Model the problem of determining the equilibrium position

Question

1. A small mass m is suspended by three springs as shown in the figure. The springs all have the same spring constant k, and the fixed ends of the spring are located as listed in the table. When the mass is at the coordinates (0, 0, 0) the springs are all at zero load. The y direction of the coordinate system is vertical.

Point	x coordinate	y coordinate	z coordinate
o	0	0	0
A	20	15	0
B	0	15	-20
C	0	15	20

(a) Model the problem of determining the equilibrium position of the mass m when loaded only by the weight of gravity. The model should be expressed as 3 equations in 3 unknowns.

(b) Assuming the springs are stiff, and therefore the displacements are small, develop a set of ODE's for the free motion of the mass and solve for the eigenvalues and vectors in terms of k and m.

(c) If the mass m is initially at zero displacement and is given a small initial velocity v^→ = x₀i^{^} + y₀j^{^} + 0K^, develop a solution for the subsequent motion.

2. Metal castings can sometimes develop a gap between the casting and mold as the molten metal hardens. The result can be a substantially altered cooling process.

(a) Set up a simple heat-­-transfer model to let you look at the effect of the gap on the cooling process. For the sake of simplicity, assume the problem to be two-­- dimensional with a square cross section for the casting, a square outer shape for the mold, and an air gap of thickness δ as shown in the figure. The mold sits in some ambient room temperature. State clearly all assumptions and develop a corresponding model of the heat transfer problem.

(b) Assuming that the casting is aluminum, the mold is diatomaceous earth and the gap is filled with air, see if you can arrive at and summarize some insight into the effect of the gap and its size on the heat transfer process.

3. The Duffing equation describes a nonlinear oscillator with a nonlinear spring (energy storage component) and possibly a damping component.

m(d²x/dt²) + C(dx/dt) + kx + εx³ = f(t),  where c, k and ε are constants.

(a) Show that the Duffing equation fails to meet the requirements for a linear operator equation.

(b) For the undamped (c = 0), homogeneous case (f = 0), the homogeneous DE falls into one of the categories for second order ODE's that can be integrated in general terms using a transformation of variables. Identify that transformation and show the general form that results.