EP0485443B1

EP0485443B1 - Tension band centrifuge rotor

Info

Publication number: EP0485443B1
Application number: EP90911620A
Authority: EP
Inventors: William Andrew Romanauska
Original assignee: EI Du Pont de Nemours and Co
Current assignee: EIDP Inc
Priority date: 1989-08-02
Filing date: 1990-07-17
Publication date: 1996-10-16
Anticipated expiration: 2010-07-17
Also published as: ATE144168T1; JPH04506927A; CA2022095A1; EP0485443A4; DE69028921D1; DE69028921T2; WO1991002302A1; EP0485443A1; IE902779A1

Abstract

The present invention relates to an applied load accepting band (12) for a centrifuge rotor (10) that is configured such that, while rotating, the applied loads on the band (12) are balanced by the tension in the band (12), so that during rotation the band (12) is subjected only to a tensile force.

Description

The present invention relates to a band for a centrifuge rotor.
The manufacture of rotating structures, such as centrifuge rotors and energy storage flywheels, has evolved from the use of homogeneous materials, such as aluminum and titanium, toward the use of composite materials. The use of such materials is believed advantageous because it permits the attainment of increased centrifugal load carrying capability. The increased load carrying capability is achieved because the lighter weight of the composite rotor permits it to spin faster for a given motive input, thus resulting in a greater relative centrifugal force.
FR-A-2.082.274 discloses a continuous peripheral band adaptable for use as an applied load accepting band in a centrifuge rotor. The band of FR-A-2.082.274 consists of fibers wound around the spokes of the rotor.
Further US-A-4,817,453 discribes a centrifuge rotor which includes a central disk formed of a plurality of laminates each having an array of fibers. A rim having a higher stiffness than that of the disc surrounds the disc so that a radially inwardly compressive stress is imposed on the disc by the rim when the rotor is rotated at any speed.
The rotating structures of the prior art believed relevant to the present invention each have some form of band that, while at rest, exhibits a predetermined arbitrary shape. However, such a band is subjected during operation to a load due to the tendency of the band to change from the arbitrary rest shape to some equilibrium rotating shape. This phemonenon may be understood from the following simplified example.
Consider an applied load accepting band for a centrifuge rotor that in the rest (i.e., non-spinning) condition is circular in shape. Assume that this band accepts three applied loads corresponding to three equiangularly spaced sample carriers. When such a rotor is spun the effects of centrifugal force on the sample carriers apply loads that act radially outward, tending to pull the band to form "corners". The perimeter of the band generally intermediate the applied loads will thus deflect radially inwardly from their original circular shape. Since the band has some predetermined stiffness associated with it, the deflection of the band from its rest shape to its equilibrium shape while rotating imposes a bending stress on the band. This bending stress in the band does not contribute to its load carrying capability, and in fact, is deleterious to the band since it results in reduced rotor life.
It is the object of the of the present invention to provide a centrifuge rotor and an applied load accepting band for the rotor which are long-lasting. This object is solved according to the invention with the features of claim 1, 2, 10, 12 and 13.
According to the present invention the applied load accepting band for a centrifuge rotor is configured such that, while rotating, the applied loads on the band are balanced by the tension in the band, so that during rotation the band is subjected only to a tensile force.
The intention will be more fully understood from the following detailed description thereof, taken in connection with the accompanying drawings, which form a part of this application, and in which:

Figure 1 is a plan view of a generalized centrifuge rotor (with the sample carriers ommitted for clarity) having an applied load accepting band in accordance with the present invention, while Figure 2 is an isometric view of the rotor of Figure 1;
Figure 1A is an enlarged view of a portion of Figure 1 illustrating the attachment of the strut to the band;
Figure 3A is a free body diagram of a portion of a band for a centrifuge rotor in accordance with the present invention in which the applied load accepting band is realized using a wound band formed of a composite material that has a constant thickness dimension from which the equation describing the shape of such a band may be derived, while Figures 3B through 3D illustrate the mathematical relationships used in the derivation of the Equations;
Figure 4 and 5 are, respectively, a plan view and a side elevational view taken along section lines 5-5 in Figure 4 illustrating a fixed angle centrifuge rotor having an applied load accepting band in accordance with the present invention;
Figures 6 and 7 are, respectively, a plan view and a side elevational view taken along section lines 7-7 in Figure 6 illustrating a vertical centrifuge rotor having an applied load accepting band in accordance with the present invention;
Figure 6A is an enlarged view of a portion of Figure 6 illustrating the attachment of the sample carrier to the band and the structure of the load transition pad;
Figure 8 and 9 are, respectively, a plan view and an isometric view illustrating a swinging bucket centrifuge rotor having an applied load accepting band in accordance with the present invention; and
Figure 10 is a plan view similar to Figure 1 showing an applied load accepting band having a variable cross sectional area in accordance with the present invention.

Throughout the following detailed description similar reference numerals refer to similar elements in all figures of the drawings.
Shown in Figures 1 and 2 are, respectively, a plan and an isometric view of a centrifuge rotor 10 having a peripheral applied load accepting band 12 in accordance with the present invention. The band 12 has a predetermined thickness associated therewith and has interior surface 12I and an exterior surface 12E thereon.
The rotor 10 includes a central hub 14 which may be connected, as diagrammatically illustrated in Figure 2, to a suitable motive source M whereby the rotor 10 may rotate about its axis of rotation 10A. The central hub 14 has a plurality of radially outwardly extending struts 16. The hub and the struts may be formed of metal, although it may be desireable to make the hub and the struts from a composite or a reinforced plastic material.
The peripheral band 12 is mounted to the struts 16 and surrounds the hub 14. The band 12 may be connected to the struts 16 by any suitable means, as will be described. The band 12 has a predetermined thickness associated therewith and has interior surface 12I and an exterior surface 12E thereon.
The band 12 has a plurality of angularly spaced, applied load accepting regions 18 defined thereon. These regions 18 are those locations on the band 12 where sample carriers 30 to be described (Figures 4 to 7) are attached to the band 12 or those locations where swinging bucket sample carriers 30 to be described (Figures 8 to 9) abut against the interior surface 12I of the band 12. Although for purposes of analysis the loads imposed on the band can be analyzed in terms of a single point on the band through which the load can be said to act, it should be appreciated that the load accepting regions 18 in actuality extend some predetermined finite distance about the periphery of the band 12. Adjacent applied load accepting regions 18 are, in a plane perpendicular to the axis 10A (that is, the plane of Figure 1), spaced apart a predetermined angular distance (2θ), depending upon the number of the sample carriers 30 on the rotor 10. The angle (2θ) is related to the number N of sample carriers disposed on the rotor 10, with (2θ) (in degrees) being equal to 360 divided by N.
As will be developed the applied load accepting band 12 in accordance with the present invention is, during centrifugation, subjected to only tensile force, thereby eliminating therefrom regions of high stress concentration which may reduce band life.
The applied load accepting band 12 may be fabricated either from a composite material or from a metal, such as aluminum or titanium. A band formed of a composite material is discussed first. Considerations of economy of manufacture using a composite material dictate that the band formed therefrom exhibits a constant cross sectional area. Accordingly, in the discussion that follows, the composite band exhibits a cross section area that is constant along its entire periphery.
In accordance with the present invention, in a plane perpendicular to the axis 10A of rotation of the rotor 10, the applied load accepting band 12 has a predetermined equilibrium curve 22, indicated by the dashed line, defined therein between adjacent applied load accepting regions 18. The equilibrium curve 22 is used herein as a definition of the shape of the band. Preferably the equilibrium curve is construed to extend centrally through the thickness of the band 12, that is, midway between the interior surface 12I and the exterior surface 12E thereof. However, it should be understood that the equilibrium curve 22 may be defined as extending through any radial location within the thickness of the band 12. The equilibrium curve 22 has a predetermined center point 22C therealong. Preferably, the strut 16 is attached to the band 12 at the center point 22C of the equilibrium curve 22 therein.
As seen in the enlarged view of Figure 1A, proceeding radially inwardly from the interior surface 12I of the band at the desired mounting location to the radially outer surface of the strut 16 is an elastomeric sheet 17 and a layer 19 of a composite material. A suitable adhesive layer 21 is disposed between the interior surface 12I of the band 12 and the elastomeric sheet 17, between the elastomeric sheet 17 and the layer 19 of composite material, and between the layer 19 of composite material and the strut 16. The elastomeric sheet 17 is provided to accomodate shear to limit strain in the adhesive layers 21, while the composite layer 19 is provided to eliminate stress in the transverse direction. Any suitable adhesive compatible with the materials being adhered may be used.
Each point on the equilibrium curve 22 lies, in the plane of Figure 1 and the the free body diagram of Figure 3, a predetermined radial distance R from the axis 10A. The distance from the axis 10A to the midpoint 22C is denoted by the reference character R₀ while the distance from the axis 10A to the applied load accepting regions is denoted by the reference character R_L. Since the adjacent applied load accepting regions 18L-1 and 18L-2 are spaced angularly a distance (2 θ), the angular distance between the radius R₀ and a radius R_L is denoted by the angle θ. When the band 12 is removed from the struts 16 by which it is attached to the hub 14 and while the band 12 is at rest, the equilibrium curve 22 from the midpoint 22C to a point adjacent to either one of the applied load accepting regions 18L-1 or 18L-2 is defined by the relationship: $\begin{matrix} \begin{matrix} \begin{matrix} (1) & {d(R/R}_{0} {)/dθ = (R/R}_{0})^{2} {RAD(1-{K/2[(R/R}_{0})^{2} {-1]})}^{2} {-(R/R}_{0}) \end{matrix} \\ \begin{matrix} (2) & {K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})] \end{matrix} \end{matrix} \end{matrix}$
where R₀ is the distance from the axis 10A to the midpoint point 22C on the equilibrium curve 22 between two adjacent applied load receiving regions 18L-1 and 18L-2,
where K is a constant of curvature (shape factor) of the band that has values greater than zero and less than 1, such that 0 < K < 1.
It is noted that the symbol "RAD" is used throughout this application (including the derivation of the set of Equations) to denote the radical sign indicating the computation of square roots.
The derivation of Equations (1) and (2) is set forth in the Appendix, which is appended to and forms part of this application.
The constant K defines a shape factor K for each of the family of equations that satisfy the differential equation (1). Since the band is to be exposed only to a tensile force while spinning the shape factor K must be limited within the range 0 < K < 1. If K lies outside these limits an equilibrium condition is not possible. The physical explanation of the limits on K can be understood with reference to a consideration of the ranges of loads able to be accommodated by a band in accordance with the invention.
As seen in the drawing Figure 3D the differential equations (1) define a family of equilibrium curves. If the shape factor K = 1, the equilibrium curve takes the form of a circle. However, a circular form for the equilibrium curve would mean that a band having such an equilibrium curve has no component of band tension available to contribute to supporting a load applied to the band. A band subjected only to a tensile force while spinning would thus be able to accommodate zero load--an impractical result. Thus to support a load a circular band must necessarily be subjected to bending.
If the shape factor K = 0, the equilibrium curve takes the form of a straight line. In this instance a band having such an equilibrium curve has no component of band tension able to contribute to supporting the centrifugal force exerted on the mass of the band. Thus, a band having an equilibrium curve in the form of a straight line and being subjected only to a tensile force while spinning must have zero mass, a clearly absurd result.
In these equations the shape factor K lies within the range 0 < K < 1.
The equilibrium curve of any band in accordance with the present invention (that is, a band subjected only to tension while spinning) will exhibit an equilibrium curve between a midpoint of a band segment and a point on the band next adjacent to the applied laod accepting region that closely matches one of the family of equilibrium curves defined by Equations (1). It is again noted that since the load accepting regions 18 has some finite extent, the shape of an actual band may deviate from its equilibrium curve in the load accepting regions 18 and still remain within the contemplation of the invention.
Moreover, it should be understood that, within the portion of the band between the midpoint and the point adjacent to the applied load accepting region a band 22 may also deviate from the mathematical definition of the equilibrium curve given by, Equations (1) and (2) and still remain within the contemplation of the present invention. To this end the equilibrium curve 22 may be viewed as a reference curve that defines a nuetral or reference radial distance R_N for each value of θ. So long as the actual radial distance R of a band approximates the nuetral radial distance R_N defined by the equations for the equilibrium curve, such a band is to be construed as lying within the contemplation of the present invention. Thus, the radial distances in an actual band need not match the equilibrium curve of the equaitons point by point, so long as the band is generally loaded only by tension while spinning it is to be construed to lie within the contemplation of the invention.
Whereas the optimim performance is provided when the shape of the band matches the equilibrium curve and thus the stresses created by bending moments are equal to zero, it is recognized that some stress created from bending moments can be tolerated in the design of a centrifuge rotor which produces less than optimim performance. Consequently, bands which approximate the edquilibrium curve must also be construed as lying within the contemplation of this invention.
A band 12 having a configuration that satisfies the equilibrium curve 22 of Equations (1) and (2) will be subjected only to tensile force while spinning. The shape of the band 12 will not change while the band is accelerating to or rotating at speed. However, the band 12 may grow outwardly, and the sample carriers 30 affixed to the band may displace radially outwardly, both movements due to to centrifugal force effects. However, the loads imposed on the band 12 due both to its weight and to the weight of the load, will be balanced by the tensile force in the band. Thus, the band will undergo no bending stresses.
When the band 12 is fabricated from a composite material the preferred material is a tape formed of a plurality of uniaxial fibers surrounded by a matrix of polyether ketone ketone (PEKK). The fiber is preferably an aramid fiber such as that manufactured and sold by E. I. DuPont de Nemours and Company under the trademark "KEVLAR". The band 12 is formed by filament winding using either tow or tape on a mandrel that has a predetermined shape that corresponds to the equilibrium curve 22. As the tape is wound on the mandrel, the resulting band has imparted thereto the shape of the equilibrium curve. The band 12 so formed has a generally constant radial or thickness dimension. It should also be noted that a band having a constant cross section may also be formed from a homogeneous material, such as titanium or aluminum.
The struts 16 are preferably attached to the interior surface 12I of the band 12 at the midpoints 22C along the equilibrium curve 22. In practice, the struts 16 may preload the band slightly, in order to accommodate variations in the radial stiffness of the band 12 and the strut 16. This preload may deform the shape of the band while it is attached to the struts from the shape corresponding to the equilibrium curve. Deformation due to the preload is, however, a elastic deformation. It should thus be clearly understood that it is the shape of the band when the same is removed from the struts and is at rest that meets, as discussed above, the relationships set forth in Equations (1) and (2) and thus falls within the scope of the present invention. Due to the preload, when assembled on the struts and at rest, the band imposes a first predetermined compressive (i.e., radially inwardly directed) force on the struts. However, while the band is spinning, the band grows due to centrifugal force effects and the band imposes a predetermined lesser compressive force on the struts.
It should be recognized that the equilibrium curve can only be obtained when the bending stresses are equal to zero. In use, it is beneficial to provide some preload of the band against the strut in order to compensate for differences in radial stiffness and the associated differences in deformation when the rotor is rotated. By design, the equilibrium shape will only be obtained in this case when the rotor reaches the design speed and contains the design load. At zero speed the bending stresses due to the preload are at a maximum. As the rotor increases speed the bending stresses created by the preload decrease while the stress created by the load increase. When the rotor reaches the design speed the bending stress created by the prelolad is zero and the band is totally in tension due to the load. At this point the band obtains the equilibrium curve.
The band 12 heretofore discussed exhibits a substantially uniform cross sectional area along the equilibrium curve. However, from the standpoint of efficiency of material usage, it may be desired to provide a band that exhibits a constant stress (as opposed to a constant cross section) along its periphery. In accordance with an alternate embodiment shown in Figure 10, it lies within the contemplation of this invention that the band 12 may exhibit a constant stress, with a variable cross sectional area along the equilibrium curve. In this instance, as is seen from the derivation, the equilibrium curve corresponds to the following: $\begin{matrix} \begin{matrix} \begin{matrix} (1A) & {d(R/R}_{0} {)/dθ=(R/R}_{0} {)RAD[(R/R}_{0})^{2} {](exp{-K[(R/R}_{0})^{2} -1]}-1) \end{matrix} \\ \begin{matrix} (2A) & {K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})] \end{matrix} \\ \begin{matrix} (3A) & {(A/A}_{0} {) = exp {-(K/2)[(R/R}_{0})^{2} -1]} \end{matrix} \end{matrix} \end{matrix}$
where A₀ is the cross sectional area of the band at the radius R₀.
A band corresponding to the relationships of Equations 1A and 2A may be fabricated from a homogeneous material such as titanium or aluminum, by any suitable process, such as numerical controlled milling. It should be unserstood that a band in accordance with this alternate embodiment of the invention may be fabricated from a composite material.
A band 12 in accordance with the present invention, whether implemented in a composite material or a homogeneous material, may be used in any of a variety of centrifuge rotors, as will be appreciated from Figures 4 through 9.
Figures 4 and 5 illustrate a plan and a vertical cross section view of a rotor 10 having a band 12 in accordance with the present invention in which the sample carriers 30 are configured to define a fixed angle centrifuge rotor. In this instance each of the sample carriers 30 is attached directly to and supported by the band 12 at an applied load accepting region 18. The carrier 30 is mounted to a load transition pad 32 that is attached to the band 12 at the applied load accepting region 18. As seen in Figures 4 and 5 the sample carriers 30 have sample container receiving cavities 36 therein. Although two such cavities 36 are illustrated, it should be understood that any convenient number of cavities 36 may be so formed in the carrier 30. In the embodiment of Figures 4 and 5, the axis 36A of each cavity 36 is inclined with respect to the axis of rotation 10A. Alternatively, in Figure 6, the axis 36A of each cavity 36 is parallel to the axis of rotation 10A, and a rotor of the vertical type is thus defined.
In Figures 4 through 7 the sample carriers 30 are fabricated from a molded plastic material. In these same Figures (as well as Figures 8 and 9) the load transition pads 32 are formed from a molded elastomeric material such as polyurethane. As seen in Figure 6A the pad 32 is attached to the interior suface 12I of the band 12 using an adhesive layer 35. A composite member 33 is attached to the radially inner surface of the pad 32 by another adhesive layer 35. The radially inner surface of the composite member 33 is flat, while the radially outer surface of the pad 32 conforms in shape to the interior surface 12I of the band 12 in the load accepting region 18 where the pad is mounted. The sample carrier 30 may be attached to the member 33 using another layer 35 of adhesive, or the carrier 30 may be nested between the hub 16 and the member 33.
As yet another alternative, as seen in Figures 8 and 9, the sample carriers 30 may be of the swinging type. To this end, the carriers 30 are thus pivotally mounted to the hub 14 so that during centrifugation the axis 36A of the cavities 36 move from a first, generally vertical, position to a second position. In the second position the axis 36A of each cavity 36 in the sample carrier 30 lies in a plane generally perpendicular to the axis of rotation 10A. Moreover, means 38 are provided whereby the end of the sample carrier 30 moves radially outwardly to its supported position against the pad 32 located in the applied load receiving region 18 on the band 12.
The pivotal mounting of the carrier 30 with respect to the hub 14 may be effected in a variety of ways. In the embodiment shown in Figures 8 and 9, the hub 14 is provided with angularly spaced pairs of radially extending arms 38A, 38B. Each arm 38A, 38B has a slot 40 therein that serves to accept a trunnion pin 42 disposed on the carrier 30. Of course the arms 38A, 38B could each carry a trunnion pin that is received in the carrier 30.

Claims

A continuous peripheral band adaptable for use as an applied load accepting band (12) in a centrifuge rotor, the band (12) having at least a first and a second applied load receiving region (18) defined thereon and a central rotational axis (10A), the applied load accepting regions (18) being spaced a predetermined angular distance (2θ) in a plane perpendicular to the axis (10A), characterized in that
the band (12) has an equilibrium curve (22) defined between the applied load accepting regions (18), the equilibrium curve (22) having a midpoint (22C) therealong, the distance in the plane perpendicular to the axis (10A) between the axis (10A) and the midpoint (22C) being defined by a predetermined reference line R_o, such that, in the plane perpendicular to the axis (10A), at any angular position θ from the reference line R_o each point on the equilibrium curve between the midpoint (22C) thereof and a point adjacent to one of the applied load accepting regions (18) lies a predetermined distance R from the axis, each distance R actual approximating some reference distance R at the corresponding angular position θ defined by the relationship: ${d(R/R}_{0} {)/dθ =(R/R}_{0})^{2} {RAD(1-{K/2[(R/R}_{0})^{2} {-1]})}^{2} {-(R/R}_{0})^{2}$
where ${K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})]$

ω is angular speed,

γ the density of the band,

g is the acceleration due to gravity, and

σ₀ is the stress in the band, and
where 0 < K < 1,
the cross sectional area of the band (12) is constant at each point therealong intermediate the applied load accepting regions (18), and
the band (12) thereby has a shape such that, when the band (12) is rotated and an applied load is imposed on the band (12) at the applied load accepting regions (18) the band (12) is loaded only by a tensile stress.
A continuous peripheral band adaptable for use as an applied load accepting band (12) in a centrifuge rotor, the band (12) having at least a first and a second applied load receiving region (18) defined thereon and a central rotational axis (10A), the applied load accepting regions (18) being spaced a predetermined angular distance (2θ) in a plane perpendicular to the axis (10A),
characterized in that
the band (12) has an equilibrium curve (22) defined between the applied load receiving regions (18), the equilibrium curve (22) having a midpoint (22C) therealong, the distance in the plane perpendicular to the axis (10A) between the axis (10A) and the midpoint (22C) being defined by a predetermined reference line R₀, such that, in the plane perpendicular to the axis (10A), at any angular position θ from the reference line R₀ each point on the equilibrium curve (22) between the midpoint (22C) thereof and a point adjacent to one of the applied load accepting regions (18) lies a predetermined distance R from the axis, each distance R actual approximating some reference distance R at the corresponding angular position θ defined by the relationship: $\begin{matrix} \begin{matrix} \begin{matrix} (1A) & {d(R/R}_{0} {)/dθ=(R/R}_{0} {)RAD[(R/R}_{0})^{2} {](exp{-K[(R/R}_{0})^{2} -1]}-1) \end{matrix} \\ \begin{matrix} (2A) & {K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})] \end{matrix} \\ \begin{matrix} (3A) & {(A/A}_{0} {) = exp {-(K/2)[(R/R}_{0})^{2} -1]} \end{matrix} \end{matrix} \end{matrix}$
where
ω is angular speed,

γ the density of the band,

g is the acceleration due to gravity, and

σ₀ is the stress in the band, and

A is the cross sectional area of the band,

A_o is the cross sectional area of the band at the radius R₀ and
where 0 < K < 1,
the band (12) thereby has a shape such that, when the band (12) is rotated and an applied load is imposed on the band (12) at the applied load accepting regions (18) the band (12) is loaded only by a tensile stress.
The band of claim 2 further characterized in that the cross sectional area of the band (12) varies intermediate the applied load accepting regions (18).
The band of one of claims 1-3 further characterized in that the band is fabricated from a composite material.
The band of claim 4 wherein the composite is a tape formed of a plurality of uniaxial fibers surrounded by a matrix.
The band of one of claims 1-3 further characterized in that the band (12) is fabricated from a homogeneous material.
The band of claim 6 further characterized in that the band (12) is fabricated from metal.
The band of one of claims 1 or 4 through 7 wherein each distance R approximates some neutral distance R_N, where the distance R_N at the corresponding angular position θ is defined by the relationship: ${d(R}_{N} {/R}_{0} {)/dθ =(R}_{N} {/R}_{0})^{2} {RAD(1-{K/2[(R}_{N} {/R}_{0})^{2} {1]})}^{2} {-(R}_{N} {/R}_{0})^{2}$
where ${K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})]$

ω is angular speed,

γ the density of the band,

g is the acceleration due to gravity, and

σ_o is the stress in the band, and
where 0 < K < 1.
The band of one of claims 1 to 8 further characterized in that the band (12) has a radially inner and a radially outer surface (12I, 12E) thereon defined with respect to the axis of rotation, wherein the equilibrium curve (22) is defined substantially midway between the radially inner and outer surfaces (12I, 12E).
A centrifuge rotor comprising a hub (14) and a band (12) of one of claims 1 to 9, the hub (14) having at least a first strut (16) with the band (12) mounted to the strut (16), wherein the band, when it is removed from the strut and while the band is at rest, has an equilibrium curve defined between the applied load accepting regions.
The rotor of claim 10 wherein the strut (16) is attached to the band (12) at the midpoint (22C) of the equilibrium curve (22).
A method of fabricating a centrifuge rotor, the rotor having a hub (14) with a strut (16) thereon and a band (12) according to one of claims 1, 4, 5, 8, 9 or 10, comprising the steps of:
(a) forming a mandrel with an outer surface having a predetermined shape thereon;

(b) winding a fiber formed of a composite material on the mandrel to form a continuous peripheral band (12) that has a shape corresponding to the shape of the mandrel, the band (12) having a first and a second applied load accepting region (18) defined thereon and a central rotational axis (10A), the band (12) having an equilibrium curve (22) defined between the applied load accepting regions (18), the equilibrium curve (22) having a midpoint (22C) therealong, the distance in the plane perpendicular to the axis (10A) between the axis (10A) and the midpoint (22C) being defined by a predetermined reference line R₀, such that, in the plane perpendicular to the axis (10A), at any angular position θ from the reference line R₀ each point on the equilibrium curve (22) between the midpoint (22C) thereof and a point adjacent to one of the applied load accepting regions (18) lies a predetermined distance R from the axis (10A), the distance R at the corresponding angular position θ being defined by the relationship: ${d(R/R}_{0} {)/dθ =(R/R}_{0})^{2} {RAD(1-{K/2[(R/R}_{0})^{2} {-1]})}^{2} {-(R/R}_{0})^{2}$
where ${K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})]$

ω is angular speed,

γ the density of the band,

g is the acceleration due to gravity, and

σ_o is the stress in the band, and
where 0 < K < 1; and

(c) mounting the band (12) formed from step b) to the strut (16) on the hub (14).
A method of fabricating a centrifuge rotor, the rotor having a hub (14) with a strut (16) thereon and a band (12) according to one of claims 2, 3 or 6 through 11, comprising the steps of:
(a) machining from a homogeneous material a continuous peripheral band (12) adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a first and a second applied load receiving region (18) defined thereon and a central rotational axis, the band (12) having an equilibrium curve (22) defined between the applied load receiving regions (18), the equilibrium curve (22) having a midpoint (22C) therealong, the distance in the plane perpendicular to the axis (10A) between the axis (10A) and the midpoint (22C) being defined by a predetermined reference line R₀, such that, in the plane perpendicular to the axis (10A), at any angular position θ from the reference line R₀ each point on the equilibrium curve (22) between the midpoint (22C) thereof and a point adjacent to one of the applied load accepting regions (18) lies a predetermined distance R from the axis (10A), the distance R at the corresponding angular position θ being defined by the relationship: $\begin{matrix} \begin{matrix} \begin{matrix} (1A) & {d(R/R}_{0} {)/dθ=(R/R}_{0} {)RAD[(R/R}_{0})^{2} {](exp{-K[(R/R}_{0})^{2} -1]}-1) \end{matrix} \\ \begin{matrix} (2A) & {K = [(γω}^{2} R_{0} ^{2} {)(1/g)(1/σ}_{0})] \end{matrix} \\ \begin{matrix} (3A) & {(A/A}_{0} {) = exp {-(K/2)[(R/R}_{0})^{2} - 1]} \end{matrix} \end{matrix} \end{matrix}$

ω is angular speed,

γ the density of the band,

g is the acceleration due to gravity, and

σ_o is the stress in the band, and

A is the cross sectional area of the band,

A_o is the cross sectional area of the band at the radius R₀ and
where 0 < K < 1; and

b) mounting the band (12) formed in step a) to the strut (16) on the hub (14).