant speed. It is shown that certain critical speeds exist at which the spinning disk is unable to support arbitrary spatially fixed transverse loads. These critical speeds are in the range of rotational speeds relevant in certain floppy disk magnetic recording applications. It is then shown that the inclusion of an elastic foundation parameter into the governing equations can raise the critical speeds of rotation. In the case of the floppy disk this corresponds to the use of a baseplate located in close proximity to the spinning disk. The results presented are the effect of the foundation stiffness parameter and clamping radius ratio on the critical speeds. Also included, for the magnetic recording application, is the effect of the baseplate position on the critical speeds. NOTATION a apni
b Cvi
d D E h K P P
Q F
R S t w Z
13, 7 0 # v
P ~7r 170 , ~7~
O)
f~ V2
inner radius of disk coefficients of series solution outer radius of disk constants of integration disk-to-baseplate separation at inner radius flexural rigidity of disk elastic (Young's) modulus of disk thickness of disk foundation stiffness Fourier mode number pressure from air film net through-flow radial co-ordinate Fourier component of the transverse displacement root of the indicial equation time transverse displacement of the disk disk-to-baseplate spacing profile coefficients of ordinary differential equations disk parameter dimensionless radial co-ordinate angular co-ordinate (body-fixed) viscosity of air Poisson's ratio of the disk mass density of the disk normal stress in the radial direction due to disk rotation normal stress in the circumferential direction due to disk rotation angular co-ordinate (space-fixed) angular velocity of the disk dimensionless angular velocity of the disk Laplacian operator
INTRODUCTION
The study of the transverse deflection of spinning disks has drawn attention for many years. It
is both an important theoretical problem as well as one having engineering relevance due to such applications as the circular saw, turbine rotors and floppy disks. The first analysis of the spinning disk was by Lamb and SouthweU [1] who determined the natural frequencies and mode shapes for a solid circular disk. They included results for a very flexible disk (negligible bending stiffness), a slowly rotating disk (negligible membrane forces) and an approximate solution for the case when both bending stiffness and membrane forces MS
29/8
-
A
525
526
GEORGE G. ADAMS
are important. In [2] Southwell generalized the results of [1] by obtaining solutions for a disk which is clamped at its inner radius and free at its outer edge. A partial clamping condition is used in [2] in which the disk is constrained in the transverse, but not in the radial direction. Eversman and Dodson [3] then solved the problem of the spinning disk which is clamped at its inner edge by using a series solution. A different method is used by Barasch and Chen [4] in order to solve a similar problem. In [5], Iwan and Moeller use an approximate method in order to estimate the stability characteristics of a rotating solid disk subject to a concentrated transverse load system. In [6] Mote obtained an approximate solution for the case of a variable thickness disk subject to arbitrary in-plane loading. Other studies, such as Simmonds [7], Johnson [8] and Eversman [9] deal with the spinning disk using membrane theory. Benson and Bogy [10] studied the spinning disk problem with the objective of determining the transverse deflection due to spatially stationary loads. They showed that if the effect of bending stiffness was not included, then the eigenvalue problem for the determination of the natural frequencies of vibration was singular. Hence the disk is unable to support an arbitrary transverse pressure. They thus concluded that the effect of bending stiffness, no matter how small, still had to be included in the analysis. Analyses which treat the coupled system of a rotating flexible disk with a read/write head are applicable to magnetic recording and are given by Greenberg [l 1]~ Adams [12, 13] and White [14]. In the present analysis it is shown that even with the effect of bending stiffness included, there exist certain critical speeds at which the spinning disk is incapable of supporting arbitrary spatially fixed loads. This is shown to occur at speeds relevant in the operation of floppy disks. The analysis is generalized to include the effect of an elastic foundation which is shown to raise the critical speeds of rotation. In magnetic recording the placement of a baseplate can be treated as an elastic foundation [12] and raises the lowest critical speed above the usual operating speed. A N A L Y S I S OF T H E S P I N N I N G D I S K
The transverse deflection of the spinning disk (Fig. 1) is governed by the partial differential equation [1]
1 t~ I[dw]+tro(r~r 2~2w r fir L a'(r)r ~r ] 902
-
D V 4w h
K
- ~ w
= p
g2w , t~t2
(1)
where the in-plane membrane stresses of rotation corresponding to a spinning disk clamped in the radial direction at the inner radius and free at the outer radius are given by [l 5]
pro "3
trr(r)=~-
ao(r) =
(
+v)(b2-r 2)
l_ - V ?a2 /r2
1+3+
v
(1 + v) (a 2 + 7b 2) - (1 + 3v)r 2 - (1 - v)Ta2b2/r 2 ,
7 = [(3+v)b 2 - ( 1 +v)aa]/[(1 +v)b 2 + ( 1 - v ) a 2 ] ,
FIG. 1. Flexible disk of thickness h rotating with constant angular velocity to.
(21
Critical speeds for a flexible spinning disk
527
and w(r, O) is the transverse deflection in body fixed co-ordinates, p is the mass density, D is the plate flexural rigidity (Eha/12(1 -v2)), v is Poisson's ratio, K is the elastic foundation stiffness, co is the angular velocity and h is the thickness. The appropriate boundary conditions for the spinning disk are that it is clamped (zero deflection and slope) at the inside radius and free (zero bending moment and effective shear) at the outside radius. These conditions become [16]
Ow w(a, O) = O, ~r(a, O) = O, a2w ( l O w 1 O2w'~l ~-r2+Vkr 0r + ~- ~----0-T)J,=b = 0,
(3)
The partial differential equation of motion (1) and boundary conditions (3) are obtained under the assumption that the in-plane membrane forces of rotation are unaffected by the transverse motion of the disk [1]. Thus the membrane stresses (2) act as restoring forces which are treated using the classical theory of a membrane-plate. Through the use of the co-ordinate transformation 49 = 0 + o~t
(4)
equation (1) is rewritten as
lt3[-Ow] ;Srl ,(r)r
+a4,(r)
_~O2wD ~349z
K
t3Zw h V4W-~w=po)2ac~,
(5)
in which a steady solution [time invariant in space fixed co-ordinates (r, 49)] has been assumed. Now the angular co-ordinate is eliminated by expanding the unknown transverse deflection as an exponential Fourier series using
1 12" w(r, 49) = L R*(r) e~p*, R*(r) = ~n.]o w(r, 49)e-ipod49
(6)
p=0
resulting in an infinite set of ordinary differential equations which is then written in dimensionless form ~4 R . . . . . . .
n,,, ..p -r z~ 3 ~9 -+- (fl, + f12~2 +/33~4)~ 2 R~ + (f14 + f15 ~2 +/36~4)~R,p
+ (/37 +f18~ 2 "k-/39~4)Rp = 0,
p = 0, 1. . . .
oo,
(7)
where
= r/b,
R(() = R*(r)/b,
fll = - - ( l + 2 p 2)-f~2(1-v)ya2/b 2 = --f14, f12 = - ~ 2 [ 3 + v - ( l - v ) y a 2 / b f13 = ~'~2 (3 + v) = f 1 6 / 3 , f17 f18 = D 2 (1 + v)p 2 (y + a2/b2),
2] =fls, = p 2 ( p 2 _ 4 ) - ~,~2 (1 -
v)~p2a2/b2,
(8)
139 = - f~23(3 + v)p 2 + Kb4/D,
~2 = poj2b,,h/8D and prime (') denotes differentiation with respect to radial position ((). Each of the equations (7) can be solved using a series expansion given by Rpi(()= ~ Ap,i("+s,,
p=0,1 ....
o0,
i = 1,2,3,4.
(9)
n=O
The indicial equation, which determines the values of s and the recursion relations can be obtained by substituting (9) into (7) and equating like coefficients of r ~+". For n = 0, this results in the indicial equation s 4 -- 4s 3 + (5 + fll )S2 __ 2(1 + fl~ )s + f17 = O,
(10)
528
GEORGE G. ADAMS
whose four roots (s~, i = 1, 2, 3, 4) can be easily determined. The use of the n = 1 equation does not result in a solution which is independent of (10). The recursion relations for each root are Ap2, = -
(l&s~ + fls)Apo,/{s~[s,
+ 2)E(s, + 1)/ + fl, ] + f17 },
(11)
from n = 2, and [ (n + si) (n + si - 1 )2 (n + si - 2) + [31 (n + si) in + si - 2) + fl~ ] Ap,i
+ [ f l 2 ( n + s , - 2 ) 2+flS]Ap. 2 . + [ f l 3 i n + s , - 4 ) ( n + s , - 2 ) + f l g ] A p , - 4 , = O , n ~4,
p = O , l . . . . oo, i = 1,2,3,4.
Hence in a given Fourier mode (p) the deflection is described by a linear combination of the four functions defined by (9~(11)
Rp(~)=CplRpl(()+Cp2Rp2(()+Cp3Rp3(~)+Cp4Rp4(~),
p=0,1 ....
~.
(12)
The constants given in (12) can be determined by applying the boundary conditions R p i a / b ) = 0,
R'p(a/b) = O,
Rp(1) + vR'p(1) - v p 2 R p ( 1 ) = 0, Rp'(1)+R~(l)-[l+p2(2-v)]R'p(1)+(3-v)p2Rp(1)=0,
(13) p=O,
1. . . .
~,
which was obtained by applying (3) and (6). By using (12) and (13), a set of four linear homogenous algebraic equations results, for each value of p, which has non-trivial solutions for certain values of the dimensionless angular velocity (f~). Thus the vanishing of the determinant determines the critical speeds at which the spinning disk is unable to support arbitrary spatially stationary load distributions.
D E T E R M I N A T I O N OF T H E F O U N D A T I O N S T I F F N E S S
In some instances (e.g. floppy disks) the foundation stiffness ( K ) is not a constant but depends upon the speed of rotation. In certain applications the disk rotates in close proximity to a stationary baseplate [ 12]. Because of the viscosity of air, the spinning disk causes an air flow out of the small disk-to-baseplate air gap thus producing a negative pressure which causes the disk to deform downward toward the fixed baseplate (Fig. 2). As described in [12], this produces a reference configuration from which the total deflection can be measured. Also as described in [12], deflection away from the reference configuration cause pressure changes which can be estimated by P = -Kw, ,<
_
[ 'dx ~-
where/~ (Fig. 2). rotation of K, an
frdx fbdx ] / fbdx,
-- 3a X23 L XZ4
3aXZ43.XZ3J/J. XZ3
(14)
is the viscosity of air, Q is the net throughflow and z = z ( x ) is the spacing profile Both Q and z are calculated as described in [12] and depend upon the speed of (~o)of the disk as well as the separation distance (d). Thus by using the average value estimate of the effect of the baseplate location on the critical speed can be obtained.
z
L,a ,t.
r
b
f I
~7FIG. 2. Floppy disk rotating in close proximity to a stationary baseplate.
Critical speeds for a flexiblespinning disk
529
RESULTS AND DISCUSSION The eigenvalue problem described by (13) was solved for each Fourier mode (p). In each instance the lowest critical speed corresponded to zero nodal circles. It is noted that the Fourier mode number (p) corresponds to the number of spatial nodal diameters. In Figs 3-5 are shown the effect of the foundation stiffness parameter on the six lowest critical speeds for different values of the clamping radius ratio (a/b). The value of a/b in Fig. 4 corresponds to that used for floppy disks 1-14]. As expected an increase in the foundation stiffness always produces an increase in the critical speed, but the effect is much more pronounced for the lower Fourier mode numbers. This is because higher values of p correspond to shorter deflection wavelengths and hence proportionally higher strain energy in bending compared with the foundation energy. In Fig. 6 is shown the effect of clamping radius ratio (a/b) on the critical speed. As a/b increases the critical speed at first decreases before eventually increasing. This initial decrease may appear to be counter-intuitive because the bending stiffness of the disk increases monotonically with a/b. However the membrane stiffness decreases monotonically with a/b and it is this effect which causes the initial decline of the critical speed value. It is noted that there are no critical speeds for the p = 0 or p = 1 modes (Figs 3-6). The p = 0 mode would 4.0
....
3.5
n 3.0
2.5
2.0
1.5
0.0
200.0
400.0
600.0
Kb41D
FIG. 3. Dimensionlesscritical speed (f~)vs dimensionlessfoundation stiffness (Kb*/D)with a/b = O, for various modes p.
4.0
3,5
3.0 N 2.5
2.0
=
0.0
200.0
400.0
600.0
Kb41D
FIG. 4. Dimensionless critical speed (Q) vs dimensionless foundation stiffness (Kb4/D) with a/b = 0.1904, for various modes p.
530
GEORGEG. ADAMS
3.5 f13'0~ 2.5
~
r
2.0~
0.0
200.0
400.0
600.0
Kb4/D
FIG. 5. Dimensionless critical speed (f~) vs dimensionless foundation stiffness (Kb'~/D) with a/b = 0.4, for various modes p.
4.0~_~
3.o ~
7
~
I
J v~
-]
/
n2"5 ~
1.5<~ p 2 0.0
0.2 0.4 0.6 0.8 a/b
FIG. 6. Dimensionless critical speed vs clamping radius ratio (a/b) with Kb4/D = 0, for various modes p.
not be expected to exist because the corresponding deflection pattern would not alter the kinetic energy of the system but would increase the strain energy. The p = 1 mode does not exist because the effect of disk rotation is to stiffen the disk with increasing speed to such an extent that a critical speed does not exist for p = 1. If the stiffening effect was not included in the analysis then a critical speed would occur. This is consistent with the results of [5] in which an approximate solution for a complete disk (a/b = 0) is presented. In Fig. 7 is shown the effect of the baseplate position upon the critical speeds in an application relevant to magnetic recording. The dimensions used are those given in 1-14-1in which ~ = 62.8 rads -1. As the baseplate-to-disk distance (d) decreases the foundation stiffness coefficient increases, thus raising the critical speed. Note, however, that as d decreases the minimum critical speed occurs at progressively higher values ofp. This is once again due to higher values of p being less sensitive to increases in the foundation stiffness. In 1-5] an estimate of the critical speed is made for the special case when the foundation stiffness and the clamping radius are both equal to zero. This approximation was said to be a lower bound and, in fact, is about 5 % less than the exact solution obtained here.
Critical speeds for a flexible spinning disk
531
80.0
70,0 w 60.0
3
50,0
40.0
0.0
0.5
1.0
1.5
2,0
2.5
Fl~. 7. Critical speed oJ (rads -1) vs disk-to-baseplate separation distance (d, in mm) with
a = 12.7 ram, b = 66.7 ram, h = 0.076 ram, E = 3.45 GPa, p = 1400 k g m -3, v = 0.3.
CONCLUSIONS
The critical speeds for a spinning elastic disk, which includes the effect of an elastic foundation, have been determined. These critical speeds were found to be in the range of rotational speeds relevant in certain floppy disk magnetic recording applications. In all cases the lowest critical speed in a given circumferential mode corresponded to zero nodal cricles. Furthermore no critical speeds were found for less than two nodal diameters due to the stiffening effect of increasing speed. REFERENCES 1. H. LAMB and R. V. SOUTHWELL, The vibrations of a spinning disk. Proc. R. Soc. Lond. A99, 272-280 (1921). 2. R. V. SOUTHWELL,On the free transverse vibrations of a uniform circular disc clamped at its centre; and on the effects of rotation. Proc. R. Soc. Lond. AI01, 133-i53 (1922). 3. W. EVERSMANand R. O. DODSON, Free vibration of a centrally clamped spinning circular disk. AIAA J. 7, 2010--2012 (1969). 4. S. BARASCHand Y. CHEW, On the vibration of a rotating disk. ASME J. appl. Mech. 39, 1143-1144 (1972). 5. W. D. IWANand T. L. MOELLER, The stability of a spinning elastic disk with a transverse load system. ASME J. appl. Mech. 43, 485-490 (1976). 6. C. D. MOTE, Free vibration of initially stressed circular disks. ASME J. Engng Ind. 89, 258-265 (1965). 7. J. G. SIMMONDS, The transverse vibrations of a flat spinning membrane. J. aeronaut. Sci. 29, 16-18 (1962). 8. M.W. JOHNSON, On the dynamics of shallow elastic membranes. Proceedings of the Symposium on Thin Elastic Shells (Edited by W. KOOTER), pp. 281-300. North-Holland Publishing Co., Amsterdam (1960). 9. W. EVERSMAN, Transverse vibrations of a clamped spinning membrane. AIAA J. 6, 1395-1397 (1968). 10. R. C. BENSONand D. B. BOGY, Deflection of a very flexible spinning disk due to a stationary transverse load. ASME J. appl. Mech. 45, 636--642 (1978). 11. H. J. GREENBERG, Flexible disk--reed/write interface. IEEE Trans. Magnetics 14, 336-338 (1978). 12. G. G. ADAMS, Analysis of the flexible disk/heed interface. ASME J. lubric. Technol. 102, 86-90 (1980). 13. G. G. ADAMS, Procedures for the study of the flexible-disk to head interface. IBM J. Res. Dev. 24, 512-517 (1980). 14. J. W. WHITE, On the design of low flying heeds for floppy disk magnetic recording. Tribotogy and Mechanics of Magnetic Storage Systems, ASLE Special Publication SP-16, Vol. 1, pp. 126-131 (1984). 15. A. C. U GURALand S. K. F ENsTER• Ad•anced Stren•th and Applied Elasticity. E•sevier-N•rth H•••and• New Y•rk (1975). 16. J. W. S. RAYLEIGH, The Theory of Sound, Vol. 1. Dover Publications, New York (1945).