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Question 12.42: A shaft with 3 m span between two bearings carries two weigh...

A shaft with 3 m span between two bearings carries two weights of 100 N and 200 N acting at the extremities of arms 0.45 m and 0.60 m long, respectively. The planes in which these weights rotate are 1.2 and 2.4 m, respectively from the left end bearing supporting the shaft (Fig.12.58). The angle between these arms is 60° as indicated in the inset (a) of Fig.12.58. The speed of rotation of the shaft is 200 rpm. If the weights are balanced by two counter- weights rotating with the shaft acting at radii of 0.3 m and placed at 0.3 m from each bearing centre, estimate the magnitude of the two balance weights and their orientation with respect to the x-axis, i.e. load A.

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\text { Given: } \quad r_{1}=r_{4}=0.3 m , r_{2}=0.45 m , r_{3}=0.6 m , W_{2}=100 N , W_{3}=200 N .

l_{2}=0.9 m , l_{3}=2.1 m , l_{4}=2.4 m .

Taking 1 as the reference plane, we have

100 \times 0.45 \times 0.9 \cos 0^{\circ}+200 \times 0.6 \times 2.1 \cos 60^{\circ}+W_{4} \times 0.3 \times 2.4 \cos \theta_{4}=0 .

40.5+126+0.72 W_{4} \cos \theta_{4}=0 .

W_{4} \cos \theta_{4}=-231.25 .

100 \times 0.45 \times 0.9 \sin 0^{\circ}+200 \times 0.6 \times 2.1 \sin 60^{\circ}+W_{4} \times 0.3 \times 2.4 \sin \theta_{4}=0 .

W_{4} \sin \theta_{4}=-303 .

\tan \theta_{4}=-303 /-231.25=1.31 .

\theta_{4}=232.66^{\circ} .

W_{4}=381 N .

Taking 4 as the reference plane, we have

100 \times 0.45 \times 1.5 \cos 0^{\circ}+200 \times 0.6 \times 0.3 \cos 60^{\circ}+W_{1} \times 0.3 \times 2.4 \cos \theta_{4}=0 .

W_{1} \cos \theta_{1}=-118.75 .

100 \times 0.45 \times 1.5 \sin 0^{\circ}+200 \times 0.6 \times 0.3 \sin 60^{\circ}+W_{1} \times 0.3 \times 2.4 \sin \theta_{4}=0 .

W_{1} \sin \theta_{1}=-43.3 .

\tan \theta_{4}=-43.3 /-118.75=0.3646 .

\theta_{1}=200^{\circ} .

W_{1}=126.6 N .

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