Transverse vibrations

Free vibration

1. Transverse vibrations

2. Effect of multiple loads/Energy (or Rayleigh’s) method To find the natural frequency of a beam on which multiple loads are acting, we find the deflection caused by each load separately and find the total effect. Multiple loads on SSB

3. Deflection in point loads

4. Effect of self weight/ Dunkerley’s method The weight of a beam is taken as an UDL acting on top of existing loads.

5. Deflection under self weight/UDL

6. A shaft 50 mm diameter and 3 metres long is simply supported at the ends and carries three loads of 1000 N, 1500 N and 750 N at 1 m, 2 m and 2.5 m from the left support. The Young's modulus for shaft material is 200 GN/m 2 . Find the frequency of transverse vibration. Given d = 50 mm = 0.05 m ; l = 3 m, W1 = 1000 N ; W2 = 1500 N ; W3 = 750 N; E = 200 GN/m 2 = 200 × 10 9 N /m 2

7. Moment of inertia of shaft Static deflections due to point loads,. 1000 N load 1500 N load

8. Static deflections due to point loads 750 N load Frequency of transverse vibrations

9. 2. Calculate the natural frequency of a shaft 20 mm diameter and 0.6 m long carrying a mass of 1 kg a its mid-point. The density of the shaft material is 40 Mg/m 3 and Young’s modulus is 200 GN/m 2 . Assume the shaft to be freely supported. Given d = 20 mm = 0.02 m ; l = 0.6 m ; m1 = 1 kg ; ρ = 40 Mg/m 3 = 40 × 106 g/m 3 = 40 × 103 kg/m 3 E = 200 GN/m 2 = 200 × 109 N/m 2

10. Moment of inertia of shaft Mass per unit length Static deflection due to point load

11. Static deflection due to self weight-UDL Natural frequency of shaft

12. Assignment Calculate the natural frequency of a shaft 3 0 mm diameter and 0.6 m long carrying a mass of 1 kg a its mid-point and 2kg at 0.4m from the left end . The density of the shaft material is 3 0 Mg/m 3 and Young’s modulus is 200 GN/m 2 . Assume the shaft to be freely supported.