Figure 2 shows views of
one of the three drums of a wire-saw to make silicon wafers. The drum is 1.328mlong,
with a cylinder of aluminum with 125mmouter
diameter and 75mminner diameter. The
shell of urethane is 25mmthick and
rests on the aluminum cylinder with no gap.
Each roller looks very much like a beam. The cutting wires are
spaced 1.260mmapart, beginning and ending 4mm
from the ends of the urethane, and each carries a tension of 5N. These wiresd spread at a 60 degree
angle to pass over three rollers.
Urethane does not possess a true Young’s modulus except in the limit
of no load. Rather its elstic modulus diminishes with increasing load, and one
usually specifies a 10%, 20%, or 30 % modulus. However, at vanishingly small
strain the modulus of this urethane is 690MPa.
Using this information calculate a load, shear, and moment diagrams for this
drum. Presume that the moment at the beam center applies to it uniformly, and
estimate both the curvature of the beam and its mid-point deflection from this
value of M. Hint: remember that to calculate the value ofIfor
a composite beam you do not have to draw the transformed section, but only
multiply the contribution to I of the urethane by a ratio of elastic moduli.
Because the drum rotates there is a cyclic stress that developes as
the lower and upper sides of the drum switch places. To cut an ingot requires
feeding 500kmof wire through the web in a cycle of 1000mforward
and 900mreverse. IOn other words each cycle makes use of only 100mof
new wire. How many stress reversals are involved?