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Counterbalancing Several Masses Located in a Single Plane Formulas and Calculator

Engineering Materials
Tolerances, Engineering Design Limits ans Fits

Counterbalancing Several Masses Located in a Single Plane Formulas and Calculator

In all balancing problems, the product of the counterbalancing mass (or weight) and its radius are calculated; it is thus necessary to select either the mass or the radius and then calculate the other value from the product of the two quantities. Design considerations usually make this decision self-evident. The angular position of the counterbalancing mass must also be calculated.

Counterbalancing Several Masses

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Counterbalancing Several Masses Located in a Single Plane can be calculated by the following formulas:

Eq :

MB rB = [ ( ∑ M r cosθ )2 + ( ∑ M r sinθ )2 ]1/2

tanθB = - ( ∑ M r sinθ) / - ( ∑ M r cosθ ) = - y / - x

Image and Table 1
Relationship of Angle Function Signs to Quadrant in Which They Occur
Relationship of Angle Function Signs to Quadrant

Angle θ
Trigonmetric
Function
0° to 90°
90° to 180°
180° to 270°
270° to 360°
Signs of the Functions
tan
+y / +x
+y / -x
-y / -x
-y / +x
sine
+y / +r
+y / +r
-y / +r
-y / +r
cosine
+x / +r
-x / +r
-x / +r
+x / +r

 

Where:

M1, M2, M3, ..., Mn = any unbalanced mass or weight, (kg, lb),
MB = counterbalancing mass or weight, (kg, lb),
r = radius to center of gravity of any unbalance mass or weitgh, (mm, in),
rB = radius to center of gravity of counterbalancing mass or weight, (mm, in),
θ = angular position of r of any unbalanced mass or weight, (degrees),
θB = angular position of rB of counterbalancing mass or weight, degrees

Reference:

Machinery's Handbook 30th edition

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