Kinematic Analysis of Cam Profiles Used in Compound Bows. A Thesis. Presented to. The Graduate Faculty of the University of Missouri

Transcription

1 Knematc Analyss of Cam Profles Used n Compound Bows A Thess Presented to The Graduate Faculty of the Unversty of Mssour In Partal Fulfllment of the Requrements for the Degree Master of Scence Andrew Joseph Hanson December 009

2 The undersgned, apponted by the Dean of the Graduate School, have examned the thess enttled: KINEMATIC ANALYSIS OF CAM PROFILES USED IN COMPOUND BOWS Presented by Andrew Joseph Hanson A canddate for the degree of Master of Scence And hereby certfy that n ther opnon t s worthy of acceptance. Professor Yuy Ln Professor Sherf El-Gzawy Professor James Noble

3 ACKNOWLEDGEMENTS To my advsor, Yuy Ln, Ph. D, P.E., for hs amcable and devoted mentorng; To my wfe, Lauren, for her patence and understandng; And to God, for Hs blessngs each day. P a g e

4 ABSTRACT The frst compound bow was nvented n Mssour n [Allen, 1968] Compound bows are unquely dfferent from other types of bows n that they use a set of cables, cams and pulley, and two elastc lmbs that act as sprngs, to create a mechancal advantage whle the bowstrng s beng drawn. In what s known as the let-off (draw force verses draw length) curve, ths allows the archer to hold the bow at a fully drawn length wth sgnfcantly less force than the maxmum draw force. Ths desgn s advantageous for huntng, where arrow speed, accuracy, and holdng weght become mportant requrements n beng successful. Snce the nventon, technology has progressed n mprovng the bow s effcency, accuracy, and arrow speed through patented emprcal methods. However, very lttle has been shown n analytcally modelng and optmal desgn of ths complex mechancal system. A preface to varous types of bows as well as subtypes of compound bows wll be ntroduced. A knematc analyss wll be shown for an eccentrc-crcular cam desgn and a onecam one-pulley desgn. By teratvely determnng the bow lmb, cam, and cable postons a relatonshp between the drawn length and drawn force wll produce a draw-force curve. In fact, ths curve represents the stran energy stored wthn the system, and upon arrow release wll be transferred nto knetc energy. Lke all mechancal systems, there s a loss n energy and effcency. A method for accurately determnng effcency wll be explaned. Experments are also conducted usng carbon fber compostes to create an adequate lmb desgn. P a g e

5 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ABSTRACT..... LIST OF FIGURES.. v. CHAPTER I. INTRODUCTION Types of Bows Characterstcs of a Compound Bow Types of Compound Bows Motvaton Lterature Survey 10 II. ANALYSIS OF SYMMETRIC CIRCULAR CAM DESIGN Introducton to Symmetrc Crcular Cam Analyss Determnng a Pont on an Eccentrc Crcle Geometry Analyss for Determnng Relevant Ponts 15.4 Force Dscusson. 19 III. ANALYSIS OF ONE CAM ONE PULLEY DESIGN Global Coordnate System Pont Analyss Force Analyss.. 5 IV. EXPERIMENTAL WORK Determnng Effcency of Compound Bow Carbon Fber Bow Lmb Manufacturng. 34 V. CONCLUSION AND FUTURE WORK REFERENCES. 41 APPENDICES.. 4 v P a g e

6 A COMPOUND BOW MANUFACTURERS 4 B FIRESTORM LITE EFFICENCY DATA. 43 C BUCKMASTER BTR EFFICIENCY DATA.. 44 D TOOLS AND PRODUCTS USED IN EXPERIMENTAL WORK.. 45 E BUCKMASTER BTR MEASURED CAM PROFILE 46 E1 SINGLE CAM (C1 BOTTOM, C MIDDLE, C3 TOP).. 46 E C1 CAM DATA. 46 E3 C CAM DATA, +110 DEGREES RELATIVE TO C1 IN DIRECTION OF ROTATION. 48 E4 C3 CAM DATA, -0 DEGREES RELATIVE TO C1 IN DIRECTION OF ROTATION.49 F BUCKMASTER BTR NUMERICAL RESULTS USING MICROSOFT EXCEL.. 51 F1 CALCULATED RESULTS.. 51 F FORCE RESULTS. 51 F3 FIRST THREE DRAW FORCE CURVE POINTS 5 v P a g e

7 LIST OF FIGURES Fgure Page 1 - Self-yew Englsh Longbow, 6.5ft Modern Recurve Bow Frst Compound Bow 4 - Let-off Curve Compound Bow Illustraton Top Hybrd Cam (Controller Cam) Bottom Hybrd Cam (Power Cam) Bottom Sngle Cam (Power Cam) Bnary Cam; No Splt-harness Connecton to Lmb Crcle Geometry Used to Calculate (Xp,Yp) Eccentrc Cam Bow n Frst Iteraton Eccentrc Cam Bow n Second Iteraton One-Cam One Pulley, Frst Iteraton One-Cam One Pulley, Second Iteraton Force Analyss; Buckmaster BTR Pont-Lne Dstance Equaton Buckmaster BTR and PSE Frestorm Lte Force/Measurement Test taken for PSE Frestorm Lte Carbon Fber Experment Experment ; Sample #1, #, and # Experment ; Vsble Deformaton n Sample #1 and # wth Foam Core 41 Experment 3; Before Surface Post-processng (Sandng, Polshng) Experment 3; Sanded Sample of Plan, 16 Undrectonal, Plan. 43 v P a g e

8 CHAPTER I INTRODUCTION 1.1 Types of Bows The use of an archery bow dates back to prehstorc tmes, but as frearms began to replace bows as a mltary weapon n the 16 th century, the bows man purpose was for huntng and recreaton. The archery bow can be classfed nto three categores: longbow, recurve bow, and compound bow. The longbow s the oldest of the three types. Long bows consst of a D-shaped body that s connected to a strng cable. The body on a long bow serves as a handle support and as an energy storng lmb. Long bows measure approxmately the heght of a person. The longbow pctured n Fg.1 runs 78 nches n total length and has a pull force of 105 lbs at a 3 nch draw. The draw length s measured from the handle poston to the pont the strng connects to the arrow. Fgure 4 - Self-yew Englsh Longbow, 6.5ft ( The second type of bow s the recurve. The recurve s drawstrng, unlke the longbow, comes n contact wth the bows lmb as the strng s beng drawn. The contact between lmb and strng allows for a dfferent relatonshp between the draw force and the length of the drawstrng pulled. In general, the force s slghtly less than lnear as the archer reaches a fully drawn 1 P a g e

9 poston[kncy, 1981]. Competton recurves run 0-50 lb and 6-70 nches n total length, huntng recurves run lb and 58-6nches. Fgure 5 - Modern Recurve Bow ( The thrd type of bow, and the topc of ths study, s the compound bow. The compound bow uses a cam or pulley pnned to the end of the lmb along wth a seres of connected cables to provde a mechancal advantage allowng the bow to store more energy whle requrng less force from the archer to hold the bow at a fully drawn poston. Fg.3 s an Fgure 6 - Frst Compound Bow (US Patent ) P a g e

10 example of the basc desgn of the frst compound bow patented by Holless Allen n [Allen, 1969] 1. Characterstcs of a Compound Bow The compound bow s a technologcally more complex and advanced system that s unquely dfferent than the other two prevous bow types. Ths system uses the lmbs, lke other bow types, to store energy to propel the arrow nto moton. The bow s strng cables are wrapped around the cams, and each cable performs a dfferent functon and carres a dfferent load requred to hold the lmb n ts deformed poston. Refer to Fg.5 for an llustraton of compound bow termnology. The bow cables consst of a draw strng whch s pulled by the archer and a set of buss cables that are on the non-draw sde. In some cases one of the buss cables may be more accurately descrbed as a controller cable. The dstrbuton of force for each cable strng s dependent upon the profles of the cams. In general, as the cam rotates and the bowstrng s drawn, the pull force (whch s a functon of the bowstrng tenson force) ncreases untl t reaches a maxmum draw force. Then, t begns to relax to a holdng draw weght when the archer s n a fully drawn poston. The relatonshp between the maxmum draw force and the holdng weght s known as let-off. The percentage let-off can be descrbed as: holdngweght 1 letoff 100% max weght (1) A let off curve s commonly shown to represent ths relatonshp between the maxmum draw force and holdng force. The area under the curve, referrng to Fg.4, s the amount of potental or stran energy that s stored n the lmbs. Snce the arrow s attached to the same poston from whch the archer pulls, the forces on the draw can be converted nto knetc energy that s 3 P a g e

11 transferred to the arrow upon release. Not all stran energy stored n the lmbs wll be transferred nto knetc energy n the bow. In the market today a hgh-performance bow s around 80% effcent whle some clam to be as hgh as 86-88% effcent. [ 007] Fgure 7 - Let-off Curve ( Just a few years ago, the standard for a hgh end bow was to shoot arrows at a speed of 300 fps (feet/second). As bow desgners contnue to strde for greater effcency by mprovng cam/lmb desgns, the hgh speed record of 300 fps has become a thng of the past. Although many hunters can shoot deer at 50 fps or less, the noton of beng able to shoot faster arrows has gven hunters a desre and nterest to pay more for ths technology, and also be able to hunt larger game. For the purpose of beng able to farly compare one bow s speed to another, an IBO (Internatonal Bowhuntng Organzaton) standard s establshed. IBO standards state that an arrow must wegh 5 grans per maxmum draw force. [IBO Rules Class Defntons, 009] Snce huntng bows are tested at 70 lb maxmum draw weght, the arrow must wegh 350 grans, or 4 P a g e

12 0.8 ounces. Also, the draw length from handle to bow strng must be 30 nches. Although t s not an IBO standard, most marketed bows have a let-off between percent. In Fg.5, a compound bow of the one-cam one-pulley desgn s llustrated. All compound bows consst of a rser, whch acts as a platform to mount the lmbs. The rser also contans a handle grp and an arrow shelf. The attached lmbs are used lke a sprng to store energy. In ther ntal poston the lmbs are already pre-loaded. The bow s strngs are categorzed nto a bow strng and a set of buss cables whch are on the draw sde and non-draw sde respectvely. The cams contan a seres of stacked cams (usually or three dependng on the desgn). In most Fgure 8 - Compound Bow Illustraton ( 5 P a g e

13 cases the bow strng and buss cables are not physcally connected (except on the pulley sde of a sngle cam desgn). Although they are not connected, they are constraned to the cams rotaton and usually end at some fxed locaton on the cam. 1.3 Types of Compound Bows Compound bows offer hunters a unque challenge that rfle huntng cannot offer. Compound bows requre the hunter to get closer to ther prey. Over the last few decades, many archery companes have been establshed to compete n ths nnovatve ndustry. Wthn the branch of compound bows, there are four basc cam profle desgns that are avalable on the market today. The four desgns are the twn cam, hybrd cam, sngle cam and bnary cam desgn. The webste, offers rch nformaton n comparng these varous desgns. Each desgn offers ts own set of advantages and dsadvantages. For example, smoothness of draw, arrow speed, manufacturablty, shootng accuracy, cost and so on. A Twn Cam system s often consdered a Two Cam or Dual Cam. The Twn Cam system contans two perfectly symmetrcal round wheels or ellptcal cams attached to the lmbs. When properly synchronzed, twn cams offer exceptonal accuracy, nock travel, and overall arrow speed. A bow s sad to have exceptonal nock travel f the arrow s end attached to the strng travels n a perfectly horzontal drecton relatve to the two lmb ends. However, twn cams requre more mantenance and servce to stay n top shootng condton. Although t s desrable to have non-extensble strngs, strngs that do extend slghtly have an effect on the synchronzaton of cam rotaton. Twn cams also have a tendency to be noser when shootng than hybrd or sngle cam desgns. [ 007] The frst compound bow nventon was of the twn cam desgn. Many youth bows, competton bows and serous shooters stll prefer ths desgn over some more expensve and technologcally advanced 6 P a g e

14 systems. Another advantage s eccentrc crcular cams are more easly manufacturable then some of the more complex cam. The hybrd cam system has ganed popularty over the last few years. The hybrd cam system features two asymmetrcal ellptcal cams: a control cam on top and a power cam on the bottom. The system s set up wth a splt-harness (buss cable), a control cable, and a draw strng. Hybrd cams clam to offer the beneft of straght nock travel wthout synchronzaton ssues. Although they are much more self relant, they too need to be adjusted perodcally to perform at ther peak effcency. There are several hybrd cam models that are mpressvely fast and Fgure 9 - Top Hybrd Cam (Controller Cam) Fgure 30 - Bottom Hybrd Cam (Power Cam) quet and can perform as good, f not better, than the even more modern sngle cam bows. Fgs. 6 and 7 are an example of the hybrd cam system. In Fgs. 6 and 7, although the draw sde cams are symmetrcal, the cam n ts entrety s sad to be asymmetrcal. Therefore, t s not a twncam desgn. [ 007] The sngle cam system s often descrbed as a Solocam or One Cam system wth a round pulley wheel on the top and an ellptcal shaped power-cam on the bottom. The sngle cam s generally queter and smpler to mantan than tradtonal twn cam systems, snce there s no need for synchronzaton of the top and bottom cam. US Patent 5,368,006 explans the sngle- 7 P a g e

15 cams usefulness to solvng prevous synchronzaton ssues. However, keepng nock travel straght becomes a more complcated tunng problem. Wthn a sngle cam system there s much room for varaton allowng for some to have an aggressve let-off curve or a smoother draw. Some offer ease of adjustablty, others do not. Overall, sngle cam relablty and smoothness s well respected and they are today s popular choce on compound bows. Fg.8 shows an example of a sngle cam whch n all cases s connected to the bottom lmb. It s sometmes called the power cam because t s responsble for the force dstrbuton between cable strngs. [ 007] Fgure 31 - Bottom Sngle Cam (Power Cam) Introduced n 005 by Bowtech Archery, the bnary cam s a modfed 3-groove twn-cam system that constrans the top and bottom cam to each other, rather than to the bow s lmbs. Fg.9 shows a bnary cam. There are no splt-harness connectons to the lmb on a bnary system, lke on a sngle or hybrd system, just cam-to-cam control cables. Ths creates a free floatng system whch allows the cams to automatcally equalze any mbalances n lmb deflecton or strng and control cable lengths. Although ths desgn does not have much of a reputaton n the 8 P a g e

16 compound bow market, so far t has proven to shoot some of the fastest arrows. Those desgns that were on the market n 005 and 006 were among some of the most popular. [ 007] Fgure 3 - Bnary Cam; No Splt-harness Connecton to Lmb [ Motvaton In ths study, two separate cam systems are analyzed; the symmetrc crcular cam desgn (Chapter ) and the one cam one pulley desgn (Chapter 3). The modeled systems wll yeld a set of forcng vectors that wll form a unty between the deformed lmb, wth a known sprng constant, to the forces on the bowstrng and buss cables. The purpose of ths study s to analytcally model the compound bow s system so that there s a greater understandng and ablty to predct the effects of changes that are made to both the cam and lmb characterstc geometry. Wth the followng model equatons, one can nput a gven cam profle and lmb propertes to accurately predct the draw-force curve thus allowng the theoretcal arrow speed to be calculated. The goal s to create an accurate analytcal approach to modelng a compound 9 P a g e

17 bow s let-off, whch can save both tme and money that would be lost whle makng and testng desgn prototypes. Chapter 4 wll dscuss expermental data for determnng desgn effcency, as well as a method for manufacturng bow lmbs. One of the major marketng factors of compound bows s ther ablty to shoot fast arrows. Snce most bows are 80% effcent, there s a consderable effort for mprovng the compound bow s effcency. 1.5 Lterature Survey There have been many advances n compound bow technology to date. Patents have mproved the bow s effcency and performance through emprcal methods; however analytcal methods were not evdent. Theses have examned parts of the compound bow and some have even analyzed ther draw-force relatonshp [Kncy, 1981; Vsnor, 007], but ether the bows used were outdated or the assumed data was not accurate enough compared to actual physcal characterstcs of a compound bow. The frst patented compound bow was of the symmetrc cam type. [Allen, 1968] At that tme, usng eccentrc dscs to create a mechancal advantage was very dfferent than the tradtonal long bow and even the recurve bow. In 1977, Kudlacek nvented the bow that used eccentrc cam elements. Ths desgn s stll n producton today on some of the more basc huntng bows and youth bows for the smplcty of manufacturng crcular dscs. Chapter two s devoted to analyzng ths type of bow, because t s a good startng pont for understandng how the cam and lmb geometry affect the let-off curve. A one cam one pulley (sngle cam) desgn was frst patented n 1994 by McPherson and as of 006 accounted for 46% of compound bow sales. [huntersfrend.com, 007] The desgn uses 10 P a g e

18 a pulley on the upper end and an eccentrc layer of cams on the lower end. The purpose of the stacked cams s to approxmately feed the same amount of strng out the pull sde and te the two lmbs together durng ts draw. One advantage of ths type of desgn s that adjusted draw lengths can be acheved usng the same strng whle nock travel s unaffected. Progresses have been made for ncreasng maxmum stored energy, bow effcency, and usablty to mprove the sngle-cam bow. In Darlngton s patent [000] an adjustment allowed a flatter contour to be mantaned at the maxmum force/draw curve for adjustable draw lengths. Strother s patented desgn [007] clams 8-94% effcency can be acheved usng hs sngle-cam desgn; however, others clamed that the best tested bows have not performed over 88%. Another patent allows for the buss cable to act through the axel pn achevng nearly 100% letoff on the drawn bowstrng. [Blahnk, 006; Knttel, 008] Chapter 3 s devoted to modelng a one cam one pulley desgn. A thess [Kncy, 1981] was presented to the Unversty of Mssour Rolla whch used a method to teratvely model the lmb, strng, and eccentrc wheel mechancs. The compound bow beng modeled was a patented [Ketchum, 1976] bow. The thess used Bernoull Euler s equaton for bendng. The cantlever beam segment was teratvely broken nto segments and the force requred to bend the beam was dstrbuted among the bow s cable strngs by summng the torques about a central pont on the beam segment. In the thess, however, t dd not explan how to mprove the stran energy storage. A more recent thess explans n detal the lmb deflecton of a cantlever beam under large deflecton as related to a compound bow. [Vsnor, 007] Vsnor s thess explans an analytcal approach to accurately determne the lmb tp poston of a beam subjected to a constant, concentrated force, wth a constant angle appled to the free end. However, realstc 11 P a g e

19 characterstcs of a bow wll not have a constant force, snce the lmb tp s changng locaton, and the angle the force appled wll change as well. 1 P a g e

20 CHAPTER ANALYSIS OF A SYMMETRIC CIRCULAR CAM DESIGN.1 Introducton to Symmetrc Crcular Cam Analyss As a good startng pont for analyzng translaton and rotaton of a cam on a lmb, frst a twn eccentrc cam desgn s consdered. In ths type of desgn, there are assumptons that can be made to smplfy the analyss process. Ths type of desgn has two completely symmetrcal cams whch rotate n unson and lkewse both the top and bottom lmbs wll deform concurrently. We can also assume the rser to be completely rgd and the strngs to be nextensble. To solve ths problem we wll set up a global coordnate system wth a center at the md-pont of the rser. Ths pont s where the arrow rests on the rser. Referrng to Fgs.11 and 1, our objectve s to solve for the tangent pont, (X t,y t ), connectng the cam to the bowstrng and buss cable as well as correctly determne the exact global coordnate poston of (X c,y c ). Ths can be done by usng a cam rotaton of as an nput where =1 to n. By knowng the exact locaton of (X c,y c ) the lmb deformaton s known as well. After calculatng the buss cable sde the bow strng sde wll be determned. If we assume the lmbs to be cantlever beams wth known physcal propertes, then ther sprng force s known as well. A moment about pont (X c,y c ) and the dstance to the bow strng and buss cable create a rato that wll become mportant n calculatng the amount of force on each strng. Fbowstrng M1 Fbusscable M () In Equaton, F bowstrng s the tenson force on the bow strng, F busscable s the tenson force on the buss cable and M1 and M are the dstances prevously mentoned from (Xc,Yc) to ther respectve strngs.. Determnng a Pont on an Eccentrc Crcle Let us next look at the geometry of a crcle wth an offset center shown n Fg.10. We can determne the ponts (X p,y p ) for a rotaton, θ, about the center (X c,y c ) usng the crcle and slope equatons. R ( X p ) ( Yp ) (3) 13 P a g e

21 Y X p p Y c X c tan( ) (4) R s the known geometrc radus of the cam. The ponts X c and Y c are assumed to be known usng a set of equatons dscussed later on. So, for an ncremental rotaton, θ we can fnd the ponts on the crcle, X p and Y p. Fgure 33 - Crcle Geometry Used to Calculate (Xp,Yp) Fgure 34 Eccentrc Cam Bow n Frst Iteraton Fgure 35 Eccentrc Cam Bow n Second Iteraton.3 Geometry Analyss for Determnng Tangent Ponts 14 P a g e

22 Contnung on n our dscusson, we wll refer to Fgs. 11 and 1 to formulate a set of equatons that wll allow us to solve our new geometry for a gven cam rotaton θ. Fg.11 represents the ntal condtons of the compound bow and Fg.1 can be consdered one angle of rotaton θ. (X t,y t ) are coordnate postons for the tangent pont of the buss strng connected to the non-draw stacked cam. (X c,y c ) lkewse are coordnate postons of the cams center of rotaton and wll be the same for both the draw and non-draw sde cams. Snce the bow s modeled symmetrcally about the x-axs, whch s the arrow s draw plane, we can say that the other cam wll have a tangent pont (X t,-y t ) and a center locaton of (X c,-y c ). Varable L b represents the length of the strng from the connecton on the lmb to the tangent pont on the cam and k 1 and k represent ther slopes respectvely. L b wll be the same for both buss cables due to symmetry. Varable r wll represent the radal dstance from the center of rotaton to the tangent pont on the cam and angle α s between l 11 and l. It can be solved usng the slope of k 1 and k. Also, φ wll be the angle created by the frst tangent pont to the th tangent pont. All ntal condtons can be assumed to be known. For Fg.1 we wsh to know the coordnate poston of (X t,y t ), (X c,y c ), k 1 and k from a rotaton and forced translaton. Equatons 5, 7, 8, 1 and 15 can be used to solve for (X t,y t ), (X c,y c ), k 1 and k. The frst s the length of the strng L B. L B S ( X t X c ) ( Yt Yc ) (5) S r( ) d (6) 0 S φ s the dstance of strng rolled onto the cam. Keep n mnd that the shape of the cam wll cause the tangent pont to change; therefore θ wll not necessarly equal φ. Also, r(θ) can be related to a coordnate translaton and rotaton. The second equaton s the slope of the strng connected to the top cam and the thrd s the slope of the strng connected to the bottom cam. Y X t t Y X c c k 1 (7) Y X c c Y t t X k (8) 15 P a g e

23 The fourth equaton (Equaton 1) uses the law of cosnes to make a relatonshp between the trangular sdes l 11, l and r. 11 ( Yt ) ( X t X 0 ) l (9) l ( Yc ) ( X c X 0 ) (10) X o t t k 1 X Y (11) ( r l ) ( l ) ( l )( l )cos (1) X 0 s the X-locaton when Y=0 used to solve dstances l 11 and l. Dstance r wll need to be determned from ponts (X t,y t ) and (X c,y c ), where: r ( X t X c ) ( Yt Yc ) (13) The last equaton can be determned by fndng a local curve of the cam profle. Ths can be accurately determned f we use least squares fttng usng seven ponts from the cam geometry calculated n Equatons 3 and 4. The seven ponts that are used wll nclude: [(X p,y p ),(X +1 p,y +1 p ), (X +7 p,y +7 p )]. Once there s an equaton for the top cam, t can be made n the quadratc form: ax bx c 0 (14) Where a, b, and c are determned constants. If we take the dervatve of the prevous equaton and evaluate t at (X t,y t ) t should equal the slope k 1. k1 ( ax t ) b (15) Lkewse, seven ponts for the bottom cam, [(X p,-y p ),(X +1 p,-y +1 p ), (X +7 p,-y +7 p )], can be used to make a quadratc equaton usng least squares fttng. If the dervatve s evaluated at (X t,-y t ), an equaton for k can be found smlar to Equaton 15. The tangent pont connected to the bowstrng needs to be determned as well. Note, although n Fgs. 11 and 1 there s only one crcle, n realty there wll be two eccentrc stacked cams wth dfferent geometres and startng ponts, (X 1 p,y 1 p ). Ths can be done usng a smlar set 16 P a g e

24 of equatons to those used n calculatng the tangent pont and rotatonal center on the buss cable sde of the bow. We wll agan refer to Fgs. 11 and 1 to fnd (X t,y t ), and X h. Y h s consdered to be along the X-axs, therefore t s zero. Three equatons wll need to be solved for these three unknown varables. Smlar to Equatons 7 and 8: k b X Yt t X h (16) k a X Y t t X h (17) where k a and k b are the slopes of lnes a and b. k b s calculated from the dervatve of the local cam curve from the top cam determned at the tangent pont X t. The curve can be found by nputtng the next seven ponts, [(X p,y p ),(X +1 p,y +1 p ), (X +7 p,y +7 p )], for the draw sde of the top cam nto the least squares fttng formula. k b, lkewse, can be found usng the bottom cam on the draw sde and least squares fttng. Equaton 1 uses the law of cosnes to create a trangle wth sde s a, b and c. Where c s the dstance between (X t,y t ) and (X c,y c ). Sde s a and b are shown n Fgs. 11 and 1. a t ) ( X c X h ) ( Y (18) b t ) ( X t X h ) ( Y (19) c t Y c ) ( X t X c ) ( Y (0) c a b a b cos( ) (1) γ represents the angle between a and b. Once X h reaches ts maxmum desred draw length the calculatons stop..4 Force Dscusson In Chapter 3, a more extensve force analyss wll be performed to solve the draw force for all teratons. In ths case, symmetry wll allow us to analyze the top cam only. Snce all geometry 17 P a g e

25 ponts are known at ths tme, an equaton for torque can be appled to one sde of the bow to fnd a rato of the forces on the cam. The moment dstances can be solved as the dstance from (Xc,Yc) to the cable whle stayng perpendcular to the cable. The rato between the two dstances creates a rato between the two forces (see Equaton ). Ths rato wll allow us to fnd the drecton formed by the combnaton of forces connected to the cam, snce the drecton of each ndvdual force s known. Lkewse, the drecton of the cable connected to (Xc,Yc) s known from our prevous analyss. If we assume the sprng constant s known for the lmb, the magntude of the lmb can be determned. The drecton of the lmb force can be assumed to be opposte the path of the lmb travel, where the lmb travel s n the drecton of the last cam center to the current cam center for all teratons. A forcng vector for all of the actng forces must be equal to zero. Snce we know the magntude and drecton of the lmb force and the drecton of the combned forces connected to the cam n addton to the drecton of the cable force through the center, the magntude of the combned forces and the drecton of the cable force through the cam center can be determned. Once the combned forces magntude s known, each ndvdual force can be solved by knowng the rato that was already determned between these forces. The magntude of the drawstrng wll allow us to fnd the magntude of the draw force for all teratons. Equaton gves the relatonshp between the draw force and the force on the bowstrng, where s the angle of the bowstrng to the x-axs. Fdraw Fstrng 1 cos () 18 P a g e

26 CHAPTER 3 ANAYLSIS OF ONE CAM ONE PULLEY DESIGN 3.1 Analytcal Model A generalzed one cam one pulley desgn holds a set of two strngs that are wrapped around three cams that are stacked as shown n Fgs. 13 and 14. C1, C and C3 are n reference to the three stacked cams. For a rght handed draw, C1 s on the left sde and s connected to the bowstrng whch also wraps around the pulley and connects to C3 (the rght most cam) on the non-draw sde. The mddle cam, C, contans the controller cable, snce the cable s connected to the lmb and controls the amount of travel of the lmb tp. For the analyss of one cam one pulley desgn we wll assume that both lmbs deform equally and the lmb deforms lnearly snce the travel dstance s relatvely small. Also noted n Fg.13, (0,Yh) represents the bowstrng nock pont connected to the arrow. The nock pont, Yh, s along the y-axs. (Xp,Yp) represent the Fgure 36 - One-Cam One Pulley, Frst Iteraton Fgure 37 - One-Cam One Pulley, Second Iteraton tangent pont connectng the strng to the crcular dsk. For the frst superscrpt, 1 denotes the non-draw sde and denotes the draw sde. The second superscrpt denotes the teraton number. (Xl,Yl) s the measured lmb tp n the (X,Y) coordnate system. (Xl 0,Yl 0 ) s the frst teraton, and (Xl 1,Yl 1 ) s the last teraton. The lne formed between these two ponts s the path of (Xc,Yc) travel. (Xc,Yc) s the rotatonal center of the pulley wth the superscrpt denotng the teraton. Lkewse, (-Xl,Yl) and (-Xc,Yc) represent the rotatonal center of the cam. (Xt,Yt) 19 P a g e

27 represent the tangent ponts on the cam. The frst superscrpt denotes the cam; C1, C and C3. The second superscrpt denotes the teraton. In Fg.14, there are sub-teratons for (Xp,Yp) and (Xt, Yt). The frst would correspond to a translaton and the second to a rotaton. Therefore, for every (Xc,Yc) and (0,Yh) teraton, there are two (Xp,Yp) and (Xt,Yt) sub-teratons. Note, n Fgs.13 and 14 the llustraton s rotated 90 degrees clockwse n relaton to Fgs.11 and 1 so that a local curve can be formed near (Xt,Yt) for cams C1, C and C3. The mportance wll be seen later on n the dscusson. For analyzng the knematcs of the system the followng assumptons must be made. The frst s that there s no slppage of cable on the cams or pulley. Second, the strng s nextensble. Thrd, the strng s consdered to be taut. Fourth, as prevously mentoned, the lmbs deform lnearly and evenly. The top and bottom lmb deform the same for all teratons. And fnally, we assume to know the geometry of the pulley, cams, and lmb. The measured cam geometry s taken from the rotatonal axs of C1, C and C3 to the edge of the cam. The frst pont taken s the tangent pont of the gven cam beng measured. Then, an approprate step sze s chosen to allow for accurate results. A reference angle between C1, C and C3 must be measured. See Appendx E Buckmaster BTR Measured Cam Profle for an example of measured data. By knowng the geometry of the lmb and ts materal propertes, a close approxmaton of the sprng force of the lmb can be determned. After knematcally determnng the poston of (Xc,Yc), Yh, both (Xp,Yp) s and all (Xt,Yt) s, a forcng vector between all the strngs under tenson can be created to fnd the amount of force requred to pull Yh along the y-axs. Ths wll yeld the desred let-off curve for the gven bow system. To analyze the system, t can be broken nto three sub-routnes. All three routnes wll use the strng length as the constranng factor. The nput wll be the lmb deflecton. For the frst sub-routne, we wll analyze the length of the controller cable, CC, connected to the controller cam, C, and the upper lmb. Snce CC s connected to the upper lmb, we know (Xc, Yc) for all teratons. The total length of the strng wll nclude the length from the upper lmb tp to the tangent pont on C, L, n addton to the amount of strng wrapped on C, S. The total length of CC wll reman constant for all teratons. The cam must be rotated untl Equaton 30 s satsfed. CC L S (3) 0 P a g e

28 L ( Yc Yt ) ( Xc Xt ) (4) S 1 f ( x)' dx (5) 1 In Equaton 4, L s the length of the strng from the center of rotaton on the upper lmb to the tangent pont on C. In Equaton 5, S s the length of cable wrapped on C. S can be determned from the measured cam data. The functon, f(x), s found usng least squares fttng smlar to chapter. For all rotatons, an arbtrary number of data ponts, for whch the cam wll be rotatng onto, should be chosen to fnd a local curve that accurately represents the cam profle. Note that the data s n reference to the center of rotaton. For Equaton 5, -1 represents the last teraton tangent pont after translaton and rotaton on C. represents the current tangent pont. It s mportant to determne the amount of rotaton so that Equaton 3 s satsfed. To satsfy Equaton 3 we must fnd the tangent pont (Xt,Yt ) for each rotaton. The tangent pont can be found usng the slope equaton and the dervatve of the local curve, f(x). Y X c c Y X t t K (6) f ( x) ax f ( x)' ax b bx c (7) K ax t b (8) By solvng Equatons 6 and 8, we have the tangent pont Xt. Equaton 6 s the slope equaton through (Xc,Yc) and (Xt,Yt ), and Equaton 8 s the slope of C at Xt. a and b are determned coeffcents usng least squares fttng. By substtutng the frst part of Equaton 34 n for Yt evaluated at Xt, the two unknowns are Xt and K. Once the tangent pont s found, Equatons 4 and 5 can be solved. If the rotaton satsfes Equaton 3, then the subroutne s complete. If not, the cam must be rotated more or less. The amount C rotates s drectly related to the value, CC. CC must have an acceptable convergence 1 P a g e

29 n order to contnue. The rotaton of C wll be desgnated as. Snce C1, C and C3 are a rgd body, wll be the same. Next, the second sub-routne wll be performed to determne the tangent pont of C3 (Xt 3,Yt 3 ) and the tangent pont of the pulley on the non-draw sde (Xp 1,Yp 1 ) for all teratons. There wll be four constranng equatons and four unknowns that need to be solved smultaneously. Lke the frst sub-routne, we wll fnd a local curve of C3 after rotaton. The functon, f(x), usng least squares fttng, wll agan take the followng form: f ( x) ax bx c. (9) Note that f(x), n ths case, s for the local cam curve, C3, and the coeffcents wll be unquely dfferent. By takng the dervatve of ths functon, and evaluatng t at Xt 3, the frst equaton can be formed to determne the slope of the buss cable, where K 3 represents the slope. 3 K3 ax t b (30) The second equaton, 1 3 Yp Yt K3 1 3 X p X t, (31) s the slope equaton from the tangent pont on the pulley to the tangent pont on C3. The thrd equaton s the equaton for the crcular pulley wth a radus, r. 1 1 p X c ) ( Yp Yc ) (3) ( X r The last equaton can be formed snce the tangent pont (Xp 1,Yp 1 ) must create a 90 degree ntersecton between the slope of the buss cable and the slope from the pulley s center. Therefore, 1 1 Y p Yc K 3 1 (33) X p X c If Equatons are solved smultaneously, and for Equaton 9, a(xt 3 ) +b(xt 3 )+c= Yt 3, the four unknown varables become Xp 1, Yp 1, Xt 3, and K 3. P a g e

30 To determne the rotaton,, of the pulley after an nput translaton, the length of the nextensble buss cable wll be consdered. The followng equaton for the length of the buss cable, BC, wll be used: BC S3 L3 S p1 (34) where S 3 s the length wrapped on C3, L 3 s the length of the buss cable from the tangent pont (Xt 3,Yt 3 ) to the tangent pont (Xp 1,Yp 1 ), and S p1 s the length of strng wrapped on the non-draw sde of the pulley. The length, BC, s assumed to be constant. S 30 (ntal length) wll have ts largest value, and can be measured. As the cam rotates C3 lets out strng. S 3 can be calculated by the change n arc length. By usng Equaton 5, where -1 s the prevous tangent pont, s current tangent, and f(x) s local curve functon that s calculated usng least squares fttng for C3, the change n S 3 can be determned. The orgnal value, S 30, subtract the change n S 3 yelds S 3. Note the change n S 3 must be added to the accumulatve change on C3. L 3 s found usng the prevously known values from calculatng Equatons and the followng equaton. L ( Yp Yt ) ( X p X t ) (35) Usng Equaton 36, we can solve for S p1 to fnd the rotaton angle,, of the pulley. In Equaton 36, r s the radus of the pulley. S p1. (36) r In the fnal sub-routne, the nock pont, Yh, must be found. Snce we assume Yh to be drawn lnearly along the y-axs, the pont can be found consderng the length of the strng from the y- axs to the tangent pont on the bowstrng sde of the pulley. The ntal length, L 40, can be determned from measurement. It s: L ( X p X h0) ( Yp ). (37) Snce we know the rotaton that s requred of the pulley due to the constrant of C3 and the buss cable length, we can determne the addtonal length that s added to BS 4 ; the total length from (Xp,Yp ) to Yh for all teratons. There s an addtonal length, S p, that must be add to BS 4 3 P a g e

31 to account for the change n the strng angle,, from the y-axs. Therefore, the amount of addtonal strng s: S p r( ). (38) In Equaton 39, represents the change n the drawstrng angle, where 1. (39) From Equaton 37 and 38, BS 4 becomes, ( 1) ( 1) BS 4 ( X p X h( 1) ) ( Yp ) r( ) (40) Equaton 40 ntroduces a few other unknowns that must be found n order to solve for Xh. The above unknowns are (Xp,Yp ) and. Equatons are used n conjuncton wth 40 to solve these four unknowns: Yh, Xp, Yp, and for all teratons. X p X c r cos (41) Yp Yc rsn (4) 1 cos (43) Y h Yh0 Y p Yh X p Y p Yh0 X p Yp Yh X p Yh Yh0 Yh can also be solved from the cam sde of the bow. To get a more accurate Yh, the mean of the two values wll be used. If the Yh found on the pulley sde and the Yh found on the C1 sde s relatvely far apart, then t s a good ndcaton that the desgn needs mprovement. To fnd Yh we wll assume that the bowstrng s dsconnected at the draw axs. To fnd Yh, we wll also need to know the tangent pont (Xt 1,Yt 1 ) and the slope, K 1, of the bowstrng for all teratons. Frst, a local curve usng least squares fttng, smlar to Equaton 9, must be found after translaton of C1. If the dervatve of the equaton, f(x), s evaluated at the tangent pont (Xt 1,Yt 1 ) the frst equaton becomes: 4 P a g e

32 1 K1 ax t b. (44) The second equaton s the slope of the lne from Xh to the tangent pont (Xp 1,Yp 1 ). 1 Yh Yt K1 1 X t (45) The thrd equaton s the length of the bowstrng from Xh to (Xp 1,Yp 1 ). The length of the bowstrng s the ntal length n addton to the amount of strng that has been unwrapped. Equaton 46 can be wrtten as: f ( x)' X h0 X t Yt X h X t Yt (46) 0 The unwrapped length s evaluated from the ntal tangent pont to the current tangent pont. If the local curve s used, we can substtute Yt 1 for a(xt 1 ) +b(xt 1 )+c. Ths wll reduce the number of unknown varables and equatons that are needed, and allow us to solve for Xh, Xt 1 and Yt 1 usng Equatons Force Analyss The objectve of determnng the tangent ponts s a key part to solvng the draw force for all teratons. Ths wll allow the drectons of each force connected to a cable to be known In Fg.15, the force drectons F1, F, F3, F4, F5, F6, Fsprng, and Fdraw are depcted. Fsprng may not act along the drecton the lmb tp travels. Although not shown, T1, T,T3, T4, T5, and T6 are the torques created by the forces F1, F, F3, F4, F5, and F6 respectvely multpled by ther moment dstances D1, D, D3, D4, D5, and D6 respectvely about the cam center of rotaton. For Fg.15, the forces are not drawn to scale, but are descrptve of ther drecton. 5 P a g e

33 6 P a g e Fgure 38 - Force Analyss; Buckmaster BTR D1, D, D3, D4, D5, and D6 can be solved usng a dmensonal pont-lne dstance equaton. Referrng to Fg.16, v s perpendcular to the lne specfed by two ponts (x 1,y 1 ) and (x,y ). v s gven by ( 1 ) 1 x x y y v. (47) The vector r s gven by y y x x r. (48) Now, by projectng r onto v the shortest dstance, as well as the moment dstance becomes ) ( ) ( ) )( ( ) )( ( y y x x y y x x y y x x d r v (49) In the case of the one cam one pulley desgn, the pvot center and the tangent pont on the cam consttute r, and the other pont that forms the lne wth the tangent pont s the other calculated pont on the strng. Refer to Fgure 13 or 14.

34 Fgure 39 - Pont-Lne Dstance Equaton [ To solve for Fdraw (Fg.15) the forces actng on the cam and pulley must be solved smultaneously. The lmb force, Fsprng, s known, snce the k-value s assumed to be known. The k- value s related to the drecton of tp travel, although the force drecton may or may not act along ths lne. If t does not, the magntude can be calculated by summng the dstance traveled multpled by the k-value and dvdng by the cosne of the angular dfference between the assumed lmb travel and the actual Fsprng drecton. The drecton of Fsprng (both Fsprngx and Fsprngy) wll need to be solved. The equaton, however, can be wrtten as: Fsprng Fsprngx Fsprngy (50) Also, snce F1 and F4, F and F6, and F3 and F5 are along the same cable, they are assumed to be equal. Let us examne the followng three equatons to extract useful nformaton that wll allow Fdraw to be solved for all teratons. Referrng to Fg.15, the followng three equatons can be formed by knematcally analyzng the stable system. F 4 F5 F6 Fsprng 0 (51) T1 T3 (5) T4 T5 T6 (53) By usng Equaton 5 and referrng to Fg.15, the assumpton can be made that F1 and F3 are equal. Snce both cables are connected to the same pulley, D1 and D3 are equal, therefore, F1 must be equal to F3. In addton, F4 and F5 are equal. If the forces are broken nto ther X and Y components and the drectons of F4, F5 and F6 are known, then there are only two unknowns. 7 P a g e

35 For example, f we are solvng the X component of F4, the Y component of F4 and both X and Y components of F5 are known. Refer to the equatons below, where A4 and A5 are angular drectons of forces F4 and F5 respectvely: F4y F4x tan( A4), (54) F5x F4x F4y cos( A5), (55) F5y F4x F4y sn( A5). (56) Smlar to Equaton 54, F6y can be solved. Let us next splt Equaton 58 nto ts X and Y components. We have: F4x F5x F6x Fsprngx 0 F4y F5y F6y Fsprngy 0 (57) (58) Now, by usng Equatons 50, 54, 57, and 58; F4x, F6x, Fsprngx and Fsprngy can be solved. The angle,, between the y-axs and the force drecton of F4 wll allow for Fdraw to be solved usng Equaton. The dstance Yh along wth the magntude of Fdraw wll allow for a draw-force curve to be plotted. As prevously mentoned, ths draw-force curve can be used to determne the amount of energy that s stored wthn the system durng a specfed draw length. Beng able to shoot a fast arrow s drectly related to the amount of energy that s stored n the system. In general, A draw-force curve that stores optmal energy wll reach the maxmum draw force quckly, plateau at a shelf, and reach ts holdng draw weght over a mnmal pulled dstance from the maxmum draw force. Besdes maxmzng the amount of stored energy gven a set of constrants, effcency s another mportant factor to consder to maxmze arrow speed. Chapter 4 wll dscuss how to calculate effcency for a compound bow and explan where there are effcency losses n a compound bow. In Appendx F, numercal results are shown for the frst 3 teratons usng an excel spreadsheet. More steps can be computed untl the holdng draw length, Yh, s reached. For ths analyss, a lmb step sze of 0.1 nches s used along the y-axs. The step sze along the x-axs s determned by followng the lmb travel prevously defned by (Xl 0,Yl 0 ) and (Xl 1,Yl 1 ). 8 P a g e

36 If the results of Appendx F-F3 are compared wth Appendx B s measured data we see that both start out along a lnear slope. The numercal results of appendx F show that for every nch the force ncreases lnearly about 5.7 pounds. The measured data s comparable at approxmately 6.5 pounds per nch. The dfference between these results could be a number of varous factors. Frst, the nput step sze may need to be decreased. Also, the measured cam data may need to be much more precse. For example, the measurement may need to be taken every degree or half degree, and the startng tangent pont for each cam may need to be more accurately determned. More accurate cam data wll yeld a more accurate local equaton for equatng the slope at each tangent pont. Dfference n results may also come from human measurement error. 9 P a g e

37 CHAPTER 4 EXPERIMENTAL WORK 4.1 Determnng Effcency of Compound Bow For determnng the effcency of a compound bow, two models were tested. The frst model tested s a Frestorm Lte by PSE. Ths model s of the hybrd cam type. The second model tested s a Buckmaster BTR brand bow of the sngle-cam type. Fg.17 shows both tested bows. Fgure 40 - Buckmaster BTR and PSE Frestorm Lte The effcency of a compound bow can be expressed by η; where η s the percentage of work converted to knetc energy when frng the arrow. Theoretcally ths value s between 0 < η < 1, but commonly t ranges from 70-85%. If there s conservaton of work, then the potental energy would be equal to the knetc, but there s loss of energy n many parts of the bow due to dynamc condtons. For example, strng vbraton, lmb nerta, arrow stffness, and cam mass. To fnd the bow s effcency we wll dvde the expermentally determned knetc energy by the expermentally determned potental energy. Therefore, η equals: mv F( l) dl (59) where m s mass, v s velocty, and F(l)dl s the force ntegrated over a pulled dstance l. 30 P a g e

38 To determne the knetc energy expermentally the arrow s weght and the bow s speed were measured. The arrow s weght for both bows tested was grans, or 9.57 grams. Speed was measured usng a chronograph. An average of fve speed tests found the arrows speed to be 31.5 fps wth a standard devaton of for the Frestorm Lte. These results correspond to a knetc energy of 73.6 joules. The average speed of the Buckmaster BTR was 0 fps wth a standard devaton of The amount of knetc energy produced by ths system was 66.5 joules. It should be noted that both bows were not set at IBO standard testng. To determne the expermental potental energy we frst construct a draw-force curve by measurng the amount of force for an ncremental draw length. In Fg.15, the bowstrng s attached to a fxed scale that measures force. The dstance was measured from the front of the arrow shelf to the nock pont. The Frestorm Lte had a drawng length of 0 ⅜ nches whle the Buckmaster had a drawng length of 0 ⅞ nches. The amount of work stored wthn the system can be found by ntegratng the draw-force curve from the brace heght to the fully drawn poston. An approxmaton for the work stored, or area under the draw-force curve used, was the average force between two measured forces multpled by the dstance between those ponts. The force-pounds for each segment were then totaled. A summary of the results found the potental energy of the PSE Frestorm Lte had an ascendng (drawn back) work of 813 n-lbs or 9 joules, and a descendng (let down) work of 796 n-lbs or 90 joules. The Buckmaster bow had an ascendng work of 814 n-lbs or 9 joules, and a descendng work of 779 n-lbs or 88 joules. 31 P a g e

39 Fgure 41 - Force/Measurement Test taken for PSE Frestorm Lte The effcency results were then calculated. The effcency usng the ascendng draw-force curve of the PSE Frestorm Lte was 80%. Lkewse, the effcency of the Buckmaster BTR was 7.%. It can be seen that the amount of work put nto both systems are very smlar, however snce the Frestorm Lte shot faster arrows ts effcency s hgher. Results are shown n Appendces B and C. Through these effcency tests, there are a few thngs to note. The frst, and most obvous, s that not all bows are created equal. Though arrow speed s probably the bggest desgn concern, effcency s also very mportant and t s drectly connected to arrow speed. Greater effcency allows the archer to produce the most amount of speed wth the least amount of effort. Also, effcency can vary for a gven bow for dfferent draw forces. In addton, conclusons can be made to compare overall effcency for testng varous bows, but where effcency s lost cannot be accurately determned. As prevously mentoned, there are many actng components whle frng an arrow. The lmbs are what store energy n a compound bow. Lmb nerta, dstance the lmbs travel, ts speed, and vbraton are all connected to the amount of energy that s transferred to the cam. Then, cam rotaton tself has a mass moment of nerta and a rotatonal dampng torque between the lmb and cam. Also, the bowstrng has ts own dynamcs. There are 3 P a g e

40 vbratons after arrow release, and the strng s slghtly extensble. The arrow cannot be assumed to be completely rgd ether, and t wll absorb some energy from the bowstrng. All of these dynamcs whch affect effcency are beyond the scope of ths study, but may need to be consdered for future work. 4. Carbon Fber Bow Lmb Manufacturng Composte materals are good choces for a compound bow lmb structure. Compostes are much less dense than steel and can carry the same mechancal propertes. For example, carbon fber s densty s about fve tmes less than steel and has a slghtly hgher tensle strength [ 009]. Also, although steel s much less expensve to manufacture, composte materals are used more often for ther thermal expanson propertes. The characterstcs of steel n 0 F are notceably dfferent than n 80 F. Carbon fber and fberglass compostes are much more consstent. These are a few reasons why bow manufacturers use composte lmbs on ther bows. To understand the manufacturng process, three desgn experments were performed. In these experments, t was the goal to successfully duplcate a bow lmb that has smlar propertes to bow lmbs that are on the market today. A good way to compare two bow lmbs s to measure ther sprng force value; k. k has unts of pounds per nch, for dsplacement measured at the tp of the lmb. In all tests k can accurately be assumed as lnear. One such bow lmb that was used for comparson has a k value of 5 pounds per nch. In a composte materal, there s a combnaton of matrx and fber that must act n concert to create a structure that yelds the propertes desred. The matrx materal that was chosen for the tests was a two part epoxy resn hardener. It s a medum vscosty, lght amber resn that s desgned for demandng structural applcatons. The epoxy resns are the system 000 seres [ 009]. The fber materal used for the experments was both undrectonal and plan weave carbon fber. Snce most all of the loadng s n one drecton, undrectonal fber s prmarly used. A few layers of plan weave were used to prevent cross spltng. For our frst and second experments a foam core was used, because t was thought that very lttle bendng stress would be concentrated near the center bendng axs. Other materals that were needed to perform the experments were: scssors for cuttng the fber, latex gloves and a plastc spoon for applyng the epoxy resn to the fber, a scale and a cup to wegh the epoxy resn mxture, towels, slcone, and plastc sheet for the vacuum baggng process, upper 33 P a g e