EML 5060 Analysis in Mechanical Engineering 10/18/96
Closed book Van Dommelen 9:00-9:50am

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

In the numerical solution of the simplest one-dimensional heat conduction problem, you need to solve a linear system of equations with a `tridiagonal' matrix of the form:

$$A = \left(\begin{array} {rrrrr} 2 & -1 & 0 & 0 & 0 \ -1 ... ... 0 & 0 & -1 & 2 & -1 \ 0 & 0 & 0 & -1 & 2 \end{array}\right)$$

The work in solving this five by five system is not as much as you would expect due to all those zeros. They remain zeros. Show that this is true by solving $A {\bf x} = {\bf b}$ if ${\bf b}$ is the unit vector

$${\bf b}^T = (1,0,0,0,0).$$

Use Gaussian elimination without partial pivoting. The inverse matrix A^-1 is bad news since it does not have all these zeros. (The solution vector ${\bf x}$ is the first column of A^-1). Solution

2.

The two spring-mass systems below have identical masses m=1 and spring constants k₁. Unfortunately, they resonate since there is an external disturbance of frequency $\omega=\sqrt{k_1}$ present. In the hope of eliminating the resonance, you provide some mutual support to the masses by attaching a second spring with stiffness k₂ between them. The new equations of motion are:

$$\left(\begin{array} {c} \ddot x_1 \ \ddot x_2 \end{array}\... ...right) + = \left(\begin{array} {c} 0 \ 0 \end{array}\right).$$

Substitute in the assumptions

$$\begin{array} {c} x_1 = A \cos(\omega t + \phi), \ x_2 = B \cos(\omega t + \phi),\end{array}$$

where A, B, and $\phi$ are constants, to get an eigenvalue problem for the frequency $\omega$. Answer the question whether the additional spring eliminates the resonating frequency. If not, give the physical reason why not.

(lost picture showing from left to right: wall, spring k1, mass m, spring k2, mass m, spring k1, wall)

Solution

3.

In a slowly rotating coordinate system, the relative velocity vector ${\bf v}^T = (u,v,w)$ of a mass satisfies the equations of motion

$$\frac{d{\bf v}}{dt} = \left(\begin{array} {ccc} 0 & -\omega... ...mega_1 \ -\omega_2 & \omega_1 & 0 \end{array}\right){\bf v},$$

assuming that no forces act on the mass. Diagonalize the matrix and then solve the above differential equations for the diagonal matrix, in other words, solve

$$\frac{d{\bf v}'}{dt} = \left(\begin{array} {ccc} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{array}\right){\bf v}',$$

Show that the solution is periodic in time. Explain why you are not (or should not be) surprised for physical reasons. Solution