Question

For the following system (a) write the equation of motion in terms of the vertical displacement of mass that both pulleys are massless and frictionless m and (b) find the natural frequency of the system. Assume X 4h аЕ во

Answer 1

国(en: HR 34k VS &ke - 2K ke = ktk-2k Je =2k| 水泉 (key=2x)

t o pellega let ing be the tension in the lord and x, and x2 thu clisplacement of pulley I and a reputinely. Now if pulley 2 is find if pulley tis fined x=282 sinaif bothers are not fixed (x= 2x2 + 2x2 = 22, Consider FBD of pally a & pulley2 ke,X,

to zero For equilibrium, sum of fous must be equal & mg - kez h₂=0 12 - ventil the time aguards Similarly, kex I pulley 1 Honce [x = 22, 42 x2] *** if * = bong Terminalet spring L 4mg X ker kez

ake where key = ke, a 2k l steps) Natural forgwary of vibration for those in AZ 4.2 an um

mx.tkqx=0] Egen ofmotion where KE Kier [x = 2x, +222 ***

summary

$f_{n}=(1/2\Pi)\sqrt{K/4m}$  

$\omega _{n}=\sqrt{K/4m}$

$m\ddot{x}+Kx=0$ equation of motion

where

K = k/2

x = 2:11 + 2.12

Refer images

Thank you