Tuesday, June 9, 2009

Introduction To Single Plane Balancing

(The following is the generic introduction cited for single plane balancing in most standard American University Physics textbooks.)

The equation governing the principle behind Single Plane Balancing is simply that the sum of all forces on a moving system, including the d'Alembert intertial forces is zero.

ΣF – ma = 0

Notice that it is essentially a restatement of Newton's Second Law.

The masses generating the inertial forces are assumed to be in or nearly in the same plane.

From turbine blade wheels to automobile tyres, this kind of analysis is valid for all spinning objects whose radius is significantly longer than the width.

The figure above shows a situation where there are two unbalanced masses determined to be at two location w.r.t. the center of rotation. We can model this dynamically as two point masses concentrated at two locations as shown. We can solve for the required amount and location of a third mass that will balance the system.

The system is rotating at a fixed angular velocity. The acceleration of the masses is assumed to be strictly centripetal. The inertial forces will then be strictly centrifugal. Since the system is rotating, the drawing shown is a snapshot of a specific moment.

The above equation reduces to:

-m1*R1*ω2 -m2*R2*ω2 – m3*R3*ω2 = 0

Thus all the forces acting are inertial. Canceling the angular velocity,

-m1*R1 -m2*R2 – m3*R3 = 0

Separating into X and Y components,

-m1*R1x -m2*R2x – m3*R3x = 0
-m1*R1y -m2*R2y – m3*R3y = 0

We can readily solve for m3*R3x and m3*R3y when all other values are known.

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