The Physics Behind Garage Door Springs

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University of Nebrsk - Lincoln DigitlCommons@University of Nebrsk - Lincoln UReCA: The NCHC Journl of Undergrdute Reserch & Cretive Activity Ntionl Collegite Honors Council 017 The Physics Behind

1 University of Nebrsk - Lincoln DigitlCommons@University of Nebrsk - Lincoln UReCA: The NCHC Journl of Undergrdute Reserch & Cretive Activity Ntionl Collegite Honors Council 017 The Physics Behind Grge Door Springs Tl Joseph Mdison Are Technicl College Follow this nd dditionl works t: Prt of the Eductionl Methods Commons, Gifted Eduction Commons, nd the Higher Eduction Commons Joseph, Tl, "The Physics Behind Grge Door Springs" (017). UReCA: The NCHC Journl of Undergrdute Reserch & Cretive Activity.. This Article is brought to you for free nd open ccess by the Ntionl Collegite Honors Council t DigitlCommons@University of Nebrsk - Lincoln. It hs been ccepted for inclusion in UReCA: The NCHC Journl of Undergrdute Reserch & Cretive Activity by n uthorized dministrtor of DigitlCommons@University of Nebrsk - Lincoln.

2 Running hed: THE PHYSICS BEHIND GARAGE DOOR SPRINGS 1 The Physics Behind Grge Door Springs Tl Joseph Mdison Are Technicl College

3 THE PHYSICS BEHIND GARAGE DOOR SPRINGS The Physics Behind Grge Door Springs Abstrct Pushing button to open your grge door before going to work seems trivil until the motor fils to lift the door. There re vrious explntions for grge door mlfunction, but this pper will focus on the most common of them ll broken spring. Inspired by the fct tht grge door technicins must mtch the right spring to the pproprite grge doors, this project produced spring conversion clcultor. This pper provides prefce nd comprehensive explntion of the mechnisms of overhed grge doors. Initilly, it introduces the grge door nd its components, then provides detiled explntion of door s lifting mechnisms, nd lstly, elbortes on spring theory, focusing on the physics nd clcultions of spring properties. Introduction At pproximtely 7 to 8 feet tll nd 9 to 16 feet wide, common residentil grge doors, known s overhed doors, re lrge enough to fit one or two crs. The doors re typiclly mde from four to five horizontl pnels ttched to one nother by hinges. Grge doors re usully severl hundred pounds, but they re reltively esy to lift mnully becuse of one or more torsion springs, which re ttched to rotting shfts bove the doors. When door is closed, the torsion springs re under tension, nd the energy stored in the wound spring(s) does most of the work of rising the door. There re six principl components of grge door: cbles, drums, torsion shft, trcks, rollers, nd springs (see Figure 1).

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 3 Opertor/Opener Drum Torsion Shft Trck Torsion Spring Trck Rollers Figure 1. Grge Door Components. Adpted from Populr mechnics, by M.

4 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 3 Opertor/Opener Drum Torsion Shft Trck Torsion Spring Trck Rollers Figure 1. Grge Door Components. Adpted from Populr mechnics, by M. Iglesis, 015, Retrieved April 11, 017, from Copyright 017 by Herst Communictions, Inc. On ech side of the door there is thin cble tht is connected to the bottom of the door nd extends ll the wy to the drum bove it (see Figure ). The drums re pulleys with grooves tht ccommodte the cbles. They sit on ech side of torsion shft, which is freely-rotting metl rod tht runs horizontlly cross the top of the door. At the sides of the door re the rollers; wheel-like structures tht keep the door ligned inside the side-trcks, nd llow the door to freely roll up nd down. The torsion spring is instlled on the torsion shft with one side brcketed to the wll of the grge nd the other side locked ginst the torsion shft vi winding cones (see Figure 11). While the door is closed nd before the torsion spring is physiclly ttched to the torsion shft during instlltion, the spring is wound by rods T times to torsion sufficient enough to generte n upwrd force equl to the weight of the door. For exmple, in the cse of 00-pound door with two torsion springs, ech spring t mximum torsion should supply bout 100 pounds of force. The spring is fully loded when the door is closed; when the door is lifted, the spring unwinds nd loses its power grdully. As the door lifts, the horizontl trck compenstes for the door s immense weight. Although the spring becomes

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 4 grdully weker with the opening of the door, the door s weight decreses s it moves verticlly nd horizontlly, which enbles the spring to continue pushing the

Right illustrtion ws dpted from DDM Grge Doors, Retrieved April 11, 017, from ddmgrgedoors.com/diy-instructions/intro-tocounterblnce.php. Copyright 016 by DDM Web Services. Inc.

5 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 4 grdully weker with the opening of the door, the door s weight decreses s it moves verticlly nd horizontlly, which enbles the spring to continue pushing the door (see Figure ). Figure. Grge Door Side nd Inside Views. Left illustrtion ws dpted from Home nd Dollrs, By Chris, 016, Retrieved nd modified April 11, 017, from Right illustrtion ws dpted from DDM Grge Doors, Retrieved April 11, 017, from ddmgrgedoors.com/diy-instructions/intro-tocounterblnce.php. Copyright 016 by DDM Web Services. Inc. Lifting Mechnisms Before delving further into the clcultions of the forementioned lifting mechnism, lterntive lifting mechnisms will be introduced. A verticl-lift grge door rises verticlly nd hs no horizontl trcks (see Figure 3). Verticl-lift mechnisms re typicl of industril-sized grges, but they re uncommon for residentil homes becuse it is unusul for residence to hve enough spce for upwrd movement of the required height (7-8 ft). The specilized component of this mechnism is the cone-shped drum (see Figure 4): Figure 4. Coneshped Drum. Picture Figure 3. Verticl-Lift Door. Mde with Microsoft-Pint.

6 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 5 The lifting cpbility of spring is relted to the dimeter of the drum. A typicl, evenly-shped drum would only be ble to lift verticl-lift door short distnce becuse the spring would lose tension nd the weight of the door remin constnt, since there is no horizontl trck to crry its weight. Specil cone-shped drums, compenste for the loss of spring tension nd the constnt weight of the door. When the spring strts spinning, the cbles reels on the wider side of the drums, nd pproch the nrrow sides s the door lifts ( Introduction to Grge Door Counterblnce ). A high-lift grge door utilizes third lifting mechnism. The high-lift mechnism is hybrid mechnism tht encpsultes qulities of the verticl lift nd the over-hed lift. In residences tht hve high ceilings, tht re not high enough for the verticl-lift, high-lift grge doors which lift higher thn the typicl over door grge door, but still rise horizontlly cn be instlled. The high-lift mechnism relies on hybrid drum tht combines elements from both previously discussed mechnisms (see Figure 5): Figure 5. Hybrid Drum. Picture Spring Geometry nd Essentil Theory The revolution of circle bout n xis under the condition tht the revolution never intersects the originl circle genertes donut or tire shpe clled torus (see Figure 6). Figure 6. Spring Specs & Cross Section of Torus. Left prt ws illustrted with Winplot. Right section ws dpted from Fox Vlley Spring Compny. Retrieved April 1, 017, from Copyright 017 Fox Vlley Springs.

7 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 6 The torus hs volume given by V=π b (see ppendix A,1). Becuse most springs in prctice hve very smll ngle of pitch between coils, the spring cn be modeled s stcked column of closely spced djcent tori. The men dimeter, D, of the spring is twice the torus dimension nd the spring wire size, d, is twice the torus dimension b (see Figure 6). Although grge door springs re clled torsion spring, they do not work bsed on torsion i.e., torque generted by twisting br bout its longitudinl xis. The torque the bility to rotte n object in torsion spring results from curvture or bending. When solid object is forced to bend round center, stresses re generted on both sides Figure 7. Neutrl Surfce in Coil. Illustrted with Winplot. of the solid (see Figure 7). The side on the outside dimeter from the center is stretched nd undergoes tension. In cross section ner the side furthest from the center, there re forces tht originte from the rest of the solid nd tend to pull the solid prt. The side on the inner dimeter of the solid closest to the center is shortened nd experiences compressive stress. In cross section ner the side closest to the center there re forces tht originte from the rest of the solid nd tend to push the solid inwrd. Evidently, there must be plce within the solid where these internl stresses re neither pulling out nor pushing in. This re is the neutrl surfce, nd in torus, the neutrl surfce is section of cylinder of the rdius,, nd height, b = d (see Figure 7). The coil is strined t ny point in circulr cross section tht does not lie on the neutrl surfce. The mount of strin t distnce y from the wire center the neutrl surfce due to bending round the center is given by y. The stress experienced t this point is given by Ey Ey S D nd is directed perpendiculr to the circulr cross section. E represents Young's Modulus of Elsticity, which mesures the stiffness of solid mteril, nd reltes stress to strin. The stress is positive regrding tension for y > 0 nd negtive regrding compression for y < 0. (see Figure 8).

This stress is referred to s the bending sher stress, nd frctures of spring coil re most likely to begin t the inner nd outer dimeters of the bending sher (see Figure 9).

8 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 7 Figure 8. Stresses Above nd Under the Neutrl Surfce. Illustrted with Winplot. The gretest stress due to bending occurs when y = b nd y = -b nd hs vlue of S Ed. This stress is referred to s the bending sher stress, nd frctures of spring coil re most likely to begin t the inner nd outer dimeters of the bending sher (see Figure 9). Eventully, ny torsion spring will fil fter repeted use (see Figure 10). Figure 9. Typicl Ftigue Filure. Adpted from Mechnicl Springs (p. 31), by A.M Whl, 1944, New York: McGrw-Hill. Figure 10. Broken Spring. Adpted from Mdison Locl Grge Door Pros, retrieved April 0, 017, from by Locl Grge Door Pros.

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 8 The stresses on circulr cross section of the wire re of opposite direction bove nd below the neutrl surfce, hence, they develop torque tht cts on ech circulr

9 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 8 The stresses on circulr cross section of the wire re of opposite direction bove nd below the neutrl surfce, hence, they develop torque tht cts on ech circulr fce due to the bending. The 4 Ed mgnitude of this torque on one circulr fce of single coil is given by (see Appendix B). But 3D this is blnced by counter torque on the other side of the wire element, so the net torque cting on ech Ed 3 circulr wire-element is zero. The bending sher stress is expressed by the eqution S. The 3 D d unwound spring exerts no torque to the torsion shft. However, when the spring is wound T turns, or full revolutions, by n externl torque (due to technicin rotting the winding cones), work is done on ech ctive coil of the spring nd the resulting energy is stored in the wound spring. The coils, which re pinned ginst the winding cones, re clled ded coils, nd the remining coils re ctive coils which re represented by N (see Winding Cone Unttched Winding Cone Attched to Spring Figure 11). The mount of energy per ctive coil for single turn of the winding cones is given by πk, where K is the Winding rods Attched to Winding Cones 4 spring rte given by K Ed (Whl 1944). When the 3DN N torsion shft is free to rotte the energy stored in the spring genertes torque which cuses the torsion shft nd the Figure 11. Winding Cones & Winding Rods. Adpted from Mdison Locl Grge Door Pros, by stff. Retrieved April 0, 017, from Copyright 017 by Locl Grge Door Pros. ttched pulley to rotte. The rottion of the torsion shft nd the pulley pulls on the cbles ttched to the 4 Ed T bottom of the closed door. The initil torque is computed s i KT. Becuse torque is the 3DN product of force multiplied by rdius t perpendiculr ngles to the force, the initil lifting force exerted by the spring on the grge door is given by the following eqution, 4 i Ed T F, where r is the rdius r 3DNr of the pulley ttched to the torsion shft. Due to friction, some dditionl force is required to fully lift the door. The dditionl force is pplied by n utomtic grge door opener or humn pulling up on the hndle within mnul system.

10 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 9 Lstly, in order to discover the lifespn of spring, the Whl correction fctor, W c is used to obtin more ccurte estimte of the bending sher stress, WS c Tble provides n estimte of the spring s lifespn (Krw 006):. Tble 1 summrizes the bove equtions nd Tble 1. Summry of Equtions Tble. Estimted Lifespn of Spring V = π b τ = πed4 3D S = 3τ πd 3 K = πed4 3DN = τ N τ i = KT = πed4 T 3DN F = τ r W c S < lb in ~10,000 cycles W c S < lb in ~5,000 cycles W c S < lb in ~50,000 cycles W c = 4D d 4(D d) d D D = ID + d W c S < lb in ~100,000 cycles N = L "ded coils" d T (turns) Note. Adpted from Clculting Spring Properties by Richrd J. Kinch, 015. Retrieved 0 April 017 from Copyright 015 by Richrd J Kinch. In the following clcultions supplied is the common mteril torsion springs re mde of ASTM (Americn Society for Testing nd Mterils) A9 oil tempered steel wire which hs men weight 0.8 lb density of 3 in nd Young s modulus of E lb in ("ASTM A9 Oil-tempered Steel Wire."). Additionlly, the clcultions re bsed on n verge of five ded coils, which will be deducted from the totl number of coils, ccording to the rule of thumb for residentil size springs. All the cble on the pulley is effectively.00 inches from the center of the torsion shft. Becuse the cble stretches very little, it needs to be lifted to the height of the door. The greter the vlue of T (spring revolutions), the greter the torque delivered by the wound spring. There must be some force exerted on the cble by the spring when the door is fully rised in order to keep the cble on the pulley; therefore, dding one-fourth of turn to the geometric estimte is necessry.

11 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 10 The clcultion of the mximum force required to mnully wind the spring is computed s follows: The mximum torque needed to wind the spring occurs t the end of the winding process nd equls the mximum torque exerted by the spring. Using the grge door industry stndrd of 18-inch winding rods, which re inserted into the ttched winding cones (see Figure 11), the mximum force is the mximum torque divided by 18 inches. Clcultions The following is n exmple of clcultions mde for n rbitrry spring: Spring Rte nd Torque Spring rte nd torque: let s pick spring with wire size d = 0.43 inches, length (L) of 30.5 inches, nd ID of inches. Its men dimeter D =.43 inches (ID+d=D). The number of coils is pprox. L/d = 30.5 inches / 0.43 inches = 16 coils. 16 minus 5 ded coils, or 11 ctive coils (N), is tken into considertion. Thus, the spring rte is K = (π*.9*10^7 * (0.43) ^4) / (3 * 11 *.43) = 36.6 in/lb. (K=τ/N). Winding 7.5 turns * 36.6 in/lb. produce torque of pprox. 65 in/lbs. per spring. Lifting Weight The 4 lift drums hve rdius of, so the lift of one spring is 6/ = 131 lbs. Stress nd lifetime Clculting the mximl stress of the spring s wire will ssist us to estimte the lifetime of the spring. The bending stress S in the spring wire is 3*65/(π*0.43^3) = 188 Kpsi. The Whl correction fctor is Wc = (4* )/ [4*( )] *0.43/.43 = nd the Whl-corrected stress is Wc * S = * 188 Kpsi = 18 Kpsi. This predicts bout 15,000-cycle lifetime.

12 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 11 A. Volume of Torus Appendix: Mthemticl Derivtions Mke cut t distnce r, with b r b, inside the circle of rdius b whose center is distnce bove the xis of rottion (see figure 1). Let dr = the thickness of the cut. Upon rottion of the circle bout the xis this cut genertes cylindricl shell of rdius r, height x nd thickness dr. From the Pythgoren Theorem x b r dv 4 r b r dr. The element of volume of the shell is. Integrting this expression over the domin of r gives the volume of the torus. with y = r -. b V 4 r b r dr 4 y b y dy b The trigonometric substitution y bsin b b, results in the reltions tht 1 b sin, b y b cos, nd cos dy b d. Figure 1. Cross Section of Torus. Illustrted with Winplot. The volume is now expressed s: V / 4 sin cos cos 4 / 3 / / / b / 1 cos 3 b b d b d, where from the hlf ngle formul cos the torus is given by the following formul 1 cos. Since cos cos 0, the volume of V / 1 sin 0 0 b b b / (Thoms 005) This result cn lso be derived quite esily by using the second centroid theorem of Pppus (Thoms 005).

13 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 1 B. The strin prllel to the neutrl surfce for pure bending of single coil (see figure 13). Consider the rc QP which is distnce y from the center of the coil with b y b. The length of QP is y where is the centrl ngle mesured in rdins. If y > 0 QP is stretched compred to the rc AB on the neutrl surfce, while if y < 0 QP is compressed compred to AB. The strin of QP is therefore y y (Shigley, Joseph, nd Mischke 011). The resulting stress is perpendiculr to the circulr cross section nd is directed out (tension) if y > 0 nd is directed in (compression) if y < 0. From Hook's Lw which sttes liner reltion between force nd strin from equilibrium, the stress is given by S Ey, where E is the Young's modulus of elsticity. Figure 13. The Neutrl Surfce of Coil. Illustrted with Winplot.

14 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 13 C. The moment for pure bending of single coil (see figure 14). Since the stress is of opposite direction bove nd below the neutrl surfce there is bending moment or torque generted ginst the fce of the circulr element. The element of force on strip with verticl displcement of y is df SdA, where da is the re of the strip with verticl coordinte of y on circle, x y b, centered t the origin. Expressing this strip s length times width gives da = xdy = b y dy, hence. The element of bending df S xdy Exydy Ey b y dy moment is given by. d ydf E y b y dy Assuming tht the mteril is uniform so tht the Young's modulus is constnt, the bending moment ginst the circulr fce due to stress from the rest of the solid is given by the following b 4E b definite integrl. E y b y dy y b y dy b 0. Figure 14. Cross Section of the Coil. Illustrted with Winplot. Using the sme trigonometric substitution of y bsin 1 y so tht sin b, cos dy b d, nd cos, the moment becomes sin cos d. From the double ngle nd b y b hlf ngle formuls: sin cos 4Eb sin cn be evluted s 1 cos 4 4 / 0 nd sin 1 cos 4, so the moment integrl / / sin 4 d Eb Eb Eb Finlly, using tht Ed D = nd d = b the formul for the bending moment of single coil, 3D 4, is obtined (Whl 1944).

15 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 14 References ASTM A9 Oil-tempered Steel Wire. (n.d.). Retrieved April 0, 017, from =1 DDM Grge Doors Since 198. (n.d.). Retrieved April 11, 017, from Grge Door Repir Mdison WI. (016, July 04). Retrieved April 11, 017, from Iglesis, M. (15, October 0). [Grge Door Components]. Retrieved April 11, 017, from Krw, R. (006). A text book of mchine design. New Delhi: Lxmi Publictions LTD. Kinch, R. J. (015, June). Clculting Spring Properties. Retrieved April 0, 017, from Shigley, J. E., Nisbett, J. K., & Budyns, R. G. (011). Shigleys mechnicl engineering design. New York: McGrw-Hill. Thoms, G. B., Heil, C., Weir, M. D., & Hss, J. (010). Thoms clculus. United Sttes: Person. Whl, A. M. (1944). Mechnicl springs (First ed.). Clevelnd, OH: Penton Pub. Co. Addendum: A specific grphing softwre, provided by the college, ws used to generte ll figures displyed in this rticle. The grphing softwre tht ws used is the newest version of WinPlot, which cn be downloded from MATC s web site: fculty.mdisoncollege.edu/lehnen/winptut/instll_winplot.html.

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 1. The Physics Behind Garage Door Springs

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 1 The Physics Behind Grge Door Springs Clicking button to open your grge door before going to work seems like trivil thing. However, on one bright dy the door will