A spring and a damper are connected to a mass as shown deriv

A spring and a damper are connected to a mass as shown; derive the equation of motion using Lagrange\'s equations. Ignore gravity. If m = 5 kg, k = 500N/m and c = 40N.S, and a driving frequency of 1 kHz is applied, what is the damping ratio? Is is under/over/critically damped, state why.

Solution

solution

Mass (m) is attached to Hook’s law spring

Fs = kx

Damping Force Fd = c (dx/dt)

From Newton

f = ma (1)

mx¨ = cx kx (2)

rearranging

mx¨ + cx + kx = 0 (3)

dividing by m

x¨ + c m x + k m x (4)

if we let 0 = ( k /m)^1/2 , be the natural frequency of the system and = c/ 2 (km)^1/2 , be the damping ratio, we get x¨ + 20x + ₀²x = 0

damping ratio =   = c / 2 (km)^1/2

= 40 / 2( 500 X 5 )^1/2

= 0.4 < 1

under damped