Question

A blade is fixed to a rigid rotor of radius R spinning at ωrad/s around the vertical z-axis (see figure). Neglect the effects of gravity. The x-axis rotates with the blade, and the origin x=0 of the axis is located at the "root" of the rotating blade (i.e., at the outer radius of the rotor as indicated in the figure.)

Given:

Young's modulus E, mass density ρ
Constant cross sectional area, A
Rotor radius R, blade length L
Angular velocity ω
Hint: if you work in the non-inertial frame of the rotating blade, where the x-axis is defined, the d’Alembert (inertial) force per unit volume is ρω2r=ρω2(R+x) along the +x direction. You can treat this force per unit volume just as you treated ρg in HW2_1. Also, the fact that the rotor is rigid means that you can consider the blade as "fixed" at x=0.

Note that we've given you a convenient MATLAB window at the bottom of this page; it may prove helpful for doing some of the integration needed in this problem, but you will not lose points if you choose not to use it.

Find a symbolic expression for the peak stress σnmax in the blade. Express your answer in terms of R, L, ρ, and ω (enter the last two as rho and omega, respectively).

Find a symbolic expression for the blade elongation δ in terms of R, L, ρ, ω, and E:

Find a symbolic expression for the displacement of the blade mid-section, ux(L/2), in terms of R, L, ρ, ω, and E: