The deflection w of a clamped circular membrane of radius r d subjected to pressure Pis given by (small deformation theory)

$$w ( r ) = \frac { P r _ { d } ^ { 4 } } { 64 K } \left[ 1 - \left( \frac { r } { r _ { d } } \right) ^ { 2 } \right] ^ { 2 }$$

where r is the radial coordinate, and $K = \frac { E t ^ { 3 } } { 12 \left( 1 - v ^ { 2 } \right) }$ , where E, t, and u are the elastic modulus, thickness, and Poisson's ratio of the membrane, respectively. Consider a membrane with P = 15 psi, $r_d$ = 15 in., E = 18 x 10$^6$ psi, t = 0.08 in., and $v$ = 0.3 . Make a surface plot of the membrane.

Molecules of a gas in a container are moving around at different speeds. Maxwell's speed distribution law gives the probability distribution P(v) as a function of temperature and speed:

$$P ( v ) = 4 \pi \left( \frac { M } { 2 \pi R T } \right) ^ { 3 / 2 } v ^ { 2 } e ^ { \left( - M v ^ { 2 } \right) / ( 2 R 7 ) }$$

where M is the molar mass of the gas in kg/mol, R = 8.31 J/(mol K), is the gas constant, Tis the temperature in kelvins, and vis the molecule's speed in m/s. Make a $3 - \mathrm { D }$ plot of $P ( v )$ as a function of $v$ and $T$ for $0 \leq v \leq 1000 \mathrm { m } / \mathrm { s }$ and $70 \leq T \leq 320 \mathrm { K }$ for oxygen (molar mass 0.032$\mathrm { kg } / \mathrm { mol } )$ .

An important derivative operation in many applications is called the Laplacian; in Cartesian coordinates, for z=f(x, y), the Laplacian is

$$z _ { x x } + z _ { y y }$$

. Determine the Laplacian in polar coordinates using the following steps. Use the Chain Rule to find

$$z _ { x x } = \frac { \partial } { \partial x } \left( z _ { x } \right)$$

. There should be two major terms, which, when expanded and simplified, result in five terms.

Find the deflection u(x, y, t) of the square membrane of side π and c²=1 for initial velocity 0 and initial deflection 0.1 xy(π-x)(π-y)

In the analysis of the bending of a uniformly loaded circular plate, the equation $w(r)$ of the deflection curve of the plate can be shown to satisfy the third-order differential equation

$$\frac { d ^ { 3 } w } { d r ^ { 3 } } + \frac { 1 } { r } \frac { d ^ { 2 } w } { d r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { d w } { d r } = \frac { q } { 2 D } r,$$

where $q$ and $D$ are constants. Here $r$ is the radial distance from a point on the circular plate to its center. (a) Find the general solution of the equation.