M_{friction} = (M/2)x/L = (15 kipft/2)
x / 100 ft
=
0.075x kipft
The drill head has a point moment of half the applied moment or
M_{drill head} = M/2 = 15 kipft / 2
= 7.5 kipft
The actual internal moment (or torque) is not the same as the applied moment.
A simple internal moment diagram can be constructed that shows the change of moment
as a function of depth. At the top, there is a 15 kipft moment, but is slowly
reduced by the friction. At the drill head end of the drill pipe, there remains
7.5 kipft internal moment. This moment is used to do the actual drilling. This
result can be modeled as function, giving
T(x) = 15  0.075x kipft (where
x is in feet)
Now that the internal moment function has been determined, the
total angle of twist can be calculated using
However, the shear modulus, G, and polar moment of inertia, J, are constant
and can be moved outside of the integral.
Both G and J can be calculated as
Also, the moment needs to be converted from kipft to kipin and the integrated length, ft, converted to in. Combining gives,
= (1.232 × 10^{3})(1,500
 375)
= 1.386 radians
or
θ = 79.40 degrees
