HW2: Free Vibration of SDOF Systems 1. The packaging for an instrument can be modeled as shown in Fig. 4, in which the instrument of mass m is restrained by springs of total stiffness k inside a container; m=10lb/g and k = 50lb/in. The container is accidentally dropped from a height of 3 ft above the ground.
Assuming that it does not bounce on contact, determine the maximum deformation of the packaging within the box and the maximum acceleration of the instrument.
Solution:
1. Compute the maximum deformation u (0) sin t By using u (t ) u (0) cos t n
n
n
We have : m = 10 lb/g , k = 50 lb/in ,h=3 ft.=36 in. , g =32.2 ft/s 2=386.4 in/s 2 mg 10 u (0) 0.2in k 50
Determine the natural frequency.
n
k
m
(50)(386.4)
kg
w
10
43.95rad / sec
By using the law of energy conservation E1
E 2
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1
m u (0)
2
u(0) u (t )
2
2 gh
mgh u (0) 2 gh
2(386.4)(36)
u(0) cos t
u (0)
sin t
n
n
166.795in / sec
n
u (t )
166.795 (0.2) cos 43.95t sin 43.95t 43.95
u (0) [u(0) ] n
2
2
where is the max imum deformation
0.22 ( 3.795) 2 3.8in
Therefore the maximum deformation is
3.8in
2. Compute the maximum acceleration From
u t
u (t ) u
max
u t
cos
n
sin
n
n
2 cos
n
n
t
t t
2 n
2
u0
n
u0
19 g
k m
50
10 /
g
3.8 19 g
Therefore the maximum deformation is u0
7340in / sec
2
19 g
2. You are hired by Shanghai Highway Corporation to perform a system identification study (i.e. experimentally determine the dynamic structural properties) of a new bridge they recently constructed in Shanghai. The Qi Dong Bridge is designed in two parts that together are 274 m long. The first part consists of slab on precast concrete I-beams; this section is 152 m long. The second part is a concrete box girder bridge spanning 122 m in total.
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You fly to Shanghai and install an accelerometer (that measures the vertical acceleration response of the bridge) at the center of the bridge (as shown above). You employ a truck of a known weight, 40 metric tons, to drive over the bridge at the fixed speed of 40 km/hr. The following time-history of response of the 122 m long box girder portion bridge, as measured by a wireless sensor with the accelerometer attached is shown below (the y-axis represents acceleration in units of “g”):
Using the data collected (which will be given in a file named “HW2QDB.mat”), please perform a system identification study of the bridge. In other words, please determine a primary natural frequency of the bridge as well as its amount of damping.
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Solution
a.
Determine the period
As the data of Matlab we can plot the graph like below :
After looking the harmonic we found 2 top peak of harmonic (see the figure) We got n-period = 6 b. Log decrement to determine damping
u ln 2n u ln 1.636 /1.154 0.925% 2 6 i
i k
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c.
Find the natural frequency
Tn
n
(167.5 165.5) 2
Tn
6 2
0.333
n
0.333
18.87rad / s
3. An SDOF system consisting of a weight, spring, and friction device is shown in Fig. 5. This device slips at a force equal to 10% of the weight, and the natural vibration period of the system is 0.25sec. If this system is given an initial displacement of 2 in. and released, what will be the displacement amplitude after six cycles? In how many cycles will the system come to rest?
Solution 1. Calculate displacement amplitude after 6 cycles We have: F = 0.1w; T n = 0.25 sec, k F
w
u F
u F
F k
u0
2 in
F
0.1w k
0.1 386.4 (8 ) 2
0.1mg k
0.1g
n
2
0.1g
2 T n
2
0.0612in.
The reduction in displacement amplitude per cycle is 4u F
0.2448 in
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The displacement amplitude after 6 cycles is
u6
u0 6 4u f
2.0 6 4 0.0612
2.0 1.488 0.5312 in
2. How many cycles will the system come to rest? Motion stops at the end of the half cycle for which the displacement amplitude is less than
u F
. Displacement amplitude at the end of the 7 th cycle is
u7
u0 7 4u f 2 7 0.2448 0.2864 is bigger than u f so the motions don’t stop; at the
end of the 8 th cycle is u7
u0 8 4u f 2 8 0.2448 0.0416 ; which is less than u F .