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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.

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

Eq :

M_{B} r_{B} = [ ( ∑ M r cosθ )^{2} + ( ∑ M r sinθ )^{2} ]^{1/2}

tanθ_{B} = - ( ∑ M r sinθ) / - ( ∑ M r cosθ ) = - y / - x

Image and Table 1

Relationship of Angle Function Signs to Quadrant in Which They Occur

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:

M_{1}, M_{2}, M_{3}, ..., M_{n} = any unbalanced mass or weight, (kg, lb),

M_{B} = counterbalancing mass or weight, (kg, lb),

r = radius to center of gravity of any unbalance mass or weitgh, (mm, in),

r_{B} = radius to center of gravity of counterbalancing mass or weight, (mm, in),

θ = angular position of r of any unbalanced mass or weight, (degrees),

θ_{B} = angular position of r_{B} of counterbalancing mass or weight, degrees

Reference:

Machinery's Handbook 30th edition

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