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A shaft carries four rotating masses A,B,C,D in the order along its axis.

$r_A \ = \ 18 cm, \ r_B \ = \ 24 cm, \ r_C \ = \ 12 cm $ and $r_D \ = \ 15 cm, \ m_B \ = \ 30 kg, \ m_C \ = \ 50 kg $ and $m_D \ = \ 40 kg.$

The plane containing 'B' and 'C' are 30 cm apart, the angular spacing of planes containing 'C' and 'D' are 90 $^{\circ}$ and 210 $^{\circ}$ respectively. relative to 'B' measured in the same plane. If the shaft and masses are to be in completely dynamic balance,

**Find:**

Mass and angular position of mass 'A'

Position of planes 'A' and 'D'.

$r_A = 0.18 \ m$ | $M_A \ = \ ?$ |
---|---|

$r_B = 0.24 \ m$ | $M_B = 30 \ kg$ |

$r_C = 0.12 \ m$ | $M_C = 50 \ kg$ |

$r_D = 0.15 \ m$ | $M_D = 40 \ kg$ |

Let, $M_A$ = Magnitude of mass A,

X = Distance between planes B and D,

Y = Distance between planes A and D

Plane | Mass(m) kg | Radius(r) m | cent.force$\div \omega^2$ (mr) kg-m | Dist from plane B(L)m | couple $\div \omega^2$ (mrL) $kg-m^2$ |
---|---|---|---|---|---|

A | $m_A$ | 0.18 | 0.18 $M_A$ | -Y | -0.18 MA$\cdot$Y |

B(R.P.) | 30 | 0.24 | 7.2 | 0 | 0 |

C | 50 | 0.12 | 6 | 0.3 | 1.8 |

D | 40 | 0.15 | 6 | X | 6X |

**Force Polygon :** 1cm = 1 kg - m

0.18 MA = 3.6 kg - m

MA = 20 kg

- The magnitude and angular position of mass Ma may be determined by drawing the force polygon to some suitable scale
- Positions of planes A and D may be obtained by drawing couple polygon.

**Couple Polygon :**

- From point c' and o' draw lines parallel to OD and OA respectively. such as they intersect at ' point d'

By measurement

6X = Vector c'd' = 2.3 $kg-m^2$

$\therefore X = 0.383 \ m$

We see from the couple polygon that the direction of vector c'd' is opposite to the direction of mass D. Therefore the planes of mass D is 0.383 n towards left of plane B and not towards right of plane B as already assumed.

Again by measurement,

-0.18MA$\cdot$Y = Vector o'd' = 3.6 $kg-m^2$

-0.18$\times$20$\times$Y = 3.6, Y = -1m

The -ve sign indicates that plane A is not towards left of B as assumed but it is 1 m towards right of plane B.