EML 5060 Analysis in Mechanical Engineering 12/14/99
Closed book Van Dommelen 5:30-7:30 pm

Show all reasoning and intermediate results leading to your answer, or credit will be lost. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator and a single A4-size handwritten formula sheet.

1.

Solve unsteady viscous flow in a duct of constant width. The coordinate across the duct is x, so that after normalizing the duct width to unity, $0\le x\le 1$. The flow velocity through the duct, u(x,t), is governed by the Navier-Stokes P.D.E.

$$u_t = \nu u_{xx} - p_x/\rho,$$

where we can normalize things so that the coefficient of viscosity $\nu=1$ and the pressure gradient $p_x/\rho=1$ too. Assume the motion starts from rest, u(x,0)=0, and that the fluid is at rest at the walls of the duct, u(0,t)=u(1,t)=0. Solve for u(x,t) by formulating an eigenfunction expansion for u and then deriving and solving ordinary differential equations for the coefficients. (In other words, do not define another variable instead of u, by, say, substracting a steady solution, as you might have seen done in some books.) Solution.

2.

The temperature at the end of a semi-infinite bar is suddenly a bit increased. The resulting small temperature increase u(x,t) satisfies the heat equation with an additional radiation term:

$$u_t = \kappa u_{xx} - \alpha u.$$

At time zero, the incremental temperature is still zero away from the end, u(x,0)=0 for x>0. The boundary condition at the end of the bar is $u(0,t)=\epsilon$, where $\epsilon$ is a constant. Solve for u(x,t). Hint: it may actually be simpler first to solve for a general boundary condition u(0,t)=f(t) and then put in the given value for f at the last moment. Solution.

3.

The initial transverse displacement u(x,t) of a guitar string of length 2 is given by:

$$u(x,0) = x \hbox{ for } 0 \le x \le 1 \qquad u(x,0) = 2-x \hbox{ for } 1 \le x \le 2.$$

and the initial transverse velocity is zero. Find the transverse displacement in the middle of the string at time t=19 if the wave propagation speed in the string a=0.25. Solution.

4.

Solve the unsteady vibrations of the membrane on a circular drum. The membrane displacement $u(r,\theta,t)$ satisfies the wave equation

$$u_{tt} = a^2 \nabla^2 u.$$

The boundary condition at the edge is

$$u(1,\theta,t)= 0.$$

For initial condition, assume that the membrane is struck at the point r=0.5 and $\theta=0$. In other words assume that $u(r,\theta,0)=0$ and that $u_t(r,\theta,0)$ is a delta function at r=0.5 and $\theta=0$, so that

$$\int_0^1 \int_0^{2\pi} u_t(r,\theta,0) f(r,\theta)\/ r \/{\rm d} r\/ {\rm d}\theta = f(0.5,0)$$

for any function $f(r,\theta)$.Solution.