Patent Publication Number: US-10786886-B2

Title: Pneumatic crankshaft clamp assembly

Description:
FIELD 
     The present disclosure relates to a crankshaft clamp assembly and, more particularly, relates to a pneumatic crankshaft clamp assembly. 
     BACKGROUND AND SUMMARY 
     This section provides background information related to the present disclosure which is not necessarily prior art. This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features. 
     It is common practice in automotive manufacturing settings to employ a crankshaft assembly system to assembly and/or balance a crankshaft for an engine. Generally, these crankshaft assembly systems include a pedestal support having a retractable arm to releasably retain a crankshaft on the pedestal support during processing. Such processing of the crankshaft can include drilling, shaving, mounting, tapping, and the like. Moreover, the processing of the crankshaft can be part of a balancing system for balancing the rotational mass of the crankshaft assembly. 
     According to the principles of the present teachings, a pneumatic crankshaft clamp assembly for retaining a crankshaft during processing is provided having advantageous constructions and method of operation. The pneumatic crankshaft clamp assembly includes a housing for supporting the crankshaft and a clamp arm pivotally coupled to the housing for movement between a clamp position and an unclamp position. The clamp arm engaging the crankshaft to retain the crankshaft in contact with the housing in the clamp position and be spaced apart from the crankshaft in the unclamp position. The proximal end of the clamp arm having a cam follower. The assembly further having a pneumatic piston disposed within a piston cylinder of the housing. The pneumatic piston is moveable in response to pneumatic pressure. The piston having a cam formed thereon engaging the cam follower of the lifting arm such that the lifting arm moves in response to pneumatic pressure. 
     Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 
    
    
     
       DRAWINGS 
       The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure. 
         FIG. 1  is a front left perspective view of a crankshaft balancer and suspension system assembly according to the principles of the present teachings; 
         FIG. 2  is a rear right perspective view of the crankshaft balancer and suspension system assembly according to the principles of the present teachings; 
         FIG. 3  is a front view of the crankshaft balancer and suspension system assembly according to the principles of the present teachings; 
         FIG. 4  is a right side view of the crankshaft balancer and suspension system assembly according to the principles of the present teachings; 
         FIG. 5  is a rear view of the crankshaft balancer and suspension system assembly according to the principles of the present teachings; 
         FIG. 6  is a top view of the crankshaft balancer and suspension system assembly according to the principles of the present teachings; 
         FIG. 7  is a perspective view of a measurement station according to the principles of the present teachings; 
         FIG. 8  is a right view of the measurement station according to the principles of the present teachings; 
         FIG. 9  is a front view of the measurement station according to the principles of the present teachings; 
         FIG. 10  is a top view of the measurement station according to the principles of the present teachings; 
         FIG. 11  is a cross-sectional view of a portion of the measurement station taken along line  11 - 11  of  FIG. 9  according to the principles of the present teachings; 
         FIG. 12  is a cross-sectional view of a portion of the measurement station taken along line  12 - 12  of  FIG. 9  according to the principles of the present teachings; 
         FIG. 13  is a cross-sectional view of a portion of the measurement station taken along line  13 - 13  of  FIG. 10  according to the principles of the present teachings; 
         FIG. 14  is a cross-sectional view of a portion of the measurement station according to the principles of the present teachings; 
         FIG. 15  is a cross-sectional view of a portion of the measurement station taken along line  15 - 15  of  FIG. 9  according to the principles of the present teachings; 
         FIG. 16  is a cross-sectional view of a portion of the measurement station taken along line  16 - 16  of  FIG. 15  according to the principles of the present teachings; 
         FIG. 17  is a cross-sectional view of a portion of the measurement station taken along line  17 - 17  of  FIG. 9  according to the principles of the present teachings; 
         FIG. 18  is a cross-sectional view of a portion of the measurement station taken along line  18 - 18  of  FIG. 9  according to the principles of the present teachings; 
         FIG. 19  is a cross-sectional view of a portion of the measurement station taken along line  19 - 19  of  FIG. 9  according to the principles of the present teachings; 
         FIG. 20  is a cross-sectional view of a portion of the measurement station taken along line  20 - 20  of  FIG. 10  according to the principles of the present teachings; 
         FIG. 21  is a first perspective view of a pneumatic crankshaft clamp assembly according to the principles of the present teachings; 
         FIG. 22  is a second perspective view of the pneumatic crankshaft clamp assembly according to the principles of the present teachings; 
         FIG. 23  is a side view of the pneumatic crankshaft clamp assembly according to the principles of the present teachings; 
         FIG. 24  is a front view of the pneumatic crankshaft clamp assembly according to the principles of the present teachings; 
         FIG. 25  is a cross-sectional view of the pneumatic crankshaft clamp assembly taken along line  25 - 25  of  FIG. 23  according to the principles of the present teachings; 
         FIG. 26  is a composite front view of the pneumatic crankshaft clamp in an unclamp position assembly according to the principles of the present teachings; 
         FIG. 27  is a top view of the pneumatic crankshaft clamp assembly according to the principles of the present teachings; 
         FIG. 28  is a cross-sectional view of the pneumatic crankshaft clamp assembly taken along line  28 - 28  of  FIG. 26  according to the principles of the present teachings; 
         FIG. 29  is a cross-sectional view of the pneumatic crankshaft clamp assembly taken along line  29 - 29  of  FIG. 26  according to the principles of the present teachings; 
         FIG. 30  is a diagram useful in understanding the dynamic vibrational modes of an elongated workpiece; 
         FIG. 31  is a view of an exemplary crankshaft, showing potential drill sites; 
         FIG. 32  is an electronic circuit diagram illustrating the processor-based circuit for calculating optimal drill data; 
         FIG. 33  is a flowchart diagram illustrating how the processor of  FIG. 32  is programmed; 
         FIG. 34  is a diagram illustrating a presently preferred model by which the processor of  FIG. 32  is programmed; 
         FIG. 35  is a diagram illustrating how the drill site solutions referenced to the respective first and second parallel planes are blended to distribute the solution across the longitudinal axis of the workpiece; 
         FIG. 36  is a vector diagram useful in understanding the theory behind the general weight splitting concept; and 
         FIG. 37  is a graphical depiction of one of the model constrains, with components labeled to aid in understanding how the data structure of the disclosed constraint model is configured in memory. 
     
    
    
     Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings. 
     DETAILED DESCRIPTION 
     Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments are provided so that this disclosure will be thorough, and will fully convey the scope to those who are skilled in the art. Numerous specific details are set forth such as examples of specific components, devices, and methods, to provide a thorough understanding of embodiments of the present disclosure. It will be apparent to those skilled in the art that specific details need not be employed, that example embodiments may be embodied in many different forms and that neither should be construed to limit the scope of the disclosure. In some example embodiments, well-known processes, well-known device structures, and well-known technologies are not described in detail. 
     The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. As used herein, the singular forms “a,” “an,” and “the” may be intended to include the plural forms as well, unless the context clearly indicates otherwise. The terms “comprises,” “comprising,” “including,” and “having,” are inclusive and therefore specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. The method steps, processes, and operations described herein are not to be construed as necessarily requiring their performance in the particular order discussed or illustrated, unless specifically identified as an order of performance. It is also to be understood that additional or alternative steps may be employed. 
     When an element or layer is referred to as being “on,” “engaged to,” “connected to,” or “coupled to” another element or layer, it may be directly on, engaged, connected or coupled to the other element or layer, or intervening elements or layers may be present. In contrast, when an element is referred to as being “directly on,” “directly engaged to,” “directly connected to,” or “directly coupled to” another element or layer, there may be no intervening elements or layers present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between,” “adjacent” versus “directly adjacent,” etc.). As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. 
     Although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers and/or sections, these elements, components, regions, layers and/or sections should not be limited by these terms. These terms may be only used to distinguish one element, component, region, layer or section from another region, layer or section. Terms such as “first,” “second,” and other numerical terms when used herein do not imply a sequence or order unless clearly indicated by the context. Thus, a first element, component, region, layer, or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the example embodiments. 
     Spatially relative terms, such as “inner,” “outer,” “beneath,” “below,” “lower,” “above,” “upper,” and the like, may be used herein for ease of description to describe one element or feature&#39;s relationship to another element(s) or feature(s) as illustrated in the figures. Spatially relative terms may be intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as “below” or “beneath” other elements or features would then be oriented “above” the other elements or features. Thus, the example term “below” can encompass both an orientation of above and below. The device may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly. 
     According to the principles of the present teachings, a crankshaft balancer and suspension system assembly  10  is illustrated having advantageous construction and method of operation. It should be understood that the specific orientation and configuration of many of the components and systems of crankshaft balancer and suspension system assembly  10  can vary unless otherwise claimed herein. Therefore, the following disclosure should be regarded as providing some embodiments of the present invention. 
     With particular reference to  FIGS. 1-6 , in some embodiments, crankshaft balancer and suspension system assembly  10  can comprise a plurality of substations each tailored to manipulate, test, and/or process a crankshaft  1000  of an engine during manufacturing. To this end, crankshaft balancer and suspension system assembly  10  can generally comprises one or more of a plurality of stations, such as, but not limited to, a measurement station  200 , a transfer station  400 , and a correction station  600 . In some embodiments, the major components of crankshaft balancer and suspension system assembly  10  can comprise a guard structure  802  (not shown) enclosing the stations for safety, noise, and/or operational considerations. Moreover, in some embodiments, measurement station  200 , a transfer station  400 , and a correction station  600 , and their associated subcomponents and structure, can be positioned upon elevated structures  804 . Each of the elevated structures  804  is configured and sized to raise the associated station above a floor level for viewing, operation, maintenance, and processing purposes, as desired. It should be understood that structures  804  can be independently formed, integrally formed, and/or operably coupled, such as via supports  806 , as desired. 
     Measurement System 
     Generally, as illustrated in  FIGS. 7-20 , measurement station  200  is configured to rotate crankshaft  1000  to obtain data relevant to determining a rotational balance of crankshaft  1000 . To this end, measurement station  200  can comprise a system operable to spin crankshaft  1000  about the longitudinal axis of crankshaft  1000 . During such spinning of crankshaft  1000 , an out-of-balance vibration may be present that results in a vibration in measurement station  200  sufficient to be measured by electronic means, thereby outputting vibration-related data. In some embodiments, this vibration-related data is used to determine a crankshaft processing protocol (e.g. determination of crankshaft production properties in connection with specification limits and the like). In some embodiments, the crankshaft processing protocol can define treatment and/or modification requirements to modify crankshaft  1000  such that the resultant rotational balance of crankshaft  1000  is within the specification limits. To this end, crankshaft processing protocol can call for removal and/or addition of material from counterweights installed on crankshaft  1000  at correction station  600 . Crankshaft  1000  can be moved from measurement station  200  to correction station  600  via transfer station  400 , as will be discussed herein. 
     In some embodiments, measurement station  200  comprises a base structure  202  connectable to elevated structure  804 . Base structure  202  can be substantially planar for supporting the remaining components of measurement station  200  thereon. In some embodiments, base structure  202  is operably coupled to a plurality of vertically extending flexural support legs  204  disposed at opposing corners of base structure  202 . In some embodiments, flexural support legs  204  are generally cylindrical in shape and sized to permit a vibration to operably occur in a measurement bridge structure  206  operably coupled to each of the plurality of flexural support legs  204 . In this way, vibrations produced within measurement bridge structure  206  can be detected, measured, and output as data relevant to determining the rotational balance of crankshaft  1000 . In some embodiments, measurement bridge structure  206  can comprise a pair of end bracket supports  208  each being coupled to a pair of the plurality of flexural support legs  204 , such as through clamping connection. Each of the end bracket supports  208  can comprise clamping or fastening structure for receiving and securing opposing, parallel support tubes  210 . In some embodiments, support tubes  210  are each generally cylindrical in shape and sized to support the weight of the remaining componentry and crankshaft  1000  during operation. In some embodiments, support tubes  210  can be parallel to each other and generally orthogonal to end bracket supports  208 . In some embodiments, end bracket supports  208  and support tubes  210  together define a generally rigid frame for supporting a drive system  212  and crankshaft  1000  upon the plurality of flexural support legs  204 . 
     In some embodiments, drive system  212  can comprise a motor  214 , such as a servo motor, operably coupled to a coupling system  216 , such as a belt, via a sprocket  218  operably coupled to a drive spindle  220 . Drive spindle  220  is connectable to crankshaft  1000  to rotate crankshaft  1000  in a direction about the longitudinal axis of crankshaft  1000  during testing and measurement. In some embodiments, drive system  212  can comprise one or more compensating plates  222 . Compensating plates  222  can be used to compensate for final weights and/or conditions that will be later applied to crankshaft  1000  that are not currently present and/or compensating plates  222  can be used to compensate for the weight of crankshaft  1000  during testing. 
     As will be appreciated from the figures, in some embodiments, coupling system  216  of drive system  212  is coupled generally adjacent to drive spindle  220 . This proximal arrangement (rather than at an end of the assembly) positions the motor  214  closer to the center of gravity the system  200 , thereby minimizing and/or eliminating the parasitic mass of the system. 
     In some embodiments, measurement station  200  can comprise a roller bridge assembly  224  operably coupled to one or more support tubes  210  of measurement bridge structure  206  for rotatably supporting crankshaft  1000  during testing and measurement. Roller bridge assembly  224  can comprise one or more rollers  226  rotatably mounted on a bridge support  228  that is clamp to one or more support tubes  210  via clamps  230 . 
     Similarly, in some embodiments, measurement station  200  can comprise one or more thrust locators  232  operably coupled to one or more support tubes  210  of measurement bridge structure  206  for ensuring proper location of crankshaft  1000  upon measurement station  200  during testing and measurement. 
     Still further, in some embodiments, measurement station  200  can comprise one or more vibration sensors  234  operably coupled to at least a portion of measurement bridge structure  206  for detecting and measuring vibration force produced during rotation of crankshaft  1000 . As has been discussed herein, this vibration force may be indicative of an out-of-balance condition in crankshaft  1000  relative to a predefined vibration limit. In some embodiments, this vibration force can be measured by one or more sensors  234 , such as an accelerometer, two-axis accelerometer, three-axis accelerometer, velocity sensors, proximity sensors, magnetic sensors, and the like. Vibration sensors  234  can output data relevant to determining a rotational balance of crankshaft  1000 . As discussed herein, this vibration data can be is used to determine a crankshaft processing protocol (e.g. determination of crankshaft production properties in connection with specification limits and the like). In some embodiments, the crankshaft processing protocol can define treatment and/or modification requirements to modify crankshaft  1000  such that the resultant rotational balance of crankshaft  1000  is within the specification limits. To this end, crankshaft processing protocol can call for removal and/or addition of material from counterweights installed on crankshaft  1000  at correction station  600 . 
     Transfer Station 
     As illustrated in  FIGS. 1-6 , in order to process crankshaft  1000  after determination of vibration data and the resultant processing protocol, crankshaft  1000  can be transferred from measurement station  200  to correction station  600  using transfer station  400 . To this end, in some embodiments as illustrated in  FIGS. 1-6 , transfer station  400  can comprise a rotatable transfer table  402  supported upon a structure, such as elevated structure  804 , and positioned between measurement station  200  and correction station  600 . Rotatable transfer table  402  is configured to permit rotation about a vertical axis of a plurality of lifting arms  404  sized and configured to support crankshaft  1000 , as will be described. In some embodiments, rotatable transfer table  402  is power-driven rotation table capable of rotating a first pair of lifting arms  404 , which support a first crankshaft  1000 , about the vertical axis from measurement station  200  to correction station  600 . Likewise and, in some embodiments, simultaneously, a second pair of lifting arms  404 , which support a second crankshaft  1000 , can be rotated about the vertical axis from correction station  600  back to measurement station  200  for testing following balance correction and/or to a load/offload station (not shown) for loading new crankshafts for measurement and correction and offloading corrected crankshafts. 
     In some embodiments, transfer station  400  further comprises a tower portion  406  operably coupled and supported by rotatable transfer table  402  to permit rotation of tower portion  406  about the vertical axis. Tower portion  406  can comprise a cam/ball screw system  408  having an internal cam follower  410  operably coupled to lift bridge  412  via a camming interface. Internal cam follower  410  can be being rotatably driven to lift and/or lower lift bridge  412 . Lift bridge  412  can be operably coupled to the plurality of lifting arms  404 , thereby permitting transfer station  400 , via tower portion  406 , cam follower  410 , and the plurality of lifting arms  404 , to lift and/or lower crankshaft  1000  into position on measurement station  200  and correction station  600 . In some embodiments, internal cam/ball screw system  408  is driven by a cam motor  414  via a cam transmission  416 . 
     It should be understood that lift bridge  412  can comprise a pair of channel support members  418  defining channel or other features  420  sized and shaped to receive complementary features formed on lifting arms  404 . In this regard, lifting arms  404  can be quickly and conveniently installed and/or repositioned along channel support members  418  to accommodate size and shape variations of crankshaft  1000 . 
     Correction Station 
     With reference to  FIGS. 1-6 , to permit vibration correction of crankshaft  1000 , correction station  600  is provided to receive, retain, and horizontally drill crankshaft  1000  to remove material at custom locations according to vibration data received from measurement station  200  and computed by a central processing unit. Each crankshaft  1000  can be corrected at correction station  600  based on a custom processing protocol determined in connection with data representative of the present crankshaft  1000 . 
     In some embodiments, correction station  600  is configured to support, rotate, and retain crankshaft  1000  to permit correction processing (e.g. rotational balancing) of crankshaft  1000 . To this end, correction station  600  can comprise a system operable to spin crankshaft  1000  about the longitudinal axis of crankshaft  1000  to position crankshaft  1000  in a predetermined orientation to permit horizontal drilling of portions thereof. 
     In some embodiments, correction station  600  comprises a base structure  602  connectable to elevated structure  804 . Base structure  602  can be substantially planar for supporting the remaining components of correction station  600  thereon. In some embodiments, base structure  602  is operably coupled to a correction bridge structure  604  for supporting crankshaft  1000  during processing. In some embodiments, correction bridge structure  604  can comprise a drive system  606  driving crankshaft  1000  to position crankshaft  1000  into various positions during drilling and processing. 
     In some embodiments, drive system  606  can comprise a motor  614 , such as a servo motor, operably coupled to a coupling system  616 , such as a belt, operably coupled to a drive spindle  618 . Drive spindle  618  is connectable to crankshaft  1000  to rotate crankshaft  1000  in a direction about the longitudinal axis of crankshaft  1000  during drilling and processing. In order to hold and retain crankshaft during drilling and processing, and accommodate the enormous forces exerted on crankshaft  1000 , correction station  600  can comprise one or more pneumatic crankshaft clamp assemblies  100 . 
     Pneumatic Crankshaft Clamp Assembly 
     As illustrated in  FIGS. 21-29 , pneumatic crankshaft clamp assembly  100  is provided having an advantageous construction and method of operation. In some embodiments, pneumatic crankshaft clamp assembly  100  can comprise a housing  110  and a clamp mechanism  112 . Clamp mechanism  112  can be configured to be pneumatically actuated between a clamp position and an unclamp position. To this end, in some embodiments, clamp mechanism  112  can comprise a clamp arm  114  having a distal end  116  and a proximal end  118 . Clamp arm  114  can be pivotally coupled to housing  110  at a clamp pivot  120  to move between the clamp position and the unclamp position, as will be described herein. 
     In some embodiments, clamp arm  114  includes an elongated grasping portion  122  extending from an enlarged central hub portion  124 . Central hub portion  124  generally surrounds and contains clamp pivot  120 . In some embodiments, a bearing or other member  126  can be disposed between central hub portion  124  and clamp pivot  120  to facilitate reduce friction operation and improved durability. Elongated grasping portion  122  can be shaped to include an angled distal portion  128  to facilitate grasping a crankshaft  1000  ( FIG. 26 ) for processing. As will be described herein, one or more spacers can be disposed at or along elongated grasping portion  122  to engage crankshaft  1000  during processing. In some embodiments, clamp arm  114  includes an elongated cam portion  130 . In some embodiments, elongated cam portion  130  can be positioned to opposingly extend from central hub portion  124  oppose of elongated grasping portion  122 . Elongated cam portion  130  can terminate at a cam end  132  having a cam follower member  134  operably coupled thereto. Cam follower member  134  can comprise a circular member, or other shaped member, that is configured to cammingly follow a cam  136  formed on a piston member  138 . In some embodiments, cam follower member  134  comprises a bearing member. 
     With particular reference to  FIG. 25 , in some embodiments, piston member  138  is slidably disposed within a piston cylinder  140  formed in housing  110 . In some embodiments, piston cylinder  140  is generally cylindrical in shape and sized to closely conform to piston member  138  for sliding movement therein in response to pneumatic actuation. A cap member  141  can be used to enclose piston cylinder  140  can contain piston member  138  therein. Cap member  141  can be coupled to housing  110  using any conventional members. In some embodiments, a seal member is disposed between cap member  141  and housing  110  to maintain a pneumatic seal. A sidewall section  142  of piston cylinder  140  is removed to form a slot or opening  144  to permit cam follower member  134  to extend therethrough and contact cam  136  formed on piston member  138  for camming operation therewith. Piston member  138  is configured to slide within piston cylinder  140  between a clamp position (down in  FIG. 25 ) and an unclamp position (up in  FIG. 25 ). 
     With continued reference to  FIG. 25 , piston member  138  can comprise one or more slots  146  formed about the sidewall thereof to receive seal members. The seal members are operable to define a first pressurizable chamber  148  extending between a top surface  150  of piston cylinder  140  and a first slot  146   a  of piston member  138 . As will be described herein, the first pressurizable chamber  148  will be pressurized in response to introduction of pneumatic pressure from clamp inlet port  150 . Clamp inlet port  150  extends through housing  110  and is fluidly coupled to first pressurizable chamber  148  such that the pneumatic pressure exerts a pressure upon a top surface  152  of piston member  138  thereby urging piston member  138  to move downward in  FIG. 25  into the clamp position. Similarly, the seal members are operable to define a second pressurizable chamber  154  extending between a second slot  146   b  and a third slot  146   c  of cap member  141 . As will be described herein, the second pressurizable chamber  154  will be pressurized in response to introduction of pneumatic pressure from unclamp inlet port  156 . Unclamp inlet port  156  extends through housing  110  and is fluidly coupled to second pressurizable chamber  154  such that the pneumatic pressure exerts a pressure upon a bottom surface  158  of piston member  138  thereby urging piston member  138  to move upward in  FIG. 25  into the unclamp position (illustrated in  FIG. 25 ). It should be recognized that first pressurizable chamber  148  and second pressurizable chamber  154  are located on opposing sides of piston member  138  such that a central portion of piston member  138 , where cam follower member  134  is positioned is contained within an unpressurized zone between first slot  146   a  and second slot  146   b , thereby enabling central portion of piston member  138  to be exposed to atmosphere. 
     In some embodiments, a bumper member  160  can be disposed at top surface  152  of piston member  138  to cushion or otherwise minimize destructive contact between piston member  138  and piston cylinder  140 . Bumper member  160  can be affixed to piston member  138  via conventional means, such as a fastener or other retaining method. 
     With continued reference to  FIG. 25 , in some embodiments, pneumatic crankshaft clamp assembly  100  comprises a counter bearing member  162  generally coupled to housing  110 . Counter bearing member  162 , in some embodiments, can be positioned opposite of cam follower member  134  to provide an opposing force on piston member  138 . In some embodiments, counter bearing member  162  rotatably mounted to a counter bearing support  164 . Counter bearing support  164  and counter bearing member  162  are disposed within an opening  166  formed in housing  110 . In some embodiments, the relative position of and/or opposing force exerted by counter bearing member  162  can be varied via one or more spacers or other adjustment means between counter bearing support  164  and housing  110 . It should be understood that alternative adjustment techniques are anticipated. 
     In some embodiments, pneumatic crankshaft clamp assembly  100  comprises an override system  168 . Override system  168  is configured to provide a manual override feature, such as for safety purposes, for urging piston member  138  upward into the unclamp position. To this end, an override cam member  170  is rotatably mounted along a guide bracket  172  extending from housing  110 . Override cam member  170  is sized to engage a cam follower rod  174  extending from bottom surface  158  of piston member  138  and through a slot  175  formed in cap member  141 . Override cam member  170  and cam follower rod  174  are sized and shaped to permit free movement of piston member  138  during normal operation; however, upon manual rotation of override cam member  170  from a first normal position to a second override position ( FIG. 25 ), override cam member  170  contacts a distal end of cam follower rod  174  of piston member  138  and mechanically urges piston member  138  against the biasing force of pneumatic pressure within first pressurizable chamber  148 . Override cam member  170  can be rotated via a handle member (not shown) disposed within an interior slot  176  and a key member  178  engaging the handle member. It should be understood that alternative manual and/or automated systems can be used for actuating override system  168 . 
     Still further, in some embodiments, pneumatic crankshaft clamp assembly  100  comprises a cover member  180  disposed over clamp arm  114 . Cover member  180 , together with sidewalls  182  of housing  110  and coverplate  183  and shields  185 , can contain and protect clamp arm  114  and further maintain a safe working area for an operator. In some embodiments, a biasing assembly  184  can be positioned within cover member  180  to contact elongated cam portion  130  of clamp arm  114  to exert a biasing force against clamp arm  114  to maintain engagement of cam follower  134  to cam  136 . Biasing assembly  184  can comprise a set screw  186  having a distal tip urging a contact member  188  into contact with clamp arm  114 . A biasing member  190 , such as a spring, can urge the contact member  188  into contact as described. 
     Housing  110  can comprise a cradle portion  192  sized and shaped to received crankshaft  1000  therein. Cradle portion  192  can comprise one or more spacer members  194  mounted thereto for direct contact with crankshaft  1000 . Spacer members  194  permit standoff spacing of crankshaft  1000  to ensure proper and exact positioning of crankshaft  1000  and improve tolerance adherence. One or more spacer members  194  can further be disposed on angled distal portion  128  of elongated grasping portion  122  to engage crankshaft  1000  during processing. 
     In some embodiments, pneumatic crankshaft clamp assembly  100  can be mounted for operation via a positioning block  196  extending from housing  110 . Positioning block  196  can comprise any one of a number of designs sufficient to safely and reliably coupled pneumatic crankshaft clamp assembly  100  to a supporting structure for operation of pneumatic crankshaft clamp assembly  100 . 
     During a clamping operation—from the unclamp position to the clamp position—pneumatic pressure is introduced into clamp inlet port  150  thereby increasing a pneumatic pressure within first pressurizable chamber  148 . This pneumatic pressure within first pressurizable chamber  148  urges piston member  138  downward. Downward movement of piston member  138  causes cam follower member  134  to cammingly follow cam  136  formed in piston member  138  and specifically along an inclined portion of cam  136 , thereby causing cam follower member  134  to be urged outwardly (arcuately to the right in  FIG. 25 ) against the biasing force of biasing assembly  184 . Counter bearing  162  opposes a force exerted on piston member  138  from cam follower member  134 . This movement of cam follower member  134  is translated along cam arm  114 , about pivot  120 , resulting in the inward movement (arcuately to the left in  FIG. 25 ) of angled distal portion  128  of elongated grasping portion  122  of crank arm  114  resulting in spacers  194  contacting crankshaft  1000 , allowing processing of crankshaft  1000 . 
     During an unclamping operation—from the clamp position to the unclamp position—pneumatic pressure is introduced into unclamp inlet port  156  thereby increasing a pneumatic pressure within second pressurizable chamber  154 . This pneumatic pressure within second pressurizable chamber  154  urges piston member  138  upward. Upward movement of piston member  138  causes cam follower member  134  to cammingly follow cam  136  formed in piston member  138  and specifically along the declined portion of cam  136 , thereby causing cam follower member  134  to be urged inwardly (arcuately to the left in  FIG. 25 ) due to the biasing force of biasing assembly  184 . Counter bearing  162  opposes a force exerted on piston member  138  from cam follower member  134 . This movement of cam follower member  134  is translated along cam arm  114 , about pivot  120 , resulting in the outward movement (arcuately to the right in  FIG. 25 ) of angled distal portion  128  of elongated grasping portion  122  of crank arm  114  resulting in spacers  194  being spaced apart from crankshaft  1000 , allowing removal of crankshaft  1000 . 
     During an override operation, which presumes pneumatic crankshaft clamp assembly  100  is in the clamp position, an operator or override machine can actuate override system  168  such that override cam member  170  engages cam follower rod  174  extending from bottom surface  158  of piston member  138  thereby mechanically urging piston member  138  against the biasing force of pneumatic pressure within first pressurizable chamber  148 . Override cam member  170  can be rotated via a manual and/or automated system. 
     Drilling System 
     Referring to  FIGS. 1-6 , in some embodiments, correction station  600  comprises a drilling system  700  for drilling and removing material from crankshaft  1000  in accordance with the teachings of the present disclosure. In some embodiments, drilling system  700  comprises a drilling device  702  being supported by a positioning system  704 . Positioning system  704  is configured to position drilling device  702  in various positioned relative to crankshaft  1000 . In some embodiments, positioning system  704  is configured to move drilling device  702  horizontally along various longitudinal positions of crankshaft  1000 . In some embodiments, positioning system  704  is configured to move drilling device  702  horizontally between an engaged drilling position (e.g. where a drill bit contacts and cuts a portion of crankshaft  1000 ) and a disengaged drilling position (e.g. where the drill bit is spaced apart from crankshaft  1000 ). In some embodiments, positioning system  704  is configured to move drilling device  702  vertically and/or angularly relative to crankshaft  1000 . 
     In some embodiments, drilling system  702  comprises a drilling chuck  706  for retaining a conventional drilling bit (not shown) operably coupled to a drilling spindle  708 . Drilling spindle  708  is coupled to a drilling motor  710  via a drilling transmission  712 . 
     In some embodiments, drilling system  702  is configured such that drilling chuck  706  and the associated drilling bit are oriented in a horizontal position. It has been found that such horizontal drilling orientation provides a number of distinct benefits not realized by the prior art. For instance, horizontal drilling provides reduce structural sizing requirements. Conventional systems often employ an angular drilling orientation that requires enormous structure to support the drilling motor and associate systems due to gravitational forces and bending moments. However, a horizontal configuration provides a simple solution by reducing the structural requirements due to the reduced bending moments and forces. Moreover, the horizontal configuration permits a more rapid cycle time because movement of the drilling system  702  can be more quickly achieved as all linear movement results in an equal movement horizontally away from the crankshaft. Therefore, when the drilling system  702  is moved out of engagement with crankshaft  1000 , a direct horizontal movement will occur more rapidly than angular movement (which includes only a reduced horizontal movement component). Therefore, the horizontal stroke of the positioning system  704  is reduced and the associated size, complexity, and cost of positioning system  704  are improved. 
     In some embodiments, positioning system  704  can comprise a first or longitudinal drive system  720  having a drive motor  722  operably coupled to rotationally-driven drive shaft  724 . Drive shaft  724  can be supported for rotation by one or more bearing supports  726 . A corresponding drive connection  728  can extend from a support platform  730  supporting drilling device  702 . Support platform  730  can be generally planar having drive connection  728  mounted thereto. Actuation of longitudinal drive system  720  enables longitudinal movement of drilling device  702  relative to crankshaft  1000  for drilling at multiple longitudinal positions along crankshaft  1000 . It should be understood that alternative drive systems can be used to move drilling device  702  to various longitudinal positions relative to crankshaft  1000 , including belt drives, cam drives, gear drives, and the like. 
     Similarly, in some embodiments, positioning system  704  can comprise a second or transverse drive system  740  having a drive motor  742  operably coupled to rotationally-driven drive shaft  744 . Drive shaft  744  can be supported for rotation by one or more bearing supports  746  mounted to support platform  730 . A corresponding drive connection  748  can extend from drilling device  702 . Actuation of transverse drive system  740  enables transverse movement of drilling device  702  relative to crankshaft  1000  and support platform  730  to drill at varying depths into crankshaft  1000 . It should be understood that alternative drive systems can be used to move drilling device  702  to various transverse positions relative to crankshaft  1000 , including belt drives, cam drives, gear drives, and the like. 
     In some embodiments, a cable guide system  760  can be provided to protect the communication and/or drive cables and other components routed to drilling device  702  and/or transverse drive system  740  to protect such communication and drive cables during movement of drilling device  702  and/or transverse drive system  740  relative to elevated structure  804 . 
     Crankshaft Processing Protocol 
     Correcting for imbalances in crankshafts or other rotating members has traditionally been somewhat of a trial and error process. Although it is possible to measure the overall imbalance of a rotating member, correcting for this imbalance by adding or subtracting weight at offsetting locations can be difficult because often there may be more than one solution for where to add or subtract material. Ideally, one would like to find an optimal solution that requires minimal invasive adding or subtracting of material. However, in conventional systems finding this optimal solution has proven difficult. 
     In the case of crankshaft  1000 , for example, there are multiple sites radially disposed along the longitudinal axis. Each of these sites can potentially be used as drill sites from which to remove weight in order to offset an imbalance. However, a machinist has only his personal judgment regarding where to drill and how deep. The goal, of course, is to remove material to counteract the measured imbalance. 
     This is not as easy as it might appear at first blush. Because the crankshaft has a significant longitudinal dimension, drilling to offset a static imbalance can introduce dynamic imbalances. This is because a crankshaft, like other elongated rotating members, can exhibit vibration in multiple modes, just as a plucked string can vibrate at the primary frequency and also at integer multiples of the primary frequency. Thus drilling to remove the primary mode of imbalance or vibration could possibly introduce unwanted vibrations at higher order modes. 
     The weight splitting control system will be illustrated in conjunction with a measurement station  200 , shown in  FIGS. 7-20 , and a correction station  600 , shown in  FIGS. 1-6 . Crankshaft  1000  to be balanced will be illustrated as crankshaft  1000 ; however, it should be understood that the principles of the present teachings are equally applicable to balancing any rotating mass, such as a prop, propeller, drive shaft, and the like. The measuring machine is disposed on elongated flexural support legs  204  that allow the body of the measuring machine  200  to vibrate as crankshaft  1000  is spun. Motion sensors  234  disposed in the body of the measuring machine provide electrical signals indicative of the vibrations exhibited by the measuring machine. Such vibrations occur when crankshaft  1000  has an imbalance. 
     In the preferred embodiment, the measurement station  200  is calibrated with a test fixture or workpiece of known axial symmetry. During calibration the motion sensor signals are referenced to two parallel and spaced apart reference planes that orthogonally intersect the measurement station  200 &#39;s axis of rotation. These reference planes are shown in  FIG. 13 . Using two reference planes allows the motion sensors to provide dynamic data reflecting imbalances in a workpiece. That is, while a single reference plane would be sufficient to measure the first order vibrational mode of crankshaft  1000 ; two reference planes also capture the second order vibrational mode. See  FIG. 30 , which illustrates these first and second vibrational modes. Because crankshaft  1000  is clamp at its ends, vibration is zero at the clamping points. Vibration reaches a single maximum in the first vibrational mode, as at 32; whereas vibration exhibits two maxima 34 in the second vibrational mode. 
     Drill Sites 
     Crankshaft  1000  typically will have several different locations where material can be removed, as by drilling, to counteract any measured imbalance. In the exemplary crankshaft, shown in  FIG. 31 , there are N potential drill sites, where N is an integer number. For each of the N sites, the following values are determined and stored in memory: radius, angle, axial location, and maximum drill depth. 
       FIG. 31  shows how these values are related. Essentially, the entry point of each drill site may be specified by a radius and an angle, measured from a common reference system to which the imbalance vector is also measured. The axial location corresponds to the location along the central axis of the crankshaft  1000  where the drill site is located. Thus the radius, angle and axial location specify a point in three dimensional space. The maximum drill depth is the depth beyond which the drill should not descend. This value is determined based on engineering strength of materials standards. 
     Processor 
     To determine the optimal drill sites and drill depths, a preferred embodiment uses an electronic circuit, as illustrated in  FIG. 32 , having a processor  940  (e.g. microprocessor or computer) that has associated computer memory  942 . The processor is programmed to perform a series of computational steps that determine the optimal drill sites and drill depths that will reduce the measured imbalance to substantially zero or to within a predefined range of substantially zero. The computed optimal drill sites and drill depths are fed as computed values to the correction station  600 , which uses the computed values to robotically or automatically control an electrically activated workpiece rotator  606  to rotate crankshaft  1000  to the correct angle, whereupon an automated drilling device  702  removes material to a certain calculated depth. 
     The processor is programmed according to the flowchart shown in  FIG. 33 . Prior to running the program shown in  FIG. 33 , a set of data are stored for the given workpiece to specify where the potential drill sites are physically located. These data specify: a radius, an angle, and an axial location for each potential drill site. These three values specify a unique point in three-dimensional space using a cylindrical coordinate system. See  FIG. 31 . If multiple different kinds of workpieces are to be balanced using the disclosed system, an array may be allocated in computer memory, to accommodate values for each different type of workpiece. Alternatively, these drill site data can be stored in a database, which the processor can query to retrieve the drill site data for crankshaft  1000 . 
     As shown in  FIG. 33 , the processor, at step  900 , retrieves the drill site data for crankshaft  1000  and stores that data in memory for subsequent use in performing the described calculations. Next, at step  902 , the processor ingests imbalance data from sensors  234 . This can be done in real time, as crankshaft  1000  is being rotated on the measurement station  200 , or imbalance data can be collected ahead of time and then fed to the processor at step  902 . 
     In the case where the data from sensors  234  represents raw vibrational data, the processor, at step  904 , processes this data to obtain plural imbalance values, each corresponding to the imbalance measured with respect to a different reference frame. In the presently preferred embodiment two parallel reference frames are defined during calibration of measurement station  200 . As discussed above, multiple separate planes are used, so that the system can measure and correct for first-order, second-order and potentially higher-order vibrational (imbalance) modes. Each imbalance measurement is a vector in weight-radius units, such as gram-centimeter units or the like. 
     In the preferred embodiment where two parallel reference planes are employed, two imbalance measurements are produced, one from the vantage point of the first parallel reference frame, and one from the vantage point of the second parallel reference frame. As will be discussed below, these two imbalance measurements are processed separately (in parallel) and are then distributed across the length of crankshaft  1000  on a ratio or percentage basis. 
     Once the imbalance data are ingested, the processor populates a predetermined data structure representing a model of the imbalance problem, as at step  906 . A further discussion of the precise details of this model is provided below. Essentially, the model represents a series of constraints, expressed in a form suitable to be manipulated by a computational solver program. In the presently preferred embodiment, the processor performs a linear programming solver program. For this linear programming solver, the model represents a series of constraints expressed as algebraic statements that are populated with values obtained from the retrieved drill site data and from the imbalance measurements taken. Other types of solvers may also be used. 
     The processor runs the solver program, at step  908 , resulting in the computational discovery of one or more solutions that satisfy all of the modeled constraints. In the presently preferred embodiment, solutions that are outside predefined limits are excluded, as shown at step  910 . Specifically, solutions specifying negligible drill depth (e.g, a drill depth of zero to a few millimeters) are excluded. After such exclusion, a single optimal solution is obtained. 
     The optimal solution so obtained is fed to the correction station  600 , as at step  912 , where the values are exported as three-dimensional vectors or ordered triplets (angle, axial location, depth) used to position crankshaft  1000 , align the drill and control the drilling depth for each drill applicable site. 
     The Model 
     As will be explained in the Theory section below, the presently preferred model represents the imbalance relative to each of the two parallel planes separately. For each plane the imbalance is a vector quantity, measured in suitable units, such a gm-cm, oz-in, or the like. For explanation purposes here, the letter Z shall be used to represent such imbalance vector. In the presently preferred embodiment, there would actually be two such vector values, one for each parallel reference plane, thus Z 1  and Z 2 . For simplicity, at this point of the discussion, only one vector Z shall be referred to, unless otherwise stated. 
     The objective of the solver program is to find the optimal set of offsetting drill site vectors to effectively negate the imbalance represented by Z. While it is theoretically possible to solve to fully offset the imbalance, in most practical applications it is sufficient, and less costly to offset the imbalance to a predefined tolerance. When represented in vector space, a predefined tolerance can be represented as a circle, with radius r, as shown in  FIG. 34 . From the solver&#39;s perspective, a circle represents a nonlinear problem that, while solvable, takes a lot of processor cycles. Therefore, to simplify the problem, the present embodiment uses an approximation of the circle corresponding to a regular polygon, inscribed within the tolerance circle. For illustration purposes, a square is disclosed here. As shown in  FIG. 34  the inscribed polygon (e.g., square) represents approximations that all fall within the tolerance radius. Moreover, because the polygon (square) is made up of straight lines, this represents a linear problem for the solver. 
     Thus in place of the tolerance circle of radius r, the model defines constraints for the solver in terms of the vector positions (x, y positions) of the corners of the inscribed polygon (square). From  FIG. 34 , it will be appreciated that the (x, y) positions of the four corners of the inscribed square all correspond to offsets from the position of the imbalance vector Z. With this observation in mind, we can now turn to how the model represents the relationship between potential drill site locations and the imbalance vector Z. 
     Specifically, each drill site corresponds to a radius, an angle and an axial location, as discussed above. For explanation purposes here, the letter V 1  shall be used to represent a drill site radius and Θv i  shall be used to represent the drill site angle. The drill site angle for each drill site is known from the angular data retrieved and stored in memory at step  900 ,  FIG. 33 . The drill site radius corresponds to a drill depth, where the vector V i  points to or terminates at the point where the drill bit stops. In sites that are not drilled, the vector V i  simply points to the surface of the drill site in its non-drilled state. 
     Thus once the model data structure has been populated with measured values, the solver is solving to determine the drill depth for each site that requires drilling according to the optimal solution discovered by the solver. The following set of equation constraints represents the constraints needed to specify solutions falling within the inscribed square (tolerance zone) shown in  FIG. 34 . Note that this set of constraints uses N to represent the number of drill sites as referenced to one of the reference planes. In this case two reference planes are used, so another identical set of equation constraints will be defined and solved, for the other plane. In the equations below, V i  is a vector quantity.
 
minΣ i=1   N   |V   i |cos θ≤ Q   1x   (Eq. 1)
 
minΣ i=1   N   |V   i |sin θ≤ Q   1y   (Eq. 2)
 
minΣ i=1   N   |V   i |cos θ≤ Q   2x   (Eq. 3)
 
minΣ i=1   N   |V   i |sin θ≤ Q   2y   (Eq. 4)
 
minΣ i=1   N   |V   i |cos θ≤ Q   3   (Eq. 5)
 
minΣ i=1   N   |V   i |sin θ≤ Q   3y   (Eq. 6)
 
minΣ i=1   N   |V   i |cos θ≤ Q   4x   (Eq. 7)
 
minΣ i=1   N   |V   i |sin θ≤ Q   4y   (Eq. 8)
 
     When the solver computes the solution to these constraints, it employs a solver algorithm that finds the minimum amount of drilling required to achieve a solution to this set of simultaneous equations. This can be understood from the fact that each constraint equation seeks the min iterative sum for each of the respective (x, y) square corner vector locations. 
     The presently preferred solver is a linear programming solver that utilizes the simplex technique, the details of which can be found in the literature. By way of implementation example, the Microsoft Foundation Solver may be used. However, it will be appreciated that a variety of different solvers can be used, so long as they can calculate a solution to the set of equation constraints outlined here. 
     The foregoing constraints are referenced to a particular reference plane. However, as will be appreciated from the example workpiece featured in  FIG. 31 , the potential drill sits are distributed longitudinally along crankshaft  1000 . In other words, each drill site has an axial location along the longitudinal axis of crankshaft  1000 . These locations are specified in the data retrieved in step  900 ,  FIG. 33 . 
     To take the axial location into account, the preferred embodiment uses a blending algorithm that assigns a drill depth for each drill site that is a blend of the resultant solver outputs using a percentage calculation that accounts for where each drill site happens to reside relative to each of the two parallel reference planes. As shown in  FIG. 35  a drill site that lies fully within one reference plane receives a 100% contribution from the solver output for that reference plane. Likewise, a drill site that lies fully within the other reference plane receives a 100% contribution from the solver output for that other reference plane. A drill site that lies half-way between the two reference planes receives a 50% contribution from the solver outputs of each of the two reference planes. Other drill sites are computed proportionally in the same fashion. 
     Theory 
     The objective of the disclosed model and computer process is to find equivalent vectors Vi that offset the imbalance vector Z. Consider for the moment a simple two-dimensional case where equivalent vectors V 1  and V 2  are selected to offset imbalance vector Z. Referring to  FIG. 36 , these equivalent vectors can be related by trigonometric relationships, using the respective angles, as illustrated. In  FIG. 36  it will be understood that vectors V 1  and V 2  represent the vectors where drilling would be applied to offset the imbalance vector. Alternatively, weight can be added corresponding to vectors that are mirror images (180 degrees offset) of vectors V 1  and V 2 . 
     Using the law of sines, the following relationships among V 1  and V 2  and Z can be expressed. 
     
       
         
           
             
               
                 V 
                 2 
               
               
                 sin 
                 ⁡ 
                 
                   ( 
                   
                      
                     
                       θ 
                       - 
                       
                         θ 
                         1 
                       
                     
                      
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   V 
                   1 
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                        
                       
                         
                           θ 
                           2 
                         
                         - 
                         θ 
                       
                        
                     
                     ) 
                   
                 
               
               = 
               
                 Z 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       180 
                       - 
                       
                         θ 
                         2 
                       
                       - 
                       
                         θ 
                         1 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Therefore, 
     
       
         
           
             
               V 
               2 
             
             = 
             
               Z 
               ⁢ 
               
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                        
                       
                         θ 
                         - 
                         
                           θ 
                           1 
                         
                       
                        
                     
                     ) 
                   
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                        
                       
                         
                           θ 
                           2 
                         
                         - 
                         
                           θ 
                           1 
                         
                       
                        
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               V 
               1 
             
             = 
             
               Z 
               ⁢ 
               
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                        
                       
                         θ 
                         - 
                         
                           θ 
                           2 
                         
                       
                        
                     
                     ) 
                   
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                        
                       
                         
                           θ 
                           2 
                         
                         - 
                         
                           θ 
                           1 
                         
                       
                        
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Weight splitting at N locations thus becomes a process of finding the absolute values of V 1 , V 2 , . . . V N  that are equivalent to Z, given Z and the respective angles θ 1 , θ 2 , . . . θ N . 
     A general theorem of equivalence can be determined when the problem is resolved into Cartesian coordinates as follows. 
     
       
         
           
             
               
                  
                 Z 
                  
               
               ⁢ 
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 2 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                    
                   
                     V 
                     i 
                   
                    
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   θ 
                   vi 
                 
               
             
           
         
       
       
         
           
             
               
                  
                 Z 
                  
               
               ⁢ 
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 2 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                    
                   
                     V 
                     i 
                   
                    
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   θ 
                   vi 
                 
               
             
           
         
       
     
     This results in an infinite number of solutions. A constraint is required to narrow the solutions. Accordingly, we apply the following constraints, the former seeking the minimum value and the latter placing a lower limit L i  and an upper limit U i  to the solutions. 
     
       
         
           
             min 
             ⁢ 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 N 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                  
                 
                   V 
                   i 
                 
                  
               
             
           
         
       
       
         
           
             0 
             &gt; 
             
               L 
               i 
             
             ≥ 
             
               V 
               i 
             
             ≥ 
             
               U 
               i 
             
           
         
       
     
     These constraints will minimize the total number of vectors required to reach Z. 
     In many practical applications it may not be necessary to exactly offset the imbalance vector Z. Rather, an acceptable tolerance range can be defined in terms of a quality radius Q, shown in  FIG. 34 . Thus the processor is programmed to find a solution that is anywhere within the acceptable region of Z, namely within the circle of radius Q. 
     In order to find a solution within a circle of radius Q, the solver will need to process nonlinear conditions, as the circle defines a curved, nonlinear locus. It is possible to solve such a nonlinear constraint using a nonlinear linear programming solver (NLPS). However, the processing cycle time required to perform the NLPS algorithm can be unacceptably long in some instances. Thus the disclosed embodiment employs a simplifying approximation that results in linear constraints that can be solved using a linear programming solver (LPS). 
     Specifically, the disclosed program process uses an inscribed polygon, in this case a square, to represent an approximation of the quality circle of radius Q. It will be appreciated that any solution that lies on any of the four sides of this square naturally fall within the acceptable radius Q. Indeed, the solutions correspond to the radius Q at the four vertices of the inscribed square. Elsewhere, the sides of the square represent better tolerance than defined by the circle of radius Q. 
     It will thus be appreciated that the model defined by Eq. 1-8 above are seeing to minimize the vector V associated with each of the four vertices of the inscribed square approximation. Note there are eight equations (Eq. 1-8) because each of the four vertices has an x-component and a y-component when represented in Cartesian coordinates. If a higher order polygon is used in place of the square, a correspondingly larger number of equations would be used to define the constraint model. 
       FIG. 37  shows one of the constraint model equations, with the equation elements labeled for better understanding. 
     The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.