The Hanging Chain Kirk Gordon, Torey Seward 6th of May, 2011 1
Contents 1 Introduction 4 2 Deriving the Catenary Curve from a Chain of Uniform Mass Density 4 2.1 Setting up the Equation....................... 4 2.2 Solving the Differential Equation of Constant Density...... 7 2.3 Graphing the Equation........................ 8 3 Deriving and Solving an Equation for a Chain of Varying Mass Density 9 3.1 Constants and Symbols....................... 9 3.2 Deriving an Equation for Varying Mass Density.......... 10 3.3 Solving the Differential Equation for a Chain of Varying Mass Density................................ 11 3.4 Graphing the Solution........................ 11 4 Applications Beyond Math 12 4.1 Marine Anchoring.......................... 12 4.2 Inverted Catenary Arch....................... 13 4.3 Suspension Bridges.......................... 14 5 Conclusion 15 6 Appendix 16 2
Abstract A differential equation modeling a hanging chain of either uniform or variable density will procure the catenary curve. This paper will first analyze a hanging chain in order to find a differential equation modeling its shape, then the equation will be solved. Furthermore, a chain of varying mass density will also be explored. It will be found that the shape of a chain of uniform mass will be a hyperbolic cosine function, which is a catenary curve. However, the solution for a chain of varying mass density will not be a catenary curve. The approach to solving the problem of a chain of varying mass density could be used to minimize the material used in cable production for power-lines and similar, non-load-bearing, hanging structures. Both solutions are important in their applications to architecture, marine anchoring and other problems. 3
1 Introduction If a chain, rope, or other string-like object of uniform density is hung between two parallel points and allowed to reach static equilibrium it takes on a unique shape. This shape is called the catenary curve, and deriving the catenary is a popular problem in the fields of physics, math, and engineering. The curve is a solution to a second order differential equation that models the change in incline of the hanging chain with respect to its change in height. The shape is seen in a variety of applications like architecture and marine anchoring. One variance on the classic problem that is explored in this paper involves a chain of nonuniform mass density. The solutions of both the classic catenary problem and varying mass density problem will be found using the technique of separation of variables and then integration. Matlab will be used to create plots of both solutions for comparison. 2 Deriving the Catenary Curve from a Chain of Uniform Mass Density In the case of uniform mass density, the rope will be symmetric about its lowest point. The origin for all equations in the case of the chain of uniform mass is at the lowest point on the chain. The only external forces acting on the chain are gravity; in other words the chain carries no load besides its own weight. 2.1 Setting up the Equation w = Weight per unit length T 1 = The horizontal component of Tension s = Length of chain 4
Figure 1: The tangential, vertical, and horizontal components of forces acting on the chain at any given point A few assumptions will be necessary in order to derive an equation. First, since there is no load on the chain and only the force of gravity acting on its mass, the horizontal tension will be the same at all points. The value of ws is equal to the weight of any section s. This analysis using Newtonian physics gives equation (2.1). This derivation is similar to that of [1]. tan θ = ws T 1 (2.1) If the shape of the hanging chain is treated as a function y(x) where y is the vertical distance of the chain from its lowest point and x is the horizontal distance from the middle of the chain (also the lowest point), then by similar triangles tan θ becomes y where y = dy/dx. 5
Figure 2: Similar triangles show the relation between y and tan θ y = ws T 1 y (x) = w T 1 s(x) Then s and y are divided into many tiny increments called d(y ) and ds. Using Pythagorean s Theorem, d(y ) = w T 1 ds (2.2) where ds = (dx) 2 + (y dx) 2 (2.3) dy = (. y )dx 6
Figure 3: The relation between ds and y Substituting (2.3) for ds in equation (2.2) gives, d(y ) = w T 1 1 + (y )2 dx (2.4) Using separation of variables, dx is divided from both sides of the equation. d(y ) dx = w T 1 1 + (y ) 2 y = w T 1 1 + (y ) 2 (2.5) Equation (2.5) is the second order ordinary differential equation modeling a hanging chain of uniform density. It can now be solved to find an equation which when plotted will directly model the shape of the hanging chain. 2.2 Solving the Differential Equation of Constant Density Here, equation (2.5) is solved using separation of variables, where the equation is rearranged so that the y variable is on the same side as d(y ) and the remaining constants are on the same side as dx. The equation is then integrated. d(y ) 1 + (y ) 2 = w T 1 dx dy 1 + (y ) 2 = w T 1 dx (2.6) The integration of the left-hand side of equation (2.6) is rather complex to do by hand, but an integration table can be used to easily find the solution. 7
The solution is an inverse hyperbolic sine function [3]. The C on the right hand side represents an unknown constant of integration. sinh 1 (y ) = w T 1 x + C (2.7) Here, x = 0 is the point at the bottom of the curve of the chain and since there is only a horizontal tension at this point, y (0) = 0 (The chain is flat at the very bottom of its curve, therefore having no slope). Thus, the constant of integration, C, is equal to zero. Furthermore, y prime can be isolated, giving the following equation which can be integrated again by separation of variables. y = sinh ( w T 1 x) dy dx = sinh ( w x) T 1 dy = sinh ( w x) T 1 This integration gives the equation for the shape of the hanging chain, again a table of integrals was used and D is another constant of integration. y(x) = T 1 w cosh w T 1 x + D As a reminder, the origin was at the center and lowest point of the chain. From this, a boundary condition can be set so that D is solved. y(0) = 0 D = T 1 w y(x) = T 1 w cosh ( w x) T 1 T 1 w (2.8) When plotted, equation (2.8) directly models the shape of a hanging chain of uniform mass density. 2.3 Graphing the Equation Despite outward appearances, the shape of the chain is not parabolic. The figure below is a plot of the solution for a chain of uniform mass density. In the plot, T 1 /w = 0.9. 8
Figure 4: A plot of Equation (2.8): Shape of a Hanging Chain with Uniform Mass Density 3 Deriving and Solving an Equation for a Chain of Varying Mass Density In the case of variable mass density, the initial analysis is similar, but the weight of the chain varies with the mass per unit length. The mass density of the chain increases proportionally with the tangential tension of the chain. The equation used here for the mass density is δ = c W (x) 2 + T 2, where δ is the mass density, c is a constant, W (x) is the one half the total weight of the chain below that point, and T is the tangential tension of the chain at that point x [2]. The equation for mass density could be almost anything, but for the sake of simplicity this is the equation used. 3.1 Constants and Symbols W (x) = Weight of half the Chain Below Point x δ = Mass density T (x) = Tension Tangential to the Chain at Point x c = Constant g = Gravitational Constant 9
3.2 Deriving an Equation for Varying Mass Density Weight equals mass times the gravitational constant g. The mass density equation is multiplied by g to find the weight W equation. From Figure 5, W (x) = T y. W = g δ = g δ x 1 + (y ) 2 Figure 5: The internal forces on either side of section AB And since δ = c W (x) 2 + T 2, Substituting W (x) = T y, T y = g x δ 1 + (y ) 2 T y = g x c W (x) 2 + T 2 1 + (y ) 2 T y = g x c (T y ) 2 + T 2 1 + (y ) 2 T y = g x c T 2 (y ) 2 + T 2 1 + (y ) 2 T y = g x c T (y ) 2 + 1 1 + (y ) 2 T y = g x c T [(y ) 2 + 1] y = g x c [(y ) 2 + 1] (3.1) Equation (3.1) is a differential equation which relates the change in slope of the chain to its change in height. The solution to this equation gives the shape of a chain of varying mass density. 10
3.3 Solving the Differential Equation for a Chain of Varying Mass Density Equation (3.1) can be solved using the technique of separation of variables and then integrating. d(y ) (y ) 2 + 1 = gc x dx tan 1 (y ) = gc x + C 1 The origin is in the middle and lowest point of the chain, where it is flat and the slope is zero. This means that at this point y (0) = 0, thus C 1 = 0. Equation (3.2) is integrated again to solve for y. dy = tan gc x y = tan(gc x) (3.2) y(x) = 1 gc ln sec(gc x) + C 2 Again, because of location of origin y(0) = 0, therefore C 2 is also equal to zero. y(x) = 1 gc ln (sec (gc x)) (3.3) Equation (3.3) is an equation modeling the shape of a chain of varying mass density. Unlike the first solution, this is not a catenary curve. 3.4 Graphing the Solution The result of this equation is shown here. 11
Figure 6: Solution for a Graph of Variable Mass Density, in this graph gc=19.6 4 Applications Beyond Math Both solutions are applicable to real-world situations. The inverted catenary curve serves as an arch in architecture and the curve appears in many everyday situations. The parameters for varying mass density of the problem in section three could be used in the production of cable for power lines to reduce the amount of material used because it only requires as much material as needed to hold up the section of cable below it. 4.1 Marine Anchoring In this paper s derivation of equation (2.5), Earth s gravitational constant was used for g, but it is not necessary to use this particular value. The solution can therefore be applicable to any situation where there is a homogeneous fluid, such as air or water. In the case of water, a drag constant can be substituted for g. This is seen in marine anchoring [2]. The efficiency of an anchor increases if excess line has been let out because it 12
causes the anchor to drag along the bottom. The force of gravity on the rope can be neglected because it is relatively smaller than the drag force, and the attachment to the boat and anchor are similar to the two suspension points of the hanging chain despite being misaligned. As a result, the excess line forms a catenary curve. Figure 7: Before excess line has been let out Figure 8: Letting out excess line in anchoring creates an underwater catenary curve 4.2 Inverted Catenary Arch The result for the chain of varying mass density is very similar to the Gateway Arch in St. Louis. The architect who designed the arch began with the inspiration to invert a hanging chain that had uniform mass density, but he was unsatisfied and wanted to alter it in some way. He achieved this by putting smaller, lighter chain links in the center of the chain and therefore altering the chain s mass per unit length [4]. The equation for mass density of the chain analyzed in this paper also had the smallest density at the center of the chain. 13
Figure 9: The Gateway Arch is similar to an inverted chain of variable mass density 4.3 Suspension Bridges A simple suspension bridge is in the shape of a catenary curve. This is because there are no supports besides the two suspension points at either end. If there is someone walking on the bridge, the bridge will still retain the shape of at catenary curve as long as the weight of the the load is relatively smaller than the weight per unit length of the bridge. Figure 10: A Simple Suspension Bridge 14
5 Conclusion The graphical results of both solutions are very similar, even though the solution to the varying mass density problem is quite a bit different from the catenary curve solution derived in section two. The main difference in equation (3.3) is its more flattened appearance in comparison to equation (2.8) due to changing vertical forces. A future project could be to solve for a chain of varying mass density using the equation δ = m x + b where b and c are constants. It would have to be done using a numerical method such as Matlab s ODE45 because of its asymmetry. 15
6 Appendix Plotting the Curve of a Chain of Uniform Mass Density x=linspace(-2,2); a=0.9; y=a*cosh(1/a*x)-a; plot(x,y) Plotting the Curve of a Chain of Variable Mass Density a=19.6; x=linspace(-0.05,0.05); y=1/a*log(sec(a*x)); plot(x,y) References [1] Simmons, George, and Steven Krantz. Differential Equations. New York: McGraw Hill, 2007. [2] Susanka, Larry.The Shape of a Hanging Rope. Bellevue College, Nov. 2005. Web. May 2011. http://scidiv.bellevuecollege.edu/ls/notes/cat1.pdf [3] OYoung, Josh J.K. Integral Table. Josh Jen Ken OYoung. 07 July 2010. Web. 16 May 2011. http://www.math.ucdavis.edu/ oyounggo/21b10/integral-table.pdf. [4] Kaza, Roger. No. 2645: Arch. University of Houston. Engines of Our Ingenuity, 2010. Web. 16 May 2011. http://www.uh.edu/engines/epi2645.htm. 16