Chapter 5.2, Problem 31E

Rotation of a Shaft Suppose the x-axis on the interval [0, L] is the geometric center of a long straight shaft, such as the propeller shaft of a ship. See Figure 5.2.12. When the shaft is rotating at a constant angular speed ω about this axis the deflection y(x) of the shaft satisfies the differential equation

$EI\frac{d^{4}y}{d{x}^4}-\rho {\omega }^{2}y=0,$

where ρ is its density per unit length. If the shaft is simply supported, or hinged, at both ends the boundary conditions are then

y(0) = 0, y″(0) = 0, y(L) = 0, y″(L) = 0.

1. (a) If λ = α⁴ = ρω²/EI, then find the eigenvalues and eigenfunctions for this boundary-value problem.
2. (b) Use the eigenvalues λ_n in part (a) to find corresponding angular speeds ω_n. The values ω_n are called critical speeds. The value ω₁ is called the fundamental critical speed and, analogous to Example 4, at this speed the shaft changes shape from y = 0 to a deflection given by y₁(x).