CA2246152A1 - Method and apparatus for improving spinning disk behavior using speed dependent clamping - Google Patents

Abstract

A stress induction method can be used to increase the operational speed range of rotating disks such as industrial circular saws and computer disk drives. In this method, stresses are generated in the disk by producing displacements or tractions at the inner radius of the disk. The magnitude of the displacements or tractions depends on the rotation speed. For example, by allowing concentrated masses to rest along the inner radius of the disk in-plane stresses proportional to the square of the rotation speed are generated. This technique can double the maximum speed for hydrodynamically uncoupled disks and can increase the maximum speed of disks with substantial hydrodynamic coupling by an order of magnitude.

Description

CA 022461~2 1998-08-27 3 ~ A31016 - 50/36246 PATENT

-BRUMBAUGH, GRAVES, DONOHUE & RAYMOND
30 RO(~ ;LLER PLAZA
NEW YORK, NEW YORK 101 12 TO ALL WHOM IT MAY CONCERN:

Be it known that I, ANTHONY A. RENSHAW, a citizen of the United States, residing in the City of New York, County of New York, State of New York,whose post office address is 315 West 23rd Street, Apt. #9F, New York, New York 10011, have invented an improvement in METHOD AND APPARATUS FOR
IMPROVING SPINNING DISK BEHAVIOR
USING SPEED DEPENDENT CLAMPING

of which the following is a SPECIFICATION

BACKGROUND OF THE INVENTION

This invention relates to a method and apparatus for increasing the maximum operable rotation speed of a thin rotating disk. This invention also relates to a method and apparatus for decreasing vibration of a rotating disk at a fixed rotation speed. This invention further relates to a method and apparatus which enables the 10 thickness of a spinning disk to be reduced without increasing the vibration of the disk.
Thin high speed rotating disks are the principal mechanical components of industrial circular saws and computer disk drives. In each of these technologies, thinner ~ CA 022461~2 1998-08-27 PATENT

disks and faster rotation are desirable to either increase production or reduce data acquisition times. However, the useful operational speed range and disk thinness in these devices are usually limited or substantially influenced by a critical speed phenomenon in which the propagation of a circumferentially traveling wave is equal and 5 opposite the rotation of the disk, H. Lamb and R.V. Southwell, 1921, Proceedings of the Royal Society of London, Vol. A99, No. 699, pp. 272-280, "The Vibrations of a Spinning Disk"; R.V. Southwell, 1922, Proceedings of the Royal Society of London, Vol. 101, pp. 133-153, "On the Free Transverse Vibrations of a Uniform Circular Disc Clamped at Its Centre; and on the Effects of Rotation"; S.A. Tobias and R.N. AInold, 1957, Proceedings of the Institute of Mechanical Engineers, Vol. 171, pp. 669-690, "The ~fluence of Dynamic Imperfections on the Vibration of Rotating Disks"; C.D.
Mote, Jr.,1965, Journal of Engineering for Industry, Vol. 87, pp. 285-264, "Free Vibration of Initially Stressed Circular Disks"; W. D. Iwan and T. L. Moeller, 1976, Journal of Applied Mechanics, Vol. 43, pp. 485-490, "The Stability of a Spinning 15 Elastic Disk with a Transverse Load System"; A. A. Renshaw, 1996, Proceedings of the Seventh Annual Inforrnation Storage and Processing Systems Symposium, ASME, 1996, pp. 175- 184, "The Stability of Flexible Spinning Disks Supported by Incompressible Hydrodynamic Lubrication". Neither industrial saws nor computer disk drives can tolerate the large transverse deflections that occur near critical speed, and, consequently, 20 these devices generally operate at a fraction of the lowest critical speed as described in . ~ CA 022461~2 1998-08-27 PATENT

C. D' Angelo m and C. D. Mote, Jr., 1993, Jr., Journal of Sound and Yibration, Vol.
168, pp. 15-30, "Aerodynarnically Excited Vibration and Flutter of a Thin Disk Rotating at Supercritical Speed."
Conventionally, in-plane residual stresses have been used in saw blades to S counteract the therrnal stresses that arise at the periphery of the saw and to increase the saw's operational speed range. E. Lindholm, 1953, Ark~vfor Fysik, Vol. 6, pp. 223-242, "The Vibrations and Bending of Pre-Stressed Circular Plates"; C. D. Mote, Jr. and R.
Syzmani, 1978, The Shock and Vibration Digest, Vol. 10, pp. 15-30, "Circular Saw Vibration Research." These residual stresses are normally produced using a technique 10 referred to as roll-tensioning in which a thin, circurnferential ring of the disk is plastically deformed by repeatedly rolling it between two loaded wheels as described in C. D. Mote, Jr., 1965, Journal of EngineeringforIndustry, Vol. 87, pp. 285-26, '~ree Vibration of Initially Stressed Circular Disks;" D. S. Dugdale, 1963, International Journal of Engineering Sciences, Vol. 1, pp. 89-100, "Effect of Intemal Stress on the 15 Flexural Stiffness of Discs;" D. S. Dugdale 1966, International Journal of Production Research, Vol. 4, pp. 237-248, "Theory of Circular Saw Tensioning;" C. D. Mote, Jr.
and L. T. Nieh, 1973, Wood Fiber, Vol. 5, pp. 160-169, "On the Foundations of Circular-Saw Stability Theory;" J. F. Carlin, F. C. Appl, H. C. Bridwell, and ~. P.
Dubois, 1975, Journal of Engineering for Industry, Vol. 97, pp. 37-48, "Effects of 20 Tensioning on Buckling and Vibration of Circular Saw Blades;" G. S. Schajer and C. D.

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PATENT

Mote, Jr., 1983, Wood Science and Technology, Vol. 17, pp. 287-302, "Analysis of Roll Tensioning and Its Influence on Circular Saw Stability."
For example, roll tensioning is recommended in U.S. Patent No. 4,979,417. Other stress induction methods have also been proposed. C. D. Mote, S Jr., 1967, Journal of EngineeringforIndus~ry, Vol. 89, pp. 611-618, "Natural Frequencies in Annuli with Induced Thermal Membrane Stresses;" C. D. Mote, Jr., and A. Rahimi, 1984, Journal of Dynamic Systems, Measurements and Control, Vol. 106,pp. l 23-128, "Real Time Vibration Control of Rotating Circular Plates by Temperature Control and System Identification;" R. G. Parker and C.D. Mote, Jr., 1991, Journal of Sound and Yibration, Vol. 145, pp. 95- 110, "Tuning of the Natural Frequency Spectrum of a Circular Plate By In-Plane Stress."
Unfortunately, the previously described methods for inducing in-plane stress in a rotating disk suffer from two major disadvantages.
First, because most methods induce a fixed magnitude of in-plane stress in the disk, the benefit derived from ple~LIes~ g is limited. This is because the magnitude of stress is limited by buckling or instability of the disk when the stress is induced in the stationary disk. In conventional roll tensioning, the critical speed of the rotating disk can be increased 30-40% at most before the resultant residual stress causes instability G. S. Schajer and C. D. Mote, Jr., 1983, Wood Science and Technology, Vol. 17, pp. 287-302, "Analysis of Roll Tensioning and Its Influence on Circular Saw CA 022461~2 1998-08-27 ¦1 r~ A31016 - 50/36246 PATENT

Stability." Actual increases in critical speed can be substantially less than 30~/O-in practice.
Second, all prior methods either apply loads, pressures, or tractions or deform the disk plastically or thermally in the central region of the disk. The central 5 region of the disk is the region between the inner radius of the disk where it is clamped and the outer periphery of the disk. ~l~int~ining the integrity of the central region of the disk is crucial in applications such as computer disk drives which cannot tolerate loads or deformations. Consequently, the prior art methods cannot be used in applications such as computer disk drives where m~int~ining disk integrity is essential.

SUMMARY OF TH~ INVENTTON
Accordingly, it is an object of the invention to provide a method of inducing stresses in a rotating disk which does not affect the integrity of the central region of the disk and which can be applied to rotating disk systems including saws and 15 computer disk drives.
Another object of the invention is to provide a method of increasing the operational speed range of a rotating disk.
A further object of the invention is to decrease the vibration of a rotating disk at a fixed speed.

~ CA 022461~2 1998-08-27 PATENl Another object of the invention is to reduce the thickness of a rotating disk without increasing disk vibration under normal operating conditions.
These and other objects of the invention are achieved by applying a nonfrictional and non-thermal substantially radial force at an inner radius of a rotating 5 disk whose magnitude depends on the rotation speed such that stress is induced in the rotating disk without affecting disk integrity. The term nonfrictional substantially radial force means and includes forces which produce radial outward displacements in the disk in excess of those that are produced by superposition of any displacements caused by preloading the disk prior to rotation, traction-free centripetal expansion of the disk and 10 any frictional forces produced by rotating the disk at or below a fixed speed, i.e., frictional forces produced by rotating the disk above a fixed speed and then decelerating the disk.
The novel stress induction method can be used in both circular saws and computer disk drives. In this method, stresses are generated in the disk by a central 15 clarnp which is designed to produce radial tractions or in-plane radial displacement at the inner radius that are proportional to the square of the rotation speed. One way of generating such stresses, for example, is by allowing freely sliding, centripetally accelerating, concentrated masses to rest along the inner radius of the disk.
In this regard, regulating the induced stress with speed-dependent 20 boundary tractions or displacements has two advantages. First, this method can be CA 022461~2 1998-08-27 9 PATENT

applied without affecting the integrity of the disk, which would perrnit its use in computer disk drives. Second, significantly higher rotation speeds are achievable than heretofore produced by other techniques. This is because the induced stresses are generated while the disk is rotating, and the destabilizing effects of the induced stress 5 can be counteracted by tensile centripetal stress.
In conventional roll tensioning, the critical speed of rotation can be increased 30-40% at best before the resultant residual stresses cause instability. G.S.
Schajer and C.D. Mote, Jr., 1983, Wood Science and Technology, Vol. 17, pp. 287-302, "Analysis of Roll Tensioning and Its Influence on Circular Saw Stability." In contrast 10 the invention described herein can double critical speed for rigid rotating disk such as circular saws and hard disk drives and can increase the critical speed in flexible, hydrodynamically coupled, rotating disks such as floppy disk drives by an order of magnitude.

BRIEF DESCR~'TION OF TH~ DRAWINGS
Further objects and advantages of the invention will be more fully appreciated from a reading of the detailed description when considered with the accompanying drawings wherein:

CA 022461~2 1998-08-27 PATENT

Fig. 1 is a graph illustrating disk critical speed of a rotating hydrodynamically uncoupled disk as a function of the clamping ratio for three different magnitudes of speed dependent stress defined by m = 0, m = 1 and m = 2;
Fig. 2(a) and (b) are graphs illustrating optimal mass values and the 5 critical disk rotation speed, respectively, as a function of the clamping ratio for a rotating hydrodynamically uncoupled disk;
Fig. 3 is a graph illustrating the ratio of total optimum clamping mass to total disk mass as a function of the clamping ratio for a rotating hydrodynamically uncoupled disk;
Figs. 4(a) and (b) are graphs illustrating disk critical speed and the number of nodal diameters, respectively as a function of clamping ratio for d = 0,0.005,0.01, 0.015 and d = 0.02 for a rotating hydrodynamically coupled disk;
Fig. 5 is an illustration of a central disk clamp for positioning a rotating disk on a rotating drive and inducing variable magnitude radial force in the disk during 15 rotation in accordance with the invention;
Fig. 6 is a perspective view of a speed dependent rotating disk clamp including a plurality of movable and fixed portions in accordance with the invention;
Figs. 7(a) and (b) are a side view and a top view, respectively, of a rotating disk with rods attached to its inner radius;

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Figs. 8(a) and (b) are a side view and a top view, respectively, of a retaining collar for the disk and rods shown in Figs. 7(a) and (b);
Figs. 9(a) and (b) are a side perspective view of a sectioned drive shaft and the top of a floppy disk, respectively;
Fig. 10 is a sectional side view of a disk and a speed dependent clamp assembly in accordance with the invention;
Figs. 1 l(a) and 1 l(b) are a top and side view, respectively, of the mounting portion of the assembly shown in Fig. 10;
Figs. 12(a) and 12(b) are top and side views, respectively, of a mounting cap in accordance with the invention; and Figs. 13(a) and 13(b) are top and side views, respectively, of sectional wedges for use in an assembly in accordance with the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
This invention described herein can be applied to any substantially axisymmetric rotating disk. The thickness of the disk may be nonuniform. The disk may be of any size, for example, the invention described herein may be applied to two-inch diameter floppy disks made from a flexible, plastic polymer as well as to three feet diameter industrial circular saws made from high strength steel.

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- In some cases, such as industrial circular saws and hard computer memory drives, the hydrodynamic forces on the rotating disk caused by the air, gas, or fluid surrounding the disk are sufficiently small that they can be neglected in analyzing the dynamic behavior of the disk. Such hydrodynamically uncoupled disks are herein 5 referred to as a hard disk system and results pertaining to these systems are indicated herein below with subscript H. In other applications, such as computer floppy disks, the hydrodynamic forces on the disk are large and cannot be neglected in analyzing the dynamic behavior of the disk. Such hydrodynamically coupled disks are herein referred to as flexible disk systems, and results pertaining to these systems are indicated herein 10 below with subscript F.
Both the hard and flexible disk systems may be mathematically described by considering the dynamic behavior of a rotating disk with in-plane stress using standard, well known and accepted models for the spinning disk, the in-plane stress, and the interaction of the disk with the surrounding medium whether it be air, gas or fluid.
A thin, axisymmetric, circular disk clamped at inner radius R, and free at outer radius Ro spins about its axis of symmetry at a constant angular speed Q-. The disk may be modeled as having a uniform thickness h, density p, i.e., mass per unit volume, Young's modulus E, and Poisson's ratio v. Polar coordinates (R,~) are fixed in the stationary frarne of reference with the center of the disk at the origin. The in-plane, 20 radial displacement of the disk is Ur, the transverse displacement of the disk is W, and n CA 022461S2 1998-08-27 PATENT

the in-plane, axisymmetric stresses a~, and a'~ are the sum of the centrifugal and boundary-induced stresses. Dimensionless variables describing the disk are defined by r = R I R w = W I h Q = Q ,~phR4 I D

u =UR /h2 a =a hR21D a~=a~hR21D

D = E h3 / [12(1 - v2)]. The clamping ratio is lC = R~
The transverse vibration and stability of a hard disk such as an industrial 5 saw and hard computer memory drives can be modeled using classical ~Circhhoffplate theory with in-plane stresses. C. D. Mote, Jr., 1965, ~ournal of Engineeringfor Industry, Vol. 87, pp. 285-264, '~ree Vibration of Initially Stresses Circular Disks"; W.
D. Iwan and T.L. Moeller, 1976, Journal of Applied Mechanics, Vol. 43, pp. 495-490, "The Stability of a Spinning Elastic Disk with a Transverse Load System"; J. L.
Nowinski, 1964, Journal of Applied Mechanics, Vol. 31, pp. 72-78, "Nonlinear Transverse Vibrations of a Spinning Disk."
For hard disk systems such as industrial saws and hard disk drives, the maximum stable rotation speed QR~ is given by the maximum value of Q for which the functional J" is positive definite A. A. Renshaw, 1996, Proceedings of the Seventh Annual ~nforrnation Stor~ge and Processing Systems Symposium, ASME, 1996, pp. 175-~ CA 022461S2 1998-08-27 ~

PATENI

184, "The Stability of Flexible Spinning Disks Supported by Incompressible Hydrodynamic Lubrication."
JH[W] = U[W]_--JQ W ~d4 (2) U is the potential energy of the disk U[w] = --I(V2w)2 -2(1 -v)[w, (w, Ir +w,~lr2) -[(w,~lr),rl2]

+ a w~2 + a~w~lr2d4 V2 is the Laplacian operator. A comma indicates partial differentiation, and dA is the 5 differential planar area of the disk.
For flexible spinning disks such as computer floppy disks, the disk is strongly coupled to the motion of the surrounding medium. For a suitable model in which a disk is enclosed in a housing similar to that of a floppy disk which is sealed to prevent radial flow at the outer edge of the disk and whose clearance is sufficiently small 10 to justify modeling the air flow using hydrodynamic lubrication theory, the maximum stable rotation speed QF m",~ is the maximum value of Q for which the functional JF is positive definite, A. A. Renshaw, 1996, Proceedings of Seventh Annual Information Storage and Processing Systems Symposium, ASME, 1996, pp. 175-184, "The Stability of Flexible Spinning Disks Supported by Incolnpres~ible Hydrodynamic Lubrication".

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PATENT

~F[W] =U[W]-- ¦~Q w,~/4}~4 (4) Admissible functions for JH, JF and U must satisfy the clarnped-free boundary conditions w = O andw" = 0 atr = lC
w", + v(w,k + w,~ 2) = O at r = 1 (5) (V2w)" + (1-v)(w,~ - w,~/r3) = O at r = 1 In a preferred embodiment, the clamp of the disk is designed to produce the speed-dependent traction at the inner boundary ar= -mQ2 at r = lC (6) The outer edge of the disk is free.
o,= 0 at r= 1 (7) m is a constant that represents the total dimensionless added mass assumed to be the concentrated in a line around the inner radius r = lC, although the traction could be produced in some other manner such as electromagnetically or hydraulically. The specific functional dependence, namely, a traction proportional to the square of the rotation is used herein merely for illustrative purposes. Any function which substantially increases with increasing rotation speed may be used.
The situation in which m = O represents an annular disk with no traction at r = lC which is often used to model rotating disk systems C. D' Angelo m and C. D.

PATENT

Mote, 1993, Journal of Sound and Vibr~rtion, Vol. 168, pp. 1-14, '~atural Frequencies of a Thin Disk, Clamped By Thick Collars with Friction at the Contacting Surfaces, Spinning at High Rotation Speed." The axisyrnmetric solutions of the generalized plane stress equations of linear elasticity with a centripetal body force and single-valued 5 displacements that satisfy (6) and (7) are ~ =Q2[c /r2 +cz +c3r2] C~ =Q2[-C /r2 +C2 +c4r2] (8) where c~ = -(3 + V)1C2 / 8 - mK2 / (1 lc2) c2= (1 + lc2)(3 + v) / 8 + mlc2 / (1-lc2) (9) c3=-(3 +v)/8 c4=-(1 +3v)/8 A mathematically equivalent description of the stress field given by (6) -(9) which uses a speed dependent displacement instead of a speed dependent traction is the following in which (6) is replaced with u,=dQ2 atr=K (10) 15 The stress field is still given by (8) and the radial displacement is u = 2 [-cl(l +v)/r+c2(1 -v)r-(l -v2)r3/8] (I l) o CA 02246152 1998-08-27 r~

PATENT

The constants c, and c2 are given by lc2(1 -v)[3 +v-(l +V)K2] 12dlc(1 -v ) c I = 8 [ I +v +( I -v) ~C2] [ 1 +v +( I -v)~C ]

(12) (1 +v)[3 +v+(l +V)lC4] 12dlc(1 -v2) 8 [ 1 +v +( 1 v)*] [ 1 +v +( 1 -V)lC ]

C3 and C4 are the same as in (9).

The traction and displacement boundary conditions (6) and (10) are S formally equivalent with (1 -*)[48d(1 -v )-(3 +3V+V2)K+(l+v)K] (~3 4~[l+V+(I-V)K ]

Hence the results in Figs. 1-4 can be directly translated into values of m instead of d and vice versa.
Since JH and JF are separable in ~, QH m"~ and QF m ", can be determined numerically by solving the symmetric eigenvalue problem for Q defining the extrema of JH and JF for the number of nodal diameter n = 0,1,2,... using the substitution w=u(r)cos(n~).

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Fig. 1 shows a plot of QH-max as a function of lC for m = 0,1 and 2. These values were deterrnined using the Galerkin method with 6 orthonormal, Chebyshev polynomials defining u(r). As few as three polynomials were sufficient to give convergence within 1% of the values plotted in Fig. 1, A. A. Renshaw, 1996, 5 Proceedings of the Seventh ~nnual Storage and Processing Systems Symposium, ASME, pp. 175-184, "'The Stability of Flexible Spinning Disks Supported by Incompressible Hydrodynamic Lubrication". The search was done from n = O to n = 10. The results show that QH~ can be approximately doubled from its value when m = O by an applol,liate choice of m. For lC = 0.35, increasing m from 0 to 1 increases QH m" ~ from 7.21 to 14.5, an increase of 101%. For lC = 0.24, increasing m from 0 to 2 increases QH-max from 5.91 to 10.6, an increase of 79%. The cusp-like behavior of the curves in Fig. 1 results from the fact that each side of the peak is determined by a different eigensolution.
Figs. 2(a) and ~b) show plots of, the value of mOp" which maximizes 15 QRma~ and the corresponding value of QH-max as a function of K. The dashed lines in Fig. 2(a) show the allowable variation in mOp, for which QH m"" remains within 90% of it maximurn value. The results verify that QH m "~ can be approximately doubled for all values of lC with an applopl;ate choice of m. Furthermore, 10% variation in m from its optimal value produces a 10% decrease in Q~ max ~ CA 022461~2 1998-08-27 n ! A31016 - 50/36246 PATEN~

- - The mass required to produce mOp, is about three times the mass of the rotating disk for most current rigid disk designs. For example, for the industrial circular saw studied by C. D' Angelo III and C.D. Mote, Jr., Journal of Sound and Vibration, Vol. 168, pp. 1-14, "Natural Frequencies of a Thin Disk, Clamped By Thick Collars 5 with Friction at the Contacting Surfaces, Spinning at High Rotation Speed," Ro = 0.178 m, R, = 0.0534 m, h = 0.775 mm, and p = 7700 kgtm3. The total mass of the disk is 0.54 kg. For K = 0.3, mOp, =1.35 giving QH-II",~ = 12.6, an increase of 90% over m = 0. The total dimensionless mass required at the inner edge of the disk, 2 J~m phRO2, is 1.6 kg, or three times the mass of the disk.
Similarly, for a 5 inch disk in a hard disk drive, Ro = 65 mm, Ri = 19.5 mm, h= 1.3 mm, and p = 2800 kgtrn3, A. A. Renshaw, 1996, Proceedings of Sevent~
Annual Information Storage and Processing Systems Symposium, ASME, 1996, pp. 175-184, "The Stability of Flexible Spirming Disks Supported by Incompressible Hydrodynamic Lubrication". In this case, the total mass of the disk is 44 g while the 15 total mass required at the inner edge is 130 g.
Fig. 3 shows the mass ratio, 2mOp, t (1-K2)~ the ratio of the total clarnping mass required to optimally increase QH~ to the total mass of the disk, as a function of K. As K increases, this ratio decreases monotonically to values that are more easily achieved in practice. For example, if 1C increased to 0.5, then mOp, = 0.42. The mass of CA 022461~2 1998-08-27 PATENT

the equivalent circular saw then becomes 0.45 kg while the clamping mass is only 0.50 kg.
Fig. 4(a) shows plots of QF m",~ as functions of K for d = 0, 0.005 0.01, and 0.015. Fig. 4(b) shows a plot of QF-max as a function of K for d = 0.02 plotted on a substantially different scale. The numerical procedure used was identical as to that described herein above for hard disk except that for Fig. 4(a) 13 radial polynomials were used instead of 6, and the search was done from n = 0 to n = 40, while for Fig. 4(b) the search was done from n = 0 to n = 150 using 17 radial polynomials. The number ofpolynomials in each case gives convergence to within 1% ofthe reported values.
Because of the hydrodynamic coupling, the floppy-disk base case with d = 0 has critical l O speeds that are approximately two orders of magnitude greater than the uncoupled, rigid-disk base case with m = 0. Small increases in d can raise the hydrodynamically coupled critical speed substantially above the already elevated d = 0 levels as indicated in Figs. 4(a) and ~b). At least an order of magnitude increase in critical speed is possible with proper choice of d.
The increases shown in Figs. 4(a) and (b) correspond to in-plane radial displacements at the clamp on the order of the disk thickness. For example, for a typical floppy disk with h = 0.05 mm and Ro = 42.5 mm, the ratio of the in-plane displacement at the clamp over the thickness is U, I h = dQ2h /Ro . For K = 0.31 and d = O, 0.005, 0.01 and 0.015, QF.ma~ = 69.8, 106,178, and 383. The corresponding displacement ratios are U, I h = O, 0.066,0.37, and 2.6.

CA 022461~2 1998-08-27 PATENT

The results in Figs. 4(a) and (b) can be easily related to values of m instead of d and vice versa using (13). For example, m = 0, v = 0.3, and ~c = 0.3 corresponds to d = 0.025. Hence, a floppy disk with traction free inner boundary conditions has a critical speed over four orders of magnitude higher than the same 5 rotating disk in the absence of hydrodynamic coupling and two orders of magnitude higher than the same rotating disk with hydrodynamic coupling but v~ni~hing in-plane displacements. There is a significant advantage to designing floppy disks with in-plane flexibility at the central clamp. It may also be advantageous to apply additional radial tractions to floppy disk inner radii to increase d and achieve further increases in the 10 maximum speed.
In practice, all clamps involve a combination of traction and displacement control involving stiction and transverse clamping pres~ule which is impossible to predict a priori, and which can be a nonlinear function of the disk rotation history, C.
D' Angelo III and C. D. Mote, Jr., 1993, Journal of Sound and Yibration, Vol. 168, 15 pp. 1-14, '~atural Frequencies of a Thin Disk, Clamped By Thick Collars with Function at the Contacting Surfaces, Spinning at High Rotation Speed." These practical difficulties hinder the design of a proposed stress induction clamp for a specific value of m and d.
However, an essential requirement is that the magnitude of the in-plane 20 stress increase with increasing Q. The primary motivation for choosing (6) and (10) as ~ CA 022461~2 1998-08-27 p PATENT

the speed boundary conditions instead of any other speed dependent function is - -numerical convenience, i.e., for the choice of Q2, the maximum rotation speed is the solution of a symmetric eigenvalue problem. In practice, any choice of speed dependent boundary conditions that generate low levels of stress in the stationary or slow rotating disk and high levels of stressing the high speed rotating disk should produce similar results since the centripetal stresses can still counteract the destabilizing effects of the induced stress. A closed-loop control system which varies the magnitude of in-plane force or displacements as a function of the disk vibration and rotation speed may also be used. C. D. Mote, Jr., and A. Rahimi, 1984, Journal of Dynamic Systems, Measurements 0 and Control, Vol. 106, pp. 123-128, "Real Time Vibration Control of Rotating Circular Plates by Temperature Control and System Identification." Such a control system could generate conditions corresponding to optimal values of m or d without having to predict the in-plane stiction, transverse clarnping force, or actual speed dependence.
Although inappropriate for computer disk drives, saw blades frequently include radial slots cut into the periphery of the blade. R. C. Yu and C. D. Mote, Jr., 1987. Journal of Sound and Vibration, Vol. 199, pp. 409-427. "Vibration and Pararnetric Excitation in Asymmetric Circular Plates Under Moving Loads." These slots reduce the colnplessi~e, therrnally-induced, hoop stress that occurs along the periphery during the cuKing process. In the absence of these slots, the colllplessi~e stresses can lower the maximurn rotation speed. Most stress-induction techniques for raising critical CA 022461~2 1998-08-27 PATENT

speed, including the one proposed herein, rely on the generation of tensile hoop stress in the periphery of the disk. Slots and other disk modifications, C. D. Mote, Jr., 1972, Journal of Dynamic Systems, Measurement, and Control, Vol. 94, pp. 64-70, "Stability Control Analysis of Rotating Plates by Finite element: Emphasis on Slots and Holes;"
R. G. Parker and C. D. Mote, Jr., 1991, Journal of Sound and Vibration, Vol. 145, pp. 95-110, "Turning of the Natural Frequency Spectrum of a Circular Plate By In-Plane Stress," limit the tensile hoop stress generated at the periphery of the disk. Thus, the use of such slots and modifications should be avoided as they are counterproductive to stress induction.
The novel stress induction method according to the invention can substantially increase the operational speed range of rotating disks such as industrial circular saws and computer disk drives. In addition, at a given operational speed, the invention described herein can reduce disk vibration. Moreover, disk thickness may be increased without increasing vibration at a given operational speed. While traditional stress induction methods can increase the maximum speed of rotation disks by 30-40%, the method described herein can double the maximum speed for disks with little hydrodynarnic coupling and can increase the maximum speed of hydrodynamically coupled disks by an order of magnitude.
The following examples illustrate representative embodiments of the invention described herein.

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F.~n~le 1 A preferred embodiment of the invention, including a thin disk, is shown in Fig. S. The thin disk may be a computer memory disk, saw blade or the like. The disk is substantially circular with an outer periphery 14 and inner radius or inner edge 5 16. The drive shaft 18 is rotated about its longitudinal axis by an external driving source such as a motor, belt, or gear (not shown). The free end of the drive shaft is designed with several short shafts 20 or other seating device for aligning and holding the device together in registered relation. A clamp 22 is comprised of a plurality of wedge shaped sections in an axisymmetric arrangement. Each wedge comprises a longitudinal 10 cylindrical hole 24 which engages the short shafts 20 of the drive shaft. The top of each wedge includes a ledge or projection 26 upon which the disk 12 is received and securely seated along the inner radius 16. A circular disk with alignment rods 30 extending downwards perpendicularly from the lid provides a mating lid for clamp 28. When the mating lid 28, disk 12, clamp sections 22, and drive shaft 18 are assembled together as a 15 unit, the alignment rods 30 mate with alignment holes 24. The top portion of the mating lid may be supported by a bearing or suitable rotating structure which is not shown.
A longitudinal force on the mating lid 28 derived from the supporting structure pushes the mating lid up against the disk 12 which in tum pushes up against the clamp 22. This longitudinal load is transmitted to the drive shaft through the short shafts 20 20.

CA 022461~2 1998-08-27 PATENT

Figs.6 is a perspective view of clamp 22. The clamp 22 is divided into six sections, each wedge shaped, by six cutting edges 34. In the illustrated embodiment, the clamp is comprised of 6 separate, identical pieces. The clamp need not comprise six pieces; any number greater than two or three will suffice since each piece must apply a 5 radial force to the inner edge 16 of the disk.
When the drive shaft 18 is rotated about its longitudinal axis, the sections of the clamp are also rotated. This rotation centripetally accelerates each section so that each section experiences a body force that is directed radially outwards from the axis of rotation. This force causes the mass of the clamp to pivot about shafts 20 and 30 so that 10 most of the clamp moves radially outward. In particular, the ledge 26 moves outward against the inner radius of the disk 16. This motion produces a radial force on the inner edge of the disk 16. Each section 36 supplies a similar radial force which is proportional to the square of the rotation speed. These forces approximate the axisymmetric, speed dependent boundary conditions described in the theoretical analysis herein.
Lubricants such as graphite may be employed to ensure free radial motion of each section of the clasp 36 as it pivots about the shafts 20 and 30.

rCA 022461~2 1998-08-27 1.!A31016 - 50/36246 PATENT

F,xample 2 Rotatin~ Disk with Rods A rotating disk with rods syrnmetrically positioned at its inner radius is shown in Fig. 7(a) and 7(b). During assembly, massive rods 54 are inserted between the disk 52 and the drive shaft 50. The massive rods 54 are shaped to fit in(lent~tions made in the disk. The inner edge of the disk is scalloped rather than circular shaped in order to secure rods 54 in position. The rods 54 and disk 52 are secured to the drive shaR using a two piece let~ g collar which slides over the entire arrangement. A top view of the retaining collar is shown in Fig. 8(b). A side view of the retaining collar is shown in Fig.8(a). A ret~ining collar is secured to the drive shaft 50 on opposite sides of the disk in a conventional manner with alignment rods lying between each of the massive rods 54. The two retaining collars prevent axial and rotational movement of the rods 54 and disk 52 relative to the drive shaft 50.
The ret~ining collar is designed to prevent bending of the disk between l S the rods or rotation of the assembly while simultaneously allowing the disk and rods to expand radially during rotation. When the shaR is rotated, the retaining collar transmits torque to the rods which then transmit torque to the disk through the disk protrusions between the rods. The rods are free to fly outwards into the inner edge of the disk 56 to ' generate speed dependent stresses in the disk.

CA 022461~2 1998-08-27 D ~ A31016 - 50/36246 PATENT

The rods are made of suitably heavy inflexible material and in total constitute the mass required to decrease vibration at the desired speed. The length of the rods is selected so as to fit within the environmental constraints of the housing of the disk. Since there are no radial restraints on the rod apart from the disk itself, very high 5 stresses can be generated. Any number of rods is suitable for use in the invention provided that at least two rods are used and that the entire arrangement is symmetric about the axis of rotation as shown in order not to unbalance the system. This arrangement is particularly suitable for use for stiffor rigid circular disks such as steel circular saws.

Example 3 Rotating Disk With Split Shaft A rotating disk 70 having a split drive shaft 72 is shown in Figs. 9(a) and 9(b). The drive shaft 72 is split into sections 74 along a predetermined length of the 15 shaft which are joined together at the base of the shaft. The sectional shaft may be prepared by wrapping the sections together or by partially sectioning the shaft along its length. The sectioned or free end of the shaft is designed to fit snugly into a disk such as a floppy disk.
During rotation, the split sections 74 of the drive shaft fly apart in an 20 out~vardly direction, pressing against the inner edge of the disk 76 and generating the ~ CA 022461~2 1998-08-27 ' A31016 - 50/36246 PATENl speed dependent stresses in the disk. Because the radial movement of the sections is partially restrained by the root or base of the drive shaft where all the sections are joined, less stress can be generated than in Example 2. The top portion of the drive shaft (not shown) m~tingly engages the top of the floppy disk. The drive shaft or at least the top 5 thereof should be of sufficient mass so as to apply the force required to minimi7.e vibration at the desired speed. In addition, the sectioned drive shaft should be axisymmetric so as to avoid unbalance of the rotating system.
In an alternate embodiment, hydraulic fluid may be pumped into an expandable shaft.

Example 4 Fig. 10 shows a sectional side view of a rotating disk arrangement including a thin disk 100, the upper portion of a motor 102 which drives a shaft 104 and speed dependent clamp comprised of three main elements. A mounting piece 106 fits snugly onto the shaft and is secured against rotation by the set screw 112. A cap piece 108 fits over the shaft and is directly attached to 106 using screws 114. Between the mounting piece 106 and the cap piece 108, several wedge shaped sections 110 are placed 20 symmetrically about the shaft. Elastomeric O-rings 116 are used to support the wedge CA 022461~2 1998-08-27 PATENT

sections between 106 and 108. Each wedge section 110 has a ledge 118 that makes direct contact with the inner edge of the disk 120. Each wedge l lO can also make contact with the mounting piece at location 122.
Fig. 11 (b) shows a side view of the mounting piece 124 which directly corresponds to the view 106 shown in Fig. 10. Fig. 11 (b) shows a top view of the mounting piece 126. A cylindrical hole 128 goes through the center of 124 and 126.
The shaft of a driving motor 104 fits snugly into hole 128. The mounting piece has a larger diameter disk shaped section 130 which possess grooves 132 into which theO-rings 116 rest. A single set screw can fit into the tapped hole 134 in order to prevent the mounting piece from rotating with respect to the motor shaft, and six tapped holes 136 in the top part of the mounting piece are used for attaching the cap piece 108.
As can be seen from Fig. l l(a), six thin walls 138 are symmetrically arranged about the central hole. These walls are essential to the design. The six wedge shaped sections 110 fit between the mounting piece and the cap. Each section is also separated by the thin walls 138. This ensures that each wedge section 110 rotates with the mounting piece when the motor is turned on. Furthermore, the outer edge 140 of each thin wall is at the same radius as the inner edge of the disk 120. With this arrangement, once the mounting piece has been secured to the motor shaft, the disk can be placed on the mounting piece and it will be centered by the six outer wall edges 140 as it rests on the O-rings of the mounting piece.

., ~

~ CA 022461~2 1998-08-27 D

PATENT

Figs. 12(a) and (b) show a side view of the cap 142 and a top view 144.
Fig. 13 shows a side view of one of the six sectional wedges 146. The center line of the wedge is 148. l S0 is a top view of the wedge. The inner edge of the disk makes contact with the wedge at location 152. The wedge may make contact with S the mounting piece at location 154. The approximate center of mass of the wedge is location 156.
The clamp may be made of a rigid material such as steel, alllminum, or high strength plastic. The wedge pieces should be as massive as possible. Lead may be a suitable material provided it is sufficiently rigid that it does not deform significantly 10 during use. Any number of wedges is suitable for use in the invention provided at least two wedges are used.
When the clamp is assembled about a disk 100 and mounted on a spinning motor shaft 104, each wedge section 110 feels a radial force outwards from the axis of rotation. This centripetal acceleration force is proportional to the mass of the 15 wedge, the radial distance of the center of mass of the wedge from the axis rotation, and the square of the rotation speed. Since the centers of mass of the mounting piece and the cap coincide with the axis of rotation, the centripetal acceleration force is negligible in these pieces.
The elastomeric O-rings 116 prevent motion of each wedge 110 in the 20 axial direction; the thin walls 138 prevent motion of each wedge in an angular direction CA 022461~2 1998-08-27 PATENT

about the axis of rotation. However, the wedge is prevented from moving radially only by the inner edge of the disk 120. The O-rings 116 provide negligible friction in the radial direction. Consequently, the radial centripetal force on each wedge is transferred to the inner edge of the rotation disk 120. This supplies the desired speed dependent 5 clamping.
The arrangement has several advantages. First, since the radial location of the center of mass of the wedge is larger than the radial location of the inner edge of the disk, the centripetal force in each wedge is larger than if these radii were the same.
Second, if the wedge pivots at all, the ledge 122 will form the center of rotation of the 10 pivoting. A mechanical advantage will be achieved between this pivot point 154, the inner edge of the disk 152 and the center of mass of the wedge 156. This mechanical advantage is the same as in a wheel barrow. As a consequence, the radial force on the inner edge of the disk my be significantly greater than without pivoting of the wedge.
The center of mass should, therefore, be as far away axially from the contact point of the 15 disk as possible.
In an alternate embodiment, it may be advantageous to replace part of the wedge with a hollow section into which other masses could be added as desired. For instance, a small groove on the top side of each wedge could be milled out into which steel balls may be optionally placed. The balls would be kept in place during operation 20 by the cap. Alternatively, screws of different mass could be placed into the wedges. In ~ CA 022461~2 1998-08-27 - 'J A31016- SOt36246 PATENT

this manner, the m~gnitude of the centripetal clamping effect could be easily varied.
Industrial circular saws change their vibration characteristics during their operation life.
As a result, the optimal clamping mass also varies. This modification would permit easy adjustment of the clamp to fit the characteristics of many different disks.Although the invention has been described herein with respect to specific embodiments, many modifications and variations therein will readily occur to those skilled in the art. For example, the force applied at the inner radius of the disk may be electromagnetically generated, or its magnitude may be varied using a feedback control system. The feedback control system includes a vibration sensor such as an accelerometer, eddy current probe or optical sensor. A control unit such as a personal computer or electronic circuit board can be used to process the sensor signal and determine the applopliate radial force magnitude. The control unit will then adjust the magrutude of the radial force in order to minimi7e vibration. In addition, the method of producing the radial force would also be capable of varying the magnitude of the applied radial force. Accordingly, all such variations and modifications are included within the intended scope of the invention.

Claims (14)

1. A method of inducing in-plane stress in a rotating circular disk comprising the steps of:
providing a circular disk having an inner radius and an outer radius; and applying a nonfrictional and non-thermal substantially radial force at an inner radius of the disk which varies with the rotation speed of the rotating circular disk.

2. A method according to claim 1 wherein the nonfrictional and non-thermal substantially radial force is proportional to the square of the rotation speed of the rotating circular disk.

3. A method according to claim 1 comprising affixing a mass at the inner radius of the disk to produce a nonfrictional and non-thermal substantially radial force which varies with acceleration of the mass.

4. A method according to claim 3 wherein said mass comprises a plurality of rod shaped objects.

5. A method according to claim 3 wherein said mass comprises a plurality of wedge shaped objects.

6. A method according to claim 1 wherein said force is an electromagnetic force.

7. A method according to claim 1 comprising providing a pressurized fluid to generate the nonfrictional and non-thermal substantially radial force.

8. A rotating circular disk arrangement comprising a disk having an outer radius and an inner radius and means for applying a nonfrictional and non-thermal substantially radial force to said disk located at an inner radius of said disk.

9. A rotating circular disk arrangement according to claim 8 comprising means for applying a nonfrictional and non-thermal substantially radial force which varies with the square of the rotation speed of the disk to said disk.

10. A rotating circular disk arrangement according to claim 8 wherein the means for applying force comprises a mass.

11. A rotating circular disk arrangement according to claim 8 wherein the mass comprises a plurality of rod shaped objects.

12. A rotating circular disk arrangement according to claim 8 wherein the mass comprises a plurality of wedge shaped objects.

13. A rotating circular disk arrangement according to claim 8 wherein the means for applying force is an electromagnetic force.

14. A rotating circular disk arrangement according to claim 8 wherein the means for applying force comprises a pressurized fluid.