Free vibration

2.  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

4.  The weight of a beam is taken as an UDL acting on top of existing loads.

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/m2. 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/m2 = 200 × 109 N/m2

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/m3 and Young’s modulus is 200 GN/m2.
Assume the shaft to be freely supported.
Given
d = 20 mm = 0.02 m ;
l = 0.6 m ;
m1 = 1 kg ;
ρ = 40 Mg/m3
= 40 × 106 g/m3
= 40 × 103 kg/m3
E = 200 GN/m2
= 200 × 109 N/m2

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. Calculate the natural frequency of a shaft 30 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 30
Mg/m3 and Young’s modulus is 200 GN/m2. Assume the shaft to be freely supported.