Problem $2[25$ pts$]$:

Dynamic Balancing of a compressor. Professor Wandke spent a summer working for an aerospace company's research division building robots to perform on$-$wing repairs of turbofan engines. Inside these engines, large rotors spin to compress air from the intake. The speeds of some components can be around $10,000$ rpm$.$ At these speeds, a misbalanced compressor blade will hit the side$-$wall of the engine, completely destroying the other blades and sending shards of metal through the entire mechanism. The compressor shaft has a diameter of $8$ cm $,$ and the blades can be modeled as cylinders with a diameter of $2$ cm and a length of $5$ cm long. To counteract small manufacturing variations, a proposed solution is to use in pre$-$defined attachment planes for adding balancing masses.

In an investigation to analyze the unbalanced loads acting on the top and bottom shaft bearings of a new compressor, the sensors revealed that the reaction forces due to rotation of blades are not zero and were recorded at a moment in time as the following:

At Bearing $1$: $\backslash (\backslash$mathrm$\{$F$\}\_\{2\}=200\backslash$mathrm$\{$~N$\}\backslash )$ in $-$y direction

At Bearing $2$: $\backslash (\backslash$mathrm$\{$F$\}\_\{1\}=150\backslash$mathrm$\{$~N$\}\backslash )$ in x $-$y plane with $30$ degree $\backslash (\backslash$mathrm$\{$w$\}/\backslash$mathrm$\{$r$\}\backslash )$ to the x $-$axis

The engineers need to balance the compressor by adding some masses at pre$-$defined attachment positions, defined at Plane A and Plane B to bring all reaction forces to zero. Obviously, the reaction forces due to gravity are never zero, so discard gravity in your calculation.

a$)$ What category of balancing need to be considered for resolving this problem? Explain why it is not possible to balance the entire shaft by adding mass just in one plane.

b$)$ Consider the two attachment planes $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ perpendicular to the shaft. Find the appropriate mass$-$ distance $(\backslash ($ m $\backslash$times r $\backslash ))$ and their directions with respect to x $-$axis that need to be attached at each plane in order to balance all bearing forces. Do the reaction forces remain zero all the time as the rotor rotates? Why?