2. Damped Free Vibration SDOF

EQ Eng, Fall 2005 HW #2
Damped SDOF Vibration
Answers
1. Text, Problem 2-1. ξ = 0.0529
K = 50.1 k/in
c = 0.539 k-s/in
2. Text, Problem 2-2.b
v(
1s)
= −0.
756in
v�
( 1s)
= 1.
118 in / s
3. See below. T = 0.846s
ξ = 0.0221
Problem 3. Analyze the dynamic response of the water tower described below.
a) Calculate the natural period (T) based on the following
information.
• the diameters are outside diameters
• both the column and the tank are made of ½"-thick steel
• unit weight of steel and water are 490pcf and 62.4pcf,
respectively. Assume the tank is completely full of water.
• locate the center of mass (cm) of the water tower at the
midpoint of the water tank. Include half of the mass of the
column.
• Use the deflections in the attached chart to calculate the lateral
stiffness of the water tower (PH / ∆Η at the cm).
b) Calculate the damping ratio (ξ) using the logarithmic decrement
method and the following information. The tower is pulled back
6" and released from rest. Five vibration cycles later the
displacement = 3".
c) Make a "movie" showing the forces on the center of
mass of the water tower by drawing nine free-body
diagrams (like the one at right) at t = 0, t = TD/8, t =
2TD/8, t = 3 TD/8, … , t = TD. Indicate the relative
position of the center of mass to the displacement at t =
0 (vmax).
Initial displacement = 24"
Initial velocity = 0"/s
vmax
0.71vmax
40'
T = TD / 8
30'
10'
Plan View
Elevation View
Fc = 100 k
Fk = 2319 k
20'
2219 k = -ma