A steel shaft, of diameter \(50 \mathrm{~mm}\), is supported on two bearings and carries a pulley and a motor, as shown in Fig. 5.46. The masses of the pulley and the motor are \(100 \mathrm{~kg}\) and \(250 \mathrm{~kg}\), respectively. A transverse load applied at any point along the length of the shaft results in the deflection of all points on the shaft, hence the system exhibits elastic coupling. The stiffness coefficients are given by (see Section 6.4 for the definition of stiffness influence coefficients)

\[k_{11}=\frac{1296}{5} \frac{E I}{l^{3}}, \quad k_{12}=k_{21}=\frac{324}{5} \frac{E I}{l^{3}}, \quad k_{22}=\frac{216}{5} \frac{E I}{l^{3}}\]
Determine the natural frequencies of the system in bending vibration for \(l=2.5\) meters.

Transcribed Image Text:

3 Pulley Shaft, 50-mm diameter I 3 FIGURE 5.46 Pulley and motor on a shaft. 1 3 Motor