EML 5060 Analysis in Mechanical Engineering 12/11/98
Closed book Van Dommelen 10:00-12:00am

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a standard size handwritten formula sheet.

1.

(25%) Classify the following Partial Differential Equation:

u_xx + 4 u_xy + 3 u_yy = 0

Put it in characteristic coordinates and solve it. Rewrite the solution in terms of the original coordinates x and y. Solution.

2.

(25%) A pipe that is closed at both ends contains air. Initially the air and pipe are at rest. Then, at time zero, the pipe is impulsively given a unit velocity along its axis. Find the velocity u(x,t) of the air in the pipe. From fluid mechanics, it may be derived that this velocity satisfies the following Partial Differential Equation:

u_tt = a² u_xx,

where a is a known constant (the speed of sound.) For this problem, the initial and boundary conditions are:

$$u(x,0)=u_t(x,0)=0, \qquad u(0,t)=u(\ell,t)=1,$$

where $\ell$ is the length of the pipe. Give a set of suitable functions to expand the solution u(x,t) in. Then give the solution u(x,t) in terms of these functions. Solution part 1. Solution part 2.

3.

(25%) One end of a very long bar is suddenly, at time zero, raised to a unit temperature. The further evolution of the temperature distribution u(x,t) in the bar is described by the following Partial Differential Equation:

$$u_t = \kappa u_{xx}, \quad (x\ge 0;\ t\ge 0),$$

where $\kappa$ is the conduction coefficient. The initial and boundary conditions are:

$$u(x,0)= 0, \qquad u(0,t) = 1.$$

Find the solution u(x,t) using a Laplace transform. From it, show that the temperature is constant on points that move away from the end as x = (constant) $\sqrt t$.Solution.

4.

(25%) 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

$$u(2,\theta,t)= 0.$$

For initial condition, assume that the membrane is struck at the point r=1 and $\theta=\pi/2$. In other words assume that $u(r,\theta,0)=0$ and that $u_t(r,\theta,0)$ is a delta function at r=1 and $\theta=\pi/2$. This means that

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

for any function $f(r,\theta)$. What frequencies will be present in the vibrations, and with what relative strength? Solution.