BEARING PRELOAD MEASUREMENT SYSTEM

Background

The present disclosure relates generally to assemblies with moving shafts and bearings supporting the shafts, and more particularly to systems and methods for measuring preload and endplay conditions in the bearings of the assemblies.

Examples of systems and methods related to measuring preload and endplay of rotating assemblies are found in the following patents and publications: WO07105655; WO03071246; EP1717464; CA2016469; US6951146; US6460423; US6443624; US6505972; US6357922; US6286374; US5877433; US5263372 and US3665758. The disclosures of these references are hereby incorporated by reference in their entirety for all purposes. Summary

Rotating shafts with supporting bearings are common mechanical assemblies and are inherent in everything from flywheels of watches to power generation turbine shafts. A bearing surface typically supports a shaft in these assemblies and allows the shaft to rotate with a set amount of friction. The bearings limit the movement of the shaft in an axial direction or in a lateral direction or, in some cases, both. Bearings may comprise roller, tapered roller, sleeve or ball rolling elements.

Many shaft and bearing rotating assemblies provide adjustment features that control the normal force that opposing surfaces contacting the ball or roller exert against the rolling element itself. The application the bearing is used in often specifies a normal force or preload on the rolling element. Applications with high rotating speeds may specify a relatively low preload force on the bearing. Applications with low speeds and high applied forces such as milling machines may have high preload force.

Where too much force is applied to the rolling elements, excess heat and friction during operation may be generated, reducing bearing life and increasing service costs. Too low of a force applied to the rolling elements may allow lateral and axial movement of the shaft beyond specifications. Determining the preload force actually applied to the rolling elements can be critical in many applications.

A method used for testing preload in high accuracy manufacturing applications is to apply incremental axial loads to a shaft and bearing and measure the displacement as a result of the applied force. These displacement values are then compared to a standard or reference set of displacement values from the same spindle with a different preload setting. The difference in displacement between the test sample and the standard at a fixed applied force determines the differential of the dimensional preload on the bearings. When the standard set of values is based on a known amount of preload or endplay, the test spindle preload can be derived from the difference. With this method the standard assembly must be identical to the test sample or specimen in terms of component materials, dimensions and assembly methods to be accurate.

When testing the same design of spindle but a different unit, it is not possible to eliminate all variation between assemblies. Sources contributing to variation in preload on the bearings are differences in material elasticity of the races, the bearings themselves and other components of the assembly and variations in manufactured dimensions of assembly components. Differences in the contact angle between the balls and the races of the bearings can alone be a source of significant variation and may alter the apparent stiffness or elasticity of the assembly as a whole.

Variations between assemblies may become apparent in force-displacement curves where the displacement in the assembly is plotted against the force applied to the assembly. A typical force-displacement curve will have positive and negative zones outside of inflection points on the curve. The positive side and negative side of the curves represent positive and negative values of the applied force, or push and pull in the axial direction. The inflection points correspond to applied forces of adequate magnitude to unseat one set of a double set of bearings. The zone between the inflection points correspond to applied forces which are lower than that required to unseat either of the bearings.

The slope of the curve at any point in the unseated zones is associated with the modulus of elasticity of the specimen. For two specimens that are identical except for different preloads on the bearings, these unseated zones of the curve will be parallel and offset or at a fixed distance apart from each other. The offset between force- displacement curves of differing preloads may be used to measure the preload.

The master curve is a hypothetical curve representing zero end play and zero preload. By selecting a master curve for an identical bearing, the value of the offset from the master curve to the curve of the preloaded assembly will provide the preload.

Differences in materials, assembly and dimensions between a test specimen and a reference or master curve specimen may result in a difference in apparent elasticity or stiffness of the assemblies. In this case the slopes of the test specimen and master force-displacement curves may not be parallel or equivalent. The difference in the apparent elasticity can be a significant source of error in determining the preload on the bearings. A more accurate preload magnitude may be determined using a correction factor to compensate for the difference in elasticity.

Methods for determining bearing preload will be disclosed. While specific configurations and machinery may be described in this disclosure as well as methods, these are examples for the purpose of illustration and should not be considered limitations. Many different variations and various embodiments may be imagined to implement aspects of these features disclosed herein that will be considered equivalent and will fall within the scope of this disclosure.

A rotating assembly may be equivalent to a spindle used in machine tools. An elastic constant derived from an elastic region of a force-displacement curve or from collected force and displacement data may be non linear but similar to Young's modulus or a stiffness factor for a material or assembly.

While a rotating assembly may be described here as an example, the methods and systems disclosed may be similarly applied to any bearing system. For example the methods and systems described here could be applied to a linearly translating bearing system or a threaded rotating and translating bearing system such as a ball screw. Description of the Drawings

Fig. 1 is a cross section of a rotating assembly including a housing, a shaft and two bearings where the inner races of the bearings are fitted to the shaft and the outer races are fitted in the housing.

Fig. 2 is a cross section of a bearing showing an inner race, an outer race and a ball captured between the races, where the contact points of the ball on the races define an angle of contact for the bearing.

Fig. 3 is a perspective view of a test system including a load system with a specimen mount, a load cell, a transducer, a shaft drive and a computer including a processor and memory.

Fig. 4 is a force-displacement chart showing the contribution of bearing displacement and component displacement to rotating assembly displacement.

Fig. 5 is a force-displacement chart for a spindle with zero preload and zero endplay showing displacement resulting from positive or push applied loads and negative or pull applied loads.

Fig. 6 is a chart showing force-displacement curves for a family of bearings.

Fig. 7 is a chart illustrating the variation of the ratio of stiffness with small variations in the contact angle of the bearing.

Fig. 8 is a force-displacement chart with a master curve and a test curve illustrating a method for determining preload in a rotating assembly.

Fig. 9 is a detail of the chart of Fig. 8 illustrating a method for determining the preload force using a corrected master curve.

Fig. 9A is a detail of the chart of Fig. 8 illustrating a method for determining the preload dimension using a corrected master curve.

Fig. 10 is a force-displacement chart including a master curve, a segmented master curve and a corrected master curve illustrating a method of using a correction factor.

Fig. 11 is a force-displacement chart including a master curve, a corrected master curve with tangent lines for a point on the master curve and a point on the corrected master curve illustrating differential calculus methods.

Fig. 12 is a cross section of a bearing for a linearly translating assembly.

Fig. 13 is a flow chart describing a method for determining bearing preload.

Fig. 14 is a flow chart describing a method for determining bearing preload. Specification

Bearings are critical to any kind of rotating machinery and there are many different configurations of rotating or translating components with supporting bearings. Some of the most common bearings are roller bearings and ball bearings. They are frequently supplied as an assembly with the balls or rollers retained in an inner race that accepts a rotating shaft or translating shaft or beam and an outer race that is fixed to a housing. An assembly with a rotating shaft or other component supported by bearings in a housing that remains relatively static may be referred to here as a rotating assembly.

Figure 1 is a cross section of a spindle or rotating assembly 2. Rotating assembly 2 may include a housing or structure 4 which remains fixed and supports additional components. Rotating assembly 2 includes a first bearing 6 with a first set of rolling elements or balls 8, an outer race 1OA and an inner race 1OB. A second bearing 12 similarly includes a second set of balls 14, an outer race 16A and an inner race 16B. Outer race 1OA may be retained in housing 4 by a retaining ring 18 and outer race 16A may be retained in housing 4 by a threaded collar 20 that screws into housing 4. A shaft collar 22 may contact inner races 1OB and 16B. Shaft collar 22 may also separate first bearing 6 and second bearing 12. Inner races 1OB and 16B and shaft collar 22 may accept and be fixed to shaft 24 and may rotate together as a single unit.

Rotating assembly 2 is configured to allow shaft 24 to rotate relative to housing 4 with minimal friction and minimum lateral and axial translation movement of shaft 24 as shown by dotted line shaft 24'. Applying a normal force or preload to balls 8 and 14 may minimize lateral and axial translation of shaft 24 with an applied load F.

Fig. 2 is a detail cross section of first bearing 6. Similar numbering may be used for similar components in this and later figures. Bearing 6 again includes a set of rolling elements or balls 8 shown here as a single ball for clarity, outer race 1OA and inner race 1OB. Ball 8 may contact races 1OA and 1OB at two substantially opposite points 30A and 30B. A line through these two points may be a contact line 32 that forms a contact angle 34 with the plane of the races. A bearing with a steep contact angle is normally selected for higher axial loading applications. A bearing with a shallow or small contact angle is may be selected for a radial load application which has low axial loads. Contact angle 34 may vary substantially between bearings, even with identical assembly methods. Contact angle 34 may be a significant factor in variation of preload on ball 8. While ball bearings may be used in the examples and figures, the methods and examples can be equally applied to roller bearing applications. Ball bearings are used as an example only and should not be considered a limitation. Returning to Figure 1, in this example the position of threaded collar 20 may determine the preload or endplay of first and second bearings 6 and 12. If threaded collar 20 is screwed into housing 4 and does not contact outer race 16 A, shaft 24 may be free to move a certain distance laterally and axially as balls 8 and 14 may not make contact with both mating races. This is an endplay condition with zero preload. As threaded collar 20 is screwed further into housing 4 and makes contact with outer race 16 A, race 16A may move relative to housing 4 until it contacts ball 14 and ball 14 contacts inner race 16B.

As collar 20 is screwed in further, second bearing 12 may push against shaft collar 22 which may push against inner race 1OB of bearing 6. Inner race 1OB then contacts ball 8 and ball 8 contacts outer race 1OA which stops against retaining ring 18. When all the components are in contact as described, screwing threaded collar 20 further into housing 4 against race 16A increases the normal or contact forces between components including balls 8 and 14. The increase in contact forces may cause some components to shift their contact positions against each other until they reach a position fixed by some level of normal force between components. Axial load or force F applied to shaft 24 may cause additional shifting of contact surfaces or elastic deformation of component materials. As more force is applied by screwing threaded collar 20 further into housing 4, the force is stored as elastic deformation of assembly components.

The force exerted on bearings 6 and 12 is a preload. The magnitude of this force may be expressed as a force or as a dimension when referring to a force-displacement diagram as described below. In this disclosure preload force may be used in examples for clarity, but a preload dimension can be determined in a similar manner.

Balls 8 and 14 may be composed of steel or ceramic or any solid material. Ball and race surfaces may be hard and wear resistant. Bearing raceways may be a case hardened material with a softer inner material and a hard surface.

Fig. 3 is a perspective view of a test system 40 for measuring preload on a rotating assembly or test specimen 41. Test system 40 may include a load system 42. Load system 42 may include a load system housing 44, a specimen mount 46 with specimen mount supports 46 A and 46B and a specimen connecting collar 48, a load cell 50, a transducer system 52 with transducer 52A and target plate 52B, a load setting means 54, with a load setting shaft 54A and a load setting handle 54B. Load system 42 may include a shaft drive 56. Shaft drive 56 may comprise a motor 56A, a drive system 56B, a drive shaft 56C and shaft drive housing 56D with components indicated by dotted lines as hidden and/or internal to shaft drive housing 56D. Shaft drive housing 56D may be soft mounted to load system housing 44 by a soft mount 56E to allow shaft drive 56 to float with any applied load F. Shaft drive housing 56D may instead be mounted to load cell 50 and may transfer force F to shaft 24 through drive shaft 56C.

In another example, shaft 24 may be driven by a system separate from load system 42 and test system 40 such as a separate drive system or a drive within test specimen 41.

Load system 42 may additionally include a load system computer or load system electronics 58 with a processor 58A, memory or memory storage system 58B and a data acquisition board 58C indicated as hidden and/or internal to load system electronics 58 by dotted lines. Test system 40 may further include a data collection system 60 with a computer 61 comprising a processor 62, a memory or memory storage system 64 and a data acquisition board 66 indicated as hidden and/or internal to computer 61 by dotted lines. Computer 61 may be any form of computing system including a personal data assistant, a desktop computer, a laptop or a server system.

Load system 42 may be configured to apply axial load or force F to test specimen 41. For this example rotating assembly 2 of Fig. 1 will be referenced here as a a specific example of test specimen 41, but other types or configurations of specimen may be tested such as a linear translating assembly as described below. Mount 46 may retain housing 4 of rotating assembly 2 in a fixed position relative to load system 42. Specimen connecting collar 48 may also provide a connection between shaft 24 and load system 42 so that positive and negative loads F may be applied to rotating assembly 2. Load cell 50 may measure the magnitude of load F applied to shaft 24. Transducer system 52 may measure displacement of test specimen 41 in response to applied force F. Shaft drive 56 may additionally rotate shaft 24 while axial load F is applied to rotating assembly 2.

Test specimen 41 may be held in place during testing by specimen mount 46. Specimen mount 46 may be configured to minimize movement of housing 4 in response to applied force F. Specimen mount 46 may be of any configuration that adequately secures the specimen during testing. Specimen mount 46 may be configured to cooperate with specimen connecting collar 48 to transmit positive and negative applied loads F generated by load setting system 54 to shaft 24. Drive shaft 56C may be linked to shaft 24 by specimen connecting collar 48. Drive motor 56A may spin drive shaft 56C through drive system 56B. Drive shaft 56C may spin shaft 24 through connecting collar 48 as force F is applied to shaft 24.

Load cell 50 may be positioned in line with load setting means 54 and shaft 24. Load cell 50 may be any kind of system that measures a load or applied force. Load cell 50 may comprise strain gauges and a structural beam that deflects under load. Load cell 50 may comprise a hydrostatic system.

Transducer system 52 may be any kind of system or component of suitable accuracy and range for measuring linear translation or displacement. Transducer system 52 may include transducer 52A comprising a linear transformer, a piezoelectric component, an inductive component, a capacitive component or a laser. Transducer system 52 may further include target plate 52B. Target plate 52B may be connected to shaft 24 or spin drive shaft 56C or other appropriate rotating component. Target plate 52B may be configured to work with transducer 52A as a target to measure displacement of the plate 52B. Transducer system 52 may measure axial displacement of housing 4 in relation to shaft 24.

Load setting means 54 may be a manual system such as a manual turn screw as shown or a hydraulic or manual jack. Load setting means 54 may be an automatic system controlled by a processor such as processor 62. Load system electronics 58 may control load application system 54 and may collect data from load cell 50 and transducer 52.

Data collection system 60 may include means for collecting data such as data acquisition board 66. Data acquisition board 66 may be operably connected to processor 62 and memory 64 and may be further operably connected to load system 42. Data acquisition board 66 may additionally send commands to load system 42 to control operation. Data acquisition board 66 may collect data from load system 42. Board 66 may collect displacement data from transducer 52 and applied load values from load cell 50 and store the data in memory 64. Data acquisition board 66 may collect additional data. Data acquisition board 66 may be operably connected to load system electronics 58 and may operate cooperatively with load system electronics 58. Data acquisition board 66 may not be a separate board. Board 66 may be integrated into the functionality of processor 62 or other on board electronics.

Load system 42 may have no load system electronics 58 and data collection system 60 may collect data directly from load system 42. Alternatively, load system 42 may have load system electronics 58 and test system 40 may not include data collection system 60.

Load system 42 may operate by applying incremental positive and negative axial loads F to shaft 24 and measuring the displacement of rotating assembly housing 4 in relation to shaft 24 at each incremental load. Data collection system 60 may record the load data from load cell 50 and the displacement data from transducer 52 at each incremental load. Values may be generated with shaft 24 rotating or with shaft 24 static. Each data point may be recorded as a first and second component such as a force component and a displacement component.

Incremental load here shall mean a set of load magnitude values that may be applied in any order to the specimen using load cell 50 and load setting means 54. In general the set of load values is of sufficient number to characterize test specimen 41 and are bounded at the outer limit by values which might cause plastic deformation of any component in rotating assembly 2. The incremental load may be continuously ramped and increments defined by collection of displacement data.

Generation of force-displacement curves and their use with rotating assemblies is well known and widely used by those skilled in the art. Comprehensive descriptions of the concepts are available in many textbooks and will not be repeated here. Force- displacement curves are similar to stress-strain curves of material science and may reflect mechanical property characteristics.

In operation, test system 40 may be used to determine a preload condition for test specimen 41. A force-displacement curve may be derived for test specimen 41 and compared to a hypothetical reference curve or master curve of zero preload and endplay to graphically determine the preload magnitude.

Spindle displacement under a certain load may be a combination of bearing displacement and displacement of other bearing spindle components. Fig. 4 includes bearing displacement curve BD, component displacement curve CD and spindle displacement curve SD. Spindle displacement at a set load is the sum of the displacement of all components. Here the magnitude of the spindle displacement is illustrated by line Ds. Spindle displacement Ds is the sum of the component displacement Dc and bearing displacement Db. Bearing displacement Db will contribute the greatest portion of the displacement in a force displacement curve for most tested bearings.

A force-displacement curve may be generated from a mathematical model. A mathematical model of rotating assembly 2 comprising the material characteristics of each component may be developed to describe the displacement of rotating assembly 2 or displacement of components of rotating assembly 2. The mathematical model may describe the displacement of components in response to applied forces. The mathematical model may use finite element techniques.

Fig. 5 shows a hypothetical master curve of a rotating assembly or spindle which includes spindle characteristics in both axial directions. Master curve MC is a specific kind of force displacement curve for an assembly with zero endplay and zero preload. Master curve MC may comprise a positive portion or push side P+ and a negative portion or pull side P-. Push side P+ of master curve MC corresponds to displacement of a first row of bearings in a two row bearing configuration of a rotating assembly. Pull side P- corresponds to displacement of a second row of bearings in the two row bearing configuration.

Master curve MC may be a curve fit to data points collected by test system 40. Curve MC may have values associated with the fit curve that vary from the data points. Master curve MC may be used in determining the preload force of a rotating assembly of identical design. Master curve MC may be described by an equation and may comprise a set of values.

Master curve MC may be generated based on a set of force-displacement values for a rotating assembly set to a condition of endplay. The values for master curve MC may be defined by subtracting the endplay contribution from each displacement component of the force-displacement values. The resulting force-displacement values will correspond to a rotating assembly with zero endplay and zero preload.

Bearings may be specified or grouped by bearing types, bearing series and bearing families. The bearing type reflects the construction and design of the bearing. Some bearing types include roller bearings, tapered roller bearings, spherical bearings, angular contact ball bearings, single row ball bearings and double row ball bearings.

The bearing series reflects the robustness of the bearing. The bearing series are, from lightest to heaviest: 8 extra thin section; 9 very thin section; 0 extra light; 1 extra light thrust; 2 light; 3 medium and 4 heavy. Each of these series also establishes a relationship between the bore size, outer diameter, and thickness of the bearing.

A bearing family may include a group of substantially identical bearings where members are differentiated by one or more parameters. For example, a family of bearings may be identical except for the contact angle. Another family may be identical except for the size of ball used. Other parameters that may differentiate bearing family members include the number of rotating elements, diameter of the bearing or radius of curvature of a race contacting a ball. The bearing family may be designed for a specific application such as machine spindles or high speed motors.

A bearing family may be defined by sharing a common series designation. A bearing family may be defined by a common bearing type. A bearing family may be defined by both a common series designation and a common bearing type.

Fig. 6 is a force-displacement chart showing a set of master curves for a bearing family where family members are identical except for contact angle 34. This example family includes 10 degree, 12 degree and 13 degree contact angle bearings. Fig. 6 includes force-displacement curves for 10, 12, 11 and 13 degree bearings and values MC(x) with its tangent line Tl, FC(x) with its tangent line T2, tangent normal TN to tangent line T2 and value MC(n). While only the first quadrant of the chart is shown for clarity, all of the techniques and concepts apply equally to the data of the third quadrant. Master curve MC for an 11 degree contact angle bearing, shown as a dotted line, may be derived from a limited amount of testing of an 11 degree specimen. Given the value MC(x) and slope of the curve or tangent Tl through MC(x) for the 11 degree bearing at a fixed force value such as 160 on the graph, a complete master curve MC may be derived for the 11 degree bearing using master curves for other family curves and a correction factor of the ratio of the slopes at a fixed force value.

Within a family of bearings the ratio of the slope at corresponding values of each curve will be substantially constant. Similarly the ratios of stiffness between two assemblies will be substantially equivalent at corresponding values of the curves. The stiffness of the 10 degree bearing is known at each point, therefore the slope for the 11 degree bearing at each corresponding value can be determined as the stiffness ratio will be substantially constant at each corresponding value.

Corresponding points of the curves can be defined by the curve intercepts of an isoforce or horizontal line. For example, still referring to Fig. 6, an isoforce line such as 160 as noted in the chart may define corresponding values MC(x) and FC(x). Corresponding points of the curves can instead be defined by the curve intercepts of an isodisplacement or vertical line (not shown). Corresponding points of the curves can also be defined by the curve intercepts of a normal of the tangent at each value of the known force displacement curve. For example, a normal TN of the tangent line T2 may define corresponding values MC(n) and FC(x). Alternatively any other method may be used to identify corresponding points that maintains an adequate accuracy in defining correction factor CF.

If the curve for the 10 degree bearing was described by a polynomial of the form y = k * x ³ + j * x + l (l) the curve for the 11 degree bearing may have the same form except for the values of the coefficients. Master curve MC for the bearing may be derived by selecting coefficients such that the curve passes through the origin and through point MC(x) with a slope equal to the line Tl. The coefficients may be determined using correction factor CF where

CF= slope Tl/ slope T2= K _τl /K _τ2 (2)

Even with strict dimensional tolerances during manufacture, the variation of contact angle 34 between bearing units may be plus or minus one degree. So a bearing specified as a 15 degree bearing may have a contact angle of 14 degrees or 16 degrees and still be within manufacturing specifications. Fig. 7 is a chart illustrating the effect of contact angle 34 on the apparent stiffness of a bearing. The 15 degree bearing of the chart is arbitrarily set as a ratio of one. A bearing of 14 degrees will have a ratio of 0.87 and a bearing of 16 degrees will have a ratio of 1.13. This is a significant variation. Using a master curve for a 15 degree bearing where the bearing actually has a contact angle of 14 degrees or 16 degrees will have a significant impact on the accuracy of the determined preload.

Fig. 8 is a force-displacement chart including a test curve TC corresponding to rotating assembly 2 under a preload and master curve MC. Test curve TC includes first inflection point IPl and second inflection point IP2. The force components of inflection points IPl and IP2 may correspond to the force required to unseat one set of bearings in a two row rotating assembly 2. Locating inflection points IPl and IP2 may include translating a portion of master curve MC from the positive region of the chart to overlay the portion of test curve TC beyond inflection point IPl. Displaced master curve MC+ is shown as a dotted line. At the point displaced master curve MC+ intercepts the x-axis a vertical line L2 is drawn to intercept test curve TC. This intercept defines the inflection point IP2. A similar method is used to define displaced master curve MC- with a vertical line Ll to locate inflection point IPl. The y-intercepts of curves MC+ and MC- may define the preload force as indicated by positive preload F(p+) and negative preload F(p-).

The distance between lines Ll and L2 may define dimensional preload D(p). Dimensional preload D(p) may be comprised of a positive portion D(p+) and a negative portion D(p-). Displaced master curve MC+ may be translated a distance D(p+) to overlay the portion of test curve TC beyond inflection point IPl. Displaced master curve MC- may be translated a distance D(p-) to overlay the portion of test curve TC beyond inflection point IP2.

Test curve TC may be a curve fit to a set of data points similar to data points Ap acquired using test system 40. Test curve TC may be described by an equation and may comprise a set of values.

The vertical distance between test curve TC and master curve MC at a fixed displacement value in the region beyond inflection point IPl also corresponds to a preload force F(pl) of the bearings of test specimen 41. Positive preload force F(p+), negative preload force F(p-) and preload force F(pl) may refer to the same preload force and may be equivalent, but may be derived by different graphical or mathematical methods.

The horizontal distance between test curve TC and master curve MC at a fixed force value in the same region corresponds to a preload dimension D(pl). Dimensional preload D(pl) may correspond to dimensional preload D(p-). Dimensional preload D(pl) and dimensional preload D(p-) may be derived by different graphical or mathematical methods.

Fig. 9 is a detail of the force-displacement curves of Fig. 8 indicated by a dotted line box C in Fig. 8 located beyond inflection point IPl. Fig. 9 again includes test curve TC and master curve MC. Test curve slope K(tc) of test curve TC is indicated by the ratio of the rise over run between two points of test curve TC which corresponds to the stiffness or elasticity of test specimen 41. Master curve slope K(mc) of master curve MC is indicated by the ratio of the rise over run between two points of master curve MC. Again, the magnitude of preload F(pl) measured as a force is the difference at a constant displacement value between test curve TC and master curve MC. The measurement point on each curve are indicated here by test value TC(pt) on test curve TC and master value MC(ptl) on master curve MC. For a test specimen and a master curve specimen of identical design, contact angle and materials, slopes K(mc) and K(tc) will be equal. In reality, variations in contact angle 34 and variations in materials and dimensions between individual specimens results in slopes K(mc) and K(tc) which are not equal. If slope K(mc) of master curve MC were equal to K(tc), the outer region of master curve MC may only be offset and equidistant from the outer region of master curve MC.

Where K(mc) and K(tc) are not equal the measured preload force F(pl) may not be accurate. Correction factor CF applied to master curve MC or test curve TC may produce a corrected curve that may result in a more accurate value than preload force F(pl). Referring still to Fig. 9, the magnitude of a corrected preload force F(corr) may be determined by the distance between test curve TC and a corrected master curve MC(corr) at a constant displacement. These measurement points are indicated by MC(pt2) on master curve MC and TC(pt) on test curve TC. The difference between corrected preload force F(corr) and uncorrected preload force F(pl) is indicated by F(cf) and

F(cf) = F(pl) - F(corr) (3) where F(pl) is the difference in force components of two values on test curve TC and master curve MC corresponding to measured data points. Corrected preload force F(corr) may compensate and correct for sources of error and result in significantly more accurate values than preload force F(pl).

As an example, the test specimen used to generate test curve TC in test system 40 may be identified in the family of curves of Fig. 6 as having a contact angle of 11 degrees. In comparing test curve TC to the master curve MC for a bearing with a 10 degree contact angle, it may become apparent that slopes K(mc) and K(tc) are not equal at corresponding values. Test specimen 41 may have a contact angle of 11 degrees which is within tolerance for test specimen 41. Using master curve MC for the 10 degree bearing will not provide an accurate preload value. Using correction factor CF master curve MC for the 10 degree bearing can be corrected to reflect a zero load master curve for contact angle 34 of test specimen 41 under test, in this case 11 degrees. Preload force F(corr) can then be determined.

Fig. 9A also illustrates a portion of Fig. 8 indicated by a dotted line box C. Here a dimensional preload D(pl) is determined using a similar method to Fig. 9. The measurements for dimensional preload D(pl) are taken at a constant force value, an isoforce line or horizontal line indicated here as 3.9. Preload dimension D(pl) is the horizontal distance between point MC(ptl) of the master curve MC and point TC(pt) of the test curve TC. The corrected preload dimension D(corr) is the distance between test curve point TC(pt) and corrected master curve point MC(pt2).

The difference between corrected preload dimension D(corr) and uncorrected preload dimension D(pl) is indicated by D(cf) and

D(cf) = D(pl) - D(corr) (3A) where D (pi) is the difference in dimension components of two values on test curve TC and master curve MC corresponding to measured data points. Corrected preload dimension D(corr) may compensate and correct for sources of error and result in significantly more accurate values than preload dimension D (pi).

Referring again to Fig. 8, corrected master curve MC(corr) may be developed by any number of methods. One method may include identifying the slope of test curve TC. Corrected master curve MC(corr) may comprise a line with slope K(tc) extended from an inflection point of master curve MC.

Another method for developing a corrected master curve MC(corr) may include defining correction factor CF in a similar manner to the stiffness ratio as a relation between master curve slope K(mc) and test curve K(tc) or

CF = K(tc) / K(mc). (4)

Correction factor CF may be applied to one or more values of master curve MC to define one or more corrected data values which are used to derive corrected preload force F(corr).

Correction factor CF may be applied to a value of master curve MC used to determine the magnitude of preload force F(pl). Correction factor CF may simply be applied to master curve value MC(ptl) resulting in a corrected master curve value

MC(pt2) = CF * MC(ptl). (5)

Fig. 10 illustrates another method for developing corrected master curve MC(corr). A segmented master curve MC(seg) may be defined by taking a subset of values constituting master curve MC and fitting line segments between sequential data points or values. Values of MC(seg) may include Pl(seg), P2(seg) and IP(seg) which may be defined as an inflection point of MC(seg). Master curve MC is shown as a dotted line and slightly offset from MC(seg) for clarity.

Corrected master curve MC(corr) may be developed using correction factor CF. Starting at point (0,0) of MC(seg), correction factor CF may be applied to the slope of each segment in a point to point or iterative manner. For example, the line segment between Pl(seg) and P2(seg) may have a formula P2(seg)(y)-Pl(seg)(y) =m(P2(seg)(x)-Pl(seg)(x)) where m is the slope of the line between the points and P2(seg)(y)-Pl(seg)(y) is the difference of the vertical components of sequential points Pl(seg) and P2(seg). P2(seg)(x)-Pl(seg)(x) is the difference of the horizontal components of sequential points Pl(seg) and P2(seg).

Next, value P2(corr) of corrected master curve MC(corr) is defined. Pl(corr) was determined in the previous iteration and by definition the x components of P2(corr) and P2(seg) are equal and P2(corr)(x)=P2(seg)(x). In finding the line segment between Pl(corr) and P2(corr), a similar formula is used where (P2(corr)(y)-Pl(corr)(y)) =m(corr)*(P2(corr)(x)-Pl(corr)(x)). Here the slope m(corr) will equal slope m of the line between Pl(seg) and P2(seg) multiplied by CF or m(corr) = m * CF. (6)

Iterating over all selected points and their segments results in a corrected master curve MC(corr) with a corrected inflection point IP(corr).

This specific algorithm for segments generating a corrected master curve is an example. Other similar algorithms may be used and still fall within the scope of this disclosure.

In another method of creating a corrected master curve MC(corr) curve fitting and calculus methods may be used. Fig. 11 illustrates a calculus method of creating a corrected curve and includes master curve MC and corrected master curve MC(corr). Master curve MC may be a curve and may be generated by applying best fit techniques to a set of data points acquired in testing a set of rotating assemblies 2. A subset of data points that may correspond to values of master curve MC are represented as points Ap in the dotted line oval. Displacement components and force components for this chart may be represented by x and y notation.

Master curve MC may be described by an equation that corresponds to a set of master data values that constitute the curve. Any point (xl,yl) on master curve MC may have a corresponding tangent or slope Tang(xl,yl). Master curve MC may have an inflection point IP(mc). Corrected master curve MC(corr) may also have an inflection point IP(corr) and any point (x2,y2) on corrected master curve MC(corr) may have a slope Tang(x2,y2).

Curve fitting techniques are well known by those skilled in the art. Master curve MC fit to data points Ap may be described by an equation with any number of terms and may be expressed generally as

Y = L a * x" (7) and the slope for this curve at any point may be expressed as the derivative of equation 7: y' = L n * a * x - ¹ (8)

One general example of an equation with two terms for a fit curve may be the polynomial: y = a * x ² + b * x (9) where a and b and c are constants and the curve passes through the origin. This or a similar equation with more or fewer terms may describe master curve MC. The slope at each point of this polynomial or on master curve MC may then be described by the first derivative of equation 9: y' = 2 * a * x + b (10) which may correspond to the slope or tangent Tang( xl,yl).

Creating corrected master curve MC(corr) may include applying correction factor CF to the slope of master curve MC. In one example the slope or tangent of corrected master curve MC(corr) may be described by the equation: y(corr)' = (2 * a * x + b) * CF = 2 * a * x * CF + b * CF. (11)

This may correspond to slope or tangent Tang ( x2,y2) of corrected master curve MC(corr).

Using the techniques of calculus, corrected master curve MC(corr) may be defined by the integral of equation 11 or:

MC(corr) = a * CF * x ² + b * CF * x + d. (12)

Here d is a constant that may be determined by evaluating corrected master curve MC(corr) represented by equation 12 where the value of x is equal to zero and MC(corr) is equal to zero.

Master curve MC may be approximated by more than one fit curve, where each curve approximates different subsets of data points Ap. For example the section between zero and the inflection point IP(mc) may be described by a polynomial such as equation 9 and the region beyond inflection point IP(mc) may be described by an equation for a different curve or a straight line. The equations for different sections may be substantially continuous at the boundaries to the corresponding subsets of data points Ap. Similar methods as described in Fig. 9 may be used to determine preload using this corrected master curve MC.

While a second degree polynomial has been used in equation 9, this is for the purposes of illustration only. An equation of a higher order or lower order may be used to describe a curve fit to data points Ap. Data points Ap that correspond to master curve MC may be acquired from testing multiple master assemblies and the data values of collected data averaged.

In another example one rotating assembly 2 may be used to develop both master curve MC and test curve TC. Data points Ap corresponding to master curve MC may be acquired from testing test specimen 41. Test specimen 41 may first be assembled such that the shaft is in an endplay condition. Test specimen 41 may be mounted in test system 40. Data points Ap acquired using test system 40 may be used to define curve A and master curve MC using methods illustrated in Fig. 9. Test specimen 41 may then be removed from test system 40, reassembled with a set preload and remounted in test system 40. Another set of data points acquired using test system 40 on reassembled and preloaded test specimen 41 may be used in fitting test curve TC. Similar techniques to those described above may then be used to determine preload force F(corr) or preload dimension D(corr) from the difference between test curve TC and master curve MC, where both test curve TC and master curve MC are derived from the same assembly in different preload conditions.

Correction factor CF may be applied to test curve values to derive a corrected test curve rather than applying correction factor CF to master curve values to derive corrected master curve MC(corr). Similar methods as described above can be used to determine a corrected preload value from a corrected test curve. Corrected master curve MC(corr) was used here as an example only. The preload force can similarly be derived using the negative portion of the curves in a similar manner as illustrated above. The positive section was used for example only.

Fig. 12 is a translating bearing 100 that includes first bearing 102 and second bearing 104. Translating bearing 100 also comprises a top housing 106 and bottom housing 108 that supports first and second bearings 102 and 104. Bearings 102 and 104 support shaft or beam 110. Beam 110 can move or translate between bearings 102 and 104 as indicated by dotted line beam 110'. Housings 106 and 108 may exert a force against bearings 102 and 104 as a preload.

To determine a preload value F(pl), a force F may be applied in increments to shaft or beam 110 by test system 40. Measured force-displacement values may be used to define at least a portion of test curve TC. In a similar manner to that discussed above for rotating assembly 2, a preload F(pl) can be determined by the distance between test curve TC and master curve MC.

Fig. 13 is a flow chart illustrating a method 200 for determining a preload on a set of bearings of a test spindle with a test spindle housing and a test spindle shaft. In step 202 a first set of reference values is defined for a reference spindle with a housing and a shaft. Each reference value may include a force component corresponding to a reference force applied to the reference spindle shaft and a displacement component corresponding to the displacement of the reference spindle shaft. The reference value and reference spindle may be an average of a plurality of spindle units of identical design. In step 204 a reference elastic constant is determined from the first set of reference values.

In step 206 a second set of test values is determined for the test spindle. Each value may include a test force component corresponding to a force applied to the test spindle shaft and a displacement component corresponding to displacement of the test spindle shaft in relation to the test spindle housing in response to the applied test force. In step 208 a test elastic constant is determined from the second set of test values. In step 210 a correction factor is defined. The correction value may be defined as a function of the reference elastic constant and the test elastic constant. In step 212 a third set of corrected values is defined by applying the correction factor to at least one value of a set of values selected from a values group. The values group may consist of the first set of reference values and the second set of test values. In step 214 the preload on the set of bearings of the test spindle is determined using at least one value from the third set of corrected values and at least one value from the set of values not selected from the values group in defining the third set of values. So if the corrected values are defined using at least one value from the reference values, the preload is determined using at least one value from the corrected values and the second set of test values. If the corrected values are defined using the second set of test values, then the preload is determined using at least one value from the corrected values and at least one value from the first set of reference values.

Fig. 14 is a flow chart illustrating a method 300 for generating a set of master curve values, each value including a displacement component and a force component. Method 300 comprises steps including step 302 of selecting a set of reference values for a first bearing, each value comprising a force component and a displacement component and step 304 of determining a first elasticity constant for the first set of reference values. In step 306 a second elasticity constant is determined for a second bearing at a corresponding value. In step 308 a correction factor is created from the ratio of the first elasticity constant to the second elasticity constant. In step 310 the correction factor is applied to the first set of bearing reference values. Step 312 derives the set of master curve values.

The described system and assemblies are examples and are not to be used as limitations. Other configurations of test systems which perform the same function fall within the scope of this disclosure. Similar methods which perform the same function also fall within the scope of this disclosure.

While embodiments of a test system and methods of use have been particularly shown and described, many variations may be made therein. This disclosure may include one or more independent or interdependent inventions directed to various combinations of features, functions, elements and/or properties, one or more of which may be defined in the following claims. Other combinations and sub-combinations of features, functions, elements and/or properties may be claimed later in this or a related application. Such variations, whether they are directed to different combinations or directed to the same combinations, whether different, broader, narrower or equal in scope, are also regarded as included within the subject matter of the present disclosure. An appreciation of the availability or significance of claims not presently claimed may not be presently realized. Accordingly, the foregoing embodiments are illustrative, and no single feature or element, or combination thereof, is essential to all possible combinations that may be claimed in this or a later application. Each claim defines an invention disclosed in the foregoing disclosure, but any one claim does not necessarily encompass all features or combinations that may be claimed. Where the claims recite "a" or "a first" element or the equivalent thereof, such claims include one or more such elements, neither requiring nor excluding two or more such elements. Further, ordinal indicators, such as first, second or third, for identified elements are used to distinguish between the elements, and do not indicate a required or limited number of such elements, and do not indicate a particular position or order of such elements unless otherwise specifically stated.