Question

Q1. For the system shown in Figure 1 where the beam with mass m and length L is connected to the fixed surfaces through three springs with same stiffness k, (i) Calculate the total kinetic energy and total potential energy of the system; (ii) Derive the equation of motion in terms of rotation angle 0; (iii) Find the natural frequency of the system; (iv) Calculate the natural period if the stiffness k of all springs is doubled; (v) If the natural frequency is to be reduced by 50%, is it possible to achieve this goal by changing the location of the spring in the middle? Justify your answer through calculations. Figure 1

Answer 1

SOLUTION :- Given k & After the initial deflection 'Ⓡ', we get G = centre of gravity F=k6, x= 1 / 2 / 3 = 16 Moment of Inertia about point B, Ig Iz = IG + mx² In = Ta + mn* = med + momen met mit 1) (i) Total Total kinetic energy = Ź IA Total Potential energy = {kx+ 4ke+lkan DK (*+ 25 + 23) 2 Potential Ika - IK

8, = 21 = 0 (5 S2 = 4y = 0 (44) 8 = 8z = 0 (212) (ü) Total energy = kinetic + Potential energies E = 1 10 + ${x} +23+26) = constant According to conservation of energy, {(,) 0* + +4+34) 6+=0 (ME) 0* + ( 4 ) 6=0 Differentiating we get, Mooty 6 6 + K(84)o 6 =0 mat) ö + x(54) = 0 © + (ante=0 – 0 Equation of motion in terms of 8". - 0

(üy comparing with, ö + w = 0 - we get, war = 6k Natural = Wn = 6k frequency rad, V m bo = 24 = ve Hz f = un 1 [6k 27 m (IV) If the stiffness is doubled, K = 2K = 6(2x) Path m 112k for 12t Hertz

in middle :- (D Changing the location 24 = 8, = 0(3) le Zn = 88 = 0(24) 12 x -- 22 ,2 ,2 ImL - 219, * 47 24/3 As total energy is conserved, 4 To + (x + 1)^2 + . 3 (mokyo +2444948= Differentiating we get (moje + (23) 06 = 0 moky 8 + (201) 20 =0 ö + 2160=0 on a a una vera This is not Natural frequency is not reduced by 50% - 4 m