Miscellaneous Exercises 55
that follows, which can be done in closed form.d^2 u
dx^2 =γ(^2) u (^4) , 0 <x,
u( 0 )=U, xlim→∞u(x)= 0.
26.A uniform, straight shaft exhibits violent behavior at certain frequen-
cies of rotation. Let thex-axis between 0 andarepresent the undeflected
centerline of the shaft, and letu(x)be the displacement of the actual cen-
terline of the shaft measured from thex-axis. Centrifugal force provides
a transverse loading on the shaft whenuis not identically equal to zero.
The equation for the displacement is
d^4 u
dx^4 −
ω^2 w
EIgu=^0 ,^0 <x<a,
wherewis the weight per unit length of the shaft,gis the acceleration
of gravity,Eis Young’s modulus,Iis the second moment of the cross-
sectional area of the shaft, andωis the angular velocity. If the shaft is
held in narrow bearings at the ends, these can be interpreted as simple
supports, leading to boundary conditions
u( 0 )= 0 , u′′( 0 )= 0 , u(a)= 0 , u′′(a)= 0.
Find a formula for those values of angular velocity (critical values or
whirling speeds) that permit the existence of nonzero solutions to this
boundary value problem.
27.Find the lowest critical value for the angular velocity of a steel shaft with
these specifications: diameter 1.5 in.; length 48 in.;w= 0 .5lb/in.;E=
30 × 106 lb/in.^2 ,I= 0 .5in.^4.
28.Sulphur dioxide (SO 2 ) is a common air pollutant that reacts with water
to form sulphuric acid. If the water is airborne, the result is acid rain; if
the water is in snow, the result is acid runoff when the snow melts. Use an
analysis similar to that of Section 3 to obtain a boundary value problem
for the concentrationu(x)(in units of mass per unit volume) of sulphur
dioxide in the air included in a layer of snow. Introduceq(x),theflow
rate of sulphur dioxide (in units of mass per unit time per unit of cross-
sectional area.) There are two important physical facts: (1) Diffusion is
governed by Fick’s law (similar to Fourier’s law)
q(x)=−Ddudx,