Final Analysis in Mechanical Engineering Due 12/15/93
Take HomeVan Dommelen

Show all reasoning. Solutions must be neat and logically presented. No attempt will be made to `figure it out': what I do not understand I do not give credit for. Any book can be used. Asking another student, TA, or instructor a question about a particular difficulty is allowed only if you have made all reasonable attempts to resolve the difficulty yourself. The question must be restricted to the minimum information needed to get started again. Looking at another student's solution or notes, allowing another to look at your solution or notes, or finding a complete or significant part of a solution together with another student or students is not allowed.

1.

Solve question 7.28 using series expansion to find the natural frequencies for the pressure in an acoustic duct that is closed at x=0 and open at $x=\ell$.Now approximate the pressure at x=0 using the most important 2 non-zero terms in the expansion, assuming that the speed of sound a=1, $\ell=\pi/2$, and that the initial pressure p(x,0)=0 while p_t(x,0)=1 for x<1 and 0 for x>1. Take the time for your approximation to be 2. followed by your social security number.

2.

Solve the same problem exactly using D'Alembert's formula.

3.

Find the vibrations of a very long string with at the end at x=0 a moving flexible support, by solving 7.36 using the Laplace transform. From it, find the amplitude at t=1. followed by your social security number and at x=1., assuming that a=p=1 and the motion of the end is linear, i.e. f(t)=t. Also find the displacement at the point given by interchanging the two values of x and t.

4.

Find the steady temperature distribution in a square plate with given heat flow into or out the four edges by solving problem 7.37. Assume that the left and bottom sides are insulated, p=f=0 and that there is constant heat flow through the top and right hand sides, q=constant and g=constant, find a two term approximation to the temperature when x=0.5 and y=0. followed by your social security number. The lower left corner is at zero temperature; assume q=1.

5.

Solve the steady heat conduction in a unit circle with given heat flow through the perimeter by solving problem 7.38. Find the temperature at r=0.5 and $\theta$ a decimal point followed by your social security number, if there is constant heat flow through the top half perimeter and none through the bottom, and the center is at zero temperature.

6.

Solve unsteady heat conduction on the same unit circle, if the boundary is insulated. The problem for $u(r,\theta,t)$ is $u_t = \nabla^2 u$$u_r(1,\theta,t)=0$Now assume that at time t=0, the solution has already evolved sufficiently that the only remaining term in the series expansion is the one that decays slowest in time. In that case, how long does it take for this solution to decay further so that only the fraction given by a decimal point followed by your social security number is left?