ISAT 300 wk12B 1
Second Order System (Continued)
m
Springforceky
D am p in g fo rce ,
dy
d t
F(t)
A Pressure Transducer
ISAT 300 wk12B 2
Second Order ResponsesResponse of a 2nd-order System to a Sinusoidal Input
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0 1 2 3 4 Frequency Ratio (/n)
0 01.
0 1.
0 2.
0 50.
Am
plit
ud
e R
atio
(y/
kx o)
ISAT 300 wk12B 3
Pressure Transducer - An Example of 2nd-Order System
A small tube, 0.5 mm in diameter, is connected to a pressure transducer through a volume of 3.5 m3. The tube has a length of 7.5 cm. Air at 1 atm and 20oC is the pressure-transmitting fluid. Calculate the natural frequency for this system.
ISAT 300 wk12B 4
Pressure Transducer - An Example of 2nd-Order System
n
r c
V L r
H z
2 2 1 2
3 2 2
6 3
1 2
0 5
0 25 10 343
3 5 10 0 075 0 5 0 25 10296
( . )
( . ) ( )
( . )[ . ( . ) ( . )].
/
/
w ith is th e a ir d en sity,
is th e v e lo c ity o f so u n d fo r a ir, an d , = 1 .9 1 1 0 is th e a ir d yn am ic v isco sity.-5
P
R Tk g m
c T m sk g m s
1 0 1 3 1 0
2 8 7 2 9 31 2 0 5
2 0 0 4 2 0 0 4 2 9 3 3 4 3
33.
( )( ). /
. . ( ) // .
ISAT 300 wk12B 5
Pressure Transducer - An Example of 2nd-Order System
Calculate the damping ratio and the attenuation of a 100-Hz pressure signal in the system.T h e can b e ca lcu la ted asd a m p in g ra tio
cr
L V
2 3 2 1 91 10
1 205 343 0 25 10
3 0 075 3 5 102 963
5
3 3
6 1 2( . )
( . )( )( . )
( . )( . ).
/
S in ce th a t th e freq u en cy ra tio can b e co m p u ted as = = 0 .3 3 8 ,
th u s, th e a tten u atio n o f th e p ressu re s ign al can b e ca lcu la ted asn
100
296
1
1 41
1 0 338 4 2 96 0 3380 457
2 2 2 2 1 2
2 2 2 2 1 2
P
Po n n
[ ( / ) ] ( / )
[ ( . ) ] ( . ) ( . ).
/
/
ISAT 300 wk12B 6
Cantilever Beam - Another Example of 2nd-Order System
A 1/16-in-diameter spring-steel rod is to be used for a vibration frequency measurement as shown in the figure below. The length of the rod may be varied between 1 and 4 in. The density of this material is 489 lbm/ft3, and the modulus of elasticity is 28.3106 psi. Calculate the range of frequencies that may be measured with this device.
ISAT 300 wk12B 7
Cantilever Beam - Another Example of 2nd-Order System
A t =x = in H zn1 11 028 3 10 7 49 10
8 69 10 11718
6 7
4 4, .( . )( . )
( . )( )
A t =x = in H zn1 11 028 3 10 7 49 10
8 69 10 4107 5
6 7
4 4, .( . )( . )
( . )( ).
T h e can b e ca lcu la ted as
= , w h ere , is th e Y o u n g ' s m o d u lu s fo r th e b eam ,
is th e m o m en t o f in ertia o f th e b eam , an d
, is th e m ass p er u n it len g th .
n a tu ra l freq u en cy
E I
m LE p si
Ir
in
m r lb m in
n
11 0 28 3 10
4
1 32
47 49 10
489 1 32
144
1
128 69 10
46
4 47 4
22
4
. .
( / ).
( )( / )
( ). /
=4
ISAT 300 wk12B 8
This Week in the Lab
1 22
2
2 n nend
d y
dt
dy
dty Kx y
d : the damped frequency
: the damping ration : the undamped natural frequency
Measuring Parameters
ISAT 300 wk12B 9
Measuring d and
Find Values for yn and yn + k
Perform the following calculations:
lny
yn
n k
Let:
4 2 2 2kThen find:
(Greek alphabet lambda)
Measure the period and then calculate!
d fT
22
ISAT 300 wk12B 10
Measuring n
After finding the previous two, the answer here is a simple calculation
d n 1 2
n
d1 2
ISAT 300 wk12B 11
The Solution
Only Concerned with the underdamped response.
y
ye t
end
tn
n
11
11
2
2
sin