How Can Mass Placement Balance a Rotating Shaft?

[phil555]

Homework Statement

A shaft 2 m long rotates at 1500 revs min–1 between bearings as
shown in FIGURE 2. The bearings experience forces of 5 kN and
3 kN acting in the same plane as shown. A single mass is to be used
to balance the shaft, so that the reactions are zero. The mass is to be
placed at a radius of 200 mm from the shaft centre, 180° from the
direction of the bearing reactions. Determine the size and position (a
and b) of the mass to be used.

(b) The shaft in part (a) is to be balanced using two masses (m1 and m2)
placed 0.5 m and 1.5 m from end A and 180° from the direction of
the bearing reactions, each on radius arms 100 mm long. Calculate
the sizes of m1 and m2.

Homework Equations

This is where I am struggling

The Attempt at a Solution

Not started due to the above.

 

[gneill]

If you consider a frame of reference that rotates with the shaft (thus a non-inertial frame of reference), it becomes possible to deal directly with the centrifugal pseudo-force. In such a frame the shaft appears motionless and a mass m located at a distance r from the shaft produces a force directed outward along r that varies with r. What's the formula for that force in terms of the angular velocity ω of the shaft and the distance r?

Once you've got a handle on that, the problem becomes one of balancing moments about the ends of the shaft.