ISAT 300wk12B1 Second Order System (Continued) A Pressure Transducer.

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

ISAT 300 wk12B 12

Response of 2nd-order system to a step input

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20 25 30wnt

y/yeThe Solution