A shaft is supported in bearings 2 m apart and projects 0.5 m beyond bearings at each end. The shaft carries three pulleys one at each end and one at the middle of its length. The mass of end pulleys is 50 kg and 20 kg and their centre of gravity are 20 and 15 mm, respectively from the shaft axis

Question

A shaft is supported in bearings 2 m apart and projects 0.5 m beyond bearings at each end. The shaft carries three pulleys one at each end and one at the middle of its length. The mass of end pulleys is 50 kg and 20 kg and their centre of gravity are 20 and 15 mm, respectively from the shaft axis. The centre pulley has a mass of 55 kg and its centre of gravity is 15 mm from the shaft axis. If the pulleys are arranged so as to give static balance, determine (a) relative angular positions of the pulleys, and (b) dynamic forces produced on the bearings when the shaft rotates at 300 rpm.

Accepted Answer

[latex] ext { Given: } m_{ A }=50 kg , m_{ B }=55 kg , m_{ C }=20 kg , r_{ A }=20 mm , r_{ B }=r_{ C }=15 mm , N=300 rpm [/latex].

The position of shaft and pulleys is shown in Fig.12.59(a).

[latex] ext { Let } M_{L}, M_{M}= ext { mass at the bearings } L ext { and } M [/latex].

[latex] r_{L}, r_{M}= ext { radium of rotation of masses } m_{L} ext { and } m_{M} ext {, respectively. } [/latex]

Table 12.41
Mrl (kg m²) (6)	l (m) (5)	Mr (kg m) (4)	Radius r (m) (3)	Mass m (kg) (2)	Plane (1)
– 0.5	– 0.5	10	0.020	50	A
0	0	[latex]M_{L} r_{L}[/latex]	[latex]r_{L}[/latex]	[latex]M_{L}[/latex]	L(RP)
0.825	1.0	0.825	0.015	55	B
[latex]2M_{M} r_{M}[/latex]	2.0	[latex]M_{M} r_{M}[/latex]	[latex]r_{M}[/latex]	[latex]M_{M}[/latex]	M
0.75	2.5	0.30	0.015	20	C

[latex] \omega=\frac{2 \pi N}{60}=\frac{2 \pi imes 300}{60}=31.42 rad / s [/latex].

Draw the force polygon as shown in Fig.12.59(c) from the data in column (4). ob = 0.825 kg m is a vertical line. bc = 1.0 and oc = 0.30.
In Fig.12.59(b), draw OA parallel to BC and OC parallel to OC.

[latex] \angle A O B=165^{\circ}, \angle B O C=61^{\circ}, \angle A O C=134^{\circ} [/latex].

Draw couple polygon as shown in Fig.12.59(d).

[latex] o^{\prime} b^{\prime}=0.825 kg \cdot m ^{2} ext { is vertical, } b^{\prime} a ext { ' parallel to } OA ext { and } a^{\prime} c ext { ' parallel to } O C [/latex].

[latex] c^{\prime} o^{\prime}=2 M_{M} r_{M}=9.2 cm =1.84 kg . m ^{2} [/latex].

[latex] M_{M} r_{M}=0.92 kg m [/latex].

[latex] ext { Dynamic force on bearing } M=M_{M} r_{M} w^{2}=0.92 imes(31.42)^{2}=908.24 N [/latex].

Now draw force polygon as shown in Fig.12.59(e).

[latex] O b=0.825 kg . m ext { is a vertical line. } b m \| o^{\prime} c^{\prime} ext { and } bm = M _{M} r _{M}=0.92 kg \cdot m , mc \| ext { oc, } mc =0.3 kg m [/latex].

[latex] c d \| O A ext { and } c d=1.0 kg \cdot m [/latex].

[latex] O d=M_{L} r_{L}=4.6 cm =0.92 kg m [/latex].

[latex] ext { Dynamic force on bearing } L=M_{L} r_{L} w^{2}=0.92 imes(31.42)=908.24 N [/latex].

Question 12.43: A shaft is supported in bearings 2 m apart and projects 0.5 ...

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