Problem Statement

A helicopter rotor can be modeled as three rectangular blades stretching outward at equal angles around the central rotor

shaft. Each blade has a length $R=5m,$ width $W=0\mathrm{.}2m,$ and mass $m=50kg.$ The rotor shaft itself has a mass of $100$ kg

and a radius of gyration $k=0\mathrm{.}2m.$ When the gearbox is disengaged, the friction on the rotor shaft is negligible.

Assume a constant local drag coefficient at any point on the blade of $C_D=0\mathrm{.}2($based on the surface area, not the frontal

area$).$ If the gearbox is disengaged when ${}_0=60rpm,$ obtain a solution for the rotational speed $(t)$ and plot the rotational

speed as a function of time for $(\frac{R}{2})\frac{R}{2}0s.$

Validation

Calculate the drag force using the midpoint speed $(\frac{R}{2})$ and use$\frac{R}{2}as$ the moment arm. This $is$ incorrect $-$ however,

because the velocity increases with radius, $as$ does the moment arm, the actual value should $be$ greater than the value

from this simple calculation. Repeat the calculations using the speed and moment arm $at$ the outer tip $of$ the blade.

The result $is$ also incorrect, but should provide $an$ upper $limit.$ These two answers should bracket the correct solution.

Applying the appropriate physic$\frac{s}{e}$ngineering principles should result $in$ a differential equation. You should $be$ able $to$

solve this equation $by$ hand $to$ get $an$ analytical solution $(as$ your "primary" solution$).$ Validate this solution $by$ using $a$

numerical method $(Euler\text{'}s$ Method, Matlab, EES$)tore-$solve the differential equation.

Hint

Note that the velocity changes $as$ the distance from the rotor hub increases. $To$ find the moment due $to$ aerodynamic drag,

you will need $to$ divide each blade into small segments and calculate the moment produced $by$ the airspeed over the blade $at$

that radius. You can then add the moment due $to$ each segment $to$ determine the total moment.