Post on 31-Mar-2018
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Dynamic Response of First Order SystemsDynamic Response of First Order Systems
Dr. Bishakh Bhattacharyay
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
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
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Leaking Tank: A First Order System
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
h
In standard form:
X=h, Xo = ho, d/dt (X) = K X
MFR = 1/R (p1 – p2)1/
= 1 for Re <1000A= Cross sectional area of tank
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
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
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Free‐Response of a First Order Systemp y
x(t) = eat xx(t) = e xoa= 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
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Forced Excitation (Unit Step)
d/dt (x(t)) = a x(t) + b u(t)
x (t) = t e a(t‐τ) b u(τ) dτxf(t) = o e ( ) b u(τ) dτ
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Forced Response (Unit Step)
xf(t) = (bu0T)[1‐e‐t/T] Us(t)
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Forced Excitation (Ramp Input)
xf(t) = buoT2(e‐t/T + t/T – 1)
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Forced Response (Pulse Input)p ( p )
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Pulse Response as sum of Step Responsesp p p
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
First Order SystemsA first order system has a differential equation of the formA first order system has a differential equation of the form
rkydtdy
1
1)(),()()(
ssGsRsGsY
/1)( tekty
Example:A thermocouple which has a transfer function linking its voltage output V and temperature input of T as
Traditional Thermocouple Measurement
and temperature input of T as
G(s) = CVs
06
/110
1030
Determine the response of the system when it is suddenly immersed in aDetermine 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 Module 2- Lecture 9
The output as an ‘The output as an ‘s’s’ function function isis
VV (( ) G () G ( ) * i t () * i t ( ))V V ((ss) = G () = G (ss) * input () * input (ss))
Sudden immersion of the thermometer gives a step input of size 100o Cand so the input as an s function as 100/s. Thus
ss100
1101030 6
= 1.010
1030 4
ss=
1.01.01030 4
ssV =
The fraction element of the form a/s(s+a) and so the output as a function of time is :p
The same is plotted in the following Figure You may note
teV 1.04 11030 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
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
13Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Assignmentg
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.
14Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 2- Lecture 9
Special References for this lecturep
System Dynamics for Engineering Students: Nicolae Lobontiu Academic System Dynamics for Engineering Students: Nicolae Lobontiu, Academic
Publisher
Feedback Control of Dynamic Systems: Frankline Powell and Emami Naeini Feedback Control of Dynamic Systems: Frankline, Powell and Emami-Naeini,
Pearson Publisher
C l S E i i N S Ni J h Wil & S Control Systems Engineering: Norman S Nise, John Wiley & Sons
Systems Dynamics and Response: S. Graham Kelly, Thomson Publisher
15Joint Initiative of IITs and IISc ‐ Funded by MHRD