Abstract:
A method for use in forming a molded part includes providing a mold having a cavity and a movable pin, injecting a moldable material into the cavity, biasing the movable pin to maintain an end of the movable pin in contact with the moldable material in the cavity during the curing of the moldable material and until the moldable material is cured, and monitoring movement of the biased movable pin during curing of the moldable material in the mold. Also disclosed is a sensor engageable with an end of a movable pin of a mold for monitoring the forming of a moldable part, and systems employing the same.

Description:
CLAIM TO PRIORITY 
     This application is a 371 national stage filing of PCT International Application No. PCT/US2009/040508 filed on Apr. 14, 2009, and published in English on Oct. 22, 2009, as WO 2009/129230, which claims the benefit of U.S. Provisional Application No. 61/044,698, filed Apr. 14, 2008, entitled “Methods For Forming Injected Molded Parts And In-Mold Sensors Therefor,” the entire subject matter these applications being hereby incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention relates generally to high fidelity systems for dimensional control of injection molded products, and more specifically, to sensors designed to measure in-mold part dimensions and cavity pressures, and mechanistic analyses used to predict the final molded part dimensions, perform control actions, and synthesize alternative feasible processes. 
     BACKGROUND OF THE INVENTION 
     Plastics manufacturing has made continuous gains in capability and competitiveness. Many industry advancements have been fueled by technological progress related to process analysis, instrumentation, and control.  FIG. 1  is a prior art closed-loop injection molding machine  10  with varying levels of feedback. It is generally recognized that feedback about the process may be provided by pressure and temperature sensors  12  disposed on the nozzle, as well as a barrel temperature sensor  14  placed in the machine. In addition, current molding controller technology also relies on machine feedback employing sensors such as a hydraulic pressure sensor  16  placed behind the screw, screw position sensors and screw velocity sensors  18 , clamp force sensor  20 , limit switches  22 , melt pressure and melt temperature flow-rate sensors  24 , and others sensors. 
     Polymer processing provides for the mass production of a wide range of economical yet complex products. In injection molding, thermoplastic feedstock in the form of pellets is melted through conduction and viscous dissipation to form a homogeneous melt. Once a melt is collected, it is forced into a mold to form the desired complex shape. The replication and final dimensions of the molded part relative to the mold cavity is related to the shrinkage of the polymer as it cools inside and outside of the mold. Shrinkage is a complex function of 1) the size, shape, and wall thickness of the part design, 2) the free volume, morphology, and material properties of the polymeric resin, 3) the details of the mold including feed system and cooling system design, and 4) the molding conditions such as flow rates, packing pressures, melt and mold temperatures, timings, etc. 
     The ability to predict and control shrinkage is directly related to the consistency of the molded part dimensions and the usefulness of the molded part, especially in tight tolerance applications which is often employed. For example, commercial and fine tolerances of 0.3% and 0.15% of the overall length dimension for polycarbonate (PC) is often employed. Material shrinkage is characterized by standard tests including ASTM D955-00 and ISO 294-4. However, these standards are typically applied to a tensile bar with a wall thickness of 3.2 mm and assumed process conditions. As such, the final shrinkage and part dimensions in industry applications may vary substantially from those reported. Product designers, mold designers, and molders employ methods to hedge errors in shrinkage rates, yet standard dimensional tolerances as specified by the Society of the Plastics Industry have not changed in the past thirty years. 
     Technological capabilities of the industry have improved since 1970 when many plastics molding machines still used open-loop control for most subsystems. Since the advent of programmable logic control, the majority of machine input variables have become individually controlled via single-input single-output PID (proportional-integral-derivative) algorithms. Continuing advances in machine and control system designs have greatly improved the time response and absolute repeatability of the process. Similar advances have been made with respect to mold making and polymer synthesis. As a result, tighter tolerances are possible, albeit with an uncertain amount of testing, instrumentation, and processing costs. 
     There has been increasing recognition that the measurement and control of the polymer state within the mold cavity is vital to product quality. Accordingly, there has been a proliferation of cavity pressure sensors based on load cells, strain gages, and piezoelectric materials. Concurrently, other methods have been developed for measuring melt temperature in the mold including infrared sensors and thermocouples. Ultrasonic methods have also been developed to detect the presence and solidification of the melt in the mold cavity. These sensors provide valuable information that is commonly used with statistical process control to track the process consistency. However, no single control strategy or system design has been universally successful, and defective components continue to be manufactured during high volume production. 
     To improve the capability of these sensors to predict quality, sensor fusion approaches have incorporated multiple sensor streams with on-line and/or post-molding analyses to predict the part dimensions. The approaches are most often either mechanistic or statistical. Mechanistic approaches vary in complexity from relatively simple analysis of pressure-volume-temperature relations to complex thermo viscoelastic modeling of residual stress relaxation. Statistical models frequently rely on regression, neural networks, or other methods. 
     One attempt is that of Anthony Bjur of NIST and Charles Thomas of the University of Utah, who developed an optical fiber sensor inserted into the ejector pin channel of a mold using an ejector pin sleeve with a sapphire window at its end. As shown in  FIG. 2 , a sapphire window  30  was positioned flush with a wall of the mold  32  having a mold cavity  33 . A fiber optic cable  34  is position within an ejector pin  36 . The fiber optic cable included a bundle of nineteen 100 micron diameter fibers, seven of which carried light from a helium-neon laser  40  and twelve of which transmitted reflected light back to a silicon photodiode  42 . In operation, incident light was transmitted through the resin and then reflected back to the detector from every boundary at which there was a discontinuity in the index of refraction. During the molding cycle, the detected light was analyzed to: 1) detect the arrival of the polymer melt, 2) detect separation of the resin from the mold wall upon shrinkage, and 3) monitor the molded part shrinkage as shown in  FIG. 3 . 
     More recently, Fathi et al. designed a glass mold and used a high speed camera to observe the shrinkage during the molding process (S. Fathi and A. H. Behravesh, “Visualization of In-Mold Shrinkage in Injection Molding Process,” Polymer Engineering &amp; Science, vol. 47, pp. 750-756, 2007). Angstadt et al. have also implemented a glass mold to observe the development of birefringence in injection molding (D.C. Angstadt, C. H. Gasparian, J. P. Coulter, and R. A. Pearson, “In-situ observation of birefringence during vibration-assisted injection molding,” SPE ANTEC, vol. 1, pp. 783-787, 2004). The size, cost, and maintenance issues associated with these designs prevent widespread adoption for in-mold shrinkage measurement. 
     In addition, there have been significant increases in molded part complexity due to the development and widespread implementation of design for manufacturing and assembly (DFMA) guidelines that leverage the capability of the injection molding process. One common DFMA guideline calls for the consolidation of multiple parts whenever possible, which leads to fewer but more complex components. Given such potential functionality arising from complex molded parts, it is currently not uncommon for a molded part, such as an inkjet cartridge, to specify more than thirty critical dimensions with tight tolerances. 
     There is a need for further sensors and methods for controlling the formation of injected molded parts. 
     SUMMARY OF THE INVENTION 
     The present invention, in a first aspect, is directed to a method for use in forming a molded part. The method includes providing a mold having a cavity and a movable pin, injecting a moldable material into the cavity, biasing the movable pin to maintain an end of the movable pin in contact with the moldable material in the cavity during the curing of the moldable material and until the moldable material is cured, and monitoring movement of the biased movable pin during curing of the moldable material in the mold. 
     The present invention, in a second aspect, is directed to a sensor for use in forming of a moldable part in a cavity of a mold. The sensor includes a housing connectable to the mold, a movable pin disposed in the housing, the movable pin having an end engageable with a moldable material in the cavity of the mold, an elastic member supported in the housing and engageable with the movable pin for applying a biasing force on the end of the movable pin to maintain the end of the movable pin in contact with the moldable material in the cavity during curing of the moldable material and until the moldable material is cured, and means for monitoring movement of the movable pin when the movable pin is in contact with the moldable material during curing of the moldable material in the cavity. 
     The present invention, in a third aspect, is directed to a method for controlling the forming of a plurality of molded parts in a cavity of a mold. The method includes monitoring at least one of an in-mold part dimension of a part and in-mold shrinkage of a part based on movement of a movable pin biased to maintain an end of the movable pin in contact with the moldable material in the cavity mold during curing of the moldable material until the moldable material is cured, and controlling a plurality of operating parameters for forming the plurality of molded parts based on the monitored at least one of the in-mold part dimension of the part and the in-mold shrinkage of the part. 
     The present invention, in a fourth aspect, is directed to a system for controlling the forming of a plurality of molded parts in a cavity of a mold. The system includes a sensor for monitoring at least one of an in-mold part dimension and an in-mold shrinkage of the part based on movement of a movable pin biased to maintain an end of the movable pin in contact with the moldable material in the cavity during curing of the moldable material until the moldable material is cured, and a processor operable to control a plurality of operating parameters for forming the plurality of molded parts based on the monitored at least one of the in-mold part dimension of the part and the in-mold shrinkage of the part. 
     The present invention, in a fifth aspect, is directed to an article of manufacture which includes at least one computer usable medium having computer readable program code logic to control the forming of a plurality of molded parts in a cavity of a mold. The computer readable program code logic when executing performs obtaining at least one of an in-mold dimension of the part and an in-mold shrinkage of the part based on monitoring a movable pin biased to maintain the end of the movable pin in contact with the moldable material in the cavity during curing of the moldable material until the moldable material is cured, and controlling a plurality of operating parameters for forming the plurality of molded parts based on the monitored at least one of the in-mold dimension of the part and the in-mold shrinkage of the part. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The subject matter which is regarded as the invention is particularly pointed out and distinctly claimed in the concluding portion of the specification. The invention, however, may best be understood by reference to the following detailed description of various embodiments and the accompanying drawings in which: 
         FIG. 1  is a side elevational view, partially broken away, of a prior art injection molding machine; 
         FIG. 2  is a side elevational view of a prior art fiber optical sensor disposed in an ejector pin; 
         FIG. 3  is a graph of the monitored molded part shrinkage verses time using the fiber optical sensor of  FIG. 2 ; 
         FIG. 4  is a block diagram of a shrinkage sensing system in accordance with the present invention; 
         FIG. 5  is a perspective view, partially cut-away, of a mold having dual-mode shrinkage and pressure sensor in accordance with the present invention; 
         FIG. 6  is an enlarged perspective view of the movable pin of  FIG. 5 ; 
         FIG. 7  is an enlarged perspective view of the upper end of the movable pin of  FIG. 6  which extends into the mold cavity; 
         FIG. 8  is an enlarged perspective view of the dual-mode shrinkage and pressure sensor of  FIG. 5 ; 
         FIG. 9  is a perspective view of a direct mount sensor in accordance with the present invention; 
         FIG. 10  is an enlarged cross-sectional view of the direct mount sensor of  FIG. 9 ; 
         FIG. 11  is a graph of deflection verses charge using the direct mount sensor of  FIG. 9 ; 
         FIG. 12  is a graph of charge verses stress using the direct mount sensor of  FIG. 9 ; 
         FIG. 13  is a perspective view, partially cut-away, of a mold having a dual-mode shrinkage and pressure sensor in accordance with the present invention; 
         FIG. 14  is an enlarged perspective view of the mold and the sensor of  FIG. 13 ; 
         FIG. 15  is an enlarged cross-sectional view of the sensor of  FIG. 13 ; 
         FIG. 16  is a graph of pressure verses time obtained using the sensor of  FIG. 15 ; 
         FIG. 17  is a graph of position verses time obtained using the sensor of  FIG. 15 ; 
         FIG. 18  is a perspective view of a capacitive shrinkage sensor in accordance with the present invention; 
         FIG. 19  is an enlarged cross-sectional view of the sensor of  FIG. 18 ; 
         FIG. 20  is a graph of capacitance verses displacement obtained using the sensor of  FIG. 18 ; 
         FIG. 21  is a diagrammatic illustration of the cylindrical capacitive elements of the sensor of  FIG. 18 ; 
         FIGS. 22 and 23  are diagrammatic illustrations of capacitive plate elements; 
         FIGS. 24-26  are graphs of shrinkage analysis based on pressure-volume-temperature behavior; 
         FIG. 27  is a block diagram of one embodiment of a controller design in accordance with the present invention; 
         FIG. 28  is a cross-sectional view of a test specimen and measurement locations for a molded part in accordance with the present invention; 
         FIG. 29  is a plot of the sensed displacement of the shrinkage sensor verses time during the molding of PP; 
         FIG. 30  is a graph of the results of the measured final part thickness against the observed in mold shrinkage as acquired according to the discussion of  FIG. 29 ; and 
         FIG. 31  is a graph of the measured final part thickness against the observed in-mold cavity pressure as acquired by a commercial, direct mount cavity pressure sensor. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     One of the challenges in the injection molding of plastic parts is the control required to produce parts with desired dimensions reproducibly. In-mold shrinkage is the major driver of final part dimensions, and it is a function of cavity pressures, mold and melt temperatures, cycle times, and material properties among other factors. Shrinkage can be very difficult to predict, and particularly, control for tight tolerance in multi-cavity operations. 
     While prior art sensors prove effective as process control variables, they are in fact poor estimators of the manufactured part quality and advances in machine control have not kept pace with part design requirements. Perhaps surprisingly, the final part dimensions are not precisely known during the product design and mold tooling phases. The control of shrinkage can be difficult, especially in tight tolerance and multi-cavity applications. Molding operations are greatly impaired by the lack of direct observability and controllability of the molded part dimensions since time and cost are expended to equilibrate and measure molded parts. Most molders instead rely on the use of cavity pressure traces or part weight measurements as estimators of the part dimensions. Yet, part dimensions are not exclusively identifiable with cavity pressures or part weight but are also correlated with changes in mold temperature, melt temperature, cycle time, material properties, etc. 
     Accordingly, there is a continued need for improved process sensors, on-line analysis, and control methods directed to shrinkage prediction and control. The present invention is generally directed to measuring the solidification process (the transition from viscous melt to a solid) as it&#39;s occurring within the mold to quantify the degree of shrinkage and translate that volume change to the final part dimensions. 
       FIG. 4  illustrates one embodiment of a shrinkage sensing system  100  in accordance with the present invention. Shrinkage sensing system  100  includes a molding machine  110 , a mold  120 , and a controller  130 . For example, as described in greater detail below, an embodiment a mold sensor  135  in accordance with the present invention may include dual-mode sensors that uses strain gages and piezoelectric elements to measure both in-mold part dimensions, and the forces exerted on the mold wall by the polymer. A shrinkage analysis of the dimensional and pressure data may then be conducted with transient heat analyses to predict the final molded part dimensions after cooling and annealing. The dual in-mold shrinkage and pressure sensors enable a cost, size, and usage similar to that of commercially available sensors while also advancing model-based controller development. The sensor and theory may facilitate higher quality and lower cost manufactured products. The present invention may be advantageous given the described sensor development together with the modeling, instrumentation, and experimentation. Desirably, the present invention is directed to measurement of molded products&#39; dimensions in situ prior to ejection from the mold. The controller may be a suitable processor or microprocessor or microcontroller having suitable memory and input and output devices. In addition, a suitable computer may be employed to implement the techniques of the present invention. 
     For example, the present invention in one embodiment provides a high fidelity system for controlling molded part dimensions by integrating, for example, three sub-systems including:
         1. a dual-mode sensor utilizing strain gages and piezoelectric elements to monitor both the in-mold part dimensions and melt pressure;   2. a shrinkage analysis to use the sensed dimensional and pressure data with transient analyses to predict the final molded part dimensions after cooling and annealing; and   3. an auxiliary controller to track the consistency of the molding process, perform control actions given the part quality requirements, and recommend alternative process set-points.       

     The present invention also provides a method for molders to achieve, track, and optimize molded part dimensions relative to tight tolerances. For example, the present invention includes:
         1. The creation and installation of the in-mold shrinkage and pressure sensor.   2. The implementation of an auxiliary shrinkage controller which uses the output of the shrinkage sensor with some preliminary shrinkage models to predict the final part shrinkage and recommend potential process settings to satisfy quality specifications.   3. The development and validation of more sophisticated shrinkage analyses for unfilled amorphous and semi-crystalline resins; the fidelity of this modeling activity is increased by the measured in-mold shrinkage and cavity pressure.   4. The implementation of the developed system in industrial applications.
 
Shrinkage Instrumentation
       

       FIG. 5  illustrates one embodiment of a mold  200  having a cavity  210  for forming a part which incorporates a dual-mold sensor  300  in accordance with the present invention for measuring in-mold shrinkage and cavity pressure. Sensor  300  is disposed in the mold to engage a movable pin  310 . With reference to  FIGS. 5-8 , sensor  300  is placed beneath movable pin  310  such as an ejector blade or ejector pin, and causes the movable pin to protrude slightly when the mold is opened as shown in  FIG. 7 . As best shown in  FIG. 8 , sensor  300  may include a housing  320 , which is connectable to the mold, e.g. received in a recess portion of the mold below the movable pin. An elastic member  330  is supported by the housing and engageable with an end of the movable pin for applying a biasing force on the end of the movable pin. For example, the elastic member may comprise at least one diaphragm for applying the biasing force. A displacement transducer  340  is operably connected to the elastic member for use in monitoring the position of the movable pin and the part dimension. For example, the displacement transducer may be one or more strain gage operably connected to the elastic member for use in measuring the position of the movable pin. At least one piezoelectric cell  350  for monitoring a biasing force exerted on the movable pin may be disposed between a stop  360  attached to the elastic member, and an end cap  370 . 
     In this embodiment, sensor  300  may be derived from button-type load cells in which an instrumented diaphragm provides a reaction force to the movable pin in contact with the surface of the part being molded. While the exact capabilities will vary with the detailed design of the system, the sensor may have a range of travel of 0.5 mm, which corresponds to a 0.5 microns (μm) resolution given only 10 bits of precision in the data acquisition system. The bending of the diaphragm under an imposed load causes the resistance of the associated strain gages to increase with increasing elongation while narrowing the strain elements. These changes in the strain gage geometry cause an increasing resistance with strain. The magnitude of the imposed load can be closely estimated by measuring the voltage across the strain gage(s) and subsequent scaling related to the gage factor and diaphragm stiffness. The design of the sensor may be based, for example, actuation forces, optimization with respect to sensitivity, linearity, and longevity, and subsequently validating the sensor&#39;s function in a variety of molding applications. 
     With reference to again to  FIG. 5 , after the mold  200  is closed, the melt pressure exerted on the top surface of the movable pin will cause the sensor to be fully actuated and impose stress on sensor  300 , and in particular, the strain gages and the piezoelectric cell. As the melt in the mold cavity cools and shrinks, the melt pressure will decay and the molded part will draw away from the cavity walls. The reaction force provided by the sensor diaphragm will cause the movable pin  310  to maintain contact with the face of the molded part and a measurable relaxation of the imposed stress in the diaphragm. While an ejector blade may have a square cross-section and an ejector pin may have a round cross-section, it will be appreciated by those skilled in the art that the blades and pins may have other suitable cross-sections. 
     The structural design of sensor  300  may be initially guided by plate bending theory which states that the maximum stress, σ, and deflection, δ, of the diaphragm are: 
                   σ   =       k   1     ⁢         P   melt     ⁢     ϕ   pin   2         h   diaphragm   2                 (   1   )               δ   =       k   2     ⁢         P   melt     ⁢     ϕ   pin   2     ⁢     ϕ   diaphragm   2         E   ⁢           ⁢     h   diaphragm   3                   (   2   )               
where P melt  is the melt pressure, φ pin  is the movable pin diameter, φ diaphragm  is the diaphragm diameter, h diaphragm  is the diaphragm thickness, E is the elastic modulus, and the coefficients k 1  and k 2  are related to the aspect ratio and constraints of the diaphragm. Similar analyses apply for different sensor geometries as well as non-round ejectors, such as the ejector blade.
 
     The selection of the strain gages and piezoelectric cell is also guided from established theory. The voltage output, V δ , from a Wheatstone bridge of four strain gages is a function of the movable pin deflection:
 
V δ =k 3 δS g V e   (3)
 
where S g  is the gage factor, V e  is the excitation voltage, and k 3  is a coefficient relating the diaphragm deflection to the imposed strain in the strain gages. When the piezoelectric cell contacts the end cap, the voltage output, V δ , from the piezoelectric cell is a function of the imposed stress:
 
V σ =k 4 h cell σ cell   (4)
 
where h cell  is the thickness of the piezoelectric cell, σ cell  a is the imposed stress in the piezoelectric cell, and k 4  is a coefficient related to the system capacitance and piezoelectric cell&#39;s permittivity.
 
       FIG. 9  illustrates an embodiment of a direct mount sensor  400  in accordance with the present invention. Sensor  400  may be directly mounted and extend into the mold cavity. In this illustrated embodiment, the direct mount design uses two structural members of varying compliance to control the contact force and deflection for shrinkage measurement. 
     For example, as best shown in  FIG. 10 , sensor  400  may include a generally hollow cylindrical housing  410  having a shoulder  412 , a movable pin such as a sensor head  420  which is disposed in and extends from housing  410 , a steel rod  430  having a step  432  disposed within and below sensor head  420 , an elastic member  440  such as a hollow aluminum sleeve sandwiched between a lower end of sensor head  420  and step  432  of steel rod  430 . The elastic or compliant member  440  is sized to provide a gap G between the lower end of the sensor head and the top of the steel rod. A piezoelectric cell  450  is disposed between a lower end of steel rod  430  and a plug  460  which attaches to housing  410 . A thermocouple  470  may be disposed at the upper end of sensor head  420 . The sensor head, the steel rod, the piezoelectric cell, and the plug may have respective passageways therethrough to define a channel through which a wire may be connected to the thermocouple. 
     In operation, the mold material pushes on sensor head  420 . The aluminum sleeve provides a compliant deflection. The steel rod on contact with the sensor head provides greater stiffness as shown in  FIG. 11 . The piezoelectric cell outputs a charge, Q, with stress, σ, as shown in  FIG. 12 . Deflection, d, is back calculated from the joint rod and sleeve compliance. 
     For example, a single piezoelectric element may be used if the system behavior is well known and the signal conditioning is of sufficient quality for determining deflection. Alternatively, the elastic member or compliant sleeve, e.g., the aluminum sleeve, may be instrumented with one or more strain gages for direct deflection measurements. Compared to the design of  FIG. 5 , the sensor design of  FIGS. 9 and 10  may be more accurate but may result in a higher cost. The higher cost may be justified through the addition of a thermocouple at the sensor head that acts as the movable pin, providing direct pressure-volume-temperature measurement in a single pressure-volume-temperature (PvT) sensor. 
       FIGS. 13 and 14  illustrate another embodiment of a mold  500  having a cavity for forming a part which incorporates a dual shrinkage and pressure sensor  600  in accordance with the present invention for measuring in-mold shrinkage and cavity pressure for production of automotive instrument panels. The sensor is installed as for a conventional pressure transducer but with the sensor head protruding slightly above the cavity surface. In operation, the melt pressure causes the sensor to retract until flush with the surface of the mold. As the polymer melt cools and shrinks, the cavity pressure decays to zero, as shown in  FIG. 16 , and the solidifying plastic will pull away from the cavity wall. A small biasing force within the sensor causes the sensor head to maintain contact with the molded part, such that the sensor continues to track the part shrinkage during the molding process. The sensor in this example measures the shrinkage in the thickness direction, as shown in  FIG. 17 , which is typically greater than the shrinkage in the transverse directions due to variations in material constraints. As such, the shrinkage in the thickness direction is almost universally desired in manufacturing applications. Furthermore, the proposed sensor design can be placed at the bottom of ribs or the edge of the cavity to directly measure changes in length. 
     As best shown in  FIG. 15 , sensor  600  employs different sensing elements for the applied pressure and sensor head position. For example, sensor  600  may include a hollow cylindrical housing  610  having a shoulder  612 , a movable pin such as a sensor head  620  which is disposed in and extends from housing  610 , a position transducer  630  disposed within and below sensor head  620 , a position transducer support  635  disposed around position transducer  630 , a spring  640  sandwiched between a lower end of sensor head  620  and position transducer support  635 . A piezoelectric cell  650  is disposed between a lower end of position transducer support  635  and a plug  660  which attaches to the lower end of housing  610 . 
     For example, the position transducer may be a differential variable reluctance displacement transducer (DVRT) used to measure the displacement of the sensor head. The DVRT is a non-contact transducer that contains sensing and compensation windings. When the rear surface of the sensor head is brought in close proximity to the DVRT transducer, the reluctance of the sensing coil is changed while the compensation coil acts as a reference. The two coils are excited with a high frequency voltage such that their difference provides a sensitive measure of the position signal independent of the ambient temperature. While the output must be linearized with respect to the sensor head displacement, the DVRT is otherwise ideal due its small size, wide operating temperature range, excellent precision (0.1% of 0.5 mm full scale range), high signal to noise ratio, and long term robustness. 
     The DVRT is supported and recessed within a threaded metal sleeve, which also supports the biasing spring. The sensor head will retract due to the force exerted by the melt pressure on its front surface. The biasing force is selected to be small relative to typical melt pressures in polymer processing, such that the rear surface of the sensor head will contact the DVRT support, which thereby transfers the load to the piezoelectric (PZT) disk(s). The voltage output, V, from the PZT cell is a function of the imposed stress:
 
V=kh cell σ cell   (5)
 
where h cell  is the thickness of the piezoelectric cell, σ cell  is the imposed stress in the piezoelectric cell, and k is a coefficient related to the system capacitance and piezoelectric cell&#39;s permittivity.
 
     The sensor design may be optimized using mold filling simulations to ensure suitability in a variety of molding applications with different cavity polymeric materials, cavity wall thicknesses, and operating conditions. A set of sensors may be manufactured according to a design of experiments about this design to investigate the effect of design parameters such as sensor head diameter, spring bias force, clearances, and other parameters. The set of sensors may then be used as an inner array in a larger design of experiments that uses an outer array designed to investigate polymer properties, cavity wall thicknesses, and operating conditions. 
       FIG. 18  illustrates an embodiment of a capacitance type shrinkage sensor  700  in accordance with the present invention. With reference to  FIG. 19 , sensor  700  includes a generally hollow cylindrical housing  710 , and a pin  720 . A lower end  722  of pin  720  is receivable and retained within housing  710 . A helical coil spring  770  is disposed around a portion of the lower end of pin  720 . The ends of the spring are sandwiched between a stop or shoulder  724  in pin  720  and a stop or shoulder  712  on housing  710 . Displacement transducer elements such as an inner displacement transducer element  740  is disposed on pin  720  and an outer displacement transducer element  750  is disposed on housing  710 . A cover  780  prevents ejection of pin  720  from the housing  710  by coil spring  770 . The coil spring biases the pin against the molded part to provide a measurement as the molded part shrinks away from the walls of the mold cavity. While this design does not measure cavity pressure, the design is compact and provides for a variety of movable pins with respect to different shapes and sizes. 
     In this embodiment, two co-axial cylindrical capacitance elements are used as transducer elements. The capacitance elements are instrumented on the movable pin and pin base as shown  FIG. 19 . The pin travels under melt pressure as the plastics shrinkage changes, resulting in a capacitance change and resulting output signal as shown in  FIG. 20 . The sliding co-axial cylindrical capacitance elements as shown in  FIG. 21  can be considered, for example, as two parallel plates sliding over each other as shown in  FIG. 22 , where the pin displacement may be sensed by varying the surface area of the electrodes of a flat plate capacitor. Also, the sensor output is linear correspondence to the pin displacement which enables the direct measurement of the in-mold shrinkage as shown in  FIG. 20 . The melt temperature within the cavity can be measured if thermocouple is used as a movable pin or within a movable pin. A shrinkage calculation may be defined as: 
             C   =         ɛ   0     ⁢       ɛ   r     ⁡     (     A   -   wx     )         d           
where ∈ r  is the relative permittivity of the material between the plates, ∈ 0  is the permittivity of vacuum, A is the plate area, d is the plate separation or gap and x is the displacement of the plate.
 
     With a plate size of 100 mm×100 mm and a spacing of 1 mm, the capacitance in vacuum, neglecting a small fringe effect, is 88.54 pF. With a vacuum dielectric, the relative dielectric constant ∈ r  is 1. An air dielectric increases K to 1.0006. Typical dielectric materials such as plastic or oil have dielectric constants of 3-10, and some polar fluids such as water have dielectric constants of 50 or more. 
     The design of the sensor may be optimized using the in-mold simulation to ensure suitability in a variety of molding applications with different polymeric materials, cavity wall thicknesses, and operating conditions. A set of sensors may be manufactured using a design of experiments to investigate the effect of design parameters such as diaphragm thickness, diaphragm diameter, and other dimensions. This set of sensors may be used as an inner array in a larger design of experiments that uses an outer array designed to investigate polymer properties, cavity wall thicknesses, and operating conditions. The results of this internal validation may be used to subsequently improve the sensor design and shrinkage analyses. 
     Shrinkage Analysis 
     Several prior art models for predicting shrinkage have been developed. The one exemplary model considers the shrinkage, s, as:
 
 s =α( T   eject   −T   final )  (6)
 
where α is the polymeric material&#39;s coefficient of thermal expansion, T eject  is the temperature of the molded part upon ejection from the mold, and T final  is the temperature of the molding during end use. This model will typically over predict the shrinkage since it does not consider the tensile stresses that develop in the molded part as the polymeric material cools from the solidification temperature to the ejection temperature. Furthermore, this model does not consider the expansive state of the melt caused by the melt pressure, which will tend to prevent the polymer from exhibiting any shrinkage until this pressure is relieved. This more complex shrinkage behavior is well characterized by the pressure-volume-temperature relation shown in  FIG. 9 .
 
     Other prior art models for predicting shrinkage are slightly more complex based on pressure-volume-temperature (PvT) data of characterized materials. As shown in  FIG. 24 , PvT data is typically plotted for various isobars as a function of melt temperature. Given this constitutive model for the specific volume, the material&#39;s volumetric shrinkage can be calculated between any two pressure-temperature states. While the specific volume of the molded part, ν final , at the end-use conditions are usually known, the specific volume of the melt, ν g , at solidification requires the tracking or estimation of the material&#39;s melt pressure and temperature as shown in  FIGS. 25 and 26 . If the pressure and temperature history are known, then an improved estimate of the shrinkage can be provided as: 
     
       
         
           
             
               
                 
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                   7 
                   ) 
                 
               
             
           
         
       
     
     Both of the above models may be deployed in the present invention. However, both of these models may be insufficient given that they do not explicitly consider the development of residual stress due to the compression of the melt by the injection pressure, decay of the compressive stress with adiabatic cooling or molecular relaxation, and evolution of tensile stresses with continued cooling and volumetric shrinkage. 
     The present invention may extend the existing shrinkage models in two significant ways. First, the shrinkage analysis may be use to the measured shrinkage and cavity pressure measurements as initial and boundary conditions during the packing and cooling stages of the molding process. By comparison, the previous shrinkage models did not have access to in-mold shrinkage data and so propagated faulty initial conditions throughout the shrinkage analysis. Second, with the previous work regarding the shift factor at low temperatures, which will be used to more accurately model the short-term properties and, when coupled with other material modeling data (e.g., time-aging time superposition), may capably predict the part dimensions after cooling and annealing. 
     Shrinkage Control 
     The present invention may employ a real time control system to interface between the operator, the developed shrinkage analysis, and the developed shrinkage sensor. As shown in  FIG. 29 , the auxiliary controller may receive voltages corresponding to the strain and piezoelectric stress as a function of time. Given application-specific design information about the sensor, movable pin, and mold cavity, the controller may calculate the in-mold movable pin deflection, part dimension(s), cavity pressure, and shrinkage from the transfer functions given in equations (1) to (4). Given the volume of the short shot corresponding to the sensor location and the cavity wall thickness, mechanistic relations may be used to provide internal process states such as the flow rate, melt viscosity, volumetric shrinkage, and melt solidification time. With just the sensor and no external process information or constitutive models, the controller may provide better observability than most commercial systems used in industry. 
     Given additional information including the melt&#39;s constitutive model and process conditions, the described shrinkage analyses of equations (6) and (7) may be used to predict the final part dimensions relative to the part&#39;s specifications and thereby provide a reject signal. In addition, these same models may be used to synthesize the minimum and maximum control limits for each process set-point while holding the other process set-points at their current values. This information can be used by the operator to simultaneously adjust multiple set-points while maintaining the current part dimension or otherwise satisfy new dimensional specifications. 
     Implementation and Validation 
     The present invention provides a complete instrumentation, analysis, and control system for managing the dimensions of molded parts. As previously discussed the performance of multiple sensor designs may be characterized in an inner array as a function of application characteristics in an outer array. Several sensors may be manufactured and installed in a mold according to the part design of  FIG. 28 . This configuration allows characterization of shrinkage in the thickness, width, and height directions for constrained and unconstrained geometries. 
     A button cell deflection sensor was designed, built, and wired with four strain gages connected in a full bridge. A movable pin is provided by an ejector pin positioned above the button cell similar to the configuration of  FIG. 8 . Experiments were conducted on a 100-ton hydraulic molding machine with a mold producing 2.5 mm plaques of different resins including PP and HIPS. Data was acquired from the shrinkage and other process sensors, and used to identify the arrival of melt at sensors in cavity, maximum shrinkage sensor position, shrinkage sensor position just prior to mold opening.  FIG. 29  plots the sensed displacement of the shrinkage sensor during the molding of PP. The sensor begins at a position of 0.52 mm. At a time of 2 seconds, the melt reaches the sensor pushes it slightly downwards; the maximum downward displacement of the pin is 0.1 mm (a position of 0.42 mm from a starting position of 0.52 mm) at a time of 5 seconds. This time was verified to coincide with the maximum cavity pressure at this location during the packing stage of the mold cavity. 
     The increasing position of the shrinkage sensor in  FIG. 29  after 5 seconds corresponds to later cooling and shrinkage of the polymer melt in the mold cavity. By a time of 15 seconds, the bulk of the polymer melt has solidified. At a time of 25 seconds, the mold is opened and the part is ejected. The in-mold shrinkage of the polymer melt is measured as the displacement of the shrinkage sensor from its minimum position at 5 seconds to its last position prior to mold opening at 25 seconds. 
     A design of experiments was conducted to characterize the performance of the sensor to predict final part dimensions at varying processing conditions. The results are provided in  FIG. 30 , which plots the measured final part thickness against the observed in mold shrinkage as acquired according to the discussion of  FIG. 29 . The results indicate that the described in-mold shrinkage sensor is an excellent predictor of final part thickness. 
     For comparison purposes,  FIG. 31  plots the measured final part thickness against the observed in-mold cavity pressure as acquired by a commercial, direct mount cavity pressure sensor. The results indicate that the correlation is not as good as that provided by the described shrinkage sensor. Part dimension measurements were also taken at locations remote to the in-mold shrinkage and cavity pressure sensors. In every case, it was found that the shrinkage sensor provided better correlation with the part measurements than the pressure sensor. 
     The present invention is a step forward from the current sensor designs that have been used for decades for monitoring injection molding. Direct measurement of the shrinkage, together with simple derivation of other process states such as flow rate and viscosity, enable the development and widespread implementation of improved process and quality control methods for injection molding. As a result, the time required for process set-up and stabilization is reduced, and part quality and consistency is improved. Due to savings associated with cycle time reductions, yield improvements, and related automation, there is the potential to significantly improve molding productivity. 
     It is noted that different sensor designs incorporating various shapes and sensing means may be implemented. For example, the position of the movable pin may be measured using at least one potentiometer, at least one inductance device, at least one magnetostrictive device, at least one optical encoder, and at least one laser interferometer. 
     Thus, while various embodiments of the present invention have been illustrated and described, it will be appreciated to those skilled in the art that many changes and modifications may be made thereunto without departing from the spirit and scope of the invention.