Abstract:
A tire manufacturing method includes a method for optimizing the uniformity of a tire by reducing the green tire radial runout. The green tire radial runout is modeled as a vector sum of each of the vectors representing contributions arising from the tire building steps. A set of vector coefficients is generated from the vector equation. The building steps include building the tire carcass, building the tire summit, transferring the summit onto the inflate carcass, and measuring the radial runout and tooling angles at each step in the process. After the model is built, the vector equations and coefficients are applied to subsequent tires. By adjusting the tooling angles, green tire radial runout can be optimized.

Description:
CROSS REFERENCE TO PREVIOUSLY FILED APPLICATIONS 
       [0001]    This application is a continuation-in-part of U.S. application Ser. No. 12/482,787, filed Jun. 11, 2009, which is a continuation-in-part of U.S. application Ser. No. 11/172,060, filed Jun. 30, 2005, which is a continuation in part of PCT application “Tire Manufacturing Method For Improving The Uniformity Of A Tire”, assigned PCT/US2004/039021, filed Nov. 19, 2004, which is a continuation-in-part of PCT application “Tire Manufacturing Method For Improving The Uniformity Of A Tire”, assigned PCT/IB2003/006462, filed Nov. 21, 2003. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    The present invention relates to a manufacturing method for tires, more specifically a method for improving the uniformity of a tire by reducing the green (uncured) tire radial runout. In a tire, and more precisely, a radial tire, the green tire radial runout (RRO) can be affected by many variables introduced from the process of assembly of the green tire. When the radial runout in a tire exceeds acceptable limits, the result may be unwanted vibrations affecting the ride and handling of the vehicle. For these reasons, tire manufacturers strive to minimize the level of radial runout in the tires delivered to their customers. 
         [0003]    A well-known and commonly practiced method to improve the radial runout is to grind the tread surface of the tire in the zones corresponding to excess tread. This method is effective, but has the drawback of creating an undesirable surface appearance and of removing wearable tread rubber from the product. In addition, this method requires an extra manufacturing step and uses expensive equipment. Another approach is disclosed in U.S. Pat. No. 5,882,452 where the before cure radial runout of the tire is measured, followed by a process of clamping and reshaping the uncured tire to a more circular form. 
         [0004]    Still another approach to a manufacturing method for improved uniformity involves a method where the factors relating to tire building and tire curing that contribute to after cure RRO or Radial Force Variation (RFV) are offset relative to a measured before cure RRO. An example of a typical method is given in Japanese Patent Application JP-1-145135. In these methods a sample group of tires, usually four, are placed in a given curing mold with each tire rotated an equal angular increment. The angular increment is measured between a reference location on the tire, such as a product joint, relative to a fixed location on the curing mold. Next, the tires are vulcanized and their composite RFV waveforms recorded. The term “composite waveform” means the raw waveform as recorded from the measuring device. The waveforms are then averaged by superposition of each of the recorded waveforms upon the others. Superposition is a point by point averaging of the recorded waveforms accomplished by overlaying the measured composite waveform from each tire. The effects of the vulcanization are assumed to cancel, leaving only a “formation” factor related to the building of the tire. In like manner, another set of sample tires is vulcanized in a curing mold and their respective RFV waveforms are obtained. The respective waveforms are again averaged by superposition, this time with the staring points of the waveforms offset by the respective angular increments for each tire. In this manner, the effects of tire building are assumed to cancel, leaving only a “vulcanization factor.” Finally, the average waveforms corresponding to the formation factor and the vulcanization factor are superimposed. The superimposed waveforms are offset relative to each other in an attempt to align the respective maximum of one waveform with the minimum of the other waveform. The angular offset thus determined is then transposed to the curing mold. When uncured tires arrive at the mold, each tire is then placed in the mold at the predetermined offset angle. In this manner, the formation and vulcanization contributions to after cure RFV are said to be minimized. A major drawback to this method is its assumption that the formation and vulcanization contributions to after cure RFV are equivalent for each tire. In particular, the factors contributing to the formation factor can vary considerably during a manufacturing run. In fact, these methods contain contradictory assumptions. The methodology used to determine the vulcanization factor relies on an assumption that the step of rotation of the tires in the curing mold cancels the tire building (or formation) effects. This assumption is valid only when the contribution of before cure RRO is consistent from one tire to the next tire, without random contributions. If this assumption is true, then the subsequent method for determination of the formation factor will produce a trivial result. 
         [0005]    Further improvements have been proposed in Japanese Patent Application JP-6-182903 and in U.S. Pat. No. 6,514,441. In these references, methods similar to those discussed above are used to determine formation and vulcanization factor waveforms. However, these methods add to these factors an approximate contribution of the before cure RRO to the after cure RFV. The two methods treat the measured before cure RRO somewhat differently. In the method disclosed in reference JP-6-198203 optimizes RRO effects whereas the method disclosed in U.S. Pat. No. 6,514,441 estimates RFV effects by application of a constant stiffness scaling factor to the RRO waveform to estimate an effective RFV. Both these methods continue to rely on the previously described process of overlapping or superpositioning of the respective waveforms in an attempt to optimize after cure RFV. 
         [0006]    The most important shortcoming of all the above methods is their reliance of superpositioning or overlapping of the respective waveforms. It is well known in the tire industry that the vehicle response to non-uniformity of RRO is more significant in the lower order harmonics, for example harmonics one through five. Since, the above methods use composite waveforms including all harmonics, these methods fail to optimize the RRO harmonics to which the vehicle is most sensitive. In addition, a method that attempts to optimize uniformity using the composite waveforms can be shown, in some instances, to produce RRO that actually increases the contribution of the important lower order harmonics. In this instance, the tire can cause more vehicle vibration problems than if the process were not optimized at all. Therefore, a manufacturing method that can optimize specific harmonics and that is free of the aforementioned assumptions for determining the effects of tire formation and tire vulcanization would be capable of producing tires of consistently improved uniformity. U.S. Pat. No. 6,856,929, owned in common with the present application, applies a similar approach to solving RFV non-uniformity. 
       SUMMARY OF THE INVENTION 
       [0007]    In view of the above background, the present invention provides a tire manufacturing method that can effectively reduce the before cure radial runout (RRO) of each tire produced. The method of the present invention operates to independently optimize each harmonic of RRO. A composite RRO signal, such as those described above, is a scalar quantity that is the variation of the tire&#39;s radial runout at each angular position around the tire. When this composite is decomposed into its respective harmonic components, each harmonic of RRO can be expressed in polar coordinates as a before cure RRO vector. This vector has a magnitude equal to the peak-to-peak magnitude of the distance variation of the respective harmonic and an azimuth equal to the angular difference between the measuring reference point and the point of maximum RRO. 
         [0008]    The invention provides a method for improving the uniformity of a tire comprising: gathering data to build a model of radial runout of a tire, and comprising the sub-step of extracting at least one harmonic of radial runout of said tire; deriving a vector equation as a sum of vectors corresponding to the contributors to green-tire radial runout; determining a set of vector coefficients from the vector equation; building said tire with a predetermined level of green tire radial runout; and applying the said vector equation and vector coefficients to future tires. 
         [0009]    The invention further provides wherein the step of gathering data to build the model comprises: recording a carcass building drum identification; building a tire carcass; recording an angle at which the carcass is loaded onto said building drum; inflating the tire carcass and measuring radial runout measurements of the carcass; recording identification for a Summit Building Drum; recording an angle at which the summit is loaded onto said Summit Building Drum; building a tire summit; obtaining a radial runout measurement of the tire summit; recording a Transfer Ring identification; transferring the summit from said Summit Building Drum onto the inflated tire carcass; recording a Transfer Ring angle; and obtaining green tire radial runout measurements. 
         [0010]    The method of the present invention provides a significant improvement over previous methods by employing a vectorial representation of the several factors that contribute to the measured before cure RRO for a tire produced by a given process. The before cure RRO vector is modeled as a vector sum of each of the vectors representing RRO contributions arising from the tire building steps—the “tire room effect vector.” For a series of tires, the method obtains such measurements as the before cure radial runout (RRO) at one or more stages of the building sequence and measurements of loading angles on the tire building tools and products. 
         [0011]    The present invention further improves on previously described methods since it does not rely on manipulation of the measured, composite RRO waveforms to estimate the tire room effects and does not rely on any of the previously described assumptions. The present invention uses the aforementioned measured data as input to a single analysis step. Thus, the coefficients of all the sub-vectors are simultaneously determined. Once these coefficients are known, the tire room effect vector is easily calculated. In summary, the first step of the method comprises gathering data, including carcass radial runout, summit radial runout and green tire radial runout in order to model at least one harmonic of radial runout of the tire; deriving a vector equation as a sum of vectors corresponding to the contributors to green-tire radial runout; determining a set of vector coefficients from the vector equation; and minimizing radial runout, or alternatively building intentionally out of round tires, by applying the gathered data to future tires. 
         [0012]    The method of the invention has an additional advantage owing to its simultaneous determination of the sub-vectors. Unlike previous methods, the method of the invention does not require any precise angular increments of the loading positions to determine the sub-vectors. This opens the possibility to continuously update the sub-vector coefficients using the measured data obtained during the production runs. Thus, the method will take into account production variables that arise during a high volume production run. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0013]    The invention will be better understood by means of the drawings accompanying the description, illustrating a non-limitative example of the execution of the tire manufacturing method for improving the uniformity of a tire according to the invention. 
           [0014]      FIG. 1  is a schematic representation of a tire manufacturing process equipped to practice the method of the invention 
           [0015]      FIGS. 2A-2C  depict schematic representations of radial runout of the tire showing the original composite waveform as well as several harmonic components. 
           [0016]      FIG. 3  is a vector polar plot showing the various contributors to green tire radial runout and the resulting radial runout. 
           [0017]      FIG. 4  is a vector polar plot showing the various contributors to green tire radial runout and the resulting radial runout after optimization. 
           [0018]      FIG. 5  is a vector polar plot showing the estimated summit radial runout vector as the difference between the green tire radial runout vector and the carcass radial runout vector. 
           [0019]      FIG. 6  is a vector polar plot showing the two groupings of vector contributors as well as the resulting radial runout. 
           [0020]      FIG. 7  is a vector polar plot showing the two groupings of vector contributors as well as the resulting radial runout after optimization. 
       
    
    
     DETAILED DESCRIPTION 
       [0021]    Reference will now be made in detail to exemplary versions of the invention, one or more versions of which are illustrated in the drawings. Each described example is provided as an explanation of the invention, and not meant as a limitation of the invention. Throughout the description, features illustrated or described as part of one version may be usable with another version. Features that are common to all or some versions are described using similar reference numerals as further depicted in the figures. 
         [0022]    Modern pneumatic tires are generally manufactured with great care and precision. The tire designer&#39;s goal is that the finished tire is free of non-unifom 1 ity in either the circumferential or lateral directions. However, the designer&#39;s good intentions notwithstanding, the multitude of steps in the tire manufacturing process can introduce a variety of non-uniformities. An obvious non-uniformity is that the tire may not be perfectly circular (radial runout or RRO). Another form of non-uniformity is radial force variation (RFV). Consider a tire mounted on a freely rotating hub that has been deflected a given distance and rolls on a flat surface. A certain radial force reacting on the flat surface that is a function of the design of the tire can be measured by a variety of known means. This radial force is, on average, equal to the applied load on the tire. However, as the tire rolls, that radial force will vary slightly due to variations in the internal tire geometry that lead to variations in the local radial stiffness of the tire. These variations may be caused on the green tire by localized conditions such as product joints used in the manufacture of the green tire, inaccurate placement of certain products. The process of curing the tire may introduce additional factors due to the curing presses or slippage of products during curing. An example of a tire manufacturing method for improving the radial force variation of a tire is disclosed in U.S. Pat. No. 6,856,929 and US 2011/011425, both commonly owned by the applicant of the present invention and incorporated by reference herein in their entirety and for all purposes. 
         [0023]      FIG. 1  shows a simplified depiction of the tire manufacturing process. A tire carcass  10  is formed on a building drum  15 . In a unistage manufacturing process, the carcass  10  remains on the drum  15 . In a two-stage process, the carcass  10  would be removed from the drum  15  and moved to a second stage finishing drum. In either case, the carcass  10  is inflated to receive a finished tread band  20  to produce the finished green tire  30 . In one variation of the invention, the RRO of the green tire  30  is measured by a measurement system  70  using a barcode  35  as a reference point. The RRO waveform is stored, here in a computer  80 . The green tire  30  is moved to the curing room where the tire is then loaded into a curing cavity  40  and cured. The cured tire  30 ′ is moved to a uniformity measurement machine  50  for measurement and recording of the tire RFV. 
         [0024]      FIG. 2A  shows a schematic of the measured RRO for a green tire  30 . The abscissa represents the circumference of the tire and the ordinate the radial runout variations.  FIG. 2A  is the as—measured signal and is referred to as a composite waveform. The composite waveform may comprise an infinite series of harmonics. The individual harmonics may be obtained by applying Fourier decomposition to the composite signal.  FIGS. 2B and 2C  depict the resulting first and second harmonics, respectively, extracted fom 1  the composite signal. The magnitude of the first harmonic of radial runout FRM 1  is defined as the difference between the maximum and minimum distances. The phase angle or azimuth of the first harmonic FRAI is defined as the angular offset between the reference location for the measurement and the location of maximum radial distance. Thus, the sine wave depicted by Cartesian coordinates in  FIG. 2B  can be equally shown as a vector in a polar coordinate scheme. Such a vector polar plot is shown in  FIG. 2C  immediately to the right of the sine wave plot. The RRO vector of the first harmonic FRH 1  has a length equal to FRM 1  and is rotated to an angle equal to the azimuth FRAI. In a similar manner, one can extract the second harmonic vector FRH 2  shown in  FIG. 1C  that has a force magnitude FRM 2  and an azimuth FRA 2 . The corresponding polar plot for the H 2  vector resembles the H 1  vector, except that the azimuth angle is now two times the angular coordinate. 
         [0025]    In the description of an example of the method that follows, the particular example is confined to the optimization of the first harmonic H 1 . However, it is within the scope of the present invention to apply the method to optimize a different harmonic such as H 2 , H 3 , etc. The following example describes the optimization of radial runout. 
         [0026]      FIG. 3  is a vector polar plot showing the contributors to first harmonic of the green tire radial runout when no optimization has been applied. These include the various tooling vectors, product vectors, an intercept vector and the variable magnitude vectors. The tooling vectors are the 1st (ii) and 2nd (iii) stage building drum vectors, the Summit Building Drum vector (iv) and the Transfer Ring vector (v). The building drums hold the carcass and summit as the tire is being built, while the Transfer Ring holds the summit as it is being placed onto the tire carcass. The product vectors are the belt ply vectors (vi and vii), cap vector (viii) and tread vector (ix). The belt ply is the protective steel belt, the cap is a nylon cover that goes over the belt ply and the tread is interface between the tire and the ground. The green tire radial runout is the vector sum of the other components. The remaining, unidentified factors are consolidated in the Intercept vector I 1  (i). If all factors were known, then the Intercept vector I 1  would not exist. Throughout this disclosure, the Intercept vector I 1  accounts for the unidentified effects. A unique attribute of the invention is the ability to optimize the after cure uniformity by manipulation of the tooling and product vectors. The ability to treat these effects in vector space is possible only when each harmonic has been extracted. 
         [0027]    The measurement of green tire RRO (xii) is preferably at the completion of tire building and before the green tire is removed from the building drum  15 . The Carcass gain vector (x) and Summit gain vector (xi) are also shown in  FIGS. 3-5 . In the preferred method, the measurement drum is the tire building drum  15 , whether it is the single drum of a unistage machine or the finishing drum of a two-stage machine. The green tire RRO measurement may also be performed offline in a dedicated measurement apparatus. In either case, the radial runout of the measurement drum can introduce a false contribution to the Green RRO vector. When the green tire RRO is measured, the result is the sum of true tire runout and the runout of the drum used for measurement of RRO. However, only the green tire RRO has an effect on the after cure RFV of the tire. 
         [0028]      FIG. 4  now shows a schematic of the optimization step. In this view the vectors iv-ix have been rotated as a unit to oppose the variable vectors. It is readily apparent that this optimization greatly reduces the green tire radial runout. The steps for performing the optimization are provided below. 
         [0029]      FIG. 5  is a vector plot showing the summit radial runout vector as the difference between the measured green tire radial runout vector and the measured carcass radial rm 1 out vector. This computation can be used as equivalent to a direct measurement of the summit radial runout vector and obviates the need for taking the measurements for the summit. 
         [0030]      FIG. 6  is a vector polar plot showing the grouping of contributors previously shown in  FIG. 3  to the first harmonic of the green tire radial runout when no optimization has been applied. Reference number xiii is the resultant vector sum of constant vectors iv through ix and variable vector xi. Reference number xiv is the resultant vector sum of constant vectors i through iii and variable vector x. Reference number xii is the same green tire radial runout as shown in  FIG. 3 . 
         [0031]      FIG. 7  is a vector polar plot showing the grouping of contributors previously shown in  FIG. 3  to the first harmonic of the green tire radial runout after optimization has been applied. Reference number xiii is the resultant vector sum of constant vectors iv through ix and variable vector xi. Reference number xiv is the resultant vector sum of constant vectors i through iii and variable vector x. Reference number xii is the same optimized green tire radial runout as shown in  FIG. 4 . 
         [0032]    The foregoing graphical representations in vector space can now be recast as equation (1) below where each term represents the vectors shown in the example of  FIG. 3 . The method can be applied to additional effects not depicted in  FIG. 3  nor described explicitly herein without departing from the scope of the invention. 
         [0000]        FRH 1=( FRH 1 cr  Effect vector)+( FRH 1 sr  Effect vector)+(1st Stage Building Drum RRO vector)+(2nd Stage Building Drum RRO vector)+(Summit Building Drum RRO vector)+(Transfer Ring RRO vector)+(Belt1 Ply RRO vector)+(Belt2 Ply RRO vector)+(Cap RRO vector)+(Tread RRO vector)   (1)
 
         [0000]    The preceding equation applies to modeling the 1st harmonic of radial runout of a tire, but holds for other harmonics such as FRH 2 -FRH 5  as well. Each tire, either in a sample set of tires, or in a tire to be optimized during tire manufacturing will have its individual representation of Equation (1) with its individual set of vector components. 
         [0033]    The first step in implementation of the method is to gather data to build the modeling equation (1). The Green RRO of the finished tire and of the unfinished tire and tooling are measureable quantities. The challenge is to estimate the gain vectors, the product vectors, the tooling vectors and the intercept vector. This is accomplished by vector rotation and regression analysis. As will be described below, the RRO is measured for an initial or sample set of tires during tire building, according to the effects to be modeled. In the example herein, the model would include the 1st Stage Building Drum, the 2nd Stage Building Drum, the Transfer Ring, the Summit Building Drum, the Belt Plies, the Cap Ply, and the Tread. However, the model can be built with more or less of these effects being modeled. This would expand or reduce Equation (1) above. If Equation (1) contains N unknowns, then a sample set of M tires must be built and measured in order to achieve a solution, and M must be greater than or equal to N. In practice, the model will be more accurate if there are more than N sample tires built, typically about 30 to 50 tires. Each of these sample tires will have an intentional variation of the product loading angles on the tire tooling. It is also possible to extract the sample data from a much larger data set comprising normal production tires. Here, the normal variations that occur in the tire building process will provide the variations sufficient to build the model. 
         [0034]    First, a reference point on the tire is chosen, such as a barcode that is applied to the carcass or a product joint that will be accessible through then entire process is identified. Then the loading angles are measured relative to this reference point. A loading angle is the difference in the angle between the reference point on the tire and a reference point on the manufacturing tooling being effect modeled. The 1st Stage Loading Angle is CBD_REF, the 2nd Stage Loading Angle is FBD_REF, the Tread and Belt Assembly Loading Angle is SBD_REF, and the Transfer Ring Loading Angle is TSR_REF. In the specific example described herein, the invention contains an improvement to account for the radial runout of the measurement drum itself. This effect may be significant when the tire building drum  15  is used as the measurement drum. The loading angle of the tire carcass on the measurement drum is recorded. 
         [0035]    If the model is to include the tooling effect of the 1st Stage Building Drum, then a series of tires, normally at least 30 tires, is built with an intentional or forced variation of the 1st Stage Loading Angle CBD_REF. It is also possible to achieve the same results through the normal manufacturing variation of the loading angle that is experienced from tire to tire during manufacturing. For example, it is expected that the loading angle may vary naturally over a range of about ten degrees from the specified angle. The latter approach is advantageous for updating the model coefficients during normal manufacturing runs without interfering with tire output. At the completion of the carcass on the 1st Stage Building Drum, a measurement device, such as device  70 , is used to measure the RRO of the carcass. In the case of a unistage tire building machine, the RRO of the carcass would preferably be measured as the carcass is inflated on the unistage drum. There are many known devices  70  to obtain the RRO measurement such as a non-contact system using a vision system or a laser. It has been found that systems for measurement of radial runout that are based on tangential imaging are preferred to those using radial imaging. The RRO data thus acquired are recorded in a computer  80 . 
         [0036]    Next, the carcass is moved to the 2nd Stage Building Drum. In order to model the tooling effect of the 2nd Stage Building Drum, the carcasses of the sample set of tires are loaded on this drum with either a forced variation or a natural variation as described above. The carcass is then inflated to a shape to ready it to receive the tire summit (belt plies, cap ply, and tread). At the completion of the carcass inflation on the 2nd Stage Building Drum, a measurement device, such as device  70 , is used to measure the RRO. The tooling effect of the 2nd Stage Building Drum would not apply to a unistage tire building machine. 
         [0037]    While the tire carcass is built, the tire summit is being built on the Summit Building Drum, a substantially cylindrical surface referred to as a form. As described above, the loading angle of the products on the Summit Building Drum (normally the first belt ply) is recorded using the forced variation or natural variation of the loading angle. At the completion of the tire summit on the Summit Building Drum, a measurement device, such as device  70 , is used to measure the RRO tire summit. 
         [0038]    Finally, the tire summit and the carcass are assembled to form the finished green tire. To execute this step, a Transfer Ring removes the tire summit from the Summit Building Drum and positions it in coaxial alignment with the carcass, still mounted on the 2nd Stage Building Drum. The carcass is further inflated until it contacts and become attached to the tire summit. To account for the tooling effect of the Transfer Ring, the azimuth angle between the tire carcass and the tire summit is varied either in a forced variation or a natural variation as in the previous steps. After the tire is assembled, a measurement device, such as device  70 , is used to measure the RRO of the finished tire. 
         [0039]    It is advantageous to ensure a wide variation of the loading angle within a given sample of tires to ensure accurate estimation of the tooling effects on the vector coefficients. To accomplish this, the loading angles must not repeat from one tooling element to the next. As an illustrative example, assume that the sample set comprises two tires. If the first tire is mounted on the 1st Stage Building Drum at 0 degrees and on the 2nd Stage Building Drum at 90 degrees, then the second tire must be mounted on the tooling at loading angles not equal to 0 degrees or 90 degrees. For example, the second tire may be mounted at 45 degrees and 135 degrees, respectively, on the two tooling elements. In practice when using a forced variation of the loading angle on a large sample set of tires, the pattern of loading angles can be specified using a design of experiments (DOE) method as known to those skilled in the art. Such a method can be found in the reference “Quality Engineering Using Robust Design” by Madhav S. Phadke, Prentice Hall (1989). 
         [0040]    Once these data have been acquired for a suitable sample set of tires, the harmonic data are extracted from the RRO waveforms. In the present invention the first harmonic data of the green radial runout GR 1  (magnitude FRM 1  and azimuth FRA 1 ), carcass runout (magnitude FRM 1   cr  and azimuth FRA 1   cr ) and summit runout (magnitude FRM 1   sr  and azimuth FRA 1   sr ) respectively are extracted and stored. The following table indicates the specific terminology. 
         [0000]    
       
         
               
               
               
             
           
               
                   
               
               
                 Vector 
                 Magnitude 
                 Azimuth 
               
               
                   
               
             
             
               
                 Green RRO (GRI) 
                 FRM1 
                 FRA1 
               
               
                 Carcass Gain (gn) 
                 Gcr 
                 Θ 
               
               
                 Summit Gain (gn) 
                 Gsr 
                 Θ 
               
               
                 Intercept (I1) 
                 IM1 
                 IA1 
               
               
                 1st Stage Building Drum 
                 BM1r 
                 BA1r 
               
               
                 2nd Stage Building Drum 
                 TM1r 
                 TA1r 
               
               
                 Transfer Ring 
                 RM1r 
                 RA1r 
               
               
                 Summit Building Drum 
                 SM1r 
                 SA1r 
               
               
                 Belt Ply 
                 NMlr 
                 NA1r 
               
               
                 Cap 
                 BZMlr 
                 BZAlr 
               
               
                 Tread 
                 KMlr 
                 KAlr 
               
               
                 1st Stage Loading Angle 
                 — 
                 CBD_REF 
               
               
                 2nd Stage Loading Angle 
                 — 
                 FBD_REF 
               
               
                 Summit Building Drum Loading Angle 
                   
                 SBD_REF 
               
               
                 Transfer Ring Loading Angle 
                 — 
                 TSR_REF 
               
               
                   
               
             
          
         
       
     
         [0041]    To facilitate rapid application of equation (1) in a manufacturing environment, it is advantageous to use a digital computer to solve the equation. This requires converting the vector equations above to a set of arithmetic equations in Cartesian coordinates. In Cartesian coordinates, each vector or sub-vector has an x-component and a y-component as shown in the example below: 
         [0000]        FRH 1 x =( FRM 1)·COS( FRA 1), and  FRH 1 y =( FRM 1)·SIN( FRA 1)   (2)
 
         [0000]    It is recognized that Equation (2) will be repeated for each of the tires in the sample set of tires. That is, if there are M tires, then there will be M recitations of Equation (2). The dependent vector (FRH 1   rx,  FRH 1   ry ) is the sum of the vectors in the equations below. 
         [0000]        FRH 1 rx=Gcr·FRM 1 cr  COS(Θ+ FRA 1 cr )+ GSR·FRM 1 sr  COS(Θ+ FRA 1 sr )+ BM 1 r ·COS( BA 1 r+CBD _REF)+ TM 1 r ·COS( TA 1 r+FBD _REF)+ SM 1 r ·COS( SA 1 r+SBD _REF)+ RM 1 r ·COS( RA 1 r+TSR _REF)+ NM 1 r ·COS( NA 1 r+NBD _REF)+ BZM 1 r ·COS( BZA 1 r+BBD _REF)+ KM 1 r ·COS( KA 1 r+KBD _REF)+ IM 1 r ·COS( IA 1 r )   (3)
 
         [0000]        FRH 1 ry=Gcr·FRM 1 cr  SIN(Θ+ FRA 1 cr )+ Gsr·FRM 1 sr  SIN(Θ+ FRA 1 sr )+ BM 1 r ·SIN( BA 1 r+CBD _REF)+ TM 1 r ·SIN( TA 1 r+FBD _REF)+ SM 1 r ·SIN( SA 1 r+SBD _REF)+ RM 1 r ·SIN( RA 1 r+TSR _REF)+ NM 1 r ·SIN( NA 1 r+NBD _REF)+ BZM 1 r  SIN( BZA 1 r+BBD _REF)+ KM 1 r ·SIN( KA 1 r+KBD _REF)+ IM 1 r ·SIN( IA 1 r )   (4)
 
         [0000]    Expanding these equations with standard trigonometric identities yields: 
         [0000]        FRH 1 rx=Gcr  COS(Θ) FRM 1 cr  COS( FRA 1 cr )− Gcr  SIN(Θ)· FRM 1 cr  SIN( FRA 1 cr )+ Gsr  COS(Θ)· FRM 1 sr  COS( FRA 1 sr )− Gsr  SIN(Θ)· FRM 1 sr  SIN( FRA 1 sr )+ BM 1 r  COS( BA 1 r )·COS( CBD _REF)− BM 1 r  SIN( BA 1 r )·SIN( CBD _REF)+ TM 1 r  COS( TA 1 r )·COS( FBD _REF)− TM 1 r  SIN( TA 1 r )·SIN( FBD _REF)+ SM 1 r  COS( SA 1 r )·COS( SBD _REF)− SM 1 r  SIN( SA 1 r )·SIN( sBD _REF)+ RM 1 r  COS( RA 1 r )·COS( TSR _REF)− RM 1 r  SIN( RA 1 r )·SIN( TSR _REF)+ NM 1 r  COS( NA 1 r )·COS( NBD _REF)− NM 1 r  SIN( NA 1 r )·SIN( NBD _REF)+ BZM 1 r  COS( BZA 1 r )·COS( BBD _REF)− BZM 1 r  SIN( BZA 1 r )·SIN( BBD _REF)+ KM 1 r  COS( KA 1 r )·COS( KBD _REF)− KM 1 r  SIN( KA 1 r )·SIN( KBD _REF)+ IM 1 r  COS( IA 1 r )   (5)
 
         [0000]        FRH 1 ry=Gcr  COS(Θ)· FRM 1 cr  SIN( FRA 1 cr )+ Gcr  SIN(Θ)· FRM 1 cr  COS( FRA 1 cr )+ Gsr  COS(Θ)· FRM 1 sr  SIN( FRA 1 sr )+ Gsr  SIN(Θ)· FRM 1 sr  COS( FRA 1 sr )+ BM 1 r  COS( BA 1 r )·SIN( CBD _REF)+ BM 1 r SIN( BA 1 r )·COS( CBD _REF)+ M 1 r  COS( TA 1 r )·SIN( FBD _REF)+ TM 1 r  SIN( TA 1 r )·COS( FBD _REF)+ SM 1 r  COS( SA 1 r )·SIN( SBD _REF)+ SM 1 r  SIN( SA 1 r )·COS( sBD _REF)+ RM 1 r  COS( RA 1 r )·SIN( TSR _REF)+ RM 1 r  SIN( RA 1 r )·COS( TSR _REF)+ NM 1 r  COS( NA 1 r )·SIN  NBD _REF)+ NM 1 r  SIN( NA 1 r )·COS( NBD _REF)+ BZM 1 r  COS( BZA 1 r )·SIN( BBD _REF)+ BZM 1 r  SIN( BZA 1 r )·COS( BBD _REF)+ KM 1 r  COS( KA 1 r )·SIN( KBD _REF)+ KM 1 r  SIN( KA 1 r )·COS( KBD _REF)+ IM 1 r  COS( IA 1 r )   (6)
 
         [0000]    To simplify the expanded equation, convert from polar to Cartesian coordinates and introduce the following identities: 
         [0000]        a=Gcr  COS(Θ),  b=Gcr  SIN(Θ)   (7)
 
         [0000]        c=Gsr  COS(Θ),  d=Gsr  SIN(Θ)   (8)
 
         [0000]        e=BM 1 r  COS( BA 1 r ),  f=BM 1 r  SIN( BA 1 r )   (9)
 
         [0000]        g=TM 1 r  COS( TA 1 r ),  h=TM 1 r  SIN( TA 1 r )   (10)
 
         [0000]        i=SM 1 r  COS( SA 1 r ),  j=SM 1 r  SIN( SA 1 r )   (11)
 
         [0000]        k=RM 1 r  COS( RA 1 r ),  I=RM 1 r  SIN( RA 1 r )   (12)
 
         [0000]        m=NM 1 r  COS( NA 1 r ),  n=NM 1 r  SIN( nA 1 r )   (13)
 
         [0000]        o=BZM 1 r  COS( BZA 1 r ),  p=BZM 1 r  SIN( BZA 1 r )   (14)
 
         [0000]        q=KM 1 r  COS( KA 1 r ),  r=KM 1 r  SIN( KA 1 r )   (15)
 
         [0000]    Substituting these identities into the expanded form of equations (3) and (4) yields: 
         [0000]        FRH 1 rx=a·FRM 1 crx−b·FRM 1 cry+c·FRM 1 srx−d·FRM 1 sry+e·CBD _REF x−f·CBD _REF y+g·FBD _REF x−h·FBD _REF y+i·SBD _REF x−j·SBD _REF y+k·TSR _REF x− 1 ·TSR _REF y+m·NBD _REF x−n·NBD _REF y+o·BBD _REF x−p·BBD _REF y+q·KBD _REF x−r·KBD _REF y+Ix    (16)
 
         [0000]        FRH 1 ry=a·FRM 1 cry+b·FRM 1 crx+c·FRM 1 sry+d·FRM 1 srx+e·CBD _REF y+f·CBD _REF x+g·FBD _REF y+h·FBD _REF x+I·SBD _REF y+j·SBD _REF x+K·TSR _REF y+ 1·TSR_REF x+m·NBD _REF y+n·NBD _REF x+o·BBD _REF y+p·BBD _REF x+q·KBD _REF y+r·KBD _REF x+Iy    (17)
 
         [0000]    The equations (16) and (17) immediately above can be written in matrix format. When the predictive coefficients vectors (a,b), (c,d), (e,f), (g,h), (i,j), (k,l), (m,n), (o,p), (q,r), and (Ilx,Ily) are known, the matrix equation provides a modeling equation by which the FRH 1  vector for an individual tire may be estimated. This basic formulation can also be modified to include other process elements and to account for different production organization schemes. These coefficient vectors may be obtained by various known mathematical methods to solve the matrix equation above. 
         [0042]    The number of tires in the sample set of tires normally will be larger than the number of effects being modeled, especially if the method is used for updating during regular production of the tires. In this case, a solution to the matrix equation must be obtained by regression analysis or similar methods. In a manufacturing environment and to facilitate real-time use and updating of the coefficients, the method is more easily implemented if the coefficients are determined simultaneously by a least-squares regression estimate. All coefficients for all building drums and products may be solved for in a single regression step. Finally, the vector coefficients are stored in a database for future use. The coefficients have a physical significance which can be understood from Equations (3) and (4) as follows: (a,b) is the carcass gain vector in units of mm of GTFR per mm of carcass radial runout (Green Tire False Round, i.e. green tire radial runout), (c,d) is the summit gain vector in units of mm of GTFR per mm of summit runout, (e,f). The physical significance of the gain vectors is that they provide a type of weighting to account for the relative impact on the finished tire RRO contributed by RRO from the carcass and the tire summit. In this specific example, the equation describing the Green tire RRO ignores the gain vectors that would be associated with the Belt Ply, the Cap, and the Tread. However, these can easily be included to improve the accuracy of the model. The 1st Stage Building Drum vector in units of mm of GTFR, (g,h) is the 2nd Stage Building Drum vector in units of mm of GTFR, (i,j) is the Summit Building Drum vector in units of mm of GTFR, (k,l) is the Transfer Ring vector in units of mm of GTFR, (m,n) is the belt ply vector in units of mm of GTFR, (o,p) is the cap vector in units of mm of GTFR, (q,r) is the tread vector in units of mm of GTFR and (Ix, Iv) is the Intercept vector I 1  in units of mm of GTFR. The equations listed above are for one 1st Stage Building Drum, one 2nd Stage Building Drum, one Summit Building Drum, etc. The products and tooling factors are nested factors meaning that although the actual process contains many building drums and many products, each tire will see only one of each. Thus, the complete equation may include a vector for each building drum and each product. 
         [0043]    The final step is to apply the model to optimize the RRO of individual tires as they are manufactured according to the illustration shown in  FIG. 4 . When subsequent tires are manufactured, the constant vectors are rotated to minimize the green tire RRO. The rotations will be calculated such that when combined with the variable effects coefficients (a,b) and (c,d), it is possible to minimize the estimated vector sum of all the effects. In  FIGS. 3 and 4 , it is shown that the vectors  4 - 9  are rotated as a group leading to a considerably smaller resulting green RRO. Alternatively, it is envisioned that the tire building steps would be altered so as to produce an optimization to a zero level of green tire radial runout. 
         [0044]    At this point in the process, the summit has been built and is in the Transfer Ring awaiting positioning on the carcass. Mathematically this means that the constant vectors iv, v, vi, vii, viii and ix and the variable vector xi in  FIG. 4  are combined into one resultant vector. This is shown as reference number xiii in  FIGS. 6 and 7 . The carcass has also been built and is sitting inflated on the 2nd stage building drum. Mathematically this means that the constant vectors i, ii and iii and the variable vector x are combined into a second resultant. This is shown as reference number xiv in  FIGS. 6 and 7 . We then rotate the first resultant opposite the second resultant. The rotation is achieved by rotating the 2nd Stage Building Drum under the Transfer Ring to effect positioning the resultant of iv, v, vi, vii, viii, ix, and xi opposite the resultant of ii, iii, x, and I. Each tire building drum carriers an identification and each tire carries a unique identification device, such as a barcode. These identification tags allow the information recorded for an individual tire to be retrieved and combined at a later step. At the completion of tire building, the green RRO is measured and its harmonic magnitude FRM 1  and azimuth FRA 1  are recorded along with the loading angle of the tire on the building or measurement drum. A reading device scans the unique barcode to identify the tire, to facilitate polling the database to find the measured and recorded tire information: FRM 1  and FRAI, the building drum identification, and the loading angle. Because the variable effects are changing from tire to tire, the rotation of the fixed vectors will change from tire to tire. 
         [0045]    Another advantageous and unique feature of the invention is the ability to update the predictive coefficients vectors with the data measured from each individual tire to account for the constant variations associated with a complex manufacturing process. Because the green RRO is continuously measured, the model may be updated at periodic intervals with these new production data so as to adjust the predictive equations for changes in the process. These updates may be appended to the existing data or used to calculate a new, independent set of predictive coefficient vectors which may replace the original data. 
         [0046]    It should be understood that the present invention includes various modifications that can be made to the tire manufacturing method described herein as come within the scope of the appended claims and their equivalents.