TENSION BAND CENTRIFUGE ROTOR

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: TITLE TENSION BAND CENTRIFUGE ROTOR - BACKGROUND OF THE INVENTION 10 Field of the Invention The present invention relates to a band for a centrifuge rotor, and in particular, to a band configured such that, in operation, it is subjected only to tensile forces. 15 Description of the Prior Art 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 20 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 25 in a greater relative centrifugal force. 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. 30 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. 35 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. In view of the foregoing it is believed advantageous to provide a centrifuge rotor which is not exposed to the stresses associated with the change in shape as the band is spun, thus avoiding the deleterious effects attendent therewith SUMMARY OF THE INVENTION The present invention relates to an applied load accepting band for a centrifuge rotor that 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. BRIEF DESCRIPTION OF THE DRAWINGS The invention 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 IA 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 Appendix; 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. An Appendix (pages A-l to A-l l) setting forth the derivation of the equations discussed herein is appended to and forms part of this application. DETAILED DESCRIPTION OF THE 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 121 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 121 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 20 to be described (Figures 4 to 7) are attached to the band 12 or those locations where swinging bucket sample carriers 20 to be described (Figures 8 to 9) abut against the interior surface 121 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 20 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. -o -O-o- 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 121 and the exterior surface 12E thereof. However, it should be understood that the equilibrium curve 22 may be defined as extending through any rddial 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 I A, proceeding radially inwardly from the interior surface 121 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 121 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 Ro while the distance from the axis 10A to the applied load accepting τegions is denoted by the reference character R . 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 Ro and a radius R 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: d(R/R 0 ) /dΘ = (R/R 0 ) 2 RAD(1 - { K/2 [(R/R 0 ) 2 -1 ] } ) 2 - (R/Ro) (1) K = [(γ 0) 2 Ro 2 ) (l/g) (1/σo)] (2 ) where Ro 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 O < K < l . It is noted that the symbol "RAD" is used throughout this application (including the Appendix) to denote the radical sign indicating the computation of square roots. 8 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 t 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 striaght line and being subjected only to a tensile force while spinning must have zero mass, a clearly absurd result. Thus, a band in accordance with the present invention which is subjected only to tensile force while spinning must, of necessity, have an equilibrium curve in which the shape factor K lies within the range 0 < K < 1. The equilibrium curve of any band in accordance with the present invention (thai 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_χ 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. To determine that the equilbrium curve of a band in an actual rotor the band is first removed from the struts that affix it to the hub. The contour of the actual band may then be plotted. If the band is exposed only to tension when spinning, the equilibrium curve of the band will closely match one of the family of equilibrium curves shown in Figure 3D. That is, the equilibrium curve of the band from the actual rotor will fall on one of the family of curves in the range between Ro and RL or will lie within a predetermined range of one of the family of equilbirium curves. To verify that such a band is subjected to only a tensile force, a brittle lacquer test may be performed (preferably prior to the disassembly of the rotor from the struts, as discussed above).. The brittle lacquer test is discussed in Richard C. Dove and Paul H. Adams, "Experimental Stress Analysis and Motion Measurement", Charles E. Merrill Books, Inc., Columbus, Ohio (1964). Other tests to verify that the band is subjected only to tensile forces could be performed. Such testing could include the mounting of strain gauges on inside and outside radial surfaces of the band. 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 20 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. 1. 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 121 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 oi me 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 creat-ed 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. -o-O-o- 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 Appendix, the equilibrium curve corresponds to the following: d(R/R 0 )/dΘ=(R/Ro) RAD [(R/R 0 ) 2 ] (exp{-K [(R/R 0 ) 2 -l] } -1) (IA) K = [(γ co 2 Ro ) (1/g) (1/σo)] (2A) (A/Ao) = exp {-(K/2) [(R/R 0 ) 2 - 1] } (3A) where Ao is the cross sectional area of the band at the radius Ro- A band corresponding to the relationships of Equations IA 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. -o-O-o- 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 6 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 121 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 121 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 20 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. Those skilled in the art, having the benefits of the teachings of the present invention as hereinabove set forth, may effect numerous modifications thereto. However, such modifications lie within the contemplation of the present invention, as defined by the appended claims. WHAT IS CLAIMED IS: A-l APPENDIX With reference to the free body diagram of Figure 3 A, the derivation of the sets of Equations 1 , 2 and IA, 2A and 3 A for the equilibrium curve, and thereby the shape, of a band 12 may be understood. In Figure 3A a portion of the band 12 between the midpoint 22C of the equilibrium curve 22 and a predetermined endpoint 22L-2 is shown. The reference axes for a Cartesian and a polar coordinate system are also shown. In the derivation the endpoint 22L-2 is located on the band 12 at the load accepting region 18-2 (Figure 1 ) and the load accepting region is depicted as the point through which the applied load may act. In the actual band the load accepting region extends for some finite distance on the band. The radii of the band at these respective points is indicated by the characters Ro and RL, respectively. The angular distance between any radius R and the radius Ro is indicated by the angle Θ. The portion of the equilibrium curve 22 not shown in Figure 3A between the midpoint 22C of the equilibrium curve 22 and the endpoint 22L-1 is symmetric to the portion of the equilibrium curve shown in Figure 3A. The free body diagram illustrates the forces acting on the band 12 while the same is spinning. In accordance with the present invention the band 12 has the same shape both while at rest and while spinning. The shape of the band is such that while the band is spinning it is subjected only to a tension force. Stated alternatively, when spinning the tension in the band balances the centrifugal force on the band due to its mass and the load on the band at the load accepting regions. As seen in the free body diagram each end of the segment of band has a tension force imposed thereon. The forces are indicated by the characters To and T, respectively, which designate the tension forces in the band at the midpoint 22C and the endpoint 22L-1. The magnitude of the indicated tension forces on the band inherently includes the loading on the band due to the weight of the sample and the sample carrier. The centrifugal force acting on the center of mass of the band is indicated by the character F. Summing forces in the x direction and thereafter differentiating produces the following: Σ F x = 0 F x + T x - To = 0 F x = To - T x dF x = - dT x (A) Similarly summing forces in the y direction and thereafter differentiating produces the following: F y = 0 - Ty = 0 F y = dF y = dT y (B) As seen from Figures 3B and 3C, if the mass of a differential segment ds of the band is dm, if its cross sectional area is A, if its angular speed is ω and if its density is γ , the differential centrifugal force dF on the differential segment of the band may be expressed as dF= Rω2dm (C) Substituting the expression (γ A ds) (1/g) for the differential mass dm, Equation (C) becomes dF= (γ A 0)2) (1/g) ( R ds) (D) From Figure 3C, similar triangles yields dF / dF x = R / x ; dF / dF y = R / y and (E) dF x = dF (x / R) ; dF y = dF (y / R) From Equations (D) and (E) the components of dF are: dF x = (γ A co 2 ) (1/g) x ds (F) dF y = (γ A ω2) (1/g) y ds (G) From Equations (A) and (F) dTχ/ds = -(γ A ) ) (1/g) x (H) and from Equations (B) and (G) dT y /ds= (γ A α> 2 )(l/g) y (I) From the free body diagram of Figure 3A - Differentiating and dividing by two yields T dT = T y dT y + T x dT x T dT = T, [ ( T y /Tx ) dT y + dT x J (J) Since the vectors ds and T both have the same direction (perpendicular to the endface of the segment of the band) similar triangles in Figure 3A yield ( T y /Tx ) = - (dy / dx) ; Tx = T (dx / ds) (K) Substituting Equation (K) into Equation (J) T dT = T (dx / ds) [ -( dy/dx) dT y + dT x J (L) Simplifying Equation (L) dT = - ( dTy/ds) dy + (dT x / ds) dx (M) From Equations (H) and (I) dT = - [(γ A o>2 ) (1/g) y]dy + - | (γ A ω 2 ) (1/g) x ] dx (N) an d dT = - (γ A ω 2 ) (1 /g) (y dy + x dx ) (0) Assuming a constant cross-section for the band, integrating Equation (O) over the limits To to T yields T-To = - (γ A ω~ ) (l/2g) (y2 - R 0 2 + x 2) (P ) Noting from Figure 3C that (y 2 + x 2 ) = R 2 , Equation (P) becomes T-To = - (γ A ω 2 ) (l/2g) (R 2 - Ro 2 ) (Q) and factoring Ro 2 from Equation (Q) and rearranging yields T= T 0 - f(γ A 0) 2 ) (l/2g)] Ro 2 [(R/R 0 ) 2 - 1] (R) Dividing Equation (R) by To , remembering that a constant cross section band is assumed, and noting that the stress is tension per unit area (i. e., σo = To / Ao ), Equation (R) becomes T/To = I - Kγ ) Ro ) (l/2g) (1/σo)] l(R/R 0 ) 2 - 1 ] (S) and T/To = 1 - {K/2 [(R/R 0 ) 2 - 1] } (T) where K = [(γ ω R 0 2 ) (1/g) (l/σ 0 )J (2) From the free body diagram of Figure 3A, summing moments about the origin To Ro = T β R (U) TΘ / TO = RO / R (V) In Figure 3A, the force triangle of the vector T yields, TR + T Θ 2 = T2 Rearranging and dividing by To 2 (TR / T 0 ) 2 = (T/ TO)2 - (T θ / To ) 2 (W) Inserting Equations (V) and (T) into Equation (W) yields (TR / To ) 2 = (1 - fK/2 [(R/Ro) 2 - 1 ] }) 2 - (R 0 / R) 2 a n d (TR / To) = RAD (1 - { K/2 |(R/R 0 ) 2 - 1 J }) 2 - (R 0 / R) 2 (X) Multiplying the right hand side of Equation (X) by R/Ro and the left hand side by (To/Tø) (which from Equation (V) is equal to R/Ro) yields (TR /T θ ) = (R/Ro) RAD (1 - {K/2 [(R/R 0 ) 2 - l ])) 2 - (R 0 /R) 2 (Y) Since the vector R dΘ and the vector dR respectively extend in the same direction as the vectors TQ and TR , respectively, similar triangles yields TR / T θ = dR / R dθ (Z) Multiplying the numerator and the demoninator of Equation (Z) by ( 1 /Ro) yields - TR / T θ = d(R/R 0 ) / [(R/Ro) dθl (AA) Simplifying (R/Rθ) (TR / T θ ) = d(R/R 0 ) / dΘ (BB) Therefore, for a band that exhibits a constant area, inserting Equation (Y) into Equation (BB) d(R/R 0 )/dΘ =(R/R 0 ) 2 RAD(1 {K/2 [(R/R 0 ) 2 -1] }) 2 - (R/R 0 ) 2 (1) where K * = [(γ 0) 2 R 0 2 ) (1/g) (l/σ 0 )] , 0 < K < 1 (2) 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 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 no component of band tension is available 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 bending. If the shape factor K = 0, the equilibrium curve takes the form of a straight line. In this instance there is no component of band tension able to contirbute to supporting the centrifugal force exerted on the mass of the band. Thus, a band having an equilibrium curve in the form of a striaght line and being subjected only to a tensile force while spinning must have zero mass, a clearly absurd result. Thus, a band in accordance with the present invention which is subjected only to tensile force while spinning must, of necessity, have a shape factor K that 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 the endpoint thereof (as these points are defined herein) that closely matches one of the family of equilibrium curves defined by Equations (1 ). To determine that the equilbrium curve of a band as the same is used in an actual rotor the band is first removed from the struts that affix it to the hub. The contour of the actual band may then be plotted. The equilbirium curve extends through the center of the band. In an actual rotor, the angle Θ (in degrees) that the radius from the axis of rotation through the load point (the radius RL), will be known from the relationship θ = 360/ (2N) - where N is the number of places on the rotor. Thus, one endpoint of the equilibrium curve of the actual rotor is the point on the band just adjacent to the load accepting region of the actual rotor. The midpoint of the band (the radius Ro)»is typically (but not necessarily) the point at which the strut attaches to the band. If the band is exposed only to tension when spinning, the equilibrium curve of the band will closely match one of the family of equilibrium curves shown in Figure 3D. That is, the equilibrium curve of the band from the actual rotor will fall on one of the family of curves in the range between Ro and R or will lie within a predetermined range of one of the family of equilbirium curves. To verify that such a band is subjected to only a tensile force, a brittle lacquer test may be performed (preferably prior to the disassembly of the rotor from the struts, as discussed above). The brittle lacquer test is discussed in Richard C. Dove and Paul H. Adams, "Experimental Stress Analysis and Motion Measurement", Charles E. Merrill Books, Inc., Columbus, Ohio (1964). Other tests to verify that the band is subjected only to tensile forces could be performed. Such testing could include the mounting of strain gauges on inside and outside radial surfaces of the band. -o-O-o- Whereas in the earlier derivation Equations (1 ) and (2) it is assumed that since the cross section area A of the band (Figure 3B) is constant, the tension and therefore the stress in the band varies. However, the invention can be implemented in a band in which the stress is constant and the opposite assumption, that the cross sectional area varies, is made. For a band having a variable cross section and constant stress dT = - (γ A 0 2 ) (1/g) (y dy + x dx ) (O) Multiplying the numerator and denominator by 2, and again noting that the stress is tension per unit area (i. e. f σ = T / A), dT = - (γ T 0) 2 ) (l/2g) (l/σ 0 ) (2ydy + 2xdx ) (CQ Dividing by T and recognizing the derivatives of y 2 and x 2 in Equation (CC) dT/T = - (γ 0)2 ) (l /2g) (l /σ 0 ) (dy 2 + dx 2 ) (DD) Recognizing from Figure 3C that (dy 2 + dx 2 ) = d(R 2 ) , dT/T = - (γ 0)2 ) (l/2g) (l/σ 0 ) (dR 2 ) (EE) Integration of Equation (DD) over the limits To to T (for the variable T) and Ro to R (for the variable R) and factoring Ro yields In (T/To) = - (Y ω 2 R 0 2 ) (l /2g) ( l/σ 0 ) l(R/Ro) 2 -- ) (FF) But, recognizing K from Equation (2), Equation (FF) bec ome s In (T/To) = - K/2 [(R/Ro) 2 -U (GG) Taking the natural logarithm, (T/To) = exp {- K/2 |(R/Ro) 2 -H J (HH) Summing moments, and from Equations (W) and (V) (TR/TO) = RAD (T/To) 2 - (R/Ro) 2 (ID From Equation (HH) (TR TO) = RAD exp {- K [(R/R 0 ) 2 -1] ) - (R/Ro) 2 (JJ) Multiplying the right hand side of Equation (JJ) by R/Ro and the left hand side by (To/Tø) (which from Equation (V) is equal to R/Ro) yields (TR/TΘ) =(R/Ro) RAD exp {-K l(R/R 0 ) 2 -1] ) - (R/Ro) 2 (KK) which is equivalent to (TR/TΘ) = RAD [(R/Ro) 2 ] (exp {-K [(R/R 0 ) 2 - 1] } - 1) (LL) From similar triangles Equations (Z) and (AA) become (TR /T θ ) = d(R)/R dΘ = d(R/R 0 ) / (R/Ro) dΘ d(R/R 0 ) /dΘ = (R/Ro) (TR /T θ ) (BB) Therefore, for a band exposed to a constant stress, but having a variable cross sectional area d(R/R 0 )/dθ =(R/R 0 ) RAD [(R/Ro) 2 ] (exp{ -K [(R/R 0 ) 2 -l ] )-l ) (IA) K = {(γ ώ- Ro 2 ) 0/g) (l/σ 0 )l (2A) (A/Ao) = exp {-(K/2) [(R/R 0 ) 2 - 1 ] } (3A)

Claims: 16 1. A continuous band adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, characterized in that the band has an equilibrium curve defined between the applied load accepting regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ fτom the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, 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 (1) where K = [(γ )2 R 0 2 ) (1/g) (l/σ 0 )] (2) where 0 < K < 1. 2. The band of claim 1 further characterized in that the band has a radially inner and a radially outer surface thereon, wherein the equilibrium curve is defined substantially midway between the radially inner and outer surfaces. 3. The band of claim 2 further characterized in that the band is fabricated from a composite material. 17 4. The band of claim 3 wherein the composite is a tape formed of a plurality of uniaxial fibers surrounded by a matrix. 5. The band of claim 1 wherein the band has a cross sectional area associated therewith between the applied load accepting regions, further characterized in that the cross sectional area of the band between the applied load receiving regions is substantially constant. 6. The band of claim 5 further characterized in that the band is fabricated from a composite material. 7. The band of claim 6 wherein the composite material is a tape formed of a plurality of uniaxial fibers surrounded by a matrix. 8. A continuous band adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a central axis extending therethrough and a first and a second applied load receiving region defined thereon, characterized in that the band has an equilibrium curve defined between the applied load receiving regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, the distance R at the corresponding angular position Θ being defined by the relationship: d(R/R 0 )/dΘ=(R/R 0 ) RAD [(R/R 0 ) 2 ) (exp {-K [(R/R 0 ) 2 -1]) -1) (IA) K = [(γ 0) 2 Ro 2 ) (1/g) (1/σo)] (2A) (A/Ao) = exp {-(K/2) [(R/R 0 ) 2 - 1] } (3A) where 0 < K < 1. 9. The band of claim 8 further characterized in that the band has a radially inner and a radially outer surface thereon, wherein the equilibrium curve is defined substantially midway between the radially inner and outer surfaces. 10. The band of claim 9 further characterized in that the band is fabricated from a homogeneous material. 11. The band of claim 10 further characterized in that the band is fabricated from metal. 12. A continuous band adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, characterized in that the band has an equilibrium curve defined between the applied load accepting regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, each distance R 19 approximating some nuetral distance RN, where the distance RN at the corresponding angular position Θ is defined by the relationship: d(R/R 0 )/dΘ =(R/Ro) 2 RAD(1 - {K/2 [(R/R 0 ) 2 1] }) 2 - (R/R 0 ) 2 (1) where K = [(γ ω 2 R 0 2 ) (1/g) (l/σ 0 )] (2) where 0 < K < 1. 13. A continuous band adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a central axis extending therethrough and a first and a second applied load receiving region defined thereon, the applied load accepting regions being spaced a predetermined angular distance (2 Θ ) in a plane perpendicular to the axis, each point on the band intermediate the applied load accepting regions having a cross sectional area measured in a plane extending radially from the axis, characterized in that the cross sectional area of the band is constant at each point therealong intermediate the applied load accepting regions such that, when the band is rotated and an applied load is imposed on the band at the applied load accepting regions the band is loaded only by a tensile stress. 14. A continuous band adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, 20 the applied load accepting regions being spaced a predetermined angular distance (2 Θ) in a plane perpendicular to the axis, each point on the band intermediate the applied load accepting regions having a cross sectional area measured in a plane extending radially from the axis, characterized in that the cross sectional area of the band varies at each point therealong intermediate the applied load accepting regions such that, when the band is rotated and an applied load is imposed on the band at the applied load accepting regions the band is loaded on by a tensile stress. 15. A centrifuge rotor comprising: a hub having at least a first strut; a continuous band adaptable for use as an applied load accepting band mounted to the strut, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, characterized in that the band, when it is removed from the strut by which it is attached to the hub and while the band is at rest, has an equilibrium curve defined between the applied load accepting regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium cuTve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, the distance R at the corresponding angular position Θ being defined by the relationship: d(R/R 0 )/dΘ =(R/Rθ) 2 RAD(1 - {K/2 [(R/R 0 ) 2 -1] }) 2 - (R/Ro) 2 0) where K = [(γ a R 0 2 ) (1/g) (l/σ 0 )] (2) where 0 < K < 1. 16. The rotor of claim 15 further characterized in that the band has a radially inner and a radially outer surface thereon, wherein the equilibrium curve is defined substantially midway between the radially inner and outer surfaces. 17. The rotor of claim 15 further characterized in that the band is fabricated from a composite material. 18. The rotor of claim 17 wherein the composite material is a tape forrrϊ&d of a plurality of uniaxial fibers surrounded by a matrix. 19. The rotor of claim 16 wherein the band has a cross sectional area associated therewith between the applied load accepting regions, further characterized in that the cross sectional area of the band between the applied load receiving regions is substantially constant. 20. The rotor of claim 19 further characterized in that the band is fabricated from a composite material. 21. The rotor of claim 20 wherein the composite is a tape formed of a plurality of uniaxial fibers surrounded by a matrix. 22. The rotor of claim 15 wherein the strut is attached to the band at the midpoint of the equilibrium curve. 23. A centrifuge rotor comprising: a hub having at least a first strut; and a continuous band adaptable for use as an applied load accepting band mounted to the strut, the band having a central axis extending therethrough and a first and a second applied load receiving region defined thereon, characterized in that the band, when it is removed from the strut by which it is attached to the hub and while the band is at rest, has an equilibrium curve defined between the applied load receiving regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, the distance R at the corresponding angular position Θ being defined by the relationship: d(R/R 0 )/dΘ =(R/R 0 )RAD [(R/R(,) 2 ] (exp {-K [(R/Ro) 2 -1]}-1) (IA) K = [(γ 0 2 Ro 2 ) (l/g) (l/σ 0 )] (2A) (A/Ao) = exp {-(K/2) |(R/R 0 ) 2 - 1 ] } (3A) where 0 < K < 1. 24. The rotor of claim 23 further characterized in that the band has a radially inner and a radially outer surface thereon, wherein the equilibrium curve is defined substantially midway between the radially inner and outer surfaces. 25. The rotor of claim 24 further characterized in that the band is fabricated from a homogeneous material. 26. The rotor of claim 25 further characterized in that the band is fabricated from metal. 27. The rotor of claim 23 wherein the strut is attached to the band at the midpoint of the equilibrium curve. 28. A centrifuge rotor comprising: a hub having at least a first strut; and a continuous band adaptable for use as an applied load accepting band mounted to the strut, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, characterized in that the band, when it is removed from the strut by which it is attached to the hub and while the band is at rest, has an equilibrium curve defined between the applied load accepting regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point a jacent to one o t e app e oa accept ng regions lies a predetermined distance R from the axis, each distance R approximating some nuetral distance RN , where the distance RN at the corresponding angular position Θ is defined by the relationship: d(R/R 0 )/dΘ =(R/Ro) 2 RAD(1 - (K/2 [(R/R 0 ) 2 -1] )) 2 - (R/R 0 ) 2 (1) where K = [(γ 0) 2 R 0 2 ) (1/g) (l/σ 0 )] (2) where 0 < K < 1. 29. The rotor of claim 28 wherein the strut is attached to the band at the midpoint of the equilibrium curve. 30. A centrifuge rotor comprising: a hub having at least a first strut; and a continuous band adaptable for use as an applied load accepting band mounted to the strut, the band having a central axis extending therethrough and a first and a second applied load receiving region defined thereon, the applied load accepting regions being spaced a predetermined angular distance (2 Θ ) in a plane perpendicular to the axis, each point on the band intermediate the applied load accepting regions having a cross sectional area measured in a plane extending radially from the axis, characterized in that the cross sectional area of . the band is constant at each point therealong intermediate the applied load accepting regions such that, when the band is rotated and an applied load is imposed on the band at the applied load accepting regions the band is loaded only by a tensile stress. 31. The rotor of claim 30 wherein the strut is attached to the band at the midpoint of the equilibrium curve. 32. A centrifuge rotor comprising: a hub having at least a first strut; and a continuous band adaptable for use as an applied load accepting band mounted to the strut, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, the applied load accepting regions being spaced a predetermined angular distance (2 Θ ) in a plane perpendicular to the axis, each point on the band intermediate the applied load accepting regions having a cross sectional area measured in a plane extending radially from the axis, characterized in that the cross sectional area of the band varies at each point therealong intermediate the applied load accepting regions such that, when the band is rotated and an applied load is imposed on the band at the applied load accepting regions the band is loaded on by a tensile stress. 33. The rotor of claim 32 wherein the strut is attached to the band at the midpoint of the equilibrium curve. 34. A method of fabricating a centrifuge rotor, the rotor having a hub with a strut thereon, comprising the steps of: 26 (a) forming a mandrel with an outer surface having a predetermined shape thereon; (b) winding a tape formed of a composite material on the mandrel to form a continuous band that has a shape corresponds to the shape of the mandrel, the band having a central axis extending therethrough and a first and a second applied load accepting region defined thereon, characterized in that the band has an equilibrium curve defined between the applied load accepting regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, 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 0 ) • where K = [(γ ω- R {) - ) (1/g) (l/σ 0 )] (2) where 0 < K < 1 ; and c) mounting the band formed from step b) to the strut on the hub. 35.. A method of fabricating a centrifuge rotor comprising the steps of: a) machining from a homogeneous material a continuous band adaptable for use as an applied load accepting band in a centrifuge rotor, the band having a central axis extending therethrough and a first and a second applied load receiving region defined thereon, characterized in that the band has an equilibrium curve defined between the applied load receiving regions, the equilibrium curve having a midpoint therealong, the distance in the plane perpendicular to the axis between the axis and the midpoint being defined by a predetermined reference line Ro, such that, in the plane perpendicular to the axis, at any angular position Θ from the reference line Ro each point on the equilibrium curve between the midpoint thereof and a point adjacent to one of the applied load accepting regions lies a predetermined distance R from the axis, the distance R at the corresponding angular position Θ being defined by the relationship: d(R/Ro)/dΘ=(R/R 0 ) RAD [(R/R 0 ) 2 ] (exp {-K [(R/R 0 ) 2 -l] }-l) (IA) K = Kγ o> 2 Ro 2 ) (l/g) (l/σ 0 )] . (2A) (A/Ao) = exp { -(K/2) {(R/R 0 ) 2 - 1] } (3A) where 0 < K < 1 ; and b) mounting the band formed in step a) to the strut on the hub.

Inventors: ROMANAUSKA WILLIAM ANDREW, ROMANAUSKA, WILLIAM, ANDREW
Applicants: DU PONT, E.I. DU PONT DE NEMOURS AND COMPANY
Application year: 1990
Main classification: B04B5
IPC codes: B04B5, B04B7