Question

I need complete and accurate solution.i am stuck here,I have to submit tommorow.
Pr.86.7A platform of weight W 4000 lb is being supported by four equal columns which are clamped to
the foundation as well as to the platform. Experimentally it has been determined that a static force of F=
1000 lb applied horizontally to the platform produces a displacement of A = 0.10 in. It is
estimated that damping in the structures is of the order of 5% of the critical damping. Determine
for this structure the following (a) undamped natural frequency, (b) absolute damping coefficient,
(c) logarithmic decrement
(d) the number of cycles and the time require for the amplitude of
motion to be reduced from an initial value of 0.1 in to 0.01 in.

Answer 1

Fav W 2D-representation 3D-representation As the system is symmetric, it can be represented by equivalent 2. dimensional figure as shown. K K , Kg & kq are the stiffnesses of column 1,2. 3 and a respectively, Let us consider the frame with column 1 and 2.

1 Kzkitka Equivalent lumped mass representations where, Kz stiffness of equivalent lumped mass system. Now, 2 K * U. N F K N 2x u N 10oo x 0.1 2 M 4000 27386 z 5.18135 O ka 5000 lb/in. M=W = - 900g = 5:18135 slags. . Undamped natural frequency (wn) av k → an 2 5018125 - 100 = 31.089 radys. M , 5000

(6 Damping coefficient (c) 2 2 x Ms Wox3. 22x5:18135*31:064 % 0.05 (cz 16.095 lb-sec/in. mic de crement (8)= 2 xł x 3 qar virst → (8) 22*** 0:05 1 1-0.052 8 = 0.3145 = (d) Let the number of cycles required an. in ( 4 ) = 273 x 1 Vizł xn. N Time required (trea) 2 Damped frequency (Wo) a ond 1-32 - 31.064 / 1-0.052 Wo 2 31.025 rates.

2a .. Period of vibration (T) 2 - 2 – 2* 31.025 2 T 20.2025 2 seconds. . Required time (treg) = Tan 20.20252 * 7.32 1.482 seconds