Question

e are choosing material for a round rod. The material must be a metal or alloy. The length of the rod must be close to 1 m, and its cross-sectional diameter must be between 1 mm and 3 mm. Its natural frequency of vibration, f, should be as low as possible. (a) Clearly list the constraints, objective, and free variables. (b) What is the performance index of the material? (c) What are the best 2 material candidates?

Answer 1

a)

Constraints are:

- The material must be a metal or alloy.

- The length of the rod must be close to 1 m.

- Its cross-sectional diameter must be between 1 mm and 3 mm.

Objective: Its natural frequency of vibration, f, should be as low as possible.

Free variables can be the density rho and Young modulus E.

b)

Generally, f = K (E*I / mL⁴)^0.5........where K is a constant, E is Young's modulus, I is section moment of inertia, m is mass per unit length.

For round rod, I = pi/32 * D⁴

m = rho*A = rho * pi/4 * D²

f ~ (E / rho)^0.5 (D/L²)

The higher the performance index, the more suitable the material. Since f should be as low as possible, we can choose 1/f as the performance index.

1/f ~ (rho / E)^0.5 (L² / D)

c)

Since D and L are constrained, the best materials will be having high rho/E which implies high rho and low E.

Plotting rho / E for various metals and alloys we can see that Cast Iron and Aluminum 6061 are the closest candidates.