1 The frame shown below consists of 2 elastic columns h3m E2

1) The frame shown below consists of 2 elastic columns (h-3m, E-200GPa, l=50*106 mm) and a rigid horizontal beam. Mass of the structure is 20 tons (a) Calculate the natural vibrational frequency of this structure. (b) Calculate the damping ratio applicable for this structure. (c) Write down equation of motion for this structure. (d) If it is pulled towards right by 2cm and left to undergo free vibration, calculate its response (x( Show all your derivation steps. Provide an Excel or MatLab graph of response (i.e.t against x(t (e) What is the amount of horizontal sway of this structure after 55 seconds (i.e. x(55)) 2cm EI Before deformation After initial deformation

Solution

Solution:-

Given

E = 200 GPa = 200 *10³ N/mm²

I = 50 *10⁶ mm⁴

L =3 m = 3000 mm

K = K₁ + K₂

K = 12EI/L³ + 12EI/L³

K = 12*200*10³*50*10⁶/(3000)³ *2 = 8888.88 N/mm = 8.88888*10⁶ N/m

M = 20 tons = 20*1000 = 20000 kg

(a) Natural vibration frequency (_n) = (K/M)

= (8.88888*10⁶/20000)

= 21 radian/second   Answer

(b) Let actual damping = C

Critical damping (C_cr) = 2(KM)

= 2(8.88888 *10⁶*2*10⁴)

= 843.273 *10³ N-s/m

Damping ratio() = C/C_cr

= C/(843.373*10³)

Damping ratio () = 1.185*10^-6 C     Answer   

(c) Equation of motion

M + C + K X = 0

20000 + C + 8.88888*10⁶ X = 0            Answer