Question

A mass m, of 20 kg is attached to the free end of a steel cantilever beam of length 1000 mm having a cross-section of 25 mm × 25 mm. Assume the mass of the cantilever to be negligible and esteel = 200 gpa. If the lateral vibration of this system is critically damped using a viscous damper, find the damping constant of the damper.

Answer 1

Answer:

Given length of cantilever beam,

l = 1000 mm = 1 m

Moment of inertia of the shaft,

I =

1

bd3 =

25 × (25)3

12 12

= 3.25 × 10-8 m4

Esteel = 200 × 109 Pa.

Mass, M = 20 kg

∴ Weight W = 20 × 9.81 = 196.2

Now, deflection of free end,

δ =

Wl3

=

196.2 × (1)3

= 0.01 m

3EI 3 × 200 × 109 × 3.25 × 10-8

As the system is critically damped,

Cc

2 = ωn2

2m

Critical damping constant,

Cc = 2mωn = 2 × 20 × 31.22

= 1248.99 N-s/m2.