Numerical Analysis Systems Optimization An electric power c

Numerical Analysis: Systems & Optimization

An electric power cable is supported from two towers (of equal height) 100 meters apart. Let a coordinate system with origin (x,y)=(0,0) be located midway between the bases of the two towers. The shape of the cable (called a catenary) is described by the curve

y=a*cosh(x/a),

where a is a physical constant and x and y are measured in meters. One wishes to determine the value of a such that the cable sags no more than 10 meters at its lowest point. Convert the problem of determining a to that of finding the root of a particular function. Give this function. Do not solve for a.

Solution

The two towers are placed 100 meters apart and the origin ( 0, 0) is located midway between the bases of the towers. Thus, the distance of the base of either tower from the origin is 50 meters. Further, when x = 50, the y-value will be the height of the tower. Let h be the height of the tower. Then, h = a cosh (50/a).

Also, the height of the tower is the distance of the lowest point of the catenary from the ground(which is equal to a) plus the sag. Now, if the sag is 10 meters, then h = a + 10. On equating the 2 values of h, we have the equation a + 10 = a cosh (50/a) or, a [ 1-cosh (50/a)] +10 = 0. The root of this equation will ascertain the value of a.