Abstract:
A theoretical based winding process and control for a surface winding machine that can provide the capability to wind products with a desired wind profile, uniform sheet compression, improved compressibility, winding stability, and ease of operator control adjustment, is disclosed herein. The theoretical based surface winding process utilizes the principle of winding a log with a desired wind profile by controlling the surface speed of at least one roller of the surface winding machine.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention is directed to an improved method for operating and controlling a surface rewinding machine. This includes a method for the formation of logs or rolls of web material wound on a central core.  
       BACKGROUND OF THE INVENTION  
       [0002]     As known to those skilled in the art, a surface rewinder is generally used for producing smaller diameter logs, or rolls, of web material wound upon a central core from large diameter parent rolls. Typically, these machines are used in the paper converting industry to produce rolls of bath tissue, kitchen towels, all purpose wipes, and the like. It is known that formed logs of web material may be as long as 510 centimeters and have an outer diameter of about 10 to 15 centimeters. The formed logs of web material are then subsequently cut transversely to their axis to obtain small rolls of wound web material that may have a length ranging from 10 to 30 centimeters in length.  
         [0003]     Several types of surface rewinders are commercially available. One available type of surface rewinder is embodied as a three-drum cradle. Exemplary three-drum cradle surface rewinders are described in U.S. Pat. Nos. 4,327,877; 4,487,377; 4,723,724; 4,828,195; 5,979,818; 6,648,266; U.K. Patent No. 2,105,688; and EPO Patent EP-A-0 498 039. Another exemplary surface rewinder utilizes a speed change among a plurality of rollers to move logs of partially wound web material from one side of a pair of winding rollers to the other. Such an exemplary surface rewinder is described in U.S. Pat. No. 4,327,877. Yet still another type of surface winder utilizes a moveable winding drum. Exemplary moveable winding drums are detailed in U.S. Pat. No. 4,909,452.  
         [0004]     Even though certain of these exemplary surface rewind machines are commercially available, those of skill in the art have realized that these machines have certain drawbacks. Primary among these drawbacks is the fact that product produced from these exemplary rewind systems are known to have non-uniform wind profiles. A typical non-uniformly wound product generally exhibits a non-uniform wind profile by having visually observable tight and loose portions in the wound roll. Such tight and loose portions in the wound roll can be shown by the use of conventional measurement techniques known by those of skill in the art.  
         [0005]     Additionally, certain of these exemplary surface rewind systems are known to provide wound rolls having significant compression of the wound sheets near the core of the roll. This requires a looser wind for the rest of roll to achieve the desired product diameter when winding a product of fixed wound length, resulting in the finally wound product having a higher average compressibility than a corresponding uniformly wound web material. Additionally, certain of the exemplary surface rewind systems can cause logs to become unstable during the winding of low-density wound rolls. Such log instability can limit the speed of the rewinder as well as the rewinder throughput capability.  
         [0006]     In an attempt to deal with these winding problems, currently available surface rewinding equipment requires the operator to provide adjustment of multiple, and complex, control settings that are interdependent and not related to the theory of the winding process. This complexity adds a high degree of uncertainty in the ability to provide a process that produces a uniformly wound product.  
         [0007]     Many of the multiple control settings generally control the lower roller speed of a surface rewinder. These multiple control settings define the amount of deceleration and the duration of deceleration of the lower roller relative to the other rollers throughout the winding cycle. As the winding progresses throughout the winding cycle, the lower roller speed typically transitions linearly between these defined control settings. Thus, it should be clear that these current surface rewinder control methods are non-theoretically based and cause non-uniformly wound rolls. This approach can be particularly problematic when winding low-density products having large diameters with little total wound paper length.  
         [0008]     Thus, there is a need to provide a true theoretical winding control process with simplified operator controls that is capable of producing a desired winding profile. Such a preferred theoretical process should be based upon the principle of winding a web material uniformly about a core. It is believed that such a theoretical process can provide the unique capability to deliver a more consistent and uniform wind and can increase the capability, throughput, and product compatibility of a surface rewinding process.  
       SUMMARY OF THE INVENTION  
       [0009]     The present invention encompasses a method of controlling a roll winder for winding a product. The roll winder has at least a roller having an adjustable surface speed. The method comprises the steps of calculating a desired log diameter or radius build profile, calculating a log motion profile according to the log diameter or radius build profile, determining a roller surface speed profile according to the log motion profile, and adjusting the surface speed of the roller according to the roller surface speed profile.  
         [0010]     The present invention also provides a method of processing a web material. The method comprises the steps of providing a roll winder comprising at least a roller having an adjustable surface speed; providing the roll winder with a core, the roll winder being capable of winding the web material about the core; calculating a desired log diameter or radius build profile; calculating a log motion profile according to the log diameter or radius build profile; determining a roller surface speed profile according to the log motion profile; adjusting the surface speed of the roller according to the roller surface speed profile; and, winding the web material about the core according to the roller surface speed profile. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0011]      FIG. 1  is a schematic view of a winding apparatus suitable for use with the present invention;  
         [0012]      FIG. 2  is a graphical representation of a surface speed profile of a winding apparatus known in the prior art;  
         [0013]      FIG. 3  is a graphical representation of an exemplary theoretical log radius build according to the present invention;  
         [0014]      FIG. 4  is a schematic view of a winding apparatus showing the geometric relationships between the upper roller, lower roller, rider roller, and log;  
         [0015]      FIG. 5  is an exemplary graphic representation of the velocity vector components of a log being wound;  
         [0016]      FIG. 6  is a graphical representation of an exemplary theoretical lower roller surface speed profile;  
         [0017]      FIGS. 7   a - 7   d  are schematic views of a winding apparatus depicting the relationships between the upper roller, lower roller, rider roller, and log; and,  
         [0018]      FIG. 8  is a graphical representation of an exemplary modified theoretical lower roller surface speed profile and an exemplary adjusted lower roller surface speed profile overlaying a theoretical lower roller surface speed profile. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0019]      FIG. 1  depicts the basic elements of a surface rewinding machine  10  (also referred to herein as surface rewinder  10  or rewinder  10 ). A web material  12  is fed from a supply parent roll (not shown) to the winding region  11  of the rewinder  10 . Virtually any known process upstream of winding region  11  could process the web material  12 . Such processes may include, but not be limited to, embossing, lotioning, coating, printing, combining of two or more web materials, combinations thereof, and the like. Generally, the web material  12  is fed through a perforation assembly  13 . As is known to one of skill in the art, a perforation assembly  13  can be provided with a non-rotating support  14  and a rotating perforation roller  15  having blades  16  disposed thereon. As would also be known to one of skill in the art, the non-rotating support  14  can be provided with a blade (not shown) which is cooperatively associated with blades  16  disposed upon rotating perforation roller  15  to provide a line of perforations across web material  12 .  
         [0020]     Downstream of the perforation assembly  13  are disposed an upper roller  17  and a lower roller  18 . Generally, upper roller  17  and lower roller  18  rotate in the same direction and are spaced to form a gap  19  through which the web material  12  and/or log  21  can pass. Optional rider roller  20  can be attached to an arm (not shown) in order to provide articulable movement along axis D. Movement of rider roller  20  along axis D provides a region where the rolling and ensuing winding of each log  21  is completed and can accommodate the resulting diameter increase of web material  12  wound upon each log  21 . As would be known to one of skill in the art, a web material  12  could be wound into a finally wound product without the presence of a core  27 . In other words, a finally wound product may have the form of a wound log  21  with, or without, a core disposed therein.  
         [0021]     As shown in  FIG. 2 , lower roller  18  is generally provided with a surface speed profile relative to the surface speed K of the other winding rollers that can utilize multiple operator control set-points. By an operator setting the various and exemplary control set-points, a surface speed versus wound length profile similar to that shown in  FIG. 2  is generated. These set-points can include deceleration percent E, deceleration start point F, deceleration period G, acceleration period H, beginning speed percent I, and ending speed percent J.  
         [0022]     As shown in the graph of  FIG. 2 , deceleration percent E defines the amount of deceleration of the lower roller  18  relative to the other winding rollers at the point of transferring the web material  12  to a new winding log  21 . Deceleration start point F defines the point in the winding process prior to transfer that the lower roller  18  begins to decelerate to the deceleration percent E speed. Deceleration period G, is the period in the winding process after transfer that lower roller  18  remains at the decelerated speed defined by deceleration percent E. Acceleration period H defines the period in the winding process during which lower roller  18  accelerates from the deceleration percent E speed to the beginning speed percent I after the transfer process. Ending speed percent J defines the final decelerated speed for the lower roller  18  near the end of the log  21  winding just prior to the transfer process. These settings can allow the operator to control the winding process through the various phases of the winding process. However, these settings are somewhat arbitrary and are not based upon any theoretical control of the winding process. Experience has determined that this method of control is not adequate to produce rolls of web material  12  having a desired uniform wind profile.  
         [0023]     It was surprisingly found that controlling lower roller  18  according to the herein described theoretical process can consistently produce a uniformly wound product about a core (i.e., uniform wind or web layer thickness throughout the entire wind) as well as allow for minimized compression on the initial sheets of a wind. In other words, controlling the surface winder  10  according to the theoretical process described infra provides the capability to deliver a more uniform wind and can increase the surface winding capability of a surface rewinder  10  to wind low-density products. However, it should also be realized that this same theoretical process could also be used or adapted by one of skill in the art to control the speed and/or position of any roller within a surface rewinder system  10 . It is also believed that the herein described theoretical process for providing a uniformly wound product about a core could be adapted by one of skill in the art to virtually any type of rewinding system to wind any type of web material  12 .  
         [0000]     A Theoretical Based Surface Winding Process  
         [0024]     The theoretical surface winding process described herein is based upon winding a rolled product (i.e., log  21 ) about an optional central core with a uniform wind profile throughout the entire wind. However, this same method could be used to wind a log  21  with any desired wind profile. The first step in the described process is the calculation of the uniform diameter or radius build for a uniform wind profile.  
         [0000]     1. Calculation of the Theoretical Uniform Diameter or Radius Build  
         [0025]     The radius or diameter build of a wound product is a function of at least one characteristic of the log  21 . Exemplary characteristics of the log  21  include, but should not be limited to, the wound length, total or finished product wound length, target finished product (i.e., log  21 ) diameter or radius, core radius or diameter, combinations thereof, and the like. One of skill in the art would also realize that the radius or diameter build can be more specifically, a function of a ratio representing the amount of web material  12  wound relative to the total amount of web material  12  to be wound, the target finished product (i.e., log  21 ) radius or diameter, and optionally the core radius or diameter. The ratio representing the amount of web material  12  wound relative to the total amount of web material  12  to be wound can be determined by the ratio of wound material  12  length to total product wound length as described infra, cycle degrees while winding to total winding cycle degrees in a wind (such as a 360 degree segmented cycle), raw feedback signal increments to total feedback signal increments within a wind, or any other method known in the art which divides the total wind into segments that can be tracked and then expressed as a ratio of the in-progress segment to the total segments representing the entire wind cycle. As would be known to one skilled in the art, any of these methods should be considered as equivalent to the ratio of wound length to total product wound length.  
         [0026]     Without desiring to be bound by theory, it is believed that the calculation for radius build as a function of wound product length is the following:  
           r   w     ⁡     (     l   w     )       =             l   w       l   tw       ·     [         (     r     finished   ⁢           ⁢   log       )     2     -       (     r   core     )     2       ]       +     r   core   2             
        where: 
            r w (l w )=wound radius as a function of wound length;     l w =wound length;     l tw =total product wound length;     r finished log =target finished product radius; and,     r core =core radius.    
               
 
         [0033]     The above calculation is based upon the assumption that each layer forming log  21  is of uniform thickness. Although the radius build is desired for any of the calculations demonstrated herein, one of skill in the art could adapt the equations described herein upon the diameter build of the log  21  in order to achieve the desired winding profile.  
         [0034]     The wound length (i.e., length of paper wound on the log  21  at any point in the wind), l w , can be determined with a reasonable degree of accuracy from feedback used to relate the length of web material  12  wound into log  21 . The wound length should be considered to include factors such as overall web material  12  strain and the like. It is in this way that the feedback can be considered as a master signal from which all winding axis control may be referenced. One of skill in the art may utilize encoder feedback to relate encoder counts from the perforation assembly  13 . However, it should be realized by one of skill in the art that other feedback devices and methods, such as a resolver, Doppler laser velocimeter, tachometer, and the like, can be used to relate such a feedback signal to a measured point in the wind. Additionally, a useful master signal could be obtained by one of the process rollers or even a sensor that relates a physical property of a wound roll to the wound length.  
         [0035]     In accordance with the present invention, the wound length may be related to the feedback counts as follows:  
         l   w     =     EFC     (     EC     l   u       )           
        where: 
            EFC=Encoder Feedback Counts;     EC=Encoder Counts; and,     l u =unit length.    
               
 
         [0040]     For example, an encoder connected to the rotating perforation roller  15  of perforation assembly  13  can provide a master signal. Since the number of encoder counts per encoder revolution is known and because the encoder can be coupled to the perforation assembly  13 , the number of encoder counts per revolution of the perforation roller  15  can be known to be the same, or some known ratio thereof. If the number of blades  16  on the rotating perforation roller  15  is known, the number of individual sheets for each revolution of the rotating perforation roller  15  is also known. The relationship of encoder counts per sheet can then be found by the quotient of encoder counts per perforation roller  15  revolution and the number of sheets per perforation roller  15  revolution. Target sheet length can be determined by knowing the number and spacing of blades  16  disposed upon rotating perforation roller  15  and the surface speed of rotating perforation roller  15  relative to web material  12 . Furthermore, since the target sheet length of the product is known, the quotient of encoder counts per sheet and sheet length results in the relationship or scaling of encoder counts to a unit length wound on a log  21 . Thus, if the sheet count of the finally wound product is known, the total encoder counts comprising a wound product is known. This relationship can be expressed as follows:  
         EC   sheet     =       (     EC     rev     rotating   ⁢           ⁢   roller         )       (       #   ⁢     Blades     rotating   ⁢           ⁢   roller           rev     rotating   ⁢           ⁢   roller         )           
        where: 
            rev rotating roller =rotating perforation roller  15  revolutions; and,     #Blades rotating roller =number of blades  16  disposed upon rotating perforation roller  15  
                and,  
         EC     l   u       =       (     EC   sheet     )       l   sheet           
             where   ⁢     :     ⁢     l   sheet       =       sheet   ⁢           ⁢   length     =       l   p     sheet         ;   and     ,     
     ⁢       1   p     =     length   ⁢           ⁢   of   ⁢           ⁢   web   ⁢           ⁢   material   ⁢           ⁢   12   ⁢           ⁢   in   ⁢           ⁢   sheet           
       and   ,          ⁢       EC     Wound   ⁢           ⁢   Log       =         EC   Sheet     ·   sheet     ⁢           ⁢   count           
   
               
               
 
         [0045]     As the master signal feedback counts increment and reach a value equal to the total encoder counts per log  21 , the master signal feedback count can be reset to zero. This then establishes a cycle representing the winding process of one log  21  with zero representing the start of the log  21  at transfer and the total value of encoder counts per log  21  representing the end of the winding log  21  at target product length.  
         [0046]     One of skill in the art would realize that the master signal feedback counts can be phased or offset to align a zero count value to about the point of transfer in the log  21  transfer process which represents zero wound length. It should also be realized by one of skill in the art that the master signal feedback could also be used to determine the speed of the process rollers and the revolutions per minute of the process rollers. This can be based on the encoder counts per unit time, known roll specifications such as a given diameter, gear ratio, and the like, as well as product data for sheet length and number of sheets per perforation roller  15  revolution, as discussed supra.  
         [0047]     The total or finished product wound length is calculated as the product of sheet length and sheet count. By way of example: 
 
 l   tw   =l   sheet ·sheet count 
        where: l tw =total product wound length.        
 
         [0049]     The target finished product log  21  radius, r finished log , is calculated to be the target radius of the log  21  according to the product design for a given finished roll diameter. The target finished product log  21  radius can also be specified as a compressed radius of the finished wound product. One of skill in the art should realize that when using the compressed radius, the radius build is changed to provide a wind designed with some level of compression throughout the wind. One of skill in the art would also realize that alternative calculations could be used to determine a radius build based upon some other desired wound roll profile that is not uniform. However, any type of radius build profile can be generated based upon the desired properties of the final wound product. Exemplary alternative profiles can include tapers in which the thickness of each layer would increase or decrease from the center to the outside of the log  21 , steps, local deviations from a uniform or other desired profile, combinations thereof, and the like. An exemplary theoretical uniform radius build is shown graphically in  FIG. 3 .  
         [0050]     It should also be realized that the core radius, r core , is the known radius of the cores used for the log  21  winding process. As intended by the present invention, the core radius, r core , used in the above calculations should be zero for a coreless winding process.  
         [0000]     2. Theoretical Log Motion (Translational Position, Translational Velocity, and Radial Growth Velocity) Based Upon the Theoretical Radius Build  
         [0051]     The theoretical log  21  motion to achieve a uniform radius build can be determined by a geometric relationship of the winding process rollers and the winding log  21  itself. The theoretical log  21  motion is also based upon the assumption of no slippage between the winding log  21  and process rollers, as well as tangent point contact between the winding log  21  and the process rollers. These assumptions could be discarded and the effects of slippage and/or non-tangent point contact could be accounted for in another manner in order to determine an alternate log  21  motion and an alternative winding process control method. However, it is believed that excluding these assumptions would provide for a radius build that is not uniform and/or theoretically based.  
         [0052]     As shown in  FIG. 4 , in order to determine theoretical log  21  motion in accordance with the instant invention, the geometric relationship between the process rollers should be known, specifically, the geometric relationship between upper roller  17  and lower roller  18 . This includes the radius, or diameter, of the rollers and the coordinate location of the roller centers in XY plane coordinates. This then establishes the spatial relationship between the rollers comprising surface rewinder  10 .  
         [0053]     The origin and orientation of this X,Y plane coordinate system can be chosen arbitrarily. For example, the chosen coordinate location can be established by known coordinates or by known lengths and angles between roller centers with one of the roller centers defined as the origin. By knowing the geometric relationship between upper roller  17  and lower roller  18 , as well as the uniform radius build, the coordinate location of the winding log  21  center C can be calculated at any point throughout the winding process. In other words, the coordinate location of the winding log  21  center C can be thought of as representing the theoretical position of the winding log  21  center C throughout the wind process according to the theoretical uniform radius build. For embodiments having moveable winding drums, or rollers, the motion profile of the winding drums, or rollers, should be known when calculating the coordinate location of the winding log  21  center C.  
         [0054]     When the geometric relationship between upper roller  17  and lower roller  18  is known, a right triangle  30  with sides of known length, known internal angles, and having a hypotenuse between upper roller  17  and lower roller  18  centers is established. The adjacent sides of the right triangle are provided parallel to the direction of the axes of the chosen X,Y coordinate system (as shown).  
         [0055]     Next, the length  23  between the center of upper roller  17  to the center C of log  21  and the length  24  between the center of lower roller  18  to the center C of log  21  are determined by the sum of the respective process roller&#39;s radius and the winding log  21  radius as determined from the theoretical uniform radius build. These lengths form a triangle  31  between the centers of the upper roller  17 , the lower roller  18 , and the log  21  having varying, but known, lengths.  
         [0056]     Then, a third triangle  32  with sides of varying, but known, lengths is geometrically determined relative to the X-axis of the chosen coordinate system to the center of upper roller  17 , and relative to the Y-axis of the chosen coordinate system to the center C of log  21 , and the center of upper roller  17  to the center C of log  21 . Likewise, another triangle  33  can be established with respect to lower roller  18  and log  21 .  
         [0057]     By knowing the geometric relationships between the upper roller  17 , the lower roller  18 , the log  21 , and the theoretical uniform radius build, the lengths of the sides of each triangle are known. The internal angles of the triangles can then be calculated throughout the entire wind process. It should be realized that, the resulting internal angles and lengths data with similar geometric techniques could be used to calculate the coordinate location in the chosen X-Y plane of the log  21  center C through the entire wind process. In any regard, the chosen geometric technique should provide a coordinate location that represents the theoretical position of the log  21  center C throughout the entire wind process.  
         [0058]     The derivative of the theoretical position of the log  21  center C throughout the wind process provides the theoretical translation velocity of the log  21  through the wind process. As would be realized by one of skill in the art, a derivative function can prove difficult to calculate as the position of log  21  center C is changing in real time and a master signal used to determine the wound length of web material  12  constituting log  21  would likely be represented by a discreet, non-continuous signal. However, the derivative can be approximated by calculating the change of both the X- and Y-coordinate positions of log  21  per controller scan. In this manner, the theoretical translation velocity of the log  21  can be calculated as the square root of the sum of the squared X and Y coordinate position change. Thus: 
 
 ΔX   (log)   =X   N   −X   N-1  
 
         [0059]     where: ΔX (log) =Change of log  21  center C X-coordinate position per controller scan;  
           X   N     =     X   ⁢     -     ⁢   coordinate   ⁢           ⁢   of   ⁢           ⁢   log   ⁢           ⁢   21   ⁢           ⁢   center   ⁢           ⁢   C   ⁢           ⁢   at   ⁢           ⁢   scan   ⁢           ⁢   N       ;       
           X     N   -   1       =       X   ⁢     -     ⁢   coordinate   ⁢           ⁢   of   ⁢           ⁢   log   ⁢           ⁢   21   ⁢           ⁢   center   ⁢           ⁢   C   ⁢           ⁢   at   ⁢           ⁢   scan   ⁢           ⁢   N     -   1       ;       
       and   ,     
     ⁢       Δ   ⁢           ⁢     Y     (   log   )         =       Y   N     -     Y     N   -   1               
        where: 
            ΔY (log) =Change of log  21  center C Y-coordinate position per controller scan;     Y N =Y-coordinate of log  21  center C at scan N;     Y N-1 =Y-coordinate of log  21  center C at scan N−1; and, 
 
 V   t =√{square root over ((Δ X   (log) ) 2 +(Δ Y   (log) ) 2 )}
   
               
 
         [0064]     where: V t =Magnitude of Theoretical Translational Velocity of log  21  in length/scan.  
         [0065]     The angle of the calculated theoretical translational velocity can be then determined from the X and Y components: 
 
Θ 2 =tan −1 (Δ Y   (log)   /ΔX   (log) ) 
 
         [0066]     where: Θ 2 =Angle of the translational velocity vector.  
         [0067]     Likewise, a change of the uniform radius build per controller scan can be taken to approximate the theoretical radial growth velocity. This can be represented by the following equation: 
 
 V   r =( r   w ) N −( r   w ) N-1  
        where: 
            V r =Theoretical Radial Growth Velocity in length/scan;     (r w ) N =radius of log  21  according to radius build profile at scan N; and,     (r w ) N-1 =radius of log  21  according to radius build profile at scan N−1.    
               
 
         [0072]     The theoretical radial growth velocity and theoretical log  21  translational velocity can then be used to determine the theoretical lower roller  18  surface speed profile.  
         [0073]     The current embodiment uses approximations to calculate values that are actually derivatives for the winding process. These approximations provide velocity as a derivative of position. This approximation method may be suitable when utilizing a non-continuous, discreet, real time feedback signal used to determine wound length and the resulting uniform or other desired radius build and the corresponding theoretical log  21  translational position and radial growth. Alternatively, the derivative can utilize the actual mathematical functions for the known theoretical position and radial growth. In this instance, the derivative of these functions would establish an equation-based calculation for the respective velocities at the discreet feedback signal intervals. In yet another embodiment, the derivative calculations may be approximated or use a derivative function calculation in non-real time. The resulting values can then be stored and referenced by the surface rewinding system  10  as required. It should also be realized that approximations of the derivatives can be performed on a per controller scan basis. As would be known to one of skill in the art, alternative methods of evaluating the required derivatives are possible including, but not limited to, evaluating the derivatives on a per unit time basis, or evaluating the derivatives on a per unit wound length basis.  
         [0000]     3. Theoretical Lower Roller Surface Speed Profile Based on Theoretical Log Motion for Theoretical Radius Build  
         [0074]     As shown in  FIG. 5 , the theoretical lower roller  18  surface speed profile is determined by decomposing the theoretical log  21  motion for a uniform radius build into vector components of translational, rotational, and radial growth velocity. These vector components can be analyzed at different physical points on the log  21  where these components may have different vector values. The theoretical log translation velocity, V t , can be determined by the derivative of the theoretical log  21  center C position and can be known based on the calculations and assumptions as discussed supra. One of skill in the art should note that the log  21  translational velocity is the same vector in magnitude and angle at any physical point on the log  21 . Likewise, the theoretical log  21  radial growth velocity, V r , can also be known based upon calculations and assumptions discussed supra to provide a uniform log  21  radius build. It should be realized that the log  21  radial growth velocity can be the same magnitude at any physical point on the surface of log  21 , but have a different vector because of the angular differences.  
         [0075]     Referring again to  FIG. 5  the surface speed of upper roller  17  is known and related to the rotational speed of the upper roller  17 . This wind axis has a known angular velocity based upon a target web material  12  speed. If one were to assume that there is no slip between the winding log  21  and the winding rollers and tangent point contact for the log  21 , the log  21  is at a matched surface velocity in both magnitude and direction to upper roller  17 , V u , at point A. Point A is the point on the log  21  surface in contact with upper roller  17 . However, the surface velocity of lower roller  18 , V l , is unknown and the rotational velocity, ω, of the lower roller  18  is unknown. If one were to assume no slip and tangent point contact during the theoretical wind process, the log  21  has surface velocity matched to the lower roller  18  surface velocity, V l , at point B.  
         [0076]     The angles for all of the velocity vectors can be determined based upon the geometric relationship between the upper roller  17 , lower roller  18  and winding log  21  described supra. Just as a coordinate location of the center C of the winding log  21  can be determined, a location for the contact point A between the upper roller  17 , and the winding log  21  as well as the contact point B between the lower roller  18  and the winding log  21  can be determined via the respective geometric relationships. The angles of the velocity of the log  21  at these contact points must be tangent to the surface of the winding log  21  and in the direction of rotation of the log  21 . Therefore, the angles Θ 1  and Θ 3  can be known. Furthermore, the angle for the radial growth velocity vector must be perpendicular to a line tangent to the surface of the winding log  21  and directed radially outward from log  21 . Therefore, the angle of the radial growth velocity vector at the point of contact A between the upper roller  17  and winding log  21  is equal to the angle Θ 3 +90 degrees and the radial growth velocity vector at the point of contact B between the lower roller  18  and winding log  21  is equal to the angle Θ 1 +90 degrees.  
         [0077]     The velocity vectors at the tangent contact points with both the upper roller  17  at point A and lower roller  18  at point B can be decomposed into rotational, translational, and radial growth components: 
 
 V   u   =V   ac   +V   t   +V   rA   equation (1) 
 
 V   l   =V   bc   +V   t   +V   rB   equation (2) 
        where: 
            V u =Velocity at the tangent contact point A between the log  21  and the upper roller  17      V ac =Velocity of point A relative the center of the log  21 ;     V t =Translational velocity;     V rA =Radial growth velocity at point A;     V l =Velocity at the tangent contact point B between the log  21  and the lower roller  18 ;     V bc =Velocity of point B relative the center of the log  21 ; and,     V rB =Radial growth velocity at point B.    
               
 
         [0086]     Also note that V ac  and V bc  are the rotational velocity components and can be determined by: 
 
 V   ac   =ω×r   ca  and 
 
 V   bc   =ω×r   cb  
        where: 
            ω=the angular velocity of the log  21 ; and,     r ca =vector of the radius of the log  21  from point C to point A     r cb =vector of the radius of the log  21  from point C to point B.    
               
 
         [0091]     Noting that each of the terms in equation 1 and equation 2 are vector quantities, both equations can then be further decomposed into components using i, j, and k unit vectors. This yields two equations with only two unknowns, ω and V l . V u , V t , V rA  and V rB  are known components, as discussed supra. Therefore, all values and angles are known except for the angular velocity of the log  21 , ω, in equation 1. This equation is then solved for ω and provides the instantaneous angular velocity of the log  21 . This known angular velocity of the log  21 , ω, can be used in equation 2. Thus V l , the lower roller  18  surface velocity, can be determined. The following equations are exemplary of this method of determining the lower roller  18  surface velocity:  
         [0092]     Velocity of upper roller  17  at contact point with log  21  at Point A (equation 1):  
                   V   u     ·     cos   ⁡     (     Θ   3     )       ·     +       V   u     ·     sin   ⁡     (     Θ   3     )       ·       =       ⁢         ω   ·     k   →       ×       r   ca     →       +         V   t     ·     cos   ⁡     (     Θ   2     )         ⁢     +                     ⁢           V   t     ·     sin   ⁡     (     Θ   2     )         ⁢     +       V   rA     ·   cos                       ⁢         (       Θ   3     +   90     )     ·     +       V   rA     ·   sin                       ⁢       (       Θ   3     +   90     )     ·               
                   V   u     ·     cos   ⁡     (     Θ   3     )       ·     +       V   u     ·     sin   ⁡     (     Θ   3     )       ·       =       ⁢       ω   ·     k   →       ×     (           r   ca     ·     cos   ⁡     (       Θ   3     +   90     )         ⁢     +                           ⁢         r   ca     ·     sin   ⁡     (       Θ   3     +   90     )         ⁢     )     +       V   t     ·   cos                     ⁢         (     Θ   2     )     ⁢     +       V   t     ·   sin                       ⁢         (     Θ   2     )     ·     +       V   rA     ·     cos   ⁡     (       Θ   3     +   90     )       ·                       ⁢     +       V   rA     ·     sin   ⁡     (       Θ   3     +   90     )       ·                 
                   V   u     ·     cos   ⁡     (     Θ   3     )       ·     +       V   u     ·     sin   ⁡     (     Θ   3     )       ·       =       ⁢         ω   ·     r   ca     ·     cos   ⁡     (       Θ   3     +   90     )         ⁢   ⁢     +                     ⁢       ω   ⁢           ⁢       r   ca     ·     sin   ⁡     (       Θ   3     +   90     )       ·     -         +       V   t     ·   cos                       ⁢         (     Θ   2     )     ⁢     +       V   t     ·   sin                       ⁢         (     Θ   2     )     ·     +       V   rA     ·     cos   ⁡     (       Θ   3     +   90     )       ·                       ⁢     +       V   rA     ·     sin   ⁡     (       Θ   3     +   90     )       ·                 
                   V   u     ·     cos   ⁡     (     Θ   3     )       ·     +       V   u     ·     sin   ⁡     (     Θ   3     )       ·       =       ⁢     [       (       -   ω     ·     r   ca     ·     sin   ⁡     (       Θ   3     +   90     )         )     +       V   t     ·                             ⁢       cos   ⁡     (     Θ   2     )       +       V   rA     ·     cos   ⁡     (       Θ   3     +   90     )           ]     ·     +                   ⁢     [       (     ω   ·     r   ca     ·     cos   ⁡     (       Θ   3     +   90     )         )     +       V   t     ·     sin   ⁡     (     Θ   2     )         +                         ⁢       V   rA     ·     sin   ⁡     (       Θ   3     +   90     )         ]     ·             
 
         [0093]     Velocity of lower roller  18  at contact point with log  21  at Point B (equation 2):  
                       V   I     ·   co     _     ⁢       s   ⁡     (     Θ   I     )       ·       +       V   I     ·     sin   ⁡     (     Θ   I     )       ·       =       ⁢         ω   ·     k   →       ×       r   ca     →       +         V   t     ·     cos   ⁡     (     Θ   2     )         ⁢     +                     ⁢           V   t     ·     sin   ⁡     (     Θ   2     )         ⁢     +       V   rB     ·   cos                       ⁢         (       Θ   I     +   90     )     ·     +       V   rB     ·   sin                       ⁢       (       Θ   3     +   90     )     ·               
                   V   I     ·     cos   ⁡     (     Θ   I     )       ·     +       V   I     ·     sin   ⁡     (     Θ   I     )       ·       =       ⁢       ω   ·     k   →       ×     (           r   ca     ·     cos   ⁡     (       Θ   I     +   90     )         ⁢     +                           ⁢         r   ca     ·     sin   ⁡     (       Θ   I     +   90     )         ⁢     )     +       V   t     ·   cos                     ⁢         (     Θ   2     )     ⁢     +       V   t     ·   sin                       ⁢         (     Θ   2     )     ·     +       V   rB     ·     cos   ⁡     (       Θ   I     +   90     )       ·                       ⁢     +       V   rB     ·     sin   ⁡     (       Θ   I     +   90     )       ·                 
                   V   I     ·     cos   ⁡     (     Θ   I     )       ·     +       V   I     ·     sin   ⁡     (     Θ   I     )       ·       =       ⁢         ω   ·     r   ca     ·     cos   ⁡     (       Θ   3     +   90     )         ⁢   ⁢     +                     ⁢       ω   ⁢           ⁢       r   ca     ·     sin   ⁡     (       Θ   3     +   90     )       ·     -         +       V   t     ·   cos                       ⁢         (     Θ   2     )     ⁢     +       V   t     ·   sin                       ⁢         (     Θ   2     )     ·     +       V   rA     ·     cos   ⁡     (       Θ   3     +   90     )       ·                       ⁢     +       V   rA     ·     sin   ⁡     (       Θ   3     +   90     )       ·                 
                   V   u     ·     cos   ⁡     (     Θ   3     )       ·     +       V   u     ·     sin   ⁡     (     Θ   3     )       ·       =       ⁢     [       (       -   ω     ·     r   ca     ·     sin   ⁡     (       Θ   3     +   90     )         )     +       V   t     ·                             ⁢       cos   ⁡     (     Θ   2     )       +       V   rA     ·     cos   ⁡     (       Θ   3     +   90     )           ]     ·     +                   ⁢     [       (     ω   ·     r   ca     ·     cos   ⁡     (       Θ   3     +   90     )         )     +       V   t     ·     sin   ⁡     (     Θ   2     )         +                         ⁢       V   rA     ·     sin   ⁡     (       Θ   3     +   90     )         ]     ·             
 
         [0094]     Solving for angular velocity, ω, using only the i vector components from equation 1:  
       ω   =                 V   u     ·     cos   ⁡     (     Θ   3     )         =       (       -   ω     ·     r   ca     ·     sin   ⁡     (       Θ   3     +   90     )         )     +       V   t     ·     cos   ⁡     (     Θ   2     )         +       V   rA     ·   cos                     (       Θ   3     +     90   ·   deg       )     -                   V   u     ·     cos   ⁡     (     Θ   3     )         +       V   t     ·     cos   ⁡     (     Θ   2     )         +       V   rA     ·     cos   ⁡     (       Θ   3     +     90   ·   deg       )                     r   ca     ·     sin   ⁡     (       Θ   3     +   90     )               
 
         [0095]     Solving for V l  using the i vector components of equation 2:  
                 V   I     ⁢     cos   ⁡     (     Θ   I     )         =       ⁢       (       -   ω     ·     r   ca     ·     sin   ⁡     (       Θ   I     +   90     )         )     +                     ⁢         V   t     ·     cos   ⁡     (     Θ   2     )         +       V   rB     ·     cos   ⁡     (     Θ   +     90   ·   deg       )                     
         V   I     =                   -   ω     ·     r   ca       ⁢     sin   ⁡     (       Θ   I     +   90     )         +                   V   t     ·     cos   ⁡     (     Θ   2     )         +       V   rB     ·     cos   ⁡     (       Θ   I     +     90   ·   deg       )                   cos   ⁡     (     Θ   I     )             
 
         [0096]     Determining the lower roller  18  surface velocity through the entire winding process provides the theoretical lower roller  18  surface speed profile. It should also be realized that other methods of solving a system of two equations with two unknowns could also be used to yield an appropriate result for the theoretical lower roller  18  surface velocity V l . An exemplary theoretical lower roller  18  surface speed profile is shown in  FIG. 6 .  
         [0000]     4. A Modified Theoretical Lower Roller Surface Speed Profile Based on Surface Winder Transfer Process and Operator Control Design  
         [0097]     Referring to  FIGS. 7   a - d , during the two-roller contact (log  21  to upper roller  17  and log  21  to lower roller  18 ) and three-roller contact (log  21  to upper roller  17 , lower roller  18 , and rider roller  20 ) phases of a typical surface winder process, the theoretical lower roller  18  surface speed profile can yield uniform theoretical winding. However, as would be appreciated by one of skill in the art, the theoretical lower roller  18  surface speed profile can optionally be modified to accommodate the transfer portions of the surface winder process. This could include transition of log  21  relative to upper roller  17  and stationary apparatus  26  and into the winding region  11 . Therefore, it would be useful to modify the theoretical lower roller  18  surface speed profile prior to and/or immediately after log  21  transfer. The log  21  transfer process of the surface winder  10 , as described herein, can be provided with any user selectable modifications to the lower roller  18  surface speed profile. As would be known to one of skill in the art and without desiring to be bound by theory, other variations of surface winders may have different or modified log  21  transfer processes. However, it is believed that the requirements of providing such modifications to the lower roller  18  surface speed profile are similar.  
         [0098]     The chop off and transfer functionality can be achieved by any means known to one skilled in the art, including those referenced in U.S. Pat. Nos. 5,979,818; 5,772,149; and 6,056,229. Referring to the exemplary process depicted in  FIG. 7   a , transfer begins as the web material  12  total product length is finishing winding onto log  21 . During transfer, the web material  12  can be separated at a targeted perforation by being pinched and/or nipped against upper roller  17  by cutoff apparatus  24 . As should be known to those of skill in the art, cutoff apparatus  24  can operate at a different peripheral speed than upper roller  17 . At the same time, a new log  25  is introduced to the winding region  11 . The new log  25  can be provided with a cross-machine direction line of glue applied to the core and then pressed against the upper roller  17  causing the loose tail of web material  12  to stick to and begin to wrap the new log  25  as the web material  12  is separated at the identified perforation by cutoff apparatus  24 . The new log  25  is then driven by the upper roller  17  and rolls along stationary apparatus  26  to the point at which the new log  25  reaches lower roller  18 . During this traversal, the new log  25  is actively winding web material  12  to the new log  25  and translating at about 50% of the upper roller  17  surface speed.  
         [0099]     Referring to  FIG. 7   b , as the new log  25  contacts lower roller  18 , the new log  25  enters gap  19  between the upper roller  17  and lower roller  18  and begins two-roller contact. The motion of the new log  25 , as it traverses through gap  19 , is preferably controlled by the speed of lower roller  18  relative to the upper roller  17 . Specifically, a decelerated lower roller  18  surface speed is typically required in order for the new log  25  to progress outward through the gap  19  and minimize compression on the new log  25  resulting from a significant dwell time in the gap  19  while winding. Also, providing the lower roller  18  with a decelerated surface speed may be required to gain control of the new log  25  and rapidly decelerate the new log  25  from the high translation speed by being driven at about 50% of the upper roller  17  surface speed while moving through the stationary apparatus  26 . The lower roller  18  surface speed can be modified from the theoretical surface speed profile as described supra for these reasons in order to control the new log  25  at a decelerated speed. This is generally referred to as the lower roller  18  deceleration at transfer E′ (as shown in  FIG. 8 ). The lower roller  18  deceleration at transfer E′ can be used to control how quickly the new log  25  progresses through the surface winder  10  gap  19 . The magnitude of the lower roller  18  deceleration at transfer E′ speed and/or the duration G′ (shown in  FIG. 8 ) that the lower roller  18  is at the lower roller  18  deceleration at transfer E′ speed can also be used to control how far the new log  25  moves through and out of the gap  19  between upper roller  17  and lower roller  18 .  
         [0100]     Referring to  FIG. 7   c , it is preferred that the desired theoretical process provide that new log  25  contact the rider roller  20  with controlled contact as the rider roller  20  reaches its minimum height point and establishes control of the new log  25  to initiate three-roller contact. Without desiring to be bound by any particular theory, the duration G′ that the lower roller  18  is at the lower roller  18  deceleration at transfer E′ speed can impact how far the new log  25  moves through and out of the gap  19  which can also directly impact how the new log  25  will be delivered to contact with the rider roller  20 . Therefore, the duration G′ of the lower roller  18  deceleration at transfer E′ speed can be used to control and optimize the new log  25  contact with rider roller  20 . This should be known to one of skill in the art as a method to control movement of the log  25  from 2-roller to 3-roller contact.  
         [0101]     As the new log  25  contacts rider roller  20  and establishes the desired contact, the rider roller  20  can optionally be at a height based on the theoretical winding log  25  radius. In this optional, but preferred embodiment, the lower roller  18  speed is returned from the lower roller  18  deceleration at transfer E′ surface speed to match the theoretical lower roller  18  surface speed profile at, or about, the point  40  (shown in  FIG. 8 ) of rider roller  20  contact. This could establish that rider roller  20  growth or position and lower roller  18  surface speed are functioning in coordination and according to the theoretical radius build discussed supra. Note that the rider roller  20  contact point with the new log  25  can be provided as an operator-based control which along with the lower roller  18  duration G′ of the deceleration at transfer E′ surface speed can be used to control the new log  25  after transfer to the point of theoretical winding control. In other words, at about the point where new log  25  contacts rider roller  20 , the lower roller  18  should be at about the surface speed determined by the theoretical radius build. In order for lower roller  18  to return to it&#39;s theoretical surface speed profile, it is believed that a transition  41  (shown in  FIG. 8 ) from the deceleration at transfer E′ speed to the theoretical surface speed profile will be required. One of skill in the art would be able to perform such a transition with any type of mathematical function known to provide such transitional behavior. Such mathematical functions can include, but not be limited to, linear functions, exponential functions, power functions, filtering functions, logarithmic functions, trigonometric functions, s-curve functions, combinations thereof, and the like. However, it should be realized that a preferable mathematical function suitable for use with the present invention would be a function that resembles an exponential decay between levels of deceleration. This exponential decay-like behavior between the levels of deceleration provides for a fast and smoothed transition that is preferred for decelerating the new log  25  translational velocity and providing control between the different levels of deceleration. This can also provide a reference velocity for lower roller  18  of surface rewinder  10  that can be more readily followed by the mechanical and/or electrical drive transmission systems of surface rewinder  10 . One should realize that this transition method could provide improved capability to control the new log  25  deceleration compared to a transition method requiring instantaneous changes in lower roller  18  acceleration. Such instantaneous transitions necessitate instantaneous changes in the new log  25  deceleration that may not be possible in exemplary systems of the prior art. Thus, without desiring to be bound by theory, the herein described transition method can provide a surface rewind system  10  with the capability to more accurately control the new log  25 . This can result in a finally wound product having minimal compression on sheets near the core of log  21  due to the ability of the lower roller  18  of surface rewind system  10  to reach a greater deceleration at transfer E′ than any known prior art rewind systems, while maintaining control of the new log  25  throughout the transition.  
         [0102]     Referring to  FIG. 7   d , after three-roller contact is established, the new log  25  continues to wind based on the theoretical uniform winding process provided by the theoretical lower roller  18  surface speed profile discussed supra. The theoretical winding continues until just prior to the next transfer point. In order to eject the finished log  25  during the transfer process, the finished log  25  preferably moves outwardly and away from upper roller  17 . This can be accomplished by either speeding up the rider roller  20 , slowing down the lower roller  18 , or a combination of both. This speed differential can result in the new log  25  moving outward as it finishes winding. As the new log  25  moves outward, new core  27  is inserted and begins to move along the path between upper roller  17  and stationary apparatus  26  toward lower roller  18 . The lower roller  18  surface speed can, therefore, optionally be modified from the theoretical surface speed profile to accomplish this result. This optional modification would require that the speed of lower roller  18  be decelerated  42  (shown in  FIG. 8 ) to the deceleration at transfer speed E′ as would be known to one of skill in the art. This speed should preferably be established through the cutoff portion of the transfer process. This is the same speed as discussed supra for control of how quickly the new log  25  moves through the upper roller  17  and lower roller  18  gap  19 .  
         [0103]     In addition to modifying the theoretical lower roller  18  surface speed to achieve ejection of the finished new log  25  at transfer and translation of the newly inserted core  27  through the gap  19  between the upper roller  17  and lower roller  18 , it may also be desirable to modify the lower roller  18  surface speed from the theoretical surface speed for the entire winding process. It may be desired to wind a log  21  tighter than, or looser than, what is provided by the herein described desired theoretical build. In an exemplary embodiment, this may be achieved by adding and/or subtracting a constant offset to the theoretically determined surface speed profile in order to create an adjusted lower roller  18  surface speed profile  43  (shown in  FIG. 8 ). The modifications to the lower roller  18  surface speed profile are then relative to the adjusted lower roller  18  surface speed profile  43 . One of skill in the art would realize that other methods of adjusting the lower roller surface speed  43  profile are possible, including, but not limited to, adding and/or subtracting a constant percentage difference from the theoretical lower roller  18  surface speed profile, adding and/or subtracting a tapered offset to the theoretical lower roller  18  surface speed profile, localized deviations, combinations thereof, and the like. In any regard, it should be readily apparent to one of skill in the art that the present inventive process provides increased product compatibility and simplified operator controls while at the same time providing the desired winding profile of a wound product.  
         [0104]     All documents cited in the Detailed Description of the Invention are, in relevant part, incorporated herein by reference; the citation of any document is not to be construed as an admission that it is prior art with respect to the present invention. To the extent that any meaning or definition of a term in this written document conflicts with any meaning or definition of the term in a document incorporated by reference, the meaning or definition assigned to the term in this written document shall govern.  
         [0105]     While particular embodiments of the present invention have been illustrated and described, it would be obvious to those skilled in the art that various other changes and modifications can be made without departing from the spirit and scope of the invention. It is therefore intended to cover in the appended claims all such changes and modifications that are within the scope of this invention.