NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Dynamic Response of First Order Systems
Dr. Bishakh Bhattacharya
y
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
This Lecture Contains
 A Few Examples of First Order Mechanical & Electrical Systems
Response of a First Order system
Unit Step Response Unit Ramp response
Assignment Problem to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 9
Leaking Tank: A First Order System
h
Considering Incompressible Fluid, the Governing EOM :
the Governing EOM :
d/dt (A h(t)) = ‐Qout = (1/R)h(t)
d/dt (h) = (1/AR) h In standard form:
X=h, Xo = ho, d/dt (X) = K X
MFR = 1/R (p1 – p2)1/
 = 1 for Re <1000
Joint Initiative of IITs and IISc ‐ Funded by MHRD
A= Cross sectional area of tank
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
A Low‐Pass RC Filter
d/dt (V2) = 1/RC (V1 – V2)
V1 = 0
d/dt (V2) = (‐1/RC) V2
Low Pass RC Filter
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 9
Free‐Response of a First Order System
p
y
x(t) = e
x(t)
= eat xo
a= 0, Open circuit condition
T= 1/a time constant, time taken to reach 1/e of the initial value
Graph of eat for ranges of a
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Forced Excitation (Unit Step)
d/dt (x(t)) = a x(t) + b u(t)
( ) b u(τ) dτ
xf(t) = 
(t) = to e a(t‐τ) b u(τ) dτ
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Forced Response (Unit Step)
xf(t) = (bu0T)[1‐e‐t/T] Us(t)
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Forced Excitation (Ramp Input)
xf(t) = buoT2(e‐t/T + t/T – 1)
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Forced Response
p
(Pulse
(
Input)
p )
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Pulse Response as sum of Step Responses
p
p
p
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 9
First Order Systems
A first order system has a differential equation of the form A
first order system has a differential equation of the form
dy

 ykr
dt




 1 
Y ( s )  G ( s ) R ( s ), G ( s )    
 s  1 




y (t )  k 1  e  t /  


Traditional Thermocouple Measurement
Example:
A thermocouple which has a transfer function linking its voltage output V and temperature input of T as
and temperature input of T as
G(s) =
30  10 6 0
V/ C
10 s  1
Determine the response of the system when it is suddenly immersed in a Determine
the response of the system when it is suddenly immersed in a
water bath at 100o C
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
The output as an ‘
The output as an ‘s’
s’ function function is
is
V (s) = G (s) * input
i
t (s)
Sudden immersion of the thermometer gives a step input of size 100o C
and so the input as an s function as 100/s. Thus
V=
30  10 6 100

10 s  1
s
=
30  10 4
10 s s  0.1
=
30  10 4  0.1
s s  0.1
The fraction element of the form a/s(s+a) and so the output as a function of time is :
p

V  30 10 4 1  e 0.1t

The same is plotted in the following Figure. You may note The
same is plotted in the following Figure You may note
the nature of the first order response. The Thermocouple took about a minute to reach close to the final value (about 3mv).
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 2- Lecture 9
13
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 9
Assignment
g
Consider a first order system which can be simply modeled as a combination of a
spring of stiffness k and damper with damping constant c connected in parallel.
Find out the response of the system when it is subjected to an unit impulse
excitation.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 9
Special
p
References for this lecture
 System Dynamics for Engineering Students: Nicolae Lobontiu,
Lobontiu Academic
Publisher
 Feedback Control of Dynamic Systems: Frankline,
Frankline Powell and Emami
Emami-Naeini,
Naeini
Pearson Publisher
 Control
C
lS
Systems Engineering:
E i
i
N
Norman
S Ni
Nise, J
John
h Wil
Wiley & Sons
S
 Systems Dynamics and Response: S. Graham Kelly, Thomson Publisher
Joint Initiative of IITs and IISc ‐ Funded by MHRD
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