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### Counterbalancing Masses in Two or More Planes Formulas and Calculator

Machine Design Applications

Counterbalancing Masses Located in Two or More Planes Formulas and Calculator.

Unbalanced masses or weights rotating about a common axis in two separate planes of rotation form a couple, which must be counterbalanced by masses or weights, also located in two separate planes, call them planes A and B, and rotating about the same common axis. In addition, they must be balanced in the direction perpendicular to the axis, as before. Since two counterbalancing masses are required, two separate equations are required to calculate the product of each mass or weight and its radius, and two additional equations are required to calculate the angular positions. The planes A and B selected as balancing planes may be any two planes separated by any convenient distance c, along the axis of rotation.

Figure 1

For balancing plane A:

For balancing plane B:

Where:

MA and MB are the mass or weight of the counterbalancing masses in the balancing planes A and B, respectively;
rA and rB are the radii;
θA and θB are the angular positions of the balancing masses in these planes;
M, r, and θ are the mass or weight, radius, and angular positions of the unbalanced masses, with the subscripts defining the particular mass to which the values are assigned;
The length c, the distance between the balancing planes, is always a positive value;
The axial dimensions, a and b, may be either positive or negative, depending upon their position relative to the balancing plane; for example, in Fig. 1, the dimension b2 would be negative.

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Source: Kents Mechanical Engineering Handbook, 4th edition