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EFFECTS OF RELAXATION OF A CORE ON A WOUND ROLL By JEFFREY SCOTT HENNING Bachelor of Science Oklahoma State University Stillwater, Oklahoma 1995 Submitted to the Faculty of the Graduate College ofthe Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May 1997 EFFECTS OF RELAXATIO OF A CORE ON A WOUND ROLL Thesis Approved: Dean of the Graduate College 11 ACKNOWLEDGMENTS I wish to thank Dr. James K. Good, Professor, Mechanical and Aerospace Engineering and my advisor. His guidance throughout my research has been invaluable. Gratitude is also extended to Charles Johnson, Sonoco, Inc., for his technical support and for providing materials. His moral support is also greatly appreciated. Thanks must also go to Ron Markum and Robert Taylor of the OSU Web Handling Research Center for their expertise. For her generous patience, support and love I thank my wife, Susie. 11l Chapter TABLE OF CONTENTS Page 1. INTRODUCTION 1 II. LITERATURE REVIEW 3 Elastic Core Model 3 Hakiel's Model. 3 Thennal Analysis 8 Tube Testing Device 9 In. FINDING AN EXPERIMENTAL RELAXAnON FUNCTION 11 Pressure Vessel Design 11 Deflection Measurement 13 Miscellaneous Instrumentation 14 Data Acquisition 15 Experimental Procedure 16 Data Analysis 17 IV. ROLL MODEL 27 Computer Model 30 V. MODEL VERlFICAnON 33 Roll Testing 35 VI. CONCLUSIONS 43 VII. FUTURE WORK 44 REFERENCES 45 IV Figure LIST OF FIGURES Page 1. 2. 3. 4. 5. 6. Pressure Vessel 13 Pressure Vessel Layout 14 Strain Data for Core # I a. Strain Data for Core #1 at 100 psi 18 b. Strain Data for Core #1 at 150 psi 18 c. Strain Data for Core #1 at 200 psi 19 Average Strain Data for C re #1 20 Nonnalized Creep Data for Core #1 20 Creep Function Plot of Core #1 21 7. Strain Data for Core #2 a. Strain Data for Core #2 at 100 psi 22 b. Strain Data for Core #2 at 150 psi 23 c. Strain Data for Core #2 at 200 psi 23 8. Average Strain Data for Core #2 24 9. Normalized Creep Plot for Core #2 24 10. Creep Function Plot of Core #2 25 II. Flow Chart 32 12. Radial Pressure Through Winding 34 13. Core Pressure Through Winding 34 14. Die Set 37 v 15. Push Test Set up 37 16. Roll #4 Push Force 39 17. Role #4 Radial Pressure _ _ .40 18. Roll # 5 Push Force 41 19. Roll #5 Radial Pressure 42 NOMENCLATURE b web thickness Ee core modulus of elasticity Er radial modulus of elasticity E( tangential modulus of elasticity F punch force h incremental change in central difference approximation J core relaxation function Jo, J1, T1, J2, T2 coefficients in the relaxation function K' effective gage factor of strain gage N interlayer force P interlayer pressure R'A effective resistance of arm shunted (ohms) Real resistance ofcalibration resistor (ohms) r radius t time Tw winding tension stress w web width vii 8T incremental change in tension stress Ecal simulated strain ET radial strain E( tangential strain Vrt radial Poison's ratio Vtr tangential Poison's ratio crT radial stress crt tangential stress viii CHAPTER I INTRODUCTION Many materials used in the world today originate through a web handling process. Thin plastic films, paper, cotton fibers, and thin metals are manufactured in continuous strips called webs. For convenience, web material is wound on to a roll where it can be easily transported and stored. The web is wound on to cores made of materials such as: composite fiber, plastic, alwninum or steel. The core is a cylindrical hollow shell. It is hollow for ease of mounting the roll on to the appropriate web handling equipment. For economic reasons, the core used for a specific web has the minimum required properties, such as strength and weight. Industry desires a disposable core so the core will not have to be returned to the manufacturing plant. With these requirements, it is important to have a process for finding the best core for the specific web. In web handling processes, it is important to maintain roll integrity. Extreme stresses within the roll cause defects to occur, such as wrinkling or starring, thus yielding the damaged portion ofthe roll unusable. Also, inappropriate stresses can cause the roll to separate from the core. Ifthat occurs, problems arise in unwinding the roll, because tension is applied to the core during unwinding. This results in an unusable roll. To ensure roll integrity, an effort has been made to develop winding models which predict the stresses within the roll. Winding models incorporate web material properties, core properties and winding techniques. From the results of winding models, criteria can be generated to determine whether flaws will occur. These models reflect the dynamic effects of the winding I process. Very few models reflect how the roll will act with environmental or time effects. It has been noted that, at times, a roll has developed defects days after it has been wound even without a temperature change. This must be due to an effect that occurred after the dynamic winding response. Current models assume the core is an elastic material. It is often a viscoelastic material. A viscoelastic material is defined as a material that exhibits an elastic strain upon loading, then a slow and continuous in.crease in strain through time at a decreasing rate.[l] When the loading is removed, the inital elastic strain is immediatly recovered and then a slow continuous decrease in strain occurs. The slow continuous increase in strain under constant stress is called creep. This viscoelastic property of a paperboard core is explored in this paper in order to explain how flaws develop through time in a wound roll. The study begins with a development of a roll model that begins with the winding process and ends at a set time later. Typically, roll models consist of second order differential equations with two boundary conditions. One boundary condition is at the outside radius of the roll, which is influenced by the winding tension. The second boundary condition is at the rollcore interface in which a core defonnation dominates. An empirical function will be developed to model the rate of radial defonnation of the core due to roll stress. The roll model will be solved analytically using a finite difference approximation. Finally, experimental verification of the model is presented. 2 CHAPTER II Literature Review Elastic Core Model An elastic core model was developed to predict stresses and strains, for spiral paper tubes loaded axisymmetricaUy by Gerhardt, of Sonoco Product Co. [2]. This model did not assume the core to be isotropic or any stress distribution through the core wall. Paper, however, was assumed to be a linear elastic material. Experimentation verified the elastic core model. Some results are as followed. One, paperboard laminates exhibit nonlinear stressstrain behavior. Two, hoop stress dominates all other stress components and it was not constant across the tube wall. Three, a stress concentration occurred at the outer wall of the tube and remained constant as the tube thickness increased. Hakiel's Model The viscoelastic properties ofthe core affect the quality of a wound roll. Core creep, induced by stresses exerted by the wound roll, plays a major part in the quality of the wound roll. The wound roll stresses must be known. Hakie1 developed a nonlinear orthotropic hoop model for center wound roUs. [3] The roll stresses are calculated by solving a second order differential equation with two boundary conditions. Since the equation is nonlinear, numeric approximations are used. 3  The following assumptions are made in Hakiel's model: I. The winding roll is a geometrically perfect cylinder with the web having uniform width, thickness, and length. 2. The roll is a collection of concentric hoops. Winding is modeled by the addition of tensioned hoops. Roll properties remain constant. 3. The roll is an orthotropic, elastic cylinder with linearelastic behavior in the circumferential direction and nonlinearelastic behavior in the radial direction. The radial modulus of elasticity is known and varies as a function of radial stress. 4. The stresses within the roll are a function of radial position only. 5. The roll is under a plane stress condition and axial stresses are equal to zero. Hakiel uses three basic stress strain equations to support his model. 1. The equilibrium equation for plane stress in cylindrical coordinates in the absence of shear: acr r(_Orr)  a + a = 0 t r (2.1) 2. The linear orthotropic constitutive equations: 4 (2.2a) (2.2b) 3. The strain compatibility equation: r(&Ort) +E t ET =0 (2.3) The model begins by solving the equilibrium equation (2.1) for crt and substituting it into the constitutive equations (2.2a,b). crr vrt (fur ) Er =EE r;:+cr r r t VI and (2.4) Et ~ ;, (r a;' +cr,) (;:)cr, Then, substituting these two equations into the strain compatibility equation (2.3) Solving and utilizing Maxwell's relationship: yields: (2.6) Hakiel's second order differential equation is the governing equation in the winding model. To solve the equation, two boundary conditions are required. The [lIst boundary condition, at the core roll interface, is obtained by equating the radial deformation of the first wound on layer (2.7) and outside deformation of the core (2.8). (2.7) 5 (2.8) Where the 8 indicates incremental changes in the variable. Substituting the defonnation equations (2.7)(2.8) in the equilibrium equation (2.1) yields the first boundary condition. &a8cr r Ir=l =(EEl ) I e  1+ V 8cr r r=) (2.9) The second boundary condition, at the outside ofthe winding roll, is found by assuming the incremental winding on ofthe last lap is equal to the hoop stress of that lap. 0, I =(Twjr=s)b crrr=s S (2.10) With these two boundary conditions, a solution can be obtained for the Hakiel's governing differential equation of the elastic region of this model. Since the model is nonlinear, the solution must be found analytically. A finite difference method with a central difference approximation of the derivatives is employed to solve the governing differential equation. The central difference approximations are as follows: (2. 11 a) (2. 11 b) Substituting the approximations into the governing differential equation (2.6) and combing variables yields: 6 ( r 2 3rJ (. 1) (2r 2 1 Et ) (.) (r 2  cr 1 + +  cr 1 + +3rJ cr (,1+1) =0 h2 2h r h2 E T h2 2h r r (2.12) Where (i) is the current lap and h is the incremental step size or web thickness. Substituting the approximations into the dynamic bOWldary conditions (2.9)(2.10) and collecting terms they become: ocr,(i +1)[~ (h :: 1+ v)}cr,(i) =0 and (2.13) ocr r (i) =(T~)h With the governing differential equation written at all interior points (web laps) in the roll, a set of equations are formed for N2 variables. The two boundary conditions supply the remaining two required equations. Thus, the boundary value problem has been reduced into a simultaneous set of equations. When put in matrix form, they yield a tridiagonal system ofthe form: [A]{ocr r } =[B] (2.14) The system can be solved by a Gaussian elimination routine with N2 forward and Nl backward substitution. The incremental stresses are accumulated at each layer and added together to get the total stress at that layer. This process steps through the roll adding layer upon layer until the entire roll is analyzed. Thennal Analysis One recent study by Qualls [4] investigated the effect of a temperature change upon a roll. Qualls showed that an increase in temperature increased the stresses causing 7 defects within the roll. This was due to expansion or contraction ofboth the web material and the core. Qualls modified Hakiel's wound roll model to calculate the interlayer pressures in a wound roll which is subjected to a homogeneous temperature change. The model includes coefficients of thermal expansion ofthe web and core. The constitutive equations become: (2.15a) (2.15b) where a r is the radial coefficient of thermal expansion ofthe web at is the tangential coefficient of thermal expansion of the web t1T is th.e temperature change Solving the model in a similar method as Hakiel, the second order differential governing equation becomes: (2.16) The boundary conditions are modified. The core's coefficient of thermal expansion is added to the inner boundary condition. (2.17) where a c is the core coefficient of thermal expansion This yields an inner boundary condition of: 8 ra8a+r 8a ( 1vEI ) =E ( a a ) l!.T Or rEI c r c (2.18) Using the fInite difference approximations (eq. 2.11a) and solving, the boundary condition becomes: !:...oa (i+1)+(1!:...V E' )oa (i)=E(a a )l!.T h r h £' rIc r c (2.19) The outer boundary condition assumes a traction free outer roll surface and the stress is equal to zero. Now, with these two boundary conditions (2.12)(2.19) the governing differential equation can be solved. A tridiagonal set of simultaneous equations is produced that when solved yield incremental pressure changes due to step temperature changes. The set of equations is solved in a stepwise linear fashion. For each increment in temperature change, updated radial pressures and radial modulus are computed. Both radial pressure and modulus are pressure dependent. QuaIl's model steps through temperature change instead of through the roll as in Hakiel's model. QuaIl's thermoelastic model was studied because his approach is the basis for the approach in this paper. Whether the core deforms due to viscoelastic behavior or thermal expansion does not alter the solution procedure. Thus, the viscoelastic model will step through time as QuaIl's model stepped through temperature. Tube Testing Device A core testing vessel was developed by Salidis and Rowlands. [5] This vessel was used in a newly patented test method for measuring material properties tubular 9  samples. The method ensured that cores were tested to failure in compressive material crushing, not a structural buckling failure. The test vessel contained an annulus of 1.5 rnrn. ball bearings, that were compressed radially against the outside of a core sample. The bearings were loaded by hydraulic fluid contained in a sealed bladder. The sample, bearings, and bladder were housed in a cavity within the vessel. The cavity had a hole in the bottom for venting the interior ofthe sample. At the top a plug was used to hold the sample in place. This device allowed for external loading, while ensuring a uniform deformation of the core with the structure of the compressed ball bearings. Experimental stressstrain values were compared to theoretical values ofaluminum tubes. Tests on aluminum tubes, with strain gages mounted on the inside oftube, showed the experimental data following the theoretical values. Paperboard tubes were tested to failure in crushing. Strain gages were placed in the hoop direction on the inside of the tube. The tests were repeated on 5 different tubes that were geometrically identical, to show repeatability. The advantage of the test vessel Salidlis and Rowlands developed was a clean working environment. The hydraulic oil was contained in a sealed blatter. The vessel also allowed for variable size specimen with changing the size of the bearing annulus. One problem with the testing device was examining how the strain on the outer surface of the tube could be measured. The ball bearings would destroy any strain gage placed on the outside ofthe tube. If strain is desired on the outside of the tube another testing apparatus is needed. 10 CHAPTER III FINDING AN EXPERIMENTAL RELAXATION FUNCTION The creep function, Ie , is a function defining the defonnation of the core over time nonnalized by the pressure the roll is exerting on the core. The function is found experimentally. It is an exponential function that takes the fonn of a generalized Maxwell equation for relaxation. [1] J, ~ J o +J, ex~;:) +J 2 ex~ ;:) (26) It is found by plotting the strain, nonnalized by pressure, versus time and fitting the curve to this equation. An experimental apparatus was developed to simulate the wound roll stresses experienced by a core. The criteria of the apparatus is to apply stress on the outer surface of the core, while exerting no stress on the inside surface of the core. A pressure vessel was designed to exert radial pressure on the outside surface, while venting the inside of the core outside the vessel. Pressure Vesse! Design The pressure vessel is designed to meet general requirements outside of this project. The pressure vessel is required to withstand a maximum pressure of2000 psi. at room temperature. The core must be held inside the vessel. The vessel must house instrumentation or allow for leads for external instrumentation and output from internal 11 instrumentation. The inside ofthe core must be vented to atmospheric pressure to simulate an actual wound role. The pressure vessel design selected was a roood cylinder capped on both ends. This minimizes stress concentrations. A pipe was used to construct the pressure vessel, with a welded cap on one end and a flange with a blind on the other acting as a porthole. Sizing the pipe to meet the required 2000 psi. maximum pressure was accomplished using the following formula that estimates the wall thickness. [6] ~= PD +A 2(SE +PyJ Where 1m is the minimwn wall thickness P is the maximum internal working pressure D is the outside diameter ofthe pipe SE is the maximum allowable stress y is values ofjoint efficiency A is the mechanical behavior allowance (3.1) Given a threeinch inside diameter of the core with a maximum outside diameter of four inches. The equation parameters are: [6] P=2000 psi. D=6.625" SE=15000 Ib/in2 y=.4 A=.065 These parameters are for a 6 inch, A106 grade B pipe. This yields required a wall thickness of .4843 inches and a schedule 120 pipe. The flange and cap were designed by the manufacturer for a working pressure. The flange is stamped with the maximum working pressure. A sixinch series 900 flange with a blind flange has a working pressure of 2200 psi. to 800°F. A sixinch extra heavy 12 welded cap has a working pressure of 2100 psi. A drawing of the pressure vessel is shown in figure 1. Pictures are shown in Appendix A figure 1 and figure 2. 6" Series 900 Welded Extraheavy 6" Pipe sch. 120 6" Series 900 Blind Figure 1. Pressure Vesse} A hollow shaft through the blind flange provides mounting for the core and venting of the inside ofthe core. The shaft is secured to the blind flange by a jam nut and sealed with an aring and washer. The core is sealed on the shaft by caps that are screwed onto the shaft and sealed with arings. The pressure vessel was welded by a certified welder and met all codes and state requirements. See figure 2 for a layout of the inside of the pressure vessel. Deflection Measurement Strain was measured by a small foil patch strain gage. Two problems in using the strain gage on the paperboard core could arise because paper that makes up the core tends to absorb the glue. One, the gage is less likely to adhere to the core. Two, the glue 13 absorbed by the core may locally change the characteristics of the core. Both problems may give erroneous output. The glue chosen was an epoxy product from Measurements Group. It was thick enough that negligible amounts of epoxy was absorbed by the core. A picture of a gage installed upon a core is shown in Appendix A Figure 3. Blind Flange Jam Nut Sealing Washer with oring Figure 2. Pressure vessel layout Miscellaneous Instrumentation A bulkheadfeed through is required to route the input and output lead wires, for the strain gage, from the inside of the pressure vessel to the outside. There is a maximum of twelve wires that go through the vessel.. The bulkheadfeed through consists of a wire cluster encapsulated in epoxy. This is placed directly into the blind flange. The pressure vessel uses compressed nitrogen from bottles supplied from a local distributor. The nitrogen bottles had a capacity of 40 cubic feet at 2500 psi. To get a range of pressure inside the vessel, a Vicors pressure regulator, rated to 250 psi, was used. An analog pressure gage, mounted on the blind flange, permitted an instantaneous pressure reading. 14 Data Acquisition The strain gages, Measurements Group model CEA06250UR350, are placed in quarter bridge circuits with outputs resistance changes converted into strain by a Measurements Group 2103a strain indicator. The strain indicator is calibrated to read in microstrain. The calibration equation is as follows. _ R'A 106 Il£cal  K'(R +R' ) cal A (3.2) [7] Thus, a required 10,000 microstrain resistance, R.:al equals 174.3 Kiloohms. A cantilever beam was set up to check the calibration. The theoretical strain equation is as follows: 3Dy E= L2 (3.3) [8] A strain gage was mounted on a beam .5 inches wide (D), .125 inches thick (y) and 10 inches long (L). The resulting strain was calculated 1875 J..u:. When the strain gage was connected to the strain indicator with the appropriate calibration factor, the strain indicator read 1876 !LE. This verified the calibration setting of the strain indicator. An Omega PX931KSV pressure transducer, rated at 01000 psi with .1% drift, was used to measure the pressure inside the vessel. This was calibrated with a dead weight tester. An IBM clone personal computer, P120, is used for data acquisition. LabVIEW, a graphical program for instrumentation, and an ATMIO16XE50 I/O board from National Instruments, is used to store pressure and strain output. 15 Experimental Procedure Two different sets ofcores of the same material with a 3inch diameter and 0.29inch wall thickness were provided by Sonoco. The estimated crush pressure was 400 psi and the modulus 70,000 psi. The two different sets of cores are similar in property values, but not identical. To avoid end effects, measurements were made 7 inches from the end ofthe core. Radial deformation is measured at the center of the length of the core. Thus, the core was cut to a length of 14 inches. Severa] strain gages were mounted around the core at the center of the length, with Measurement Group MBond GA2 epoxy. At times, cores do not deform in a uniform manner, thus a number ofgages were used to get an average deformation. The end caps were press fitted into the ends of the core to insure a good seal at the ends. The core was then wrapped with a 3M Poly tape to seal the outside of the core. With the paperboard cores, there was a tendency for the compressed nitrogen to seep through the core. The nitrogen would change the moisture content of the core, changing the properties. Thus, the outer tape seal was required. After wiring the strain gages to the bulkhead feed through and connected to the strain indicator, the gages were given time to equalize. The strain indicator was zeroed and calibrated and the acquisition program was initiated. Pressure is applied to the vessel slowly and recording ofthe pressure and strain was made in five second intervals. This gives a core modulus of elasticity reading for each gage. After the maximum pressure for the experiment is achieved, the pressure and strain recording is changed to 120 seconds. The experiment runs for 48 hours, or until the strain has stabilized. 16 Data Analysis In order to achieve an accurate creep function over the pressure range of the core, three different pressures were tested, 100 psi, 150 psi, 200 psi. In preliminary examination ofthe testing procedure, it was found that testing at above 50% of crush pressure, a buckling phenomenon may have occurred. Thus, the maximum 200 psi test pressure is 50% ofthe 400 psi crush pressure rating of the cores. Pressure, time, and strain from all gages were recorded at each ofthe given pressures. The pressure and strain data taken during the initial start up, through the elastic range, was used to calculate the elastic modulus. An elastic modulus of 80,000 psi, ±1O,000 psi reading on each strain gage, was used as an assessment of the accuracy of the gage reading. If the elastic modulus was not within the 80,000 psi ±10,000 psi range it was removed from the analysis. The most likely cause of faulty gauge data is improper attachment ofthe gauge or a flaw in the core. Core analysis from the first set of cores are shown first. The data for the second set of cores will be analyzed in the same manner, following the first set. The plots in Figure 3 show the strain data at the three set pressures for the first set of cores. 17 Strain Through Time at 100 psi 0.000 1000.000 400 600 800 1000 1200 1400 C 2000.000 __Strain 2 '2; 3000.000 __Strain 3 0.. u 4000.000 __Sttain4 ! r:: 5000.000 __StrainS i.i  6000.000 '" 7000.000 8000.000 Time (min) Figure 3a. Strain Data for Core #1 at 100 psi. Strain Through TIme at 150 psi 1000 0  1000 200 400 600 600 1000 1200 1400 r:: 2000 C ~ ; 3000 0 _Gage 3 .. (,l 4000 g 5000 __Gage4 I: i.i 6000 en 7000 . 8000 9000 TIme (min) Figure 3b. Strain Data for Core #1 at 150 psi. 18 Strain Through TIme at 200 psi 0.000 2000.000 200 400 600 800 1000 1200 1400 1 :E 4000.000 :5 __strain1 0 6000.000 __strain 2 ..u I 8000.000 __strain 3 r:: i.s. 10000.000 U) 12000.000 _ 14000.000 Time (min) Figure 3c. Strain Data for Core # I at 200 psi. The plots in figure 3 show a variation in strain between individual gages within a given pressure. This is due to the core not deforming uniformly because of inconsistency within the core. Therefore, the strain from each gauge is averaged to get an average defonnation across the entire core. The elastic strain was then subtracted in order to isolate the strain caused by creep. Figure 4 shows the average strain. 19 Average Strain with Elastic Strain Removed +100 psi e150 psi .!r 200 psi 400 600 800 1000 1200 1400 1 0 1000 . ..., 0 .....+++++++_1 1000 2000 3000 4000 5000. 6000 7000 8000 9000 ·10000 ..1.... ' Time (min) Figure 4. Average Strain Data for Core #1 The family of curves produced by these tests do not follow the same path. A function was needed to associate the curves so that they may be modeled as a Maxwell modeL Normalizing the strain curves, by dividing by their associate pressures, the normalized curves can be fit with the Maxwell model. This is shown in figure 5. Normalized Creep Data 10 .: 0 I '2 400 600 800 100:> 1200 1400 "T 10 eu g 20 +100 psi c:= e 150 psi ." I! Co 30 +0200 psi iii 'C G 40 .!:! 'jij ~ 50 0 z 60 Time (min) Figure 5. Normalized Creep Plot for Core #1 20 The lines of normalized strain, the strain divided by the corresponding pressure, follow similarpaths. They were represented by a single creep function, derived from the generalized Maxwell creep function discussed earlier. The normalized strain is averaged and a creep function is estimated. The error introduced into the analysis by averaging the normalized strain and developing a creep function was insignificant, due to the magnitude ofthe strains and the size of the deviation of the normalized strains. This is reinforced in the testing results in Chapter V. The coefficients to the creep function were found by utilizing an EXCEL worksheet solver function. Within the worksheet, the creep function is compared to the data at each time point. The difference is then summed and minimized by changing the function coefficients, thus getting an appropriate creep function. The EXCEL Solver function yields the following equation. ti~ ti~ J =60.06 + 17.82e 2213 1 + 4224e 865.50 (microin/in / psi) (3.4) A plot of the creep function verses the normalized creep data is shown in Figure 6. Nonnalized Qeep Data with Relaxation Function 10 c: 0 ] 400 WJ 800 10CXJ 1200 1400 b.. 10 .100psi uE 20 ~15Opsi c:::: ~ III tr 2OO psi  Co 30 en ir Qeep fl.l'"COOn "C CD 40 ~ n; E 50 0 z 00 TIme (min) Figure 6. Creep Function Plot of Core #1 21 The Maxwell Model follows the average strain data. The average nonnalized creep data seems to be diverging as time progresses past the time shown. Thus, the Maxwell creep function is only applicable through this time period. The analysis on the second set of core data follows in the same manner as the first set. The strain data for the second set of cores are seen in Figure 7. Core #2 Strain Through Time At 100 psi סס. 0 OO סס. 1000 OO 200 400 600 800 1000 1200 1400 1 C 2000. סס OO ~ .(.,. סס. 3000 OO (J I 4000. סס OO c __Straingage 2 ~... סס. 5000 OO Cf) __Straingage 3 סס. 6000 OO __Straingage 4 סס. 7000 OO Time (min) Figure 7a. Strain Data for Core #2 at 100 psi. 22 Core #2 Strain Through Time at 150 psi 0.00Et00 1.00803 200 400 600 800 1000 1200 1400 C 2.ooE+03 :§ __Slrain1 0 ... 3.ooE+OO __Strain 2 u 4.00803 __Strain 3 gc 5.ooE+OO __Strain4 ·.i.i  6.ooE+OO en 7.ooE+OO 8.ooE+OO Time (min) Figure 7b. Strain Data for Core #2 at 150 psi. Core #2 Strain Through Time at 200 psi סס. 0 OO סס. 1000 OO 200 400 600 BOO 1000 1200 1400 סס. 2000 OO ' סס. 2 3000 OO __Strain 1 '2 סס. 4000 OO __Strain 2 "T 0... סס. 5000 OO __ Strain 4 ug סס. 6000 OO c~ סס. 7000 OO Ui 8000. סס OO סס. 9000 OO סס 1 oo. סס oo Time (min) Figure 7c. Strain Data for Core #2 at 200 psi. The strain data is averaged with respect to its pressure and shown in Figure 8. 23 Averag,e Strain with Elastic Strain Removed 0 :5 1000 I,.: .0.. 2000 lJ I c:: 3000 g en 4000 GI Cl ~ Gl 5000 . >< 6000 +100 psi ~150psi 6200 psi 600 BOO Time (min) 1000 1200 1400 1 0 Figure 8. Average Strain Data for Core #2 The average strain is nonnalized by dividing by the test pressure and is shown in Figure 9. Nonnalized creep Data ~100psi er 150 psi fr200psi e Oeep Furctl:ln 0 5 c:: ,'2. 0 10 .. lJ §. 15 c::= Ill t!o. U) 20 't:I Gl !:! 25 'jij E 30 0z 35 400 TIme (min) 100J 1200 1400 Figure 9. Nonnalized Creep Plot for Core #2 24 Using the EXCEL equation solver in the same manner as for core #1 with the average of the normalized creep data, the general Maxwell creep function is obtained. The equation is : time time J =5.96+.0892 * e 122.86 + 5951 * e 1917.88 (microin/in I psi) (3.5) A plot of the creep fimction verses the normalized creep data is shown in Figure 10. Nonnalized Q'eep Data +100p;i a150 p;i Ir 200 p;i &Qeep FurcIi::r1 0 5 .E C "T .0.. 10 u :§. 15 1::::: Ill ~Q. U) 20 j .!=! 25 iii E 0 30 z 35 400 Time (min) 1cxx) 1200 1400 Figure 10. Creep Function Plot of Core #2 The Maxwell creep function fits the average nonnalized data. However, the normalized average data diverge as the time continues. This function is only representative ofthe normalized data at shorter time intervals. 25 In a roll, the core will experience a pressure change as the core defonns. All of these tests are executed at constant pressures. But, with the Maxwell creep function modeled through a pressure range, a pressure may be calculated given the defonnation at a given pressure within the range. A mathematical model will be developed to accomplish this job. 26 CHAPTER IV ROLL MODEL The core is influenced by pressure due to the wound roll above it. This pressure does not remain constant. As the core defonns under the pressure of the roll through time, the roll pressure decreases. Thus, a model is needed to predict the defonnation of the core and the resulting pressure change through time. The model begins with a wound roll on a core. The initial pressures with in the wound roll are found by using Hakiel's Model, discussed in Chapter II. The time varying model is developed just as Hakiel's Model. Hakiel's model applies to the wound roll in that the governing differential equation (2.12) does not change. [3] (2.12) and with the central difference approximations included the equation (2.18) is: (~ ~J(J (i 1) + (2r 2 + 1 .s.Jcr (i) +(.c+ ~Jcr (i + 1) =0 h 2 2h T h 2 E T h2 2h r r (2.18) The boundary conditions are where this mod.el and Hakiel's Model differ. The boundary condition at the coreroll interface is calculated by assuming the deflection of the core is equal to the deflection ofthe first layer ofthe roll. Umat'l =Ucore divide both sides by radius ofthe core 27 (4.1) Ematl ) = Ecore (4.2) First, we will focus upon the strain in the web materiaL Starting with the equilibrium equation (2.7) and substituting it into the constitutive equation (2.8b) yields: Applying the central difference approximations for fIrst derivative, fur crJ.j(i+I)crJ.j(i) = Or h where j denotes the stress at the current point in time yields: rearranging, we have: (4.3) (2.17a) (4.4) (4.5) Since roll winding is an accretive process and the pressures which are computed affect the radial modulus, Er, the second order differential equation in radial pressure is solved several times for differences in pressure which are summed to yield total pressures in each layer. So the previous expression is typically cast in the form: ( r I) (.) (1 r 1 v tr) (.) C' . =   l)cr . 1+ I +       l)cr . 1 ~ee,matl hE r,J E h E E r,J t t t r (4.6) 28 Now, let us focus upon the strain in the core. The strain at the core is defined as: Ecore  JJc(tt ')ddadtr tI The integral will be approximated at each step in time as: Ecore,Llt =J c (t j  tjl )c;r,jl (i) (4.7) (Ref. 9) (4.8) To obtain the total core strain at any point in time, the strains at each time step must be summed as: n=j Ecore,totaJ = LJc(tj tnl)O"m(i) (4.9) n=l Equation (4.9) yields the viscoelastic strain at a given time increment. Elastic strain of the core (4.10) must also be added to the equation. 0" . (i) r,J E core,elastic = E c (4.1 0) Now, assuming the generalized Maxwell form for the relaxation function from equation (2.6) yields: (4.11) Now, equating the strain in the core and in the web material yields: (4.12) 29 The outer boundary condition is calculated assuming a traction free outer roll surface. (4.13) With these two boundary conditions, the governing differential equation can be solved for the static, time dependent region of the model. With the boundary conditions (4.12), (4.13), the governing differential equation can be solved. The set oftridiagonal simultaneous equations are set up and solved in the same manner as the QuaIl's model described in Chapter II, with the exception of stepping through time instead of through temperature. The set of equations are solved in a stepwise linear fashion through time that yields increments in pressure decay. Then, the radial pressures and radial modulus are updated each time step. The final solution yields a pressure profile of the roll through time. Computer Model The finite difference method of solving differential equations can be readily solved on a personal computer. A FORTRAN code was developed to solve the model and output the solution. The algorithm flow chart is shown in figure 11. A listing ofthe FORTRAN code is in Appendix B. The program requires the user to input some system requirements and material properties. The system requirements are: winding tension, outside radius ofthe core, outside radius ofthe roll, number of increments through the roll, total time, and number 30 of increments through time. The material properties required are: radial stiffness and Poisson's ratio of the core, the third order polynomial for the radial modulus ofthe web, tangential modulus of the web, and the directional Poisson's ratio's ofthe roll. The program outputs five columns ofradial position, radial stress, and incremental stress change. The output format lends itself to its use in a spreadsheet software, such as Microsoft EXCEL. In EXCEL, one can plot the radial stress versus time. 31 Calculate incremental radial stress at current roll lap solve bound Solve tridiagonal matrix using Gaussian Elimination Update pressure through out all laps rolled No! Calculate radial stress in the roll through time step Solve tridiagonal matrix using ,Guassian elimination Update pressure through out entire roll No! Figure 11. Flow chart. 32 CHAPTER V MODEL VERIFICAnON The roll model calculates the radial stresses due to the combination of the winding of the roll and the defonnation of the core through time. It begins at time zero and ends at a specified time. The model was developed previously in this paper. The FORTRAN code was used with the given core properties and the properties of a polyester film. The properties are as follows. Roll ill = 1.75 in Roll aD = 5.75 in Roll Iterations = 1000 Winding Tension = 2000 psi ICI "377" 200 gauge film Er = 54.97"'P.07819*P2 +.0001388*P3 (psi) Et = 600000 (psi) Core E = 80000 (psi) Creep Coefficients (microin/in / psi) Jo = 60.059 11 = 17.8]7 T1 =221.31 12 = 42.242 T2 = 865.50 The film Er property equation comes from a stack test. A stack of loose web was compressed with the nonnal pressure and strain recorded. The slope of this data was used to generate this Er third order polynomial as a function ofpressure. The film Et property is established in a tensile inplane stress verses strain test. The roll pressure builds very quickly during winding. Figures 12 and 13 show how the radial pressure develops in the roll and how the pressure varies during winding, 33 respectively. Note, the core pressure has nearly reached its maximum value after eleven minutes from a total winding time of 70 minutes. Radial Pressures 0 WT = 2.5 min o WT= 5.3 min Through Winding 200 l! WT = 11.9 min L;jjjiiiil",. ~iiiiii~~~:I11x WT = 29.1 min ,.... 150 t"""kf..+~+~,,I___'~_1_____l x WT = 51.5 min .[ 0 WT = 69 min ....... ~ 100 f''lk+""+__+'llfr+j~_+__"Q..__t___l :::l CI.l CI.l J: 50 , &'i\ctt~+___+__+___'l~_l O+~____i3I+__+~i i____l:~__! 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25 5.75 Winding Radius (in) Figure 12 Radial Pressure Through Winding Core Pressure Through Winding ..200 l:O 0.. 'u' 150 10 :::J ~ 100 u 10 P4 50 u 10 8 0 ~ ( II I o 20 40 Winding time (min) 60 80 Figure 13. Core Pressure Through Winding 34 Roll Testing Six wound rolls ofIeI 377200 film were made in accordance with the previous parameters. The first three rolls were tested by instrumenting the core with strain gauges and measuring the core strain over time. The results were found to be widely varying. This was caused by the way the strain was read. The gauges were mounted on the core and connected to the strain indicators. The gauges were unable to be connected to the indicator during winding of the roll, so they were disconnected. After winding, the gauges were reconnected. Connection ofthe gauges consisted of twisting wires together which, if connection changes occur, can change the resistance of the lead wires. The theoretical resistance change within the system using the gauge factor, was 1 to 2 ohms. This was seen in leadwire resistance change in connections alone. Thus, this method of testing was inconclusive and a new method had to be developed. Note, this problem had no impact on the pressure chamber testing. The connections were not disconnected during the test and the leadwire resistance did not change. Instrumentation was zeroed before testing began, thus taking the lead wire resistance into account. A new test method was derived from the same method used by Hakiel in his study of wound roll stresses. [3] The interlayer pressures were calculated by forcing two layers of the web, close to the core, to slip upon one another. The layers of web to be tested are not in direct contact with the core. If these layers were to be used, problems may arise. The friction coefficient between the core and web is not known and difficult to calculate. Sliding the web from a compressed core section over an uncompressed section would 35 lead to problems with binding. Finally, shearing of the tape bonding ofthe web to the core, required at the start ofthe winding process, would require an unknown force. In order to eliminate these problems, layers of web removed from the core are used in the test. By knowing the web to web static coefficient of friction, and reading the force required to slide the web, the interlayer pressure can be found. (5.1) The interlayer pressure (P) is found by the interlayer force divided by the surface area. P=~ 2mw (5.2) The force required, to cause the web layers to slip, was provided by a material testing system, INSTRON model 8502. Figure 15 shows the test setup. Dies were created to ensure the repeatability of the correct web layers sliding on each test. The bottom male die has a running fit to the outside diameter of the core. It has an outside diameter of 1.95". The top female die has a loose fit to the outside diameter ofthe male die. This is so that the web will be pushed up by the male die into the bottom die and so that the correct web layers slide in each of the tests perfonned' Figure 14 shows the die set. 36 Female die ~L:: ....;...d Wound roll Male dipe......I~:~__;.:I Figure 14. Die Set Female die Male die Wound roll ~ INSTRON Figure 15. Push Test Set Up Each roll is tested over a 1500 minute time interval. The roll was placed in the INSTRON press with the two dies placed as in previous figure. The INSTRON press was set to move 0.01" per second. The force was recorded along with the displacement. When the required web layer begins to slip past each other, the pressure levels off This 37  was the push force required. The force and displacement were recorded on a PC and the slip pressure was read from the record. RESULTS Rolls #1, #2, and #3 were tested under the strain acquisition procedure in which data received was deemed unusable. Roll #1 Core type #1 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 11.5" ill: 3.5625" Winding Tension: 1700 psi Winding Speed: 50 ft/min Note, the coefficient of static friction was found by employing tests on web material in the same fashion as described by Ducotey. [10] Roll #2 Core type #1 Web: ICI377200 gauge Coefficient ofFriction: .21 aD: 11.625" ill: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ft/min Roll #3 Core type #1 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 10.625" ill: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ft/min 38 Pictures ofRoll #2 and Roll #3 are in Appendix A, figure 4 and figure 5, respectively. Roll #4 was the first roll to undergo this push test through a step time interval. The winding parameters are the same as in the model with the exception of the following: Roll #4 Core type #1 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 10.875" ill: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ftlmin Roll #4 Push Force at 1.95" 3000 2500 ~~ :c 2000 't:l 1500 Cll 0 ..J 1000 500 0 0 200 400 e Theoretical & Theoretical at 1st layer 600 800 1000 1200 1400 1600 Time (min) Figure 16. Roll #4 Push Force In figure 16, the experimental push force is plotted along with the values predicted from the computer model. Also, the plot displays the predicted force to push the first two layers by one another. This shows the drop in pressure at one layer past the core and an indication of the core pressure. 39 The Roll #4 push force data through time follow the model well. At the first few time intervals, however, the data diverge. This may be due to the layout of the roll and the core type. The outside ofthe roll is rough and sticks out of the roll several inches. During the first few tests, the die did not slide smoothly due to the snug fit of the die against the core. After the sides ofthe core had worn a little, the data seem to correlate better. Notice that after 1500 minutes, the pressure near the core drops to zero. Then the core will no longer deform viscoelastically. A plot of the radial pressure of the roll at time = 0 and time = 1500 minutes is shown in figure 17. Roll #4 Radial Pressure ___ TIme = 0 min ~Tlme =1500 min 200.000 180.000 160.000 140.000 iii ~:..; 100.000 III VI BO.OOO .. a 60.000 40.000 .. 20.000 0.000 1.76 2.25 2.75 3.25 3.75 RJldlua (In) 4.25 4.75 5.25 5.75 Figure 17. Roll #4 Radial Pressure The figure shows that the model is accurately predicting the later time intervals and need not be estimated further. Roll #4 developed a starring flaw at the core roll interface during the winding, due to the deformation of the core and the pressure change of the roll. This flaw is shown in a picture in the Appendix A figure 6. 40 Roll #5 was the next roll to undergo this push test. The winding parameters are the same as in the model with the exception ofthe following: Roll #5 Core type #2 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 11.75" rD: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ftJmin A picture of roll #5 is found in Appendix A figure 7. Roll #5 Push Force at 1.95" ~Experirrental a Theoretical & Theoretical at 1st layer 1000.00 1500.00 2000.00 Time (min) 500.00 o+ ~_< 0.00 Figure 18. Roll #5 Push Force. This plot is set up the same as the previous plot for Roll #4. They are similar. Roll #5 correlates better at first. The core type #2 is smooth and stuck out of the roll by only a half an inch. The data seem to diverge at the end of the test period. This is due to the model creep function which did not fit the normalized averages of the core data at the longer time periods. A plot of Roll #5 radial pressure through time is shown in figure 19. 41 RoU.S Radial Pressure __Tlme =0 min __Tlme =1500 min 200.000 180.000 160.000 140.000 'ii 120.000 ~ ..!".. 100.000 ! 80.000 A 60.000 40.000 20.000 0.000 1.75 2.25 2.75 3.25 3.75 Radin/min) 4.25 4.75 5.25 5.75 Figure 19. Roll #5 Radial Pressure. 42  CHAPTER VI CONCLUSIONS This study has shown how the viscoelastic properties of the core effect a wound rolL The viscoelastic core properties are represented by a relaxation function. The relaxation function is found experimentally by submitting cores to a series of constant stress tests over time. The strain is recorded and nonnalized by dividing by the test pressure. A relaxation function is fitted to the average of the normalized strain. The function can then be used in the developed mathematical model to predict wound roD behavior. Model verification tests show that the model can accurately predict the stresses through the roll. Viscoelasticity of the core will cause the core pressure to decrease over time and finally decay to zero. This is a localized phenomenon and does not affect the entire roll. The pressure decay at the core is, however, significant. It can cause staring to occur, as seen in a picture in Appendix A figure 6. The verification tests show that an accurate creep function is needed. In core type 2, the creep function does not accurately predict the strain data at the later test times. This caused a divergence in the accuracy ofthe model. This model is a tangible way to predict the pressure decay within a wound roll in the vicinity of the core. The web handling industry will find this research useful. 43  CHAPTER VII FUTURE WORK This model was developed with the assumption that viscoelastic properties ofthe core were not affected during the winding of a roll. A new model can be developed to include the creep function during the winding process. There will be situations were the pressure would drop appreciably during winding. This study was conducted under conditions in which temperature and humidity were held constant. In industry, wound rolls are submitted to environmental changes which may affect the core. The environmental conditions need to be considered. The two cores shown in this study were similar but not the same. Their relaxation functions were different. A correlation can be developed to relate different core types. 44  REFERENCES 1. Findley, William N., Lai, James S., and Onaran, Kasif, Creep and Relaxation of Nonlinear Viscoelastic Materials, North Holland Publishing Company, New York, 1976. 2. Gerhardt, T.D., "External Pressure Loading of Spiral Paper Tubes: Theory and Experiment". Sonoco Products Company, Madison, WI. 3. Hakiel, Z., ''Nonlinear Model for Wound Roll Stresses", Tappi Journal, May 1987. Pg. 113117. 4. Qualls, William R & Good, 1. Keith, "Thermal Analysis of a Wound Roll", Accepted by Journal of Applied Mechanics, January 1997. 5. Salikis, Edmond P. & Rowlands, Robert E., "A Novel Radial Compression Testing Device for Tubes", Tappi Journal, January 1997. Pg. 234237. 6. ASME B3I.31996 Edition, Process Piping, ASME Code for Pressure Piping, B3l an American National Standard, The American Society ofMechanical Engineers, 1996. Pg. 304.1.1304.1.2, Table AI, AIA. 7. Measurements Group, "Strain Gage Conditioner and Amplifier System Instruction Manual", Raleigh, North Carolina, 1992. 8. Shigley, Joseph E., & Mischke, Charles R, Mechanical Engineering Design, 5th ed., McGrawHill, New York, 1989. 9. Qualls, William R, Hygrothermomechanical Characterization of Viscoelastic Centerwound Rolls, Ph.D. Dissertation, Oklahoma State University, May 1995. 10. Ducotey, Keith S. Traction Between Webs and Rollers in Web Handling Applications, Ph.D. Dissertation, Oklahoma State University, May 1993. 45 APPENDIX A PICTURES 46  Figure AI, Fixture Setup #1. 47  Figure A2, Fixture Setup #2. 48 Figure A3, Strain Gage Application. Figure A4, Roll #2. 49 \ Roll #2 Core Type 1 Figure A5, Roll #3. Figure A6 Roll # 4. 50 Roll # 3 Core Type 1 o yp 1 ,. Figure A6 Roll # 5. 51 Roll #5 Core Type 2 APPENDIXB COMPUTER CODE 52  ********************.****************************•••***•••**••***** ****************************••********************.**••••••••****.* **.*********************•••••***••••***••*****.***••*******••**.*** ***** ***** ***** ***** ***** ***** ***** ***** ***** CORE RELAXAnON 1st ed. ***** Written By: Jeff Henning ***** Project Coordinator: Dr. 1. Keith Good *.*** ***** ***** ***** ***** ***** Web Handling Research Laboratory (WHRC) Oklahoma State University ***** Department ofMechanical and Aerospace Engineering Stillwater, Oklahoma ***** ***** ***** ***** ***** Calculates Stresses Induced Due to the ***** Relaxation ofthe Core in Time ***** ***** ***** ***** ***************************************.*************************** ***********************.****.************************************** ******************************************************************* ** ***** Main Program Corerelax ***** opens and closes input/output files ***** make calls to subroutine ** ********************.***********.*.******************************** PROGRAM CORERELAX OPEN(1 ,FILE='CORECRP.IN') OPEN(2,FILE='CORECRP.OUT') OPEN(4,FILE='COREPRS.OUT') WRJTE(*,*)'***.*************************************************' WRJTE(*,*)'ENTER "1" FOR FILE INPUT OR "0" FOR KEYBOARD INPUT' READ(*,·)IANS WRITE(*,*)'***.*****************.*******************************' IF(IANS.EQ.O) THEN CALL UINPUTS ELSE CALL INPUTS ENDIF WRITE(4,120) WRITE(*,*)'*****Calculating*****' CALL WINDER CALL RELAX WRITE(*,*)'*****Finished*****' WRITE(*,*)'******.*.*******.************************************' 53 C CALL OUTS 120 FORMATC TIME CLOSE(2) CLOSE(4) STOP END PRESSURE DELTA B(l)') ******************************************************************* ******************************************************************* ** ***** Subroutine RELAX ******************************************************************* SUBROUTINE RELAX IMPLICIT REAL*8 (AH,J,OZ) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKIDCLOCK,U(lOOO),DTP(lOOO),NCLOCK,JO,Jl,Tl ,J2,T2 COMMON/MATLPROP/EC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMONffIMEfR(1000),TIME,TWIND,LAP,DTOLD,DTIME COMMON/MATCOEFF/A(1000),B(1000),C(1000),D(lOOO),N COMMONIPRESIP(1OOO),DP(l OOO),S,OLDP,W(1000) C WRITE(*,*)'THSTRESS IN OK' C CCCCC C R(I)=RIN LAP=1 H=(ROUTRIN)/NLAPS DO I=2,NLAPS+1 R(I)=R(1)+(11)*H END DO C CCCCC DO L=1,NLAPS VP(L)=P(L) END DO C DCLOCK=DCLOCKINCLOCK DO 999 K=l,NCLOCK TIME=K*DCLOCK WRITE(2,1OO)TIME C C cccccccccccccccccccccccccccccccccccc CCCC THIS IS STRESS FORMULATION WITH EC C CCC CORE BOUNDARY CONDITION 54  C D(1)=(IDOvrtEt/(Ec*R1N)RINIH) C D(l )=(ldOEtlEr(1 )*vtrEt/(Ec*rin» C(l)=rin/h S=P(1) OLDP=P(l) B(l)=Et*S*(JO+Jl *EXP(TIMErrl)+J2*EXP(TIMErr2» C CCC ROLL ANALISIS C DO 60 1=2, NLAPS vtr=vrt*Er(I)/Et A(Il)=(R(I)**2fH**2R(I)/(2DO*H)*(3dOEtlEr(I)*vtr+vrt» D(I)=(1DO2DO*R(I)**2fH**2+vrtEtlEr(I)*(1+vtr» C(I)=R(I)**2fH**2+R(I)/(2DO*H)*(3dOEt/Er(I)*vtr+vrt) C A(Il )=(R(I)**2fH**23DO*R(I)/(2DO*H» C D(I)=(l DO2DO*R(I)"'*2fH**2EtlEr(I» C C(I)=R(I)**21H**2+3DO*R(I)/(2DO*H) C B(I)=ODO C 60 CONTINUE C CCC OUTER BOUNDARY CONDITION C D(NLAPS+1)=lDO B(NLAPS+I)=ODO A(NLAPS)=ODO C C CCC SOLVE MATRIX FOR PRESSURES AND PRINT C CALL SOLVETRl(DTP,NLAPS+1) LAP=NLAPS+I CALL VTOTPRESS(DTP) CALL OUTS C CALL STRAIN 999 CONTINUE C 100 FORMAT('TIME =',F7.0,' (MIN)') C WRITE(*,*)'THSTRESS OUT OK' RETURN END **************************.**************************************** 55  *****Subroutine Strain *****Solves the strain at the first layer of roll ***** which is equal to the strain of the core ******************************************************************* SUBROUTINE STRAIN IMJ>LICIT REAL*8 (AH,J,OZ) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKIDCLOCK,U(1000),DTP(1000),NCLOCK,JO,Jl,Tl,J2,T2 COMMONIMATLPROP/EC,E,vc,ET,AA(0:3),ER(1000),vrt,vtr COMMON/TIME/R(1000),TIME,TWIND,LAP,DTOLD,DTIME COMMONIMATCOEFFIA(l OOO),B(1 OOO),C(1OOO),D(1 OOO),N COMMON/PRESIP(1OOO),DP(1 OOO),S,OLDP,VP(1000) COMMON/STRNNSTRN,ESTRN,TSTRN C WRITE(*,*)'STRAIN IN OK' C VSTRN=VSTRN+«J0+J1*EXP(TIMEffl)+J2*EXP(TIME/T2)*OLDP) C ' *1000000) VSTRN=VSTRN+B(I) ESTRN=(P(1)/EC)* 1000000 TSTRN={VST~+ESTRN) C WRITE(3,40) C WRITE(3,50)TIME,TSTRN,VSTRN,EST~ 50 FORMAT(F1O.2,' ',F15A,' ',FI5A,' ',FlOA) RETURN END ******************************************************************* ******************************************************************* ***** Subroutine SOLVETRI ***** SOLVES THE TRIDIAGONAL SYSTEM OF DIMENSION 1DIM ***** FOR THE SOLUTION VECTOR X(IDIM) ******************************************************************* SUBROUTINE SOLVETRI(X,IDIM) IMJ>LICIT REAL*8 (AH,J,OZ) INTEGER IDIM DIMENSION X(1000) COMMONIMATCOEFF/A(1000),B(1000),C(1000),D(1000),N C WRITE(*,*)'SOLVE IN OK' N=IDIM DO 900I=2,N D(I)=D(I)(A(Il)/I)(Il»*C(Il) B(1)=B(l)(A(ll)/1)(11 »*B(11) 900 CONTINUE X(N)=B(N)/I)(N) DO 910 I=(Nl),l,l 56 X(I)=(B(I)C(I)*X(I+1»10(1) 910 CONTINUE C WRITE(*,*)'SOLVETRl OUT OK ' RETURN END ******************************************************************* ******************************************************************* ***** Subroutine TOTPRESS ***** UPDATES THE TOTAL PRESSURE P(I) ***** UPDATES THE INITIAL VISCOELASTIC CHANGE IN PRESSURE VDP(I) ***** UPDATES THE INITAL TOTAL VISCOELASTIC CHANGE IN PRESSURE VP(I) ******************************************************************* SUBROUTINE VTOTPRESS(DELTA) IMPLICIT REAL*8 (AH,J,OZ) DIMENSION DELTA(1000) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMONIMATLPROPIEC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMON/TIME/R(lOOO),TIME,TWIND,LAP,DTOLD,DTIME COMMONIPRESfP(l000),DP(1000),S,OLDP,VP(lOOO) C DO 1=1,LAP P(I)=P(I)+DELTA(I) Er(I)=AA(3)*(ABS(p(I»)**3+AA(2)*(ABS(p(I»)**2+AA(1)* , (ABS(P(I))+AA(O) END DO C RETURN END ******************************************************************* ******************************************************************* ***** Subroutine OUTS ***** PRINTS OUPUT OF RADIAL PRESSURES TO THE FILE "OUT.DAT" ******************************************************************* SUBROUTINE OUTS IMPLICIT REAL*8 (AH,J,OZ) COMMONfPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKIDCLOCK,U(l OOO),DTP(1OOO),NCLOCK,JO,J 1,Tl ,12,T2 COMMONIMATCOEFFIA(lOOO),B(lOOO),C(lOOO),D(IOOO),N COMMONIMATLPROPIEC,E,vC,ET,AA(0:3),ER(l 000),vrt,vtr COMMONITIME/R(lOOO),TIME,TWIND,LAP,DTOLD,DTIME COMMONIPRESfP(lOOO),DP(1 OOO),S,OLDP,VP(1 000) C C C WRITE(*,*)'OUTS IN OK I 57 PHRASE='************************************************••' WRlTE(2,20)' RADIUS PRESSURE C WRlTE(5,60)TIME DO 9200 I=I,NLAPS+1 WRlTE(2,30)R(D,P(I),DTP(I) 9200 CONTINUE WRITE(4,40)TlME,P(1),DTP(1),B(1) C 10 FORMAT(lX,A50) 20 FORMAT(A47,' ',FI0.1) 30 FORMAT(FI5.4,' ',FI5.8,' ',EI5.8) 40 FORMAT(FIO.O,' ',FlS.4,' ',EIS.8,' ',E15.8) RETURN END DELTA' ******************************************************************* ******************************************************************* ***** Subroutine INPUTS ***** INITIALIZES ALL INPUTS PARAMETERS BY READING ***** INPUT FILE "IN.DAT" ******************************************************************* SUBROUTINE INPUTS IMPLICIT REAL*8 (AH,J,OZ) COMMONIPARAMIRlNC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCK/DCLOCK,U(1000),DTP(l000),NCLOCK,JO,JI,Tl,J2,T2 COMMONIMATLPROPIEC,E,vc,ET,AA(0:3),ER(IOOO),vrt,vtr CCC C WRITE(*,*)'ENTER THE WINDING TENSION' READ(l,lOO)WT C WRITE(*,*)'ENTER THE INSIDE RADIUS OF THE CORE' C READ(1,1 OO)RINC C WRITE(*,*)'ENTER THE INSIDE RADIUS OF THE ROLL' READ(1,1OO)RIN C WRITE(*,*)'ENTER THE OUTSIDE RADIUS OF THE ROLL' READ(1,1OO)ROUT C WRITE(*,*)'ENTER THE RADIAL STIFFNESS OF THE CORE I READ(1,120)EC C WRITE(*,*)'ENTER THE YOUNGS MODULUS OF THE CORE ' C READ(I,I20)E C WRITE(*,*)'ENTER POISSONS RATIO OF THE CORE ' READ(I,IOO)vc C WRITE(*,*)'ENTER THE TANGENTIAL MODULUS OF THE ROLL' READ(1 ,1 OO)ET C WRlTE(*,*)'ENTER THE COEFFICIENTS (c3,c2,cl,cO) OF THE' C WRITE(*,*)'RADIAL MODULUS ER=c3*P"'2+c2*P"'2+cl *P+cO ' READ(1, I20)AA(3),AA(2),AA(1 ),AA(O) 58 C WRlTE(*,*)'ENTER THE POISSONS RATIO vrt OF THE ROLL' READ(1,100)vrt C WRlTE(*,*)'ENTER THE POISSONS RATIO vtr OF THE ROLL' READ(1,1 OO)vtr C READ(1,120)JO C READ(l,120)Jl C READ(1,120)Tl C READ(1,120)J2 C READ(1,120)T2 C C WRlTE(*,*)'ENTER THE TOTAL NUMBER OF LAPS TO BE WOUND ' READ(l,llO)NLAPS C WRlTE(*,*)'ENTER NUMBER OF CORE SEGMENTS (FOR DISPL FORM)' C READ(I,llO)NCORE C WRITE(*,*)'ENTER THE CHANGE IN TIME ' READ(1,100)DCLOCK C WRITE(*,*)'ENTER THE NUMBER TIME INCREMENTS' READ(1,11O)NCLOCK 100 FORMAT(F12.4) 110 FORMAT(I9) 120 FORMAT(EI2.5) CLOSE(1) 130 continue RETURN END ******************************************************************* ******************************************************************* ***** Subroutine UINPUTS ***** INITIALIZES ALL INPUTS PARAMETERS BY READING ***** KEYBOARD ENTRY ******************************************************************* SUBROUTINE UINPUTS IMPUCIT REAL*8 (AH,J,OZ) COMMONIPARAMfRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKJDCLOCK,U(1000),DTP(1000),NCLOCK,JO,n,T1,12,T2 COMMONIMATLPROP/EC,E,v,ET,AA(0:3),ER(1000),vrt,vtr CCC WRlTE(*,*)'ENTER THE WINDING TENSION I READ(*,1 OO)WT WRITE(I,100)WT 59  C WRITE(*,*)'ENTER THE INSIDE RADIUS OF THE CORE ' C READ(*,1 OO)RINC C WRITE(l,lOO)RINC WRlTE(*,*)'ENTER THE INSIDE RADIUS OF THE ROLL' READ(*,1 OO)RIN WRITE(1,lOO)RIN WRlTE(*,*)'ENTER THE OUTSIDE RADIUS OF THE ROLL' READ(*,lOO)ROUT WRITE(1,100)ROUT WRITE(*,*)'ENTER THE RADIAL STIFFNESS OF THE CORE ' READ(*,120)EC WRlTE(1,120)EC C WRITE(*,*)'ENTER THE YOUNGS MODULUS OF THE CORE ' C READ(*,120)E C WRlTE(1,120)E WRlTE(*,*)'ENTER POISSONS RATIO OF THE CORE ' READ(*,1OO)vc WRlTE(1,1OO)vc WRlTE(*,*)'ENTER THE TANGENTIAL MODULUS OF THE ROLL' READ(*,100)ET WRITE(1, 1OO)ET WRITE(*,*)'ENTER THE COEFFICIENTS (c3,c2,c1,cO) OF THE' WRlTE(*,*)'RADIAL MODULUS ER=c3*PJ\3+c2*PJ\2+c1*P+cO ' READ(*,120)AA(3),AA(2),AA(1),AA(O) WRlTE(1,120)AA(3),AA(2),AA(I),AA(O) WRlTE(*,*)'ENTER THE POISSONS RATIO vrt OF THE ROLL' READ(*,lOO)vrt WRlTE(1,lOO)vrt WRITE(*,*)'ENTER THE POISSONS RATIO vtr OF THE ROLL' READ(*,1OO)vtr WRlTE(1,100)vtr WRlTE(*,*)'ENTER THE CREEP FUNCTION COEFFICIENTS' WRITE(*,*)'1=10+JI*EXP(tfn)+12*EXP(t/T2)' WRlTE(*,*)'10' READ(*,120)JO WRITE(l,120)10 WRITE(*,*)'11' READ(*,120)J1 WRITE(1,120)11 WRITE(*,*)'Tl' READ(*,120)T1 WRITE(1,120)Tl WRITE(*,*)'12' READ(*,120)12 WRITE(l,120)J2 60 WRITE(*,*)'T2' READ(*,120)T2 WRITE(1,I20)T2 WRITE(*,*)'ENTER THE TOTAL NUMBER OF LAPS TO BE WOUND < 1000' READ(*,110)~PlPS WRITE(l,IlO)~PlPS C WRITE(*,*)'ENTER NUMBER OF CORE SEGMENTS (FOR DISPL FORM)' C READ(*,11O)NCORE C WRITE(I,IlO)NCORE WRITE(*,*)'ENTER THE CHANGE IN TIME ' READ(*,100)DCLOCK WRITE(l, IOO)DCLOCK WRITE(*,*)'ENTER THE NUMBER TIME INCREMENTSI READ(*,11 O)NCLOCK WRITE(1, 11O)NCLOCK 100 FORMAT(FI2.4) 110 FORMAT(I9) 120 FORMAT(E12.5) CLOSE(l) 130 continue RETURN END ******************************************************************* ******************************************************************* ** ***** Subroutine Winder ***** similar to standard elastic finite difference winding routines ***** determines the change in radial pressure at radial increment ***** due to the addition of each layer ***** store these values in the array DP(Iayer,layer) ** ******************************************************************* SUBROUTINE WINDER IMPLICIT REAL*8 (AH,J,OZ) COMMONfPARAMIRINC,RIN,ROUT,~APS,NlNC,NCORE,WT COMMONICLOCK/DCLOCK,U(1OOO),DTP(1 OOO),NCLOCK,JO,J I,Tl ,12,T2 COMMONfMATLPROP/EC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMONffIMEIR(IOOO),TIME,TWIND,LAP,DTOLD,DTIME COMMONfMATCOEFF/A(1OOO),B(l 000),C(1OOO),D( IOOO),N COMMON/PRES/P(1 OOO),DP(1 OOO),S,OLDP,VP(I 000) COMMON/TRY/AA1,BB 1,CC1,RK C CCCCC CALCULATE INITIAL PRESSURES C C WT=lOO.O 61 C Ec=E*(rin**2rinc**2)/(rin**2+rinc**2vc*(rin**2rinc**2»/rin C C WRITE(*,*)'WINDER IN OK' H=(ROUTRIN)/NLAPS R(l)=RIN LAP=1 DO I=2,NLAPS+I R(I)=R(1 )+(11 )*H END DO C CCCCC WIND FIRST LAP C WRITE(*,*)'ONE' C LAP=1 DT=EC*WT*RIN**2/(EC*R(1)**2+H*ET) DT=WT DP(1 )=(DT*H)IR(1) CALL TOTPRESS(DP) C CCCCC WIND ON SECOND LAP C C WRITE(*,*)'TWO' LAP=LAP+l I=LAP D(1 )=(1 DOvrtEt/(Ec*RIN)RINIH) C(1)=R(l)/H B(I)=ODO DP(2)=(WT*H)IR(LAP) DP(1 )=(B(1)DP(2)*C(1 »ID(1) CALL TOTPRESS(DP) c CCCCC WIND ON ALL REMAINING LAPS C DO 50 M=3,NLAPS LAP=M C D(I )=1 DOvrtEtJ(Ec*RIN)RIN/H C(I)=R(1)/H B(1)=ODO C DO 1=2, LAPl A(Il )=(R(I)**21H**23DO*R(I)/(2DO*H» D(I)=(IDO2DO*R(I)**21H**2Et/Er(I» C(I)=R(l)**21H**2+3DO*R(I)/(2DO*H) 62 B(I)=ODO END DO C DP(LAP)=(WT*H)IR(LAP) B(LAP)=DP(LAP) D(LAP)=lDO A(LAPl )=ODO C C CALL SOLVETRI(DP,LAP) CALL TOTPRESS(DP) 50 CONTINUE CALL OUTS C WRITE(*,*)'WINDER OUT OK' RETURN END ******************************************************************* ******************************************************************* ***** Subroutine TOTPRESS ***** UPDATES THE TOTAL PRESSURE P(I) ***** UPDATES THE INITIAL VISCOELASTIC CHANGE IN PRESSURE VDP(I) ***** UPDATES THE INITAL TOTAL VISCOELASTIC CHANGE IN PRESSURE VP(I) ******************************************************************* SUBROUTINE TOTPRESS(DELTA) IMPLICIT REAL*8 (AH,J,OZ) DIMENSION DELTA(lOOO) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMONIMATLPROPIEC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMON/TIMEIR(l000),TIME,TWIND,LAP,DTOLD,DTIME COMMON/PRES/P(lOOO),DP(l OOO),S,OLDP,VP(l 000) C DO I=l,LAP P(I)=P(I)+DELTA(I) Er(I)=AA(3)*(ABS(p(I}»**3+AA(2}*(ABS(P(D»**2+AA(1)* I (ABS(p(I»)+AA(O) END DO C RETURN END 63 VITA Jeffrey Scott Henning Candidate for the Degree of Master of Science Thesis: EFFECTS OF RELAXAnON OF A CORE ON A WOUND ROLL Major Field: Mechanical Engineering Biographical: Personal Data: Born in Stillwater, Oklahoma, on August 24, 1966. Married to Rebecca Sue Worsham. Education: Graduated from Jenks High School, Jenks, Oklahoma, in May 1984; attended Oral Roberts University and Tulsa Community College; received Bachelor of Science degree in Mechanical Engineering from Oklahoma State University, Stillwater, Oklahoma, in May 1995. Completed the requirements for the Master of Science degree with a major in Mechanical Engineering at Oklahoma State University in May, 1997. Experience: Employed in industry before returning to finish degree. Research Assistant at the OSU Web Handling Research Center. Professional Memberships: Oklahoma Society of Professional Engineers, National Society ofProfessional Engineers, Certified Engineer Intern.
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Title  Effects of Relaxation of a Core on a Wound Roll 
Date  19970501 
Author  Henning, Jeffrey Scott 
Document Type  
Full Text Type  Open Access 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  EFFECTS OF RELAXATION OF A CORE ON A WOUND ROLL By JEFFREY SCOTT HENNING Bachelor of Science Oklahoma State University Stillwater, Oklahoma 1995 Submitted to the Faculty of the Graduate College ofthe Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May 1997 EFFECTS OF RELAXATIO OF A CORE ON A WOUND ROLL Thesis Approved: Dean of the Graduate College 11 ACKNOWLEDGMENTS I wish to thank Dr. James K. Good, Professor, Mechanical and Aerospace Engineering and my advisor. His guidance throughout my research has been invaluable. Gratitude is also extended to Charles Johnson, Sonoco, Inc., for his technical support and for providing materials. His moral support is also greatly appreciated. Thanks must also go to Ron Markum and Robert Taylor of the OSU Web Handling Research Center for their expertise. For her generous patience, support and love I thank my wife, Susie. 11l Chapter TABLE OF CONTENTS Page 1. INTRODUCTION 1 II. LITERATURE REVIEW 3 Elastic Core Model 3 Hakiel's Model. 3 Thennal Analysis 8 Tube Testing Device 9 In. FINDING AN EXPERIMENTAL RELAXAnON FUNCTION 11 Pressure Vessel Design 11 Deflection Measurement 13 Miscellaneous Instrumentation 14 Data Acquisition 15 Experimental Procedure 16 Data Analysis 17 IV. ROLL MODEL 27 Computer Model 30 V. MODEL VERlFICAnON 33 Roll Testing 35 VI. CONCLUSIONS 43 VII. FUTURE WORK 44 REFERENCES 45 IV Figure LIST OF FIGURES Page 1. 2. 3. 4. 5. 6. Pressure Vessel 13 Pressure Vessel Layout 14 Strain Data for Core # I a. Strain Data for Core #1 at 100 psi 18 b. Strain Data for Core #1 at 150 psi 18 c. Strain Data for Core #1 at 200 psi 19 Average Strain Data for C re #1 20 Nonnalized Creep Data for Core #1 20 Creep Function Plot of Core #1 21 7. Strain Data for Core #2 a. Strain Data for Core #2 at 100 psi 22 b. Strain Data for Core #2 at 150 psi 23 c. Strain Data for Core #2 at 200 psi 23 8. Average Strain Data for Core #2 24 9. Normalized Creep Plot for Core #2 24 10. Creep Function Plot of Core #2 25 II. Flow Chart 32 12. Radial Pressure Through Winding 34 13. Core Pressure Through Winding 34 14. Die Set 37 v 15. Push Test Set up 37 16. Roll #4 Push Force 39 17. Role #4 Radial Pressure _ _ .40 18. Roll # 5 Push Force 41 19. Roll #5 Radial Pressure 42 NOMENCLATURE b web thickness Ee core modulus of elasticity Er radial modulus of elasticity E( tangential modulus of elasticity F punch force h incremental change in central difference approximation J core relaxation function Jo, J1, T1, J2, T2 coefficients in the relaxation function K' effective gage factor of strain gage N interlayer force P interlayer pressure R'A effective resistance of arm shunted (ohms) Real resistance ofcalibration resistor (ohms) r radius t time Tw winding tension stress w web width vii 8T incremental change in tension stress Ecal simulated strain ET radial strain E( tangential strain Vrt radial Poison's ratio Vtr tangential Poison's ratio crT radial stress crt tangential stress viii CHAPTER I INTRODUCTION Many materials used in the world today originate through a web handling process. Thin plastic films, paper, cotton fibers, and thin metals are manufactured in continuous strips called webs. For convenience, web material is wound on to a roll where it can be easily transported and stored. The web is wound on to cores made of materials such as: composite fiber, plastic, alwninum or steel. The core is a cylindrical hollow shell. It is hollow for ease of mounting the roll on to the appropriate web handling equipment. For economic reasons, the core used for a specific web has the minimum required properties, such as strength and weight. Industry desires a disposable core so the core will not have to be returned to the manufacturing plant. With these requirements, it is important to have a process for finding the best core for the specific web. In web handling processes, it is important to maintain roll integrity. Extreme stresses within the roll cause defects to occur, such as wrinkling or starring, thus yielding the damaged portion ofthe roll unusable. Also, inappropriate stresses can cause the roll to separate from the core. Ifthat occurs, problems arise in unwinding the roll, because tension is applied to the core during unwinding. This results in an unusable roll. To ensure roll integrity, an effort has been made to develop winding models which predict the stresses within the roll. Winding models incorporate web material properties, core properties and winding techniques. From the results of winding models, criteria can be generated to determine whether flaws will occur. These models reflect the dynamic effects of the winding I process. Very few models reflect how the roll will act with environmental or time effects. It has been noted that, at times, a roll has developed defects days after it has been wound even without a temperature change. This must be due to an effect that occurred after the dynamic winding response. Current models assume the core is an elastic material. It is often a viscoelastic material. A viscoelastic material is defined as a material that exhibits an elastic strain upon loading, then a slow and continuous in.crease in strain through time at a decreasing rate.[l] When the loading is removed, the inital elastic strain is immediatly recovered and then a slow continuous decrease in strain occurs. The slow continuous increase in strain under constant stress is called creep. This viscoelastic property of a paperboard core is explored in this paper in order to explain how flaws develop through time in a wound roll. The study begins with a development of a roll model that begins with the winding process and ends at a set time later. Typically, roll models consist of second order differential equations with two boundary conditions. One boundary condition is at the outside radius of the roll, which is influenced by the winding tension. The second boundary condition is at the rollcore interface in which a core defonnation dominates. An empirical function will be developed to model the rate of radial defonnation of the core due to roll stress. The roll model will be solved analytically using a finite difference approximation. Finally, experimental verification of the model is presented. 2 CHAPTER II Literature Review Elastic Core Model An elastic core model was developed to predict stresses and strains, for spiral paper tubes loaded axisymmetricaUy by Gerhardt, of Sonoco Product Co. [2]. This model did not assume the core to be isotropic or any stress distribution through the core wall. Paper, however, was assumed to be a linear elastic material. Experimentation verified the elastic core model. Some results are as followed. One, paperboard laminates exhibit nonlinear stressstrain behavior. Two, hoop stress dominates all other stress components and it was not constant across the tube wall. Three, a stress concentration occurred at the outer wall of the tube and remained constant as the tube thickness increased. Hakiel's Model The viscoelastic properties ofthe core affect the quality of a wound roll. Core creep, induced by stresses exerted by the wound roll, plays a major part in the quality of the wound roll. The wound roll stresses must be known. Hakie1 developed a nonlinear orthotropic hoop model for center wound roUs. [3] The roll stresses are calculated by solving a second order differential equation with two boundary conditions. Since the equation is nonlinear, numeric approximations are used. 3  The following assumptions are made in Hakiel's model: I. The winding roll is a geometrically perfect cylinder with the web having uniform width, thickness, and length. 2. The roll is a collection of concentric hoops. Winding is modeled by the addition of tensioned hoops. Roll properties remain constant. 3. The roll is an orthotropic, elastic cylinder with linearelastic behavior in the circumferential direction and nonlinearelastic behavior in the radial direction. The radial modulus of elasticity is known and varies as a function of radial stress. 4. The stresses within the roll are a function of radial position only. 5. The roll is under a plane stress condition and axial stresses are equal to zero. Hakiel uses three basic stress strain equations to support his model. 1. The equilibrium equation for plane stress in cylindrical coordinates in the absence of shear: acr r(_Orr)  a + a = 0 t r (2.1) 2. The linear orthotropic constitutive equations: 4 (2.2a) (2.2b) 3. The strain compatibility equation: r(&Ort) +E t ET =0 (2.3) The model begins by solving the equilibrium equation (2.1) for crt and substituting it into the constitutive equations (2.2a,b). crr vrt (fur ) Er =EE r;:+cr r r t VI and (2.4) Et ~ ;, (r a;' +cr,) (;:)cr, Then, substituting these two equations into the strain compatibility equation (2.3) Solving and utilizing Maxwell's relationship: yields: (2.6) Hakiel's second order differential equation is the governing equation in the winding model. To solve the equation, two boundary conditions are required. The [lIst boundary condition, at the core roll interface, is obtained by equating the radial deformation of the first wound on layer (2.7) and outside deformation of the core (2.8). (2.7) 5 (2.8) Where the 8 indicates incremental changes in the variable. Substituting the defonnation equations (2.7)(2.8) in the equilibrium equation (2.1) yields the first boundary condition. &a8cr r Ir=l =(EEl ) I e  1+ V 8cr r r=) (2.9) The second boundary condition, at the outside ofthe winding roll, is found by assuming the incremental winding on ofthe last lap is equal to the hoop stress of that lap. 0, I =(Twjr=s)b crrr=s S (2.10) With these two boundary conditions, a solution can be obtained for the Hakiel's governing differential equation of the elastic region of this model. Since the model is nonlinear, the solution must be found analytically. A finite difference method with a central difference approximation of the derivatives is employed to solve the governing differential equation. The central difference approximations are as follows: (2. 11 a) (2. 11 b) Substituting the approximations into the governing differential equation (2.6) and combing variables yields: 6 ( r 2 3rJ (. 1) (2r 2 1 Et ) (.) (r 2  cr 1 + +  cr 1 + +3rJ cr (,1+1) =0 h2 2h r h2 E T h2 2h r r (2.12) Where (i) is the current lap and h is the incremental step size or web thickness. Substituting the approximations into the dynamic bOWldary conditions (2.9)(2.10) and collecting terms they become: ocr,(i +1)[~ (h :: 1+ v)}cr,(i) =0 and (2.13) ocr r (i) =(T~)h With the governing differential equation written at all interior points (web laps) in the roll, a set of equations are formed for N2 variables. The two boundary conditions supply the remaining two required equations. Thus, the boundary value problem has been reduced into a simultaneous set of equations. When put in matrix form, they yield a tridiagonal system ofthe form: [A]{ocr r } =[B] (2.14) The system can be solved by a Gaussian elimination routine with N2 forward and Nl backward substitution. The incremental stresses are accumulated at each layer and added together to get the total stress at that layer. This process steps through the roll adding layer upon layer until the entire roll is analyzed. Thennal Analysis One recent study by Qualls [4] investigated the effect of a temperature change upon a roll. Qualls showed that an increase in temperature increased the stresses causing 7 defects within the roll. This was due to expansion or contraction ofboth the web material and the core. Qualls modified Hakiel's wound roll model to calculate the interlayer pressures in a wound roll which is subjected to a homogeneous temperature change. The model includes coefficients of thermal expansion ofthe web and core. The constitutive equations become: (2.15a) (2.15b) where a r is the radial coefficient of thermal expansion ofthe web at is the tangential coefficient of thermal expansion of the web t1T is th.e temperature change Solving the model in a similar method as Hakiel, the second order differential governing equation becomes: (2.16) The boundary conditions are modified. The core's coefficient of thermal expansion is added to the inner boundary condition. (2.17) where a c is the core coefficient of thermal expansion This yields an inner boundary condition of: 8 ra8a+r 8a ( 1vEI ) =E ( a a ) l!.T Or rEI c r c (2.18) Using the fInite difference approximations (eq. 2.11a) and solving, the boundary condition becomes: !:...oa (i+1)+(1!:...V E' )oa (i)=E(a a )l!.T h r h £' rIc r c (2.19) The outer boundary condition assumes a traction free outer roll surface and the stress is equal to zero. Now, with these two boundary conditions (2.12)(2.19) the governing differential equation can be solved. A tridiagonal set of simultaneous equations is produced that when solved yield incremental pressure changes due to step temperature changes. The set of equations is solved in a stepwise linear fashion. For each increment in temperature change, updated radial pressures and radial modulus are computed. Both radial pressure and modulus are pressure dependent. QuaIl's model steps through temperature change instead of through the roll as in Hakiel's model. QuaIl's thermoelastic model was studied because his approach is the basis for the approach in this paper. Whether the core deforms due to viscoelastic behavior or thermal expansion does not alter the solution procedure. Thus, the viscoelastic model will step through time as QuaIl's model stepped through temperature. Tube Testing Device A core testing vessel was developed by Salidis and Rowlands. [5] This vessel was used in a newly patented test method for measuring material properties tubular 9  samples. The method ensured that cores were tested to failure in compressive material crushing, not a structural buckling failure. The test vessel contained an annulus of 1.5 rnrn. ball bearings, that were compressed radially against the outside of a core sample. The bearings were loaded by hydraulic fluid contained in a sealed bladder. The sample, bearings, and bladder were housed in a cavity within the vessel. The cavity had a hole in the bottom for venting the interior ofthe sample. At the top a plug was used to hold the sample in place. This device allowed for external loading, while ensuring a uniform deformation of the core with the structure of the compressed ball bearings. Experimental stressstrain values were compared to theoretical values ofaluminum tubes. Tests on aluminum tubes, with strain gages mounted on the inside oftube, showed the experimental data following the theoretical values. Paperboard tubes were tested to failure in crushing. Strain gages were placed in the hoop direction on the inside of the tube. The tests were repeated on 5 different tubes that were geometrically identical, to show repeatability. The advantage of the test vessel Salidlis and Rowlands developed was a clean working environment. The hydraulic oil was contained in a sealed blatter. The vessel also allowed for variable size specimen with changing the size of the bearing annulus. One problem with the testing device was examining how the strain on the outer surface of the tube could be measured. The ball bearings would destroy any strain gage placed on the outside ofthe tube. If strain is desired on the outside of the tube another testing apparatus is needed. 10 CHAPTER III FINDING AN EXPERIMENTAL RELAXATION FUNCTION The creep function, Ie , is a function defining the defonnation of the core over time nonnalized by the pressure the roll is exerting on the core. The function is found experimentally. It is an exponential function that takes the fonn of a generalized Maxwell equation for relaxation. [1] J, ~ J o +J, ex~;:) +J 2 ex~ ;:) (26) It is found by plotting the strain, nonnalized by pressure, versus time and fitting the curve to this equation. An experimental apparatus was developed to simulate the wound roll stresses experienced by a core. The criteria of the apparatus is to apply stress on the outer surface of the core, while exerting no stress on the inside surface of the core. A pressure vessel was designed to exert radial pressure on the outside surface, while venting the inside of the core outside the vessel. Pressure Vesse! Design The pressure vessel is designed to meet general requirements outside of this project. The pressure vessel is required to withstand a maximum pressure of2000 psi. at room temperature. The core must be held inside the vessel. The vessel must house instrumentation or allow for leads for external instrumentation and output from internal 11 instrumentation. The inside ofthe core must be vented to atmospheric pressure to simulate an actual wound role. The pressure vessel design selected was a roood cylinder capped on both ends. This minimizes stress concentrations. A pipe was used to construct the pressure vessel, with a welded cap on one end and a flange with a blind on the other acting as a porthole. Sizing the pipe to meet the required 2000 psi. maximum pressure was accomplished using the following formula that estimates the wall thickness. [6] ~= PD +A 2(SE +PyJ Where 1m is the minimwn wall thickness P is the maximum internal working pressure D is the outside diameter ofthe pipe SE is the maximum allowable stress y is values ofjoint efficiency A is the mechanical behavior allowance (3.1) Given a threeinch inside diameter of the core with a maximum outside diameter of four inches. The equation parameters are: [6] P=2000 psi. D=6.625" SE=15000 Ib/in2 y=.4 A=.065 These parameters are for a 6 inch, A106 grade B pipe. This yields required a wall thickness of .4843 inches and a schedule 120 pipe. The flange and cap were designed by the manufacturer for a working pressure. The flange is stamped with the maximum working pressure. A sixinch series 900 flange with a blind flange has a working pressure of 2200 psi. to 800°F. A sixinch extra heavy 12 welded cap has a working pressure of 2100 psi. A drawing of the pressure vessel is shown in figure 1. Pictures are shown in Appendix A figure 1 and figure 2. 6" Series 900 Welded Extraheavy 6" Pipe sch. 120 6" Series 900 Blind Figure 1. Pressure Vesse} A hollow shaft through the blind flange provides mounting for the core and venting of the inside ofthe core. The shaft is secured to the blind flange by a jam nut and sealed with an aring and washer. The core is sealed on the shaft by caps that are screwed onto the shaft and sealed with arings. The pressure vessel was welded by a certified welder and met all codes and state requirements. See figure 2 for a layout of the inside of the pressure vessel. Deflection Measurement Strain was measured by a small foil patch strain gage. Two problems in using the strain gage on the paperboard core could arise because paper that makes up the core tends to absorb the glue. One, the gage is less likely to adhere to the core. Two, the glue 13 absorbed by the core may locally change the characteristics of the core. Both problems may give erroneous output. The glue chosen was an epoxy product from Measurements Group. It was thick enough that negligible amounts of epoxy was absorbed by the core. A picture of a gage installed upon a core is shown in Appendix A Figure 3. Blind Flange Jam Nut Sealing Washer with oring Figure 2. Pressure vessel layout Miscellaneous Instrumentation A bulkheadfeed through is required to route the input and output lead wires, for the strain gage, from the inside of the pressure vessel to the outside. There is a maximum of twelve wires that go through the vessel.. The bulkheadfeed through consists of a wire cluster encapsulated in epoxy. This is placed directly into the blind flange. The pressure vessel uses compressed nitrogen from bottles supplied from a local distributor. The nitrogen bottles had a capacity of 40 cubic feet at 2500 psi. To get a range of pressure inside the vessel, a Vicors pressure regulator, rated to 250 psi, was used. An analog pressure gage, mounted on the blind flange, permitted an instantaneous pressure reading. 14 Data Acquisition The strain gages, Measurements Group model CEA06250UR350, are placed in quarter bridge circuits with outputs resistance changes converted into strain by a Measurements Group 2103a strain indicator. The strain indicator is calibrated to read in microstrain. The calibration equation is as follows. _ R'A 106 Il£cal  K'(R +R' ) cal A (3.2) [7] Thus, a required 10,000 microstrain resistance, R.:al equals 174.3 Kiloohms. A cantilever beam was set up to check the calibration. The theoretical strain equation is as follows: 3Dy E= L2 (3.3) [8] A strain gage was mounted on a beam .5 inches wide (D), .125 inches thick (y) and 10 inches long (L). The resulting strain was calculated 1875 J..u:. When the strain gage was connected to the strain indicator with the appropriate calibration factor, the strain indicator read 1876 !LE. This verified the calibration setting of the strain indicator. An Omega PX931KSV pressure transducer, rated at 01000 psi with .1% drift, was used to measure the pressure inside the vessel. This was calibrated with a dead weight tester. An IBM clone personal computer, P120, is used for data acquisition. LabVIEW, a graphical program for instrumentation, and an ATMIO16XE50 I/O board from National Instruments, is used to store pressure and strain output. 15 Experimental Procedure Two different sets ofcores of the same material with a 3inch diameter and 0.29inch wall thickness were provided by Sonoco. The estimated crush pressure was 400 psi and the modulus 70,000 psi. The two different sets of cores are similar in property values, but not identical. To avoid end effects, measurements were made 7 inches from the end ofthe core. Radial deformation is measured at the center of the length of the core. Thus, the core was cut to a length of 14 inches. Severa] strain gages were mounted around the core at the center of the length, with Measurement Group MBond GA2 epoxy. At times, cores do not deform in a uniform manner, thus a number ofgages were used to get an average deformation. The end caps were press fitted into the ends of the core to insure a good seal at the ends. The core was then wrapped with a 3M Poly tape to seal the outside of the core. With the paperboard cores, there was a tendency for the compressed nitrogen to seep through the core. The nitrogen would change the moisture content of the core, changing the properties. Thus, the outer tape seal was required. After wiring the strain gages to the bulkhead feed through and connected to the strain indicator, the gages were given time to equalize. The strain indicator was zeroed and calibrated and the acquisition program was initiated. Pressure is applied to the vessel slowly and recording ofthe pressure and strain was made in five second intervals. This gives a core modulus of elasticity reading for each gage. After the maximum pressure for the experiment is achieved, the pressure and strain recording is changed to 120 seconds. The experiment runs for 48 hours, or until the strain has stabilized. 16 Data Analysis In order to achieve an accurate creep function over the pressure range of the core, three different pressures were tested, 100 psi, 150 psi, 200 psi. In preliminary examination ofthe testing procedure, it was found that testing at above 50% of crush pressure, a buckling phenomenon may have occurred. Thus, the maximum 200 psi test pressure is 50% ofthe 400 psi crush pressure rating of the cores. Pressure, time, and strain from all gages were recorded at each ofthe given pressures. The pressure and strain data taken during the initial start up, through the elastic range, was used to calculate the elastic modulus. An elastic modulus of 80,000 psi, ±1O,000 psi reading on each strain gage, was used as an assessment of the accuracy of the gage reading. If the elastic modulus was not within the 80,000 psi ±10,000 psi range it was removed from the analysis. The most likely cause of faulty gauge data is improper attachment ofthe gauge or a flaw in the core. Core analysis from the first set of cores are shown first. The data for the second set of cores will be analyzed in the same manner, following the first set. The plots in Figure 3 show the strain data at the three set pressures for the first set of cores. 17 Strain Through Time at 100 psi 0.000 1000.000 400 600 800 1000 1200 1400 C 2000.000 __Strain 2 '2; 3000.000 __Strain 3 0.. u 4000.000 __Sttain4 ! r:: 5000.000 __StrainS i.i  6000.000 '" 7000.000 8000.000 Time (min) Figure 3a. Strain Data for Core #1 at 100 psi. Strain Through TIme at 150 psi 1000 0  1000 200 400 600 600 1000 1200 1400 r:: 2000 C ~ ; 3000 0 _Gage 3 .. (,l 4000 g 5000 __Gage4 I: i.i 6000 en 7000 . 8000 9000 TIme (min) Figure 3b. Strain Data for Core #1 at 150 psi. 18 Strain Through TIme at 200 psi 0.000 2000.000 200 400 600 800 1000 1200 1400 1 :E 4000.000 :5 __strain1 0 6000.000 __strain 2 ..u I 8000.000 __strain 3 r:: i.s. 10000.000 U) 12000.000 _ 14000.000 Time (min) Figure 3c. Strain Data for Core # I at 200 psi. The plots in figure 3 show a variation in strain between individual gages within a given pressure. This is due to the core not deforming uniformly because of inconsistency within the core. Therefore, the strain from each gauge is averaged to get an average defonnation across the entire core. The elastic strain was then subtracted in order to isolate the strain caused by creep. Figure 4 shows the average strain. 19 Average Strain with Elastic Strain Removed +100 psi e150 psi .!r 200 psi 400 600 800 1000 1200 1400 1 0 1000 . ..., 0 .....+++++++_1 1000 2000 3000 4000 5000. 6000 7000 8000 9000 ·10000 ..1.... ' Time (min) Figure 4. Average Strain Data for Core #1 The family of curves produced by these tests do not follow the same path. A function was needed to associate the curves so that they may be modeled as a Maxwell modeL Normalizing the strain curves, by dividing by their associate pressures, the normalized curves can be fit with the Maxwell model. This is shown in figure 5. Normalized Creep Data 10 .: 0 I '2 400 600 800 100:> 1200 1400 "T 10 eu g 20 +100 psi c:= e 150 psi ." I! Co 30 +0200 psi iii 'C G 40 .!:! 'jij ~ 50 0 z 60 Time (min) Figure 5. Normalized Creep Plot for Core #1 20 The lines of normalized strain, the strain divided by the corresponding pressure, follow similarpaths. They were represented by a single creep function, derived from the generalized Maxwell creep function discussed earlier. The normalized strain is averaged and a creep function is estimated. The error introduced into the analysis by averaging the normalized strain and developing a creep function was insignificant, due to the magnitude ofthe strains and the size of the deviation of the normalized strains. This is reinforced in the testing results in Chapter V. The coefficients to the creep function were found by utilizing an EXCEL worksheet solver function. Within the worksheet, the creep function is compared to the data at each time point. The difference is then summed and minimized by changing the function coefficients, thus getting an appropriate creep function. The EXCEL Solver function yields the following equation. ti~ ti~ J =60.06 + 17.82e 2213 1 + 4224e 865.50 (microin/in / psi) (3.4) A plot of the creep function verses the normalized creep data is shown in Figure 6. Nonnalized Qeep Data with Relaxation Function 10 c: 0 ] 400 WJ 800 10CXJ 1200 1400 b.. 10 .100psi uE 20 ~15Opsi c:::: ~ III tr 2OO psi  Co 30 en ir Qeep fl.l'"COOn "C CD 40 ~ n; E 50 0 z 00 TIme (min) Figure 6. Creep Function Plot of Core #1 21 The Maxwell Model follows the average strain data. The average nonnalized creep data seems to be diverging as time progresses past the time shown. Thus, the Maxwell creep function is only applicable through this time period. The analysis on the second set of core data follows in the same manner as the first set. The strain data for the second set of cores are seen in Figure 7. Core #2 Strain Through Time At 100 psi סס. 0 OO סס. 1000 OO 200 400 600 800 1000 1200 1400 1 C 2000. סס OO ~ .(.,. סס. 3000 OO (J I 4000. סס OO c __Straingage 2 ~... סס. 5000 OO Cf) __Straingage 3 סס. 6000 OO __Straingage 4 סס. 7000 OO Time (min) Figure 7a. Strain Data for Core #2 at 100 psi. 22 Core #2 Strain Through Time at 150 psi 0.00Et00 1.00803 200 400 600 800 1000 1200 1400 C 2.ooE+03 :§ __Slrain1 0 ... 3.ooE+OO __Strain 2 u 4.00803 __Strain 3 gc 5.ooE+OO __Strain4 ·.i.i  6.ooE+OO en 7.ooE+OO 8.ooE+OO Time (min) Figure 7b. Strain Data for Core #2 at 150 psi. Core #2 Strain Through Time at 200 psi סס. 0 OO סס. 1000 OO 200 400 600 BOO 1000 1200 1400 סס. 2000 OO ' סס. 2 3000 OO __Strain 1 '2 סס. 4000 OO __Strain 2 "T 0... סס. 5000 OO __ Strain 4 ug סס. 6000 OO c~ סס. 7000 OO Ui 8000. סס OO סס. 9000 OO סס 1 oo. סס oo Time (min) Figure 7c. Strain Data for Core #2 at 200 psi. The strain data is averaged with respect to its pressure and shown in Figure 8. 23 Averag,e Strain with Elastic Strain Removed 0 :5 1000 I,.: .0.. 2000 lJ I c:: 3000 g en 4000 GI Cl ~ Gl 5000 . >< 6000 +100 psi ~150psi 6200 psi 600 BOO Time (min) 1000 1200 1400 1 0 Figure 8. Average Strain Data for Core #2 The average strain is nonnalized by dividing by the test pressure and is shown in Figure 9. Nonnalized creep Data ~100psi er 150 psi fr200psi e Oeep Furctl:ln 0 5 c:: ,'2. 0 10 .. lJ §. 15 c::= Ill t!o. U) 20 't:I Gl !:! 25 'jij E 30 0z 35 400 TIme (min) 100J 1200 1400 Figure 9. Nonnalized Creep Plot for Core #2 24 Using the EXCEL equation solver in the same manner as for core #1 with the average of the normalized creep data, the general Maxwell creep function is obtained. The equation is : time time J =5.96+.0892 * e 122.86 + 5951 * e 1917.88 (microin/in I psi) (3.5) A plot of the creep fimction verses the normalized creep data is shown in Figure 10. Nonnalized Q'eep Data +100p;i a150 p;i Ir 200 p;i &Qeep FurcIi::r1 0 5 .E C "T .0.. 10 u :§. 15 1::::: Ill ~Q. U) 20 j .!=! 25 iii E 0 30 z 35 400 Time (min) 1cxx) 1200 1400 Figure 10. Creep Function Plot of Core #2 The Maxwell creep function fits the average nonnalized data. However, the normalized average data diverge as the time continues. This function is only representative ofthe normalized data at shorter time intervals. 25 In a roll, the core will experience a pressure change as the core defonns. All of these tests are executed at constant pressures. But, with the Maxwell creep function modeled through a pressure range, a pressure may be calculated given the defonnation at a given pressure within the range. A mathematical model will be developed to accomplish this job. 26 CHAPTER IV ROLL MODEL The core is influenced by pressure due to the wound roll above it. This pressure does not remain constant. As the core defonns under the pressure of the roll through time, the roll pressure decreases. Thus, a model is needed to predict the defonnation of the core and the resulting pressure change through time. The model begins with a wound roll on a core. The initial pressures with in the wound roll are found by using Hakiel's Model, discussed in Chapter II. The time varying model is developed just as Hakiel's Model. Hakiel's model applies to the wound roll in that the governing differential equation (2.12) does not change. [3] (2.12) and with the central difference approximations included the equation (2.18) is: (~ ~J(J (i 1) + (2r 2 + 1 .s.Jcr (i) +(.c+ ~Jcr (i + 1) =0 h 2 2h T h 2 E T h2 2h r r (2.18) The boundary conditions are where this mod.el and Hakiel's Model differ. The boundary condition at the coreroll interface is calculated by assuming the deflection of the core is equal to the deflection ofthe first layer ofthe roll. Umat'l =Ucore divide both sides by radius ofthe core 27 (4.1) Ematl ) = Ecore (4.2) First, we will focus upon the strain in the web materiaL Starting with the equilibrium equation (2.7) and substituting it into the constitutive equation (2.8b) yields: Applying the central difference approximations for fIrst derivative, fur crJ.j(i+I)crJ.j(i) = Or h where j denotes the stress at the current point in time yields: rearranging, we have: (4.3) (2.17a) (4.4) (4.5) Since roll winding is an accretive process and the pressures which are computed affect the radial modulus, Er, the second order differential equation in radial pressure is solved several times for differences in pressure which are summed to yield total pressures in each layer. So the previous expression is typically cast in the form: ( r I) (.) (1 r 1 v tr) (.) C' . =   l)cr . 1+ I +       l)cr . 1 ~ee,matl hE r,J E h E E r,J t t t r (4.6) 28 Now, let us focus upon the strain in the core. The strain at the core is defined as: Ecore  JJc(tt ')ddadtr tI The integral will be approximated at each step in time as: Ecore,Llt =J c (t j  tjl )c;r,jl (i) (4.7) (Ref. 9) (4.8) To obtain the total core strain at any point in time, the strains at each time step must be summed as: n=j Ecore,totaJ = LJc(tj tnl)O"m(i) (4.9) n=l Equation (4.9) yields the viscoelastic strain at a given time increment. Elastic strain of the core (4.10) must also be added to the equation. 0" . (i) r,J E core,elastic = E c (4.1 0) Now, assuming the generalized Maxwell form for the relaxation function from equation (2.6) yields: (4.11) Now, equating the strain in the core and in the web material yields: (4.12) 29 The outer boundary condition is calculated assuming a traction free outer roll surface. (4.13) With these two boundary conditions, the governing differential equation can be solved for the static, time dependent region of the model. With the boundary conditions (4.12), (4.13), the governing differential equation can be solved. The set oftridiagonal simultaneous equations are set up and solved in the same manner as the QuaIl's model described in Chapter II, with the exception of stepping through time instead of through temperature. The set of equations are solved in a stepwise linear fashion through time that yields increments in pressure decay. Then, the radial pressures and radial modulus are updated each time step. The final solution yields a pressure profile of the roll through time. Computer Model The finite difference method of solving differential equations can be readily solved on a personal computer. A FORTRAN code was developed to solve the model and output the solution. The algorithm flow chart is shown in figure 11. A listing ofthe FORTRAN code is in Appendix B. The program requires the user to input some system requirements and material properties. The system requirements are: winding tension, outside radius ofthe core, outside radius ofthe roll, number of increments through the roll, total time, and number 30 of increments through time. The material properties required are: radial stiffness and Poisson's ratio of the core, the third order polynomial for the radial modulus ofthe web, tangential modulus of the web, and the directional Poisson's ratio's ofthe roll. The program outputs five columns ofradial position, radial stress, and incremental stress change. The output format lends itself to its use in a spreadsheet software, such as Microsoft EXCEL. In EXCEL, one can plot the radial stress versus time. 31 Calculate incremental radial stress at current roll lap solve bound Solve tridiagonal matrix using Gaussian Elimination Update pressure through out all laps rolled No! Calculate radial stress in the roll through time step Solve tridiagonal matrix using ,Guassian elimination Update pressure through out entire roll No! Figure 11. Flow chart. 32 CHAPTER V MODEL VERIFICAnON The roll model calculates the radial stresses due to the combination of the winding of the roll and the defonnation of the core through time. It begins at time zero and ends at a specified time. The model was developed previously in this paper. The FORTRAN code was used with the given core properties and the properties of a polyester film. The properties are as follows. Roll ill = 1.75 in Roll aD = 5.75 in Roll Iterations = 1000 Winding Tension = 2000 psi ICI "377" 200 gauge film Er = 54.97"'P.07819*P2 +.0001388*P3 (psi) Et = 600000 (psi) Core E = 80000 (psi) Creep Coefficients (microin/in / psi) Jo = 60.059 11 = 17.8]7 T1 =221.31 12 = 42.242 T2 = 865.50 The film Er property equation comes from a stack test. A stack of loose web was compressed with the nonnal pressure and strain recorded. The slope of this data was used to generate this Er third order polynomial as a function ofpressure. The film Et property is established in a tensile inplane stress verses strain test. The roll pressure builds very quickly during winding. Figures 12 and 13 show how the radial pressure develops in the roll and how the pressure varies during winding, 33 respectively. Note, the core pressure has nearly reached its maximum value after eleven minutes from a total winding time of 70 minutes. Radial Pressures 0 WT = 2.5 min o WT= 5.3 min Through Winding 200 l! WT = 11.9 min L;jjjiiiil",. ~iiiiii~~~:I11x WT = 29.1 min ,.... 150 t"""kf..+~+~,,I___'~_1_____l x WT = 51.5 min .[ 0 WT = 69 min ....... ~ 100 f''lk+""+__+'llfr+j~_+__"Q..__t___l :::l CI.l CI.l J: 50 , &'i\ctt~+___+__+___'l~_l O+~____i3I+__+~i i____l:~__! 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25 5.75 Winding Radius (in) Figure 12 Radial Pressure Through Winding Core Pressure Through Winding ..200 l:O 0.. 'u' 150 10 :::J ~ 100 u 10 P4 50 u 10 8 0 ~ ( II I o 20 40 Winding time (min) 60 80 Figure 13. Core Pressure Through Winding 34 Roll Testing Six wound rolls ofIeI 377200 film were made in accordance with the previous parameters. The first three rolls were tested by instrumenting the core with strain gauges and measuring the core strain over time. The results were found to be widely varying. This was caused by the way the strain was read. The gauges were mounted on the core and connected to the strain indicators. The gauges were unable to be connected to the indicator during winding of the roll, so they were disconnected. After winding, the gauges were reconnected. Connection ofthe gauges consisted of twisting wires together which, if connection changes occur, can change the resistance of the lead wires. The theoretical resistance change within the system using the gauge factor, was 1 to 2 ohms. This was seen in leadwire resistance change in connections alone. Thus, this method of testing was inconclusive and a new method had to be developed. Note, this problem had no impact on the pressure chamber testing. The connections were not disconnected during the test and the leadwire resistance did not change. Instrumentation was zeroed before testing began, thus taking the lead wire resistance into account. A new test method was derived from the same method used by Hakiel in his study of wound roll stresses. [3] The interlayer pressures were calculated by forcing two layers of the web, close to the core, to slip upon one another. The layers of web to be tested are not in direct contact with the core. If these layers were to be used, problems may arise. The friction coefficient between the core and web is not known and difficult to calculate. Sliding the web from a compressed core section over an uncompressed section would 35 lead to problems with binding. Finally, shearing of the tape bonding ofthe web to the core, required at the start ofthe winding process, would require an unknown force. In order to eliminate these problems, layers of web removed from the core are used in the test. By knowing the web to web static coefficient of friction, and reading the force required to slide the web, the interlayer pressure can be found. (5.1) The interlayer pressure (P) is found by the interlayer force divided by the surface area. P=~ 2mw (5.2) The force required, to cause the web layers to slip, was provided by a material testing system, INSTRON model 8502. Figure 15 shows the test setup. Dies were created to ensure the repeatability of the correct web layers sliding on each test. The bottom male die has a running fit to the outside diameter of the core. It has an outside diameter of 1.95". The top female die has a loose fit to the outside diameter ofthe male die. This is so that the web will be pushed up by the male die into the bottom die and so that the correct web layers slide in each of the tests perfonned' Figure 14 shows the die set. 36 Female die ~L:: ....;...d Wound roll Male dipe......I~:~__;.:I Figure 14. Die Set Female die Male die Wound roll ~ INSTRON Figure 15. Push Test Set Up Each roll is tested over a 1500 minute time interval. The roll was placed in the INSTRON press with the two dies placed as in previous figure. The INSTRON press was set to move 0.01" per second. The force was recorded along with the displacement. When the required web layer begins to slip past each other, the pressure levels off This 37  was the push force required. The force and displacement were recorded on a PC and the slip pressure was read from the record. RESULTS Rolls #1, #2, and #3 were tested under the strain acquisition procedure in which data received was deemed unusable. Roll #1 Core type #1 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 11.5" ill: 3.5625" Winding Tension: 1700 psi Winding Speed: 50 ft/min Note, the coefficient of static friction was found by employing tests on web material in the same fashion as described by Ducotey. [10] Roll #2 Core type #1 Web: ICI377200 gauge Coefficient ofFriction: .21 aD: 11.625" ill: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ft/min Roll #3 Core type #1 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 10.625" ill: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ft/min 38 Pictures ofRoll #2 and Roll #3 are in Appendix A, figure 4 and figure 5, respectively. Roll #4 was the first roll to undergo this push test through a step time interval. The winding parameters are the same as in the model with the exception of the following: Roll #4 Core type #1 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 10.875" ill: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ftlmin Roll #4 Push Force at 1.95" 3000 2500 ~~ :c 2000 't:l 1500 Cll 0 ..J 1000 500 0 0 200 400 e Theoretical & Theoretical at 1st layer 600 800 1000 1200 1400 1600 Time (min) Figure 16. Roll #4 Push Force In figure 16, the experimental push force is plotted along with the values predicted from the computer model. Also, the plot displays the predicted force to push the first two layers by one another. This shows the drop in pressure at one layer past the core and an indication of the core pressure. 39 The Roll #4 push force data through time follow the model well. At the first few time intervals, however, the data diverge. This may be due to the layout of the roll and the core type. The outside ofthe roll is rough and sticks out of the roll several inches. During the first few tests, the die did not slide smoothly due to the snug fit of the die against the core. After the sides ofthe core had worn a little, the data seem to correlate better. Notice that after 1500 minutes, the pressure near the core drops to zero. Then the core will no longer deform viscoelastically. A plot of the radial pressure of the roll at time = 0 and time = 1500 minutes is shown in figure 17. Roll #4 Radial Pressure ___ TIme = 0 min ~Tlme =1500 min 200.000 180.000 160.000 140.000 iii ~:..; 100.000 III VI BO.OOO .. a 60.000 40.000 .. 20.000 0.000 1.76 2.25 2.75 3.25 3.75 RJldlua (In) 4.25 4.75 5.25 5.75 Figure 17. Roll #4 Radial Pressure The figure shows that the model is accurately predicting the later time intervals and need not be estimated further. Roll #4 developed a starring flaw at the core roll interface during the winding, due to the deformation of the core and the pressure change of the roll. This flaw is shown in a picture in the Appendix A figure 6. 40 Roll #5 was the next roll to undergo this push test. The winding parameters are the same as in the model with the exception ofthe following: Roll #5 Core type #2 Web: ICI377200 gauge Coefficient of Friction: .21 aD: 11.75" rD: 3.5625" Winding Tension: 2000 psi Winding Speed: 50 ftJmin A picture of roll #5 is found in Appendix A figure 7. Roll #5 Push Force at 1.95" ~Experirrental a Theoretical & Theoretical at 1st layer 1000.00 1500.00 2000.00 Time (min) 500.00 o+ ~_< 0.00 Figure 18. Roll #5 Push Force. This plot is set up the same as the previous plot for Roll #4. They are similar. Roll #5 correlates better at first. The core type #2 is smooth and stuck out of the roll by only a half an inch. The data seem to diverge at the end of the test period. This is due to the model creep function which did not fit the normalized averages of the core data at the longer time periods. A plot of Roll #5 radial pressure through time is shown in figure 19. 41 RoU.S Radial Pressure __Tlme =0 min __Tlme =1500 min 200.000 180.000 160.000 140.000 'ii 120.000 ~ ..!".. 100.000 ! 80.000 A 60.000 40.000 20.000 0.000 1.75 2.25 2.75 3.25 3.75 Radin/min) 4.25 4.75 5.25 5.75 Figure 19. Roll #5 Radial Pressure. 42  CHAPTER VI CONCLUSIONS This study has shown how the viscoelastic properties of the core effect a wound rolL The viscoelastic core properties are represented by a relaxation function. The relaxation function is found experimentally by submitting cores to a series of constant stress tests over time. The strain is recorded and nonnalized by dividing by the test pressure. A relaxation function is fitted to the average of the normalized strain. The function can then be used in the developed mathematical model to predict wound roD behavior. Model verification tests show that the model can accurately predict the stresses through the roll. Viscoelasticity of the core will cause the core pressure to decrease over time and finally decay to zero. This is a localized phenomenon and does not affect the entire roll. The pressure decay at the core is, however, significant. It can cause staring to occur, as seen in a picture in Appendix A figure 6. The verification tests show that an accurate creep function is needed. In core type 2, the creep function does not accurately predict the strain data at the later test times. This caused a divergence in the accuracy ofthe model. This model is a tangible way to predict the pressure decay within a wound roll in the vicinity of the core. The web handling industry will find this research useful. 43  CHAPTER VII FUTURE WORK This model was developed with the assumption that viscoelastic properties ofthe core were not affected during the winding of a roll. A new model can be developed to include the creep function during the winding process. There will be situations were the pressure would drop appreciably during winding. This study was conducted under conditions in which temperature and humidity were held constant. In industry, wound rolls are submitted to environmental changes which may affect the core. The environmental conditions need to be considered. The two cores shown in this study were similar but not the same. Their relaxation functions were different. A correlation can be developed to relate different core types. 44  REFERENCES 1. Findley, William N., Lai, James S., and Onaran, Kasif, Creep and Relaxation of Nonlinear Viscoelastic Materials, North Holland Publishing Company, New York, 1976. 2. Gerhardt, T.D., "External Pressure Loading of Spiral Paper Tubes: Theory and Experiment". Sonoco Products Company, Madison, WI. 3. Hakiel, Z., ''Nonlinear Model for Wound Roll Stresses", Tappi Journal, May 1987. Pg. 113117. 4. Qualls, William R & Good, 1. Keith, "Thermal Analysis of a Wound Roll", Accepted by Journal of Applied Mechanics, January 1997. 5. Salikis, Edmond P. & Rowlands, Robert E., "A Novel Radial Compression Testing Device for Tubes", Tappi Journal, January 1997. Pg. 234237. 6. ASME B3I.31996 Edition, Process Piping, ASME Code for Pressure Piping, B3l an American National Standard, The American Society ofMechanical Engineers, 1996. Pg. 304.1.1304.1.2, Table AI, AIA. 7. Measurements Group, "Strain Gage Conditioner and Amplifier System Instruction Manual", Raleigh, North Carolina, 1992. 8. Shigley, Joseph E., & Mischke, Charles R, Mechanical Engineering Design, 5th ed., McGrawHill, New York, 1989. 9. Qualls, William R, Hygrothermomechanical Characterization of Viscoelastic Centerwound Rolls, Ph.D. Dissertation, Oklahoma State University, May 1995. 10. Ducotey, Keith S. Traction Between Webs and Rollers in Web Handling Applications, Ph.D. Dissertation, Oklahoma State University, May 1993. 45 APPENDIX A PICTURES 46  Figure AI, Fixture Setup #1. 47  Figure A2, Fixture Setup #2. 48 Figure A3, Strain Gage Application. Figure A4, Roll #2. 49 \ Roll #2 Core Type 1 Figure A5, Roll #3. Figure A6 Roll # 4. 50 Roll # 3 Core Type 1 o yp 1 ,. Figure A6 Roll # 5. 51 Roll #5 Core Type 2 APPENDIXB COMPUTER CODE 52  ********************.****************************•••***•••**••***** ****************************••********************.**••••••••****.* **.*********************•••••***••••***••*****.***••*******••**.*** ***** ***** ***** ***** ***** ***** ***** ***** ***** CORE RELAXAnON 1st ed. ***** Written By: Jeff Henning ***** Project Coordinator: Dr. 1. Keith Good *.*** ***** ***** ***** ***** ***** Web Handling Research Laboratory (WHRC) Oklahoma State University ***** Department ofMechanical and Aerospace Engineering Stillwater, Oklahoma ***** ***** ***** ***** ***** Calculates Stresses Induced Due to the ***** Relaxation ofthe Core in Time ***** ***** ***** ***** ***************************************.*************************** ***********************.****.************************************** ******************************************************************* ** ***** Main Program Corerelax ***** opens and closes input/output files ***** make calls to subroutine ** ********************.***********.*.******************************** PROGRAM CORERELAX OPEN(1 ,FILE='CORECRP.IN') OPEN(2,FILE='CORECRP.OUT') OPEN(4,FILE='COREPRS.OUT') WRJTE(*,*)'***.*************************************************' WRJTE(*,*)'ENTER "1" FOR FILE INPUT OR "0" FOR KEYBOARD INPUT' READ(*,·)IANS WRITE(*,*)'***.*****************.*******************************' IF(IANS.EQ.O) THEN CALL UINPUTS ELSE CALL INPUTS ENDIF WRITE(4,120) WRITE(*,*)'*****Calculating*****' CALL WINDER CALL RELAX WRITE(*,*)'*****Finished*****' WRITE(*,*)'******.*.*******.************************************' 53 C CALL OUTS 120 FORMATC TIME CLOSE(2) CLOSE(4) STOP END PRESSURE DELTA B(l)') ******************************************************************* ******************************************************************* ** ***** Subroutine RELAX ******************************************************************* SUBROUTINE RELAX IMPLICIT REAL*8 (AH,J,OZ) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKIDCLOCK,U(lOOO),DTP(lOOO),NCLOCK,JO,Jl,Tl ,J2,T2 COMMON/MATLPROP/EC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMONffIMEfR(1000),TIME,TWIND,LAP,DTOLD,DTIME COMMON/MATCOEFF/A(1000),B(1000),C(1000),D(lOOO),N COMMONIPRESIP(1OOO),DP(l OOO),S,OLDP,W(1000) C WRITE(*,*)'THSTRESS IN OK' C CCCCC C R(I)=RIN LAP=1 H=(ROUTRIN)/NLAPS DO I=2,NLAPS+1 R(I)=R(1)+(11)*H END DO C CCCCC DO L=1,NLAPS VP(L)=P(L) END DO C DCLOCK=DCLOCKINCLOCK DO 999 K=l,NCLOCK TIME=K*DCLOCK WRITE(2,1OO)TIME C C cccccccccccccccccccccccccccccccccccc CCCC THIS IS STRESS FORMULATION WITH EC C CCC CORE BOUNDARY CONDITION 54  C D(1)=(IDOvrtEt/(Ec*R1N)RINIH) C D(l )=(ldOEtlEr(1 )*vtrEt/(Ec*rin» C(l)=rin/h S=P(1) OLDP=P(l) B(l)=Et*S*(JO+Jl *EXP(TIMErrl)+J2*EXP(TIMErr2» C CCC ROLL ANALISIS C DO 60 1=2, NLAPS vtr=vrt*Er(I)/Et A(Il)=(R(I)**2fH**2R(I)/(2DO*H)*(3dOEtlEr(I)*vtr+vrt» D(I)=(1DO2DO*R(I)**2fH**2+vrtEtlEr(I)*(1+vtr» C(I)=R(I)**2fH**2+R(I)/(2DO*H)*(3dOEt/Er(I)*vtr+vrt) C A(Il )=(R(I)**2fH**23DO*R(I)/(2DO*H» C D(I)=(l DO2DO*R(I)"'*2fH**2EtlEr(I» C C(I)=R(I)**21H**2+3DO*R(I)/(2DO*H) C B(I)=ODO C 60 CONTINUE C CCC OUTER BOUNDARY CONDITION C D(NLAPS+1)=lDO B(NLAPS+I)=ODO A(NLAPS)=ODO C C CCC SOLVE MATRIX FOR PRESSURES AND PRINT C CALL SOLVETRl(DTP,NLAPS+1) LAP=NLAPS+I CALL VTOTPRESS(DTP) CALL OUTS C CALL STRAIN 999 CONTINUE C 100 FORMAT('TIME =',F7.0,' (MIN)') C WRITE(*,*)'THSTRESS OUT OK' RETURN END **************************.**************************************** 55  *****Subroutine Strain *****Solves the strain at the first layer of roll ***** which is equal to the strain of the core ******************************************************************* SUBROUTINE STRAIN IMJ>LICIT REAL*8 (AH,J,OZ) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKIDCLOCK,U(1000),DTP(1000),NCLOCK,JO,Jl,Tl,J2,T2 COMMONIMATLPROP/EC,E,vc,ET,AA(0:3),ER(1000),vrt,vtr COMMON/TIME/R(1000),TIME,TWIND,LAP,DTOLD,DTIME COMMONIMATCOEFFIA(l OOO),B(1 OOO),C(1OOO),D(1 OOO),N COMMON/PRESIP(1OOO),DP(1 OOO),S,OLDP,VP(1000) COMMON/STRNNSTRN,ESTRN,TSTRN C WRITE(*,*)'STRAIN IN OK' C VSTRN=VSTRN+«J0+J1*EXP(TIMEffl)+J2*EXP(TIME/T2)*OLDP) C ' *1000000) VSTRN=VSTRN+B(I) ESTRN=(P(1)/EC)* 1000000 TSTRN={VST~+ESTRN) C WRITE(3,40) C WRITE(3,50)TIME,TSTRN,VSTRN,EST~ 50 FORMAT(F1O.2,' ',F15A,' ',FI5A,' ',FlOA) RETURN END ******************************************************************* ******************************************************************* ***** Subroutine SOLVETRI ***** SOLVES THE TRIDIAGONAL SYSTEM OF DIMENSION 1DIM ***** FOR THE SOLUTION VECTOR X(IDIM) ******************************************************************* SUBROUTINE SOLVETRI(X,IDIM) IMJ>LICIT REAL*8 (AH,J,OZ) INTEGER IDIM DIMENSION X(1000) COMMONIMATCOEFF/A(1000),B(1000),C(1000),D(1000),N C WRITE(*,*)'SOLVE IN OK' N=IDIM DO 900I=2,N D(I)=D(I)(A(Il)/I)(Il»*C(Il) B(1)=B(l)(A(ll)/1)(11 »*B(11) 900 CONTINUE X(N)=B(N)/I)(N) DO 910 I=(Nl),l,l 56 X(I)=(B(I)C(I)*X(I+1»10(1) 910 CONTINUE C WRITE(*,*)'SOLVETRl OUT OK ' RETURN END ******************************************************************* ******************************************************************* ***** Subroutine TOTPRESS ***** UPDATES THE TOTAL PRESSURE P(I) ***** UPDATES THE INITIAL VISCOELASTIC CHANGE IN PRESSURE VDP(I) ***** UPDATES THE INITAL TOTAL VISCOELASTIC CHANGE IN PRESSURE VP(I) ******************************************************************* SUBROUTINE VTOTPRESS(DELTA) IMPLICIT REAL*8 (AH,J,OZ) DIMENSION DELTA(1000) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMONIMATLPROPIEC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMON/TIME/R(lOOO),TIME,TWIND,LAP,DTOLD,DTIME COMMONIPRESfP(l000),DP(1000),S,OLDP,VP(lOOO) C DO 1=1,LAP P(I)=P(I)+DELTA(I) Er(I)=AA(3)*(ABS(p(I»)**3+AA(2)*(ABS(p(I»)**2+AA(1)* , (ABS(P(I))+AA(O) END DO C RETURN END ******************************************************************* ******************************************************************* ***** Subroutine OUTS ***** PRINTS OUPUT OF RADIAL PRESSURES TO THE FILE "OUT.DAT" ******************************************************************* SUBROUTINE OUTS IMPLICIT REAL*8 (AH,J,OZ) COMMONfPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKIDCLOCK,U(l OOO),DTP(1OOO),NCLOCK,JO,J 1,Tl ,12,T2 COMMONIMATCOEFFIA(lOOO),B(lOOO),C(lOOO),D(IOOO),N COMMONIMATLPROPIEC,E,vC,ET,AA(0:3),ER(l 000),vrt,vtr COMMONITIME/R(lOOO),TIME,TWIND,LAP,DTOLD,DTIME COMMONIPRESfP(lOOO),DP(1 OOO),S,OLDP,VP(1 000) C C C WRITE(*,*)'OUTS IN OK I 57 PHRASE='************************************************••' WRlTE(2,20)' RADIUS PRESSURE C WRlTE(5,60)TIME DO 9200 I=I,NLAPS+1 WRlTE(2,30)R(D,P(I),DTP(I) 9200 CONTINUE WRITE(4,40)TlME,P(1),DTP(1),B(1) C 10 FORMAT(lX,A50) 20 FORMAT(A47,' ',FI0.1) 30 FORMAT(FI5.4,' ',FI5.8,' ',EI5.8) 40 FORMAT(FIO.O,' ',FlS.4,' ',EIS.8,' ',E15.8) RETURN END DELTA' ******************************************************************* ******************************************************************* ***** Subroutine INPUTS ***** INITIALIZES ALL INPUTS PARAMETERS BY READING ***** INPUT FILE "IN.DAT" ******************************************************************* SUBROUTINE INPUTS IMPLICIT REAL*8 (AH,J,OZ) COMMONIPARAMIRlNC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCK/DCLOCK,U(1000),DTP(l000),NCLOCK,JO,JI,Tl,J2,T2 COMMONIMATLPROPIEC,E,vc,ET,AA(0:3),ER(IOOO),vrt,vtr CCC C WRITE(*,*)'ENTER THE WINDING TENSION' READ(l,lOO)WT C WRITE(*,*)'ENTER THE INSIDE RADIUS OF THE CORE' C READ(1,1 OO)RINC C WRITE(*,*)'ENTER THE INSIDE RADIUS OF THE ROLL' READ(1,1OO)RIN C WRITE(*,*)'ENTER THE OUTSIDE RADIUS OF THE ROLL' READ(1,1OO)ROUT C WRITE(*,*)'ENTER THE RADIAL STIFFNESS OF THE CORE I READ(1,120)EC C WRITE(*,*)'ENTER THE YOUNGS MODULUS OF THE CORE ' C READ(I,I20)E C WRITE(*,*)'ENTER POISSONS RATIO OF THE CORE ' READ(I,IOO)vc C WRITE(*,*)'ENTER THE TANGENTIAL MODULUS OF THE ROLL' READ(1 ,1 OO)ET C WRlTE(*,*)'ENTER THE COEFFICIENTS (c3,c2,cl,cO) OF THE' C WRITE(*,*)'RADIAL MODULUS ER=c3*P"'2+c2*P"'2+cl *P+cO ' READ(1, I20)AA(3),AA(2),AA(1 ),AA(O) 58 C WRlTE(*,*)'ENTER THE POISSONS RATIO vrt OF THE ROLL' READ(1,100)vrt C WRlTE(*,*)'ENTER THE POISSONS RATIO vtr OF THE ROLL' READ(1,1 OO)vtr C READ(1,120)JO C READ(l,120)Jl C READ(1,120)Tl C READ(1,120)J2 C READ(1,120)T2 C C WRlTE(*,*)'ENTER THE TOTAL NUMBER OF LAPS TO BE WOUND ' READ(l,llO)NLAPS C WRlTE(*,*)'ENTER NUMBER OF CORE SEGMENTS (FOR DISPL FORM)' C READ(I,llO)NCORE C WRITE(*,*)'ENTER THE CHANGE IN TIME ' READ(1,100)DCLOCK C WRITE(*,*)'ENTER THE NUMBER TIME INCREMENTS' READ(1,11O)NCLOCK 100 FORMAT(F12.4) 110 FORMAT(I9) 120 FORMAT(EI2.5) CLOSE(1) 130 continue RETURN END ******************************************************************* ******************************************************************* ***** Subroutine UINPUTS ***** INITIALIZES ALL INPUTS PARAMETERS BY READING ***** KEYBOARD ENTRY ******************************************************************* SUBROUTINE UINPUTS IMPUCIT REAL*8 (AH,J,OZ) COMMONIPARAMfRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMON/CLOCKJDCLOCK,U(1000),DTP(1000),NCLOCK,JO,n,T1,12,T2 COMMONIMATLPROP/EC,E,v,ET,AA(0:3),ER(1000),vrt,vtr CCC WRlTE(*,*)'ENTER THE WINDING TENSION I READ(*,1 OO)WT WRITE(I,100)WT 59  C WRITE(*,*)'ENTER THE INSIDE RADIUS OF THE CORE ' C READ(*,1 OO)RINC C WRITE(l,lOO)RINC WRlTE(*,*)'ENTER THE INSIDE RADIUS OF THE ROLL' READ(*,1 OO)RIN WRITE(1,lOO)RIN WRlTE(*,*)'ENTER THE OUTSIDE RADIUS OF THE ROLL' READ(*,lOO)ROUT WRITE(1,100)ROUT WRITE(*,*)'ENTER THE RADIAL STIFFNESS OF THE CORE ' READ(*,120)EC WRlTE(1,120)EC C WRITE(*,*)'ENTER THE YOUNGS MODULUS OF THE CORE ' C READ(*,120)E C WRlTE(1,120)E WRlTE(*,*)'ENTER POISSONS RATIO OF THE CORE ' READ(*,1OO)vc WRlTE(1,1OO)vc WRlTE(*,*)'ENTER THE TANGENTIAL MODULUS OF THE ROLL' READ(*,100)ET WRITE(1, 1OO)ET WRITE(*,*)'ENTER THE COEFFICIENTS (c3,c2,c1,cO) OF THE' WRlTE(*,*)'RADIAL MODULUS ER=c3*PJ\3+c2*PJ\2+c1*P+cO ' READ(*,120)AA(3),AA(2),AA(1),AA(O) WRlTE(1,120)AA(3),AA(2),AA(I),AA(O) WRlTE(*,*)'ENTER THE POISSONS RATIO vrt OF THE ROLL' READ(*,lOO)vrt WRlTE(1,lOO)vrt WRITE(*,*)'ENTER THE POISSONS RATIO vtr OF THE ROLL' READ(*,1OO)vtr WRlTE(1,100)vtr WRlTE(*,*)'ENTER THE CREEP FUNCTION COEFFICIENTS' WRITE(*,*)'1=10+JI*EXP(tfn)+12*EXP(t/T2)' WRlTE(*,*)'10' READ(*,120)JO WRITE(l,120)10 WRITE(*,*)'11' READ(*,120)J1 WRITE(1,120)11 WRITE(*,*)'Tl' READ(*,120)T1 WRITE(1,120)Tl WRITE(*,*)'12' READ(*,120)12 WRITE(l,120)J2 60 WRITE(*,*)'T2' READ(*,120)T2 WRITE(1,I20)T2 WRITE(*,*)'ENTER THE TOTAL NUMBER OF LAPS TO BE WOUND < 1000' READ(*,110)~PlPS WRITE(l,IlO)~PlPS C WRITE(*,*)'ENTER NUMBER OF CORE SEGMENTS (FOR DISPL FORM)' C READ(*,11O)NCORE C WRITE(I,IlO)NCORE WRITE(*,*)'ENTER THE CHANGE IN TIME ' READ(*,100)DCLOCK WRITE(l, IOO)DCLOCK WRITE(*,*)'ENTER THE NUMBER TIME INCREMENTSI READ(*,11 O)NCLOCK WRITE(1, 11O)NCLOCK 100 FORMAT(FI2.4) 110 FORMAT(I9) 120 FORMAT(E12.5) CLOSE(l) 130 continue RETURN END ******************************************************************* ******************************************************************* ** ***** Subroutine Winder ***** similar to standard elastic finite difference winding routines ***** determines the change in radial pressure at radial increment ***** due to the addition of each layer ***** store these values in the array DP(Iayer,layer) ** ******************************************************************* SUBROUTINE WINDER IMPLICIT REAL*8 (AH,J,OZ) COMMONfPARAMIRINC,RIN,ROUT,~APS,NlNC,NCORE,WT COMMONICLOCK/DCLOCK,U(1OOO),DTP(1 OOO),NCLOCK,JO,J I,Tl ,12,T2 COMMONfMATLPROP/EC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMONffIMEIR(IOOO),TIME,TWIND,LAP,DTOLD,DTIME COMMONfMATCOEFF/A(1OOO),B(l 000),C(1OOO),D( IOOO),N COMMON/PRES/P(1 OOO),DP(1 OOO),S,OLDP,VP(I 000) COMMON/TRY/AA1,BB 1,CC1,RK C CCCCC CALCULATE INITIAL PRESSURES C C WT=lOO.O 61 C Ec=E*(rin**2rinc**2)/(rin**2+rinc**2vc*(rin**2rinc**2»/rin C C WRITE(*,*)'WINDER IN OK' H=(ROUTRIN)/NLAPS R(l)=RIN LAP=1 DO I=2,NLAPS+I R(I)=R(1 )+(11 )*H END DO C CCCCC WIND FIRST LAP C WRITE(*,*)'ONE' C LAP=1 DT=EC*WT*RIN**2/(EC*R(1)**2+H*ET) DT=WT DP(1 )=(DT*H)IR(1) CALL TOTPRESS(DP) C CCCCC WIND ON SECOND LAP C C WRITE(*,*)'TWO' LAP=LAP+l I=LAP D(1 )=(1 DOvrtEt/(Ec*RIN)RINIH) C(1)=R(l)/H B(I)=ODO DP(2)=(WT*H)IR(LAP) DP(1 )=(B(1)DP(2)*C(1 »ID(1) CALL TOTPRESS(DP) c CCCCC WIND ON ALL REMAINING LAPS C DO 50 M=3,NLAPS LAP=M C D(I )=1 DOvrtEtJ(Ec*RIN)RIN/H C(I)=R(1)/H B(1)=ODO C DO 1=2, LAPl A(Il )=(R(I)**21H**23DO*R(I)/(2DO*H» D(I)=(IDO2DO*R(I)**21H**2Et/Er(I» C(I)=R(l)**21H**2+3DO*R(I)/(2DO*H) 62 B(I)=ODO END DO C DP(LAP)=(WT*H)IR(LAP) B(LAP)=DP(LAP) D(LAP)=lDO A(LAPl )=ODO C C CALL SOLVETRI(DP,LAP) CALL TOTPRESS(DP) 50 CONTINUE CALL OUTS C WRITE(*,*)'WINDER OUT OK' RETURN END ******************************************************************* ******************************************************************* ***** Subroutine TOTPRESS ***** UPDATES THE TOTAL PRESSURE P(I) ***** UPDATES THE INITIAL VISCOELASTIC CHANGE IN PRESSURE VDP(I) ***** UPDATES THE INITAL TOTAL VISCOELASTIC CHANGE IN PRESSURE VP(I) ******************************************************************* SUBROUTINE TOTPRESS(DELTA) IMPLICIT REAL*8 (AH,J,OZ) DIMENSION DELTA(lOOO) COMMONIPARAMIRINC,RIN,ROUT,NLAPS,NINC,NCORE,WT COMMONIMATLPROPIEC,E,vc,ET,AA(O:3),ER(1000),vrt,vtr COMMON/TIMEIR(l000),TIME,TWIND,LAP,DTOLD,DTIME COMMON/PRES/P(lOOO),DP(l OOO),S,OLDP,VP(l 000) C DO I=l,LAP P(I)=P(I)+DELTA(I) Er(I)=AA(3)*(ABS(p(I}»**3+AA(2}*(ABS(P(D»**2+AA(1)* I (ABS(p(I»)+AA(O) END DO C RETURN END 63 VITA Jeffrey Scott Henning Candidate for the Degree of Master of Science Thesis: EFFECTS OF RELAXAnON OF A CORE ON A WOUND ROLL Major Field: Mechanical Engineering Biographical: Personal Data: Born in Stillwater, Oklahoma, on August 24, 1966. Married to Rebecca Sue Worsham. Education: Graduated from Jenks High School, Jenks, Oklahoma, in May 1984; attended Oral Roberts University and Tulsa Community College; received Bachelor of Science degree in Mechanical Engineering from Oklahoma State University, Stillwater, Oklahoma, in May 1995. Completed the requirements for the Master of Science degree with a major in Mechanical Engineering at Oklahoma State University in May, 1997. Experience: Employed in industry before returning to finish degree. Research Assistant at the OSU Web Handling Research Center. Professional Memberships: Oklahoma Society of Professional Engineers, National Society ofProfessional Engineers, Certified Engineer Intern. 



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