by Tal Joseph

Madison Area Technical College

Abstract

Pushing a button to open your garage door before going to work seems trivial until the motor fails to lift the door. There are various explanations for a garage door malfunction, but this paper will focus on the most common of them all—a broken spring. Inspired by the fact that garage door technicians must match the right spring to the appropriate garage doors, this project produced a spring conversion calculator. This paper provides a preface and comprehensive explanation of the mechanisms of overhead garage doors. Initially, it introduces the garage door and its components, then provides a detailed explanation of a door’s lifting mechanisms, and lastly, elaborates on spring theory, focusing on the physics and calculations of spring properties.

Introduction

At approximately 7 to 8 feet tall and 9 to 16 feet wide, common residential garage doors, known as overhead doors, are large enough to fit one or two cars. The doors are typically made from four to five horizontal panels attached to one another by hinges. Garage doors are usually several hundred pounds, but they are relatively easy to lift manually because of one or more torsion springs, which are attached to rotating shafts above the doors. When a door is closed, the torsion springs are under tension, and the energy stored in the wound spring(s) does most of the work of raising the door.

There are six principal components of a garage door opener: cables, drums, torsion shaft, tracks, rollers, and springs (see Figure 1).

Figure 1. Garage Door Components. Adapted from Popular mechanics, by M. Iglesias, 2015, Retrieved April 11, 2017, from www.popularmechanics.com/home/outdoor-projects/how-to/a6041/garage-door-opener-how-it-works/. Copyright 2017 by Hearst Communications, Inc.

On each side of the door there is a thin cable that is connected to the bottom of the door and extends all the way to the drum above it (see Figure 2). The drums are pulleys with grooves that accommodate the cables. They sit on each side of a torsion shaft, which is a freely rotating metal rod that runs horizontally across the top of the door. At the sides of the door are rollers: wheel-like structures that keep the door aligned inside the side-tracks. The rollers allow the door to freely roll up and down. The torsion spring is installed on the torsion shaft, with one side bracketed to the wall of the garage and the other side locked against the torsion shaft by winding cones (see Figure 11).

While the door is closed and before the torsion spring is physically attached to the torsion shaft during installation, the spring is wound by rods T times to a torsion sufficient enough to generate an upward force equal to the weight of the door. For example, in the case of a 200-pound door with two torsion springs, each spring at maximum torsion should supply about 100 pounds of force. The spring is fully loaded when the door is closed; when the door is lifted, the spring unwinds and loses its power gradually. As the door lifts, the horizontal track compensates for the door’s immense weight. Although the spring becomes gradually weaker with the opening of the door, the door’s weight decreases as it moves vertically and horizontally, which enables the spring to continue pushing the door (see Figure 2).

Lifting Mechanisms

Before delving further into the calculations of the aforementioned lifting mechanism, alternative lifting mechanisms will be introduced. A vertical-lift garage door rises vertically and has no horizontal track (see Figure 3). Vertical-lift mechanisms are typical of industrial-sized garages, but they are uncommon for residential homes because it is unusual for a residence to have enough space for upward movement of the required height (7-8 ft.). The specialized component of this mechanism is the cone-shaped drum (see Figure 4).

The lifting capability of a spring is related to the diameter of the drum. A typical, evenly shaped drum would only be able to lift a vertical-lift door a short distance because the spring would lose tension and the weight of the door would remain constant, since there is no horizontal track to carry its weight. Special cone-shaped drums compensate for the loss of spring tension and the constant weight of the door. When the spring starts spinning, the cables reel on the wider sides of the drums and approach the narrow sides as the door lifts (“Introduction to Garage Door Counterbalance”).

A high-lift garage door utilizes a third lifting mechanism. The high-lift mechanism is a hybrid mechanism that encapsulates qualities of the vertical-lift and the overhead lift. In residences that have high ceilings that aren’t high enough for the vertical-lift, high-lift garage doors—which lift higher than the typical overhead garage door but still rise horizontally—can be installed. The high-lift mechanism relies on a hybrid drum that combines elements from both previously discussed mechanisms (see Figure 5).

Spring Geometry and Essential Theory

The revolution of a circle about an axis—under the condition that the revolution never intersects the original circle—generates a “donut” or “tire shape” called a torus (see Figure 6).”

The torus has a volume given by: V=2π 2b2a (see Appendix A, 1) Because most springs in practice have a very small angle of pitch between coils, the spring can be modeled as a stacked column of closely spaced adjacent tori. The mean diameter, D, of the spring is twice the torus dimension, a, and the spring wire size, d, is twice the torus dimension, b, (see Figure 6).

Although garage door springs are called torsion springs, they don’t work based on torsion —i.e., a torque generated by twisting a bar about its longitudinal axis.. The torque—(the ability to rotate an object)—in a torsion spring results from curvature or bending. When a solid object is forced to bend around a center, stresses are generated on both sides of the solid (see Figure 7). The side on the outside diameter from the center is stretched and undergoes tension. In cross sections near the side farthest from the center, there are forces that originate from the rest of the solid and tend to pull the solid apart. The side on the inner diameter of the solid closest to the center is shortened and experiences compressive stress. In cross sections near the side closest to the center, there are forces that originate from the rest of the solid and tend to push the solid inward.

Evidently, there must be a place within the solid where these internal stresses are neither pulling out nor pushing in. This area is the neutral surface, and in a torus, the neutral surface is a section of a cylinder of the radius, a, and height, 2b = d, (see Figure 7). The coil is strained at any point in a circular cross section that does not lie on the neutral surface. The amount of strain at a distance y from the wire center—the neutral surface—due to bending around the center is given by:

(2)

The stress experienced at this point is directed perpendicular to the circular cross section and is given by:

(3)

E represents Young’s Modulus of Elasticity, which measures the stiffness of a solid material, and relates stress to strain. The stress is positive regarding tension for y>0 and negative regarding compression for y<0(see Figure 8).

The greatest stress due to bending occurs when y = b and y = -b and has a value of:

This stress is referred to as the bending shear stress, and fractures of a spring coil are most likely to begin at the inner and outer diameters of the bending shear (see Figure 9). Eventually, any torsion spring will fail after repeated use (see Figure 10).

The stresses on a circular cross section are of opposite direction above and below the neutral surface, hence, they develop a torque that acts on each circular face due to the bending. The magnitude of this torque on one circular face of a single coil is given by:

(see Appendix B)

But the magnitude is balanced by a counter torque on the other side of the wire element, so the net torque acting on each circular wire element is zero. The bending shear stress is expressed by the equation:

The unwound spring exerts no torque on the torsion shaft. However, when the spring is wound T turns, or full revolutions, by an external torque (due to technician rotating the winding cons), work is done on each active coil of the spring, and the resulting energy is stored in the wound spring. The coils, which are pinned against the winding cones, are dead coils, and the remaining coils are active coils, which are represented by N (see Figure 11). The amount of energy per active coil for a single turn of the winding cones is given by 2πK, where K is the spring rate given by:

(Wahl 1944)

When the torsion shaft is free to rotate, the energy stored in the spring generates a torque, which causes the torsion shaft and the attached pulley to rotate. The rotation of the torsion shaft and the pulley pulls on the cables attached to the bottom of the closed door. The initial torque is computed as:

Because torque is the product of force multiplied by a radius at perpendicular angles to the force, the initial lifting force exerted by the spring on the garage door is given by the following equation, where r is the radius of the pulley attached to the torsion shaft:

Due to friction, some additional force is required to fully lift the door. The additional force is applied by an automatic door opener’s electric motor, which has a relatively small horsepower, or a human pulling on the handle within a manual system.

Lastly, in order to discover the lifespan of a spring, the Wahl Correction Factor is used to obtain a more accurate estimate of the bending shear stress. Before moving to calculations, Table 1 summarizes the above equations and Table 2 provides an estimate of the spring’s lifespan (Karwa 2006):

For the following calculations supplied is the common material torsion springs are made of— ASTM A229 oil tempered steel wire which has a mean weight density of and a Young’s modulus of ("ASTM A229 Oil-tempered Steel Wire."). In addition, the calculations are based on an average of 5 “dead coils” which will be deducted from the total number of coils (rule of thumb for residential size springs). All of the cable on the pulley is effectively 2.00 inches from the center of the torsion shaft. The cable stretches very little so it needs to be lifted to the height of the door. an initial estimate of, T. The greater the value of T, the greater the torque delivered by the wound spring. There still needs to be some force exerted on the cable by the spring when the door is fully raised in order to keep the cable on the pulley; therefore, adding ¼ of a turn to the geometric estimate is necessary.

The calculation of the maximum force required to manually wind the spring is computed as follows: The maximum torque needed to wind the spring occurs at the end of the winding process and equals the maximum torque exerted by the spring. Using the industry standard of 18-inch winding rods which are inserted into the attached winding cones (see figure 11), the maximum force is the maximum torque divided by 18 in.

The following is an example of calculations made for an arbitrary spring:

Spring rate and torque: let’s pick a spring with wire size d = 0.243 inches, length (L) of 30.5 inches, and ID of 2 inches. Its mean diameter D = 2.243 inches (ID+d=D). The number of coils is approx. L/d = 30.5 inches / 0.243 inches = 126 coils. 126 minus 5 dead coils, or 121 active coils (N), is taken into consideration. Thus, the spring rate is K = (π*2.9*10^7 * (0.243) ^4) / (32 * 121 * 2.243) = 36.6 in/lb. (K=τ/N). Winding 7.25 turns * 36.6 in/lb. produce a torque of approx. 265 in/lbs. per spring.

Lifting Weight: The 4’ lift drums have a radius of 2’, so the lift of one spring is 262/2 = 131 lbs.

Stress and lifetime: Calculating the maximal stress of the spring’s wire will assist us to estimate the lifetime of the spring. The bending stress S in the spring wire is 32*265/(π*0.243^3) = 188 Kpsi. The Wahl correction factor is Wc = (4*2.243-0.243)/ [4*(2.243-0.243)] + 0.615*0.243/2.243 = 1.1578 and the Wahl-corrected stress is Wc * S = 1.1578 * 188 Kpsi = 218 Kpsi. This predicts about a 15,000-cycle lifetime.

Note: The Appendix: Mathematical Derivations is available in the downloadable pdf.

References

ASTM A229 Oil-tempered Steel Wire. (n.d.). Retrieved April 20, 2017, from

matweb.com/search/DataSheet.aspx?MatGUID=417e182b8e9c42e7b84e437ee233908d&ckck=1

DDM Garage Doors Since 1982. (n.d.). Retrieved April 11, 2017, from ddmgaragedoors.com/diy-instructions/intro-to-counterbalance.php

Garage Door Repair Madison WI. (2016, July 04). Retrieved April 11, 2017, from

https://www. madisongaragerepair.com/blog/page/2/

Iglesias, M. (15, October 02). [Garage Door Components]. Retrieved April 11, 2017, from

popularmechanics.com/home/outdoor-projects/how-to/a6041/garage-door-opener-how-it-works/.

Karwa, R. (2006). A text book of machine design. New Delhi: Laxmi Publications LTD.

Kinch, R. J. (2015, June). Calculating Spring Properties. Retrieved April 20, 2017, from

truetex.com/garage.htm

Shigley, J. E., Nisbett, J. K., & Budynas, R. G. (2011). Shigleys mechanical engineering design. New

York: McGraw-Hill.

Thomas, G. B., Heil, C., Weir, M. D., & Hass, J. (2010). Thomas calculus. United States: Pearson.

Wahl, A. M. (1944). Mechanical springs (First ed.). Cleveland, OH: Penton Pub. Co.

Addendum: A specific graphing software, provided by the college, was used to generate all figures displayed in this article. The graphing software that was used is the newest version of WinPlot©, which can be downloaded from MATC’s web site: faculty.madisoncollege.edu/alehnen/winptut/Install_Winplot.html.

Figure 10. Broken Spring. Adapted from Madison Local Garage Door Pros, retrieved April 20, 2017, from www.madisongaragerepair.com/garage-door-spring/. 2017 by Local Garage Door Pros.

Table 1 Table 2

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