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 distance of the tower from the origin is 50 metres.Thus, when x =50, the value of y will give the height of the tower. Let the height of the tower be h. Then h = a cosh( 50/a). Also, since the sag is 10 metres, we have h = a +10. Therefore, a + 10 = a cosh (50/a). The solution to this equation will give the value of a.