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)

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 (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