dynamic problem

CE 3012 STRUCTURAL ANALYSIS III TUTORIAL 1 Due on 28 th Feb, 2018

1. A single story building is idealized as a rigid girder supported b y weightless columns as shown in Fig. 1. In order to evaluate the dynamic properties of this structure, a free vibration test is done by displacing the roof s ystem (rigid girder) laterally by a hydraulic  jack and then suddenly released. The force needed to displace the girder by 5 mm was 90 kN. After the instantaneous release of the force, the maximum displacement on the first return is only 4 mm and the period of this displacement cycle is found to be 1.4 sec.. Determine the following following dynamic behavioural properties of the structure; structure; a) Lumped mass of the girder  b) Undamped frequency of vibration c) Damping ratio d) Damping coefficient e) Damped frequency of vibration f) Amplitude after 6 cycles

Fig.1 2. A free vibration test is conducted on an empty elevated water tank shown in Fig. 2. A cable attached to the tank applies a force of 75 kN and pulls the tank by 50 mm. The cable is suddenly cut and the resulting free vibration is recorded. At the end of four complete cycles, the time is 2 sec and the amplitude is 25 mm. From thes data, compute the following dynamic properties. a. Damping ratio  b.  Natural period of undamped vibration c. Lateral stiffness of the structure d. Lumped mass

e. Damping coefficient f.  Number of cycles required for the displacement amplitude to decrease to 5 mm. The weight of water required to fill the tank is 350 kN. Determine the natural vibration  period and damping ratio of the structure with the tank full. The supporting system of the tank is enlarged with the objective of increasing its seismic resistance. The lateral stiffness of the modified system is doubled. If the damping coefficient is unaffected by this process, determine the natural p eriod and the damping ratio.

Fig. 2

3. a) Consider undamped forced vibration of a SDOF system of mass m and stiffness k . The total response will be given by

with usual notations. Sketch the normalized response (v(t)/v st ) with time t/T  for  = 0.2,   = 0.5 , ̇   =  where T  is the period of harmonic loading and v st  is the steady state response p0 /k.  b) Now consider the response of an underdamped structure to harmonic loading and sketch the normalized response (v(t)/v st ) again with time t/T  for  = 0.2,  = 0.05,   = 0.5 , ̇   =  where T  is the period of natural vibration and v st  is the steady state response p0 /k.

Discuss the effect of damping on forced vibration.