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Question 12.44: A shaft carries four masses in parallel planes A, B, C and D...

A shaft carries four masses in parallel planes A, B, C and D in order. The masses at B and C are 18 kg and 12.5 kg respectively and each has an eccentricity of 6 cm. The masses at A and D have an eccentricity of 8 cm. The angle between the masses at B and C is 100° and that between the masses at B and A is 190° (both angles measured in the same sense). The axial distance between the planes A and B is 10 cm and between B and C 20 cm. If the shaft is in complete dynamic balance, determine
(i) the masses at A and D
(ii) the distance between the planes C and D
(iii) the angular position of the mass at D.[IAS, 2004]

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\text { Given: } M_{ B }=18 kg , M_{ C }=12.5 kg , r_{ B }=r_{ C }=60 mm , r_{ A }=r_{ D }=80 mm .

\angle B O C=100^{\circ}, \angle BOA =190^{\circ} .

\text { Let } M_{A}, M_{D}=\text { mass at } A \text { and } D .

L = distance between A and D

Table 12.42
Mrl (kg m²)
(6)		 L (m)
(5)		 Mr (kg m)
(4)		 Eccentricity, r
(m) (3)		 Mass m
(kg) (2)		 Plane
(1)
0 0 0.08M _{A} 0.08 M _{A} A (RP)
0.108 0.1 1.08 0.06 18 B
0.225 0.3 0.75 0.06 12.5 C
0.08M _{D}L L 0.08M _{D} 0.08 M _{D} D

Draw couple polygon as shown in Fig.12.60(c) from the data in column (6) of Table 12.42

o^{\prime} c^{\prime}=0.08 M_{D} L=5.9 cm =0.236 kg \cdot m ^{2} .

\text { In Fig.12.60(b), draw } O D \text { parallel to } o^{\prime} c^{\prime} \cdot \angle A O D=71^{\circ} \text { so that } \angle B O D=251^{\circ} .

\text { 1. Draw } o b \| O B, o b=1.08 kg m .

\text { 2. Draw } b c \| O C, b c=0.75 kg m .

\text { 3. Draw } c d \| O A \text { and } o d \| O D \text { to meet at } d .

c d=0.08 M_{A}=3.9 cm =0.78 kg m , M_{A}=9.75 kg .

o d=0.08 M_{D}=3.3 cm =0.66 kg m , M_{D}=8.25 kg .

0.08 \times 8.25 \times L=0.236 .

L = 0.3576 m or 35.76 cm.

12.60a
12.60c

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