A shaft of span 1 m and diameter 25 mm is simply supported at the ends. It carries a 1.5 kN concentrated load at mid-span. If E is 200 GPa, its fundamental frequency will be nearly

Concept:

Angular frequency is given by:

where δ = deflection of the beam

Deflection of a simply supported beam when a concentrated load is placed at midspan is given by:

δ = 

Calculation:

Given:

L = 1 m, d = 25 mm, E = 200 × 10⁹ Pa, P = 1.5 × 10³ N

Deflection of the beam is:

δ = 

δ =  = 8.14 mm

Angular frequency is:

ω_n =  = 34.69 rad/s

Angular frequency can also be calculated by:

ω_n = 2πf

where f = frequency in Hz

34.69 = 2πf

∴ f = 5.52 Hz