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

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1 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 refuse to obey nd the motor will fil to lift it more thn one foot ( Grge Door Repir 016). There re vrious resons nd explntions for such thing, but this pper will focus on the most common of them ll broken spring. This pper ws inspired by the needs of grge door technicin to mtch the right spring to the pproprite grge door. This project produced spring conversion clcultor, the preprtion of which provides prefce nd comprehensive explntion of the mechnisms of opertion for over-hed grge doors. Initilly, this pper introduces the grge door nd its components, then, provides detiled explntion of door s lifting mechnisms, nd lstly, elbortes on the spring theory, focusing on the physics nd clcultions of spring properties. Common residentil grge doors re seven to eight feet tll nd pproximtely nine-to-sixteen feet wide, enough to fit one or two crs. The doors re typiclly mde from four to five horizontl pnels ttched to ech other vi hinges. A grge door is ctully quite hevy, usully severl hundred pounds. This might seem surprising since they re reltively esy to lift mnully. Wht mkes this possible is one or more torsion springs ttched to rotting shft bove the door. When the door is shut, these springs re under tension nd the energy stored in the wound spring(s) is used to do most of the work of rising the door. There re six principl components to grge door opener: cbles, drums, torsion shft, trcks, rollers, nd springs (see figure 1).

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

2 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 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). During instlltion, before the torsion spring is physiclly ttched to the torsion shft nd while the door is ll the wy down, the spring is wound T times with rods to torsion sufficient to generte n upwrd force equl to the weight of the door. For exmple, for 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. Tht is where the horizontl trck provides compenstion for the door s immense weight. The door s weight

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 3 decreses s it moves up nd more horizontlly, which enbles the spring to keep on pushing the door even though the spring becomes grdully weker with the opening

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.

3 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 3 decreses s it moves up nd more horizontlly, which enbles the spring to keep on pushing the door even though the spring becomes grdully weker with the opening of 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. Before we delve ny further into the clcultions of the forementioned mechnism, I would like to briefly introduce some lterntive lifting mechnisms. In verticl-lift (see figure 3) the door goes stright up nd hs no horizontl trck. This mechnism is uncommon for residentil homes simply becuse it is unusul to hve the spce for the door to lift tht high. This mechnism is more typicl of industril-sized grges. The specilized component of this mechnism is the coneshped drum (see figure 4): Figure 3. Verticl-Lift Door. Mde with Microsoft-Pint. Since the lifting cpbility of the spring is relted to the dimeter of the drum, norml, evenly-shped drum would only be cpble to lift the door short distnce. This is becuse the spring would lose some of its tension, menwhile the weight of the door would remin constnt becuse there is no horizontl trck to crry its weight. With the specil cone-shped drums, the spring strts spinning while the cbles re reeling on the wider side of the drums, nd get closer Figure 4. Coneshped Drum. Picture to the nrrow side s the door goes up. Tht is how this mechnism compenstes for the loss of spring tension versus the constnt weight of the door ( Introduction to Grge Door Counterblnce ).

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 4 A third mechnism is clled the high-lift. This is hybrid mechnism tht encpsultes qulities of both the verticl lift nd the over-hed lift.

4 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 4 A third mechnism is clled the high-lift. This is hybrid mechnism tht encpsultes qulities of both the verticl lift nd the over-hed lift. In households tht hve high ceilings, but not high enough for the verticl-lift, there is opportunity to cpitlize on the size of the room nd instll door tht cn lift higher thn the typicl door, but still goes horizontl. This mechnism relies on hybrid drum tht combines elements from both previously discussed techniques (see figure 5): Now tht the bsics bout grge-door pprtuses nd mechnisms hve been ddressed, we cn initite our investigtion into spring theory. Imgine circle tht is rotted bout n xis tht does not intersect the originl circle. This genertes donut or tire shpe, clled torus, s demonstrted in figure 6. Figure 5. Hybrid Drum. Picture 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. The torus hs volume given by V=π b (see ppendix A.). Most springs in prctice hve very smll ngle of pitch between coils, so the spring cn be modeled s "stcked" column of closely spced

THE PHYSICS BEHIND GARAGE DOOR SPRINGS 5 djcent toruses. 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).

5 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 5 djcent toruses. 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 spring is clled torsion spring, it relly does not work bsed on torsion (i.e. torque generted by twisting like in torsion br). The torque (the bility to rotte n object like the torsion shft) in torsion spring is ctully the result of curvture or bending. When solid object is forced to bend round center, stresses re generted on both sides of the solid (see figure 7). The side furthest from the center is stretched nd is sid to be under tension. On ech cross section ner the side furthest from the center there re forces (from the Figure 7. Neutrl Surfce in Coil. Illustrted with Winplot. rest of the solid) tending to pull it prt. The side of the solid closest to the center is shortened nd experiences compressive stress. On ech cross section ner the side closest to the center there re forces (gin from the rest of the solid) tending to push it in. Evidently, then there must be plce inside the solid where these internl stresses re neither pulling out nor pushing in. This is clled the neutrl surfce nd for torus this is section of cylinder of 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 is the Young's Modulus of Elsticity tht reltes stress to strin. The stress is positive (tension) for y > 0 nd negtive (compression) for y < 0. (see figure 8).

This 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, which is depicted in figure 9.

6 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 6 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 mgnitude of S Ed. This 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, which is depicted in 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 7 Since the stresses on circulr cross section of the wire re of opposite direction bove nd below the neutrl surfce they develop torque or moment tht cts ginst

7 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 7 Since the stresses on circulr cross section of the wire re of opposite direction bove nd below the neutrl surfce they develop torque or moment tht cts ginst ech circulr fce due to the bending. 4 Ed The mgnitude of this torque on one circulr fce of single coil is given by (see ppendix B.). 3D But this is blnced by counter torque on the other side so the net torque cting on ech circulr wireelement is zero. The bending sher stress is expressed by the eqution S. The unwound spring Ed 3 3 D d exerts no torque to the torsion shft. However, when the spring is wound T turns (full revolutions) by n externl torque (due to the force exerted on the winding rods when the winding cones re rotted), work is done on ech ctive coil of the spring nd this 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 considered ctive (see figure 11). The Winding Cone Unttched Winding Cone Attched to Spring number of ctive coils is designted s N. The mount of energy per ctive coil for single turn of the winding cones is Winding rods Attched to Winding Cones given by πk, where K is the spring rte given by K 4 Ed (Whl 1944). When the torsion shft is free 3DN N to rotte the energy stored in the spring is used to generte 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. torque which cuses the torsion shft nd the ttched pulley to rotte. This in turn pulls on the cbles 4 Ed T ttched to the bottom of the closed door. The initil torque is computed s i KT. Since torque 3DN equls force times rdius t right ngles to the force, the initil lifting force exerted by the spring on the grge door is given by 4 i Ed T F, where r is the rdius of the pulley ttched to the torsion shft. r 3DNr Due to friction, some dditionl force is required to fully lift the door. This is supplied by the reltively smll horse-power electric motor of n utomtic grge door opener or humn pulling up on the hndle in mnul system.

8 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 8 Lstly, in order to discover the lifespn of spring we use W c which is the Whl correction fctor used to obtin more ccurte estimte the bending sher stress, nmely, WS c. Before moving to clcultions, Tble 1 summrizes the bove equtions nd Tble provides n estimte of the spring s lifespn (Krw 006): 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" T d (turns) Note. Adpted from Clculting Spring Properties by Richrd J. Kinch, 015. Retrieved 0 April 017 from Copyright 015 by Richrd J Kinch. For the following clcultions supplied is the common mteril torsion springs re mde of 0.8 lb ASTM A9 oil tempered steel wire which hs men weight density of 3 in nd Young s modulus of E lb in ("ASTM A9 Oil-tempered Steel Wire."). In ddition, the clcultions re bsed on n verge of 5 ded coils which will be deducted from the totl number of coils (rule of thumb for residentil size springs). All of the cble on the pulley is effectively.00 inches from the center of the torsion shft. The cble stretches very little so it needs to be lifted to the height of the door. n initil estimte of, T. The greter the vlue of T, the greter the torque delivered by the wound spring. There still needs to 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 ¼ of turn to the geometric estimte is necessry.

9 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 9 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 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 in. The following is n exmple of clcultions mde for n rbitrry spring: 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 Whlcorrected stress is Wc * S = * 188 Kpsi = 18 Kpsi. This predicts bout 15,000-cycle lifetime.

10 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 10 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).

11 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 11 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.

12 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 1 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).

13 THE PHYSICS BEHIND GARAGE DOOR SPRINGS 13 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.

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The Physics Behind Garage Door Springs

The Physics Behind Garage Door Springs 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