ME 475Mechatronics

Semester: February 2015

Dr. Sumon SahaAssistant Professor

Department of Mechanical Engineering

Bangladesh University of Engineering and Technology

Friday, September 18, 2015

MechatronicsMechatronics

Transfer Function

Today’s topic

MechatronicsMechatronics

•A transfer function (TF) relates one input and one output:

)(

)(system

)(

)(

sY

ty

sU

tu→→

Transfer function of a system, G(s), is defined as the ratio of the Laplace

Transform (LT) of the output variable, Y(s), to the LT of the input

Transfer function

Transform (LT) of the output variable, Y(s), to the LT of the input

variable, U(s), with all the initial conditions are assumed to be zero.

where:

)(

)()(

sU

sYsG =

[ ][ ])(L)(

)(L)(

tusU

tysY

=

=

MechatronicsMechatronics Transfer function

MechatronicsMechatronics

• Additive property

– Y(s) = G1(s)U1(s)+ G2(s)U2(s)

• Multiplicative property

– Y2(s) = G1(s)G2(s)U(s)

Properties of Transfer Functions

2 1 2

• ODE equivalence

ubdt

dubya

dt

dya

dt

yda

asasa

bsbsG

sU

sY

01012

2

2

01

2

2

01

ODE Equivalent

)()(

)(functionTransfer

+=++

+++

==

MechatronicsMechatronics Example: Transfer Functions

Step Input:

Laplace transform:( ) ( ) ( ) ( )0sY s Y Y s KX sτ − + =

( )n

n

n

dxs X s

dt=

( )

( )

0 0( )

0

0 0

tx t

M t

y t t

==

>

= =

( )dyy Kx t

dtτ + =

ndt

( ) ( )( ) 1

Y s KG s

X s sτ= =

+steady state gain

time constant

K

τ

=

=

: ( )M

Step input X ss

=

( )/

( ) ( ) ( )

1

( ) 1 t

Y s G s X s

K M

s s

y t KM eτ

τ−

=

=+

⇒ = −

1

1

11

1 at

Ls

L es a

−

− −

=

= + MuPAD Notebook: ilaplace(K*M/(s*(tau*s + 1)), s, t)

MechatronicsMechatronics Example: Transfer Functions

Laplace transform:( ) ( ) ( ) ( )0sY s Y Y s KU sτ − + =

( )n

n

n

dxs X s

dt=

( )

Ramp Input:( )dy

y Ku tdt

τ + =( )

0 0( )

0

0 0

tu t

at t

y t t

==

>

= =

ndt

( ) ( )( ) 1

Y s KG s

U s sτ= =

+

2: ( )

aRamp input X s

s=

( )2

/

( ) ( ) ( )

1

( ) 1t

Y s G s U s

K a

s s

y t Ka e Katτ

ττ −

=

=+

= − +

1

2

1L t

s

− =

MuPAD Notebook: ilaplace(K*a/(s^2*(tau*s + 1)), s, t)

MechatronicsMechatronics Problem

( ) 50

50G s

s=

+

A system has a transfer function,

For step response, find the time constant, settling time and rise time.

( )Y s KAnswer: Time constant: τ = 0.02 s

Settling time: Ts = 0.08 sRise time: Tr = 0.044 s

>> c = tf([50],[1 50])

c =

50

------

s + 50

>> stepinfo(c)

ans =

RiseTime: 0.0439

SettlingTime: 0.0782

( ) ( )( ) 1

Y s KG s

X s sτ= =

+

MechatronicsMechatronics Problem

Find the transfer function G(s) of the following differential

equation:

Using G(s), find the response c(t) to an input r(t) = u(t), a unit

step, assuming zero initial condition.

Answer: Taking Laplace transform assuming zero initial

2dc

c rdt

+ =

Answer: Taking Laplace transform assuming zero initial

condition,

( ) ( ) ( )

( ) ( )( )

2

1

2

sC s C s R s

C sG s

R s s

+ =

= =+

MechatronicsMechatronics Problem

Find the transfer function G(s) of the following differential

equation:

Using G(s), find the response c(t) to an input r(t) = u(t), a unit

step, assuming zero initial condition.

Answer: Now,

2dc

c rdt

+ =

Answer: Now,

Taking inverse Laplace transformation,

( ) ( ) ( ) 1 1 1/ 2 1/ 2

2 2C s G s R s

s s s s= = = −

+ +

( ) ( ) 21

2

tc t u t e

−= −

( )1

1

11 or

1 at

L u ts

L es a

−

− −

=

= + MuPAD Notebook: ilaplace(1/(s*(s + 2)), s, t)

MechatronicsMechatronics Second order system

Write down the response equation for this system.

What is the transfer function for this system?

m

k

m

c

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ = 0 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

Taking Laplace transform: ( ) ( ) ( )2

2 22 n

n n

Ks F s sF s F s

s

ωζω ω+ + =

ζ = 0 (No damping)

( ) ( ) ( )

( ) ( )2

2 2

2

2

n n

n

n n

s F s sF s F ss

KF s

s s s

ζω ω

ωζω ω

+ + =

=+ +

( ) ( )2

2 1 2

2 2 2 2

1 22 2

1, ,

n

n

n n

n n

K C CF s K

s s s s

sHere C C

ωω

ω ω

ω ω

= = + + +

= = −

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ = 0 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

( ) 1 sF s K

= −

Taking inverse Laplace transform,

( )2 2

1

n

sF s K

s s ω

= − +

( ) ( )1 cosn

f t K tω= −1

2 2cos

sL t

sα

α− = +

MuPAD Notebook: ilaplace(wn*wn*K/(s*(s^2+wn^2)), s, t)

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ = 1 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

Taking Laplace transform: ( ) ( ) ( )2

2 22 n

n n

Ks F s sF s F s

s

ωζω ω+ + =

ζ = 1 (Critical damping)

( ) ( ) ( )

( ) ( )2

2 2

2

2

n n

n

n n

s F s sF s F ss

KF s

s s s

ζω ω

ωζω ω

+ + =

=+ +

( ) ( ) ( )

( )

2 2

22 2

2 31 2

2 1 2 32 2

2

1 1 1, , ,

n n

n n n

n

n n n nn

K KF s

s s s s s

CC CK Here C C C

s s s

ω ωω ω ω

ωω ω ω ωω

= =+ + +

= + + = = − = −

+ +

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ = 1 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

( ) 1 1nF s K

ω = − −

Taking inverse Laplace transform,

( )( )2

1 1n

n n

F s Ks s s

ωω ω

= − − + +

( ) ( )1t tn n

nf t K e te

ω ωω− −= − − ( )1

2

1 atL te

s a

− −

= +

MuPAD Notebook: ilaplace(wn*wn*K/(s*(s+wn)^2), s, t)

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ < 1 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

Taking Laplace transform: ( ) ( ) ( )2

2 22 n

n n

Ks F s sF s F s

s

ωζω ω+ + =

ζ < 1 (Under damping)

( ) ( ) ( )

( ) ( )2

2 2

2

2

n n

n

n n

s F s sF s F ss

KF s

s s s

ζω ω

ωζω ω

+ + =

=+ +

( ) ( )2

2 31

2 2 2 22 2

n

n n n n

K C s CCF s K

s s s s s s

ωζω ω ζω ω

+= = + + + + +

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ < 1 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

( ) 2

21

1n n

sζ

ζω ω ζζ

+ + − −n

a ζω= −

Taking inverse Laplace transform,

( )( )

( ) ( )

( ) ( )( )

2

2 2 2

2 2

111

1

/1

n n

n n

d d

d

s

F s Ks s

s a aK

s s a

ζω ω ζζ

ζω ω ζ

ω ω

ω

+ + − − = − + + −

− −= −

− +

( ){ }1 cos / sinat

d d df K e t a tω ω ω = − −

21d n

ω ω ζ= −

( )1

2 2cosats a

L e ts a

αα

− −

= − +

( )1

2 2sinat

L e ts a

αα

α−

= − +

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ > 1 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

Taking Laplace transform: ( ) ( ) ( )2

2 22 n

n n

Ks F s sF s F s

s

ωζω ω+ + =

ζ > 1 (Over damping)

( ) ( ) ( )

( ) ( )2

2 2

2

2

n n

n

n n

s F s sF s F ss

KF s

s s s

ζω ω

ωζω ω

+ + =

=+ +

( ) ( )( )( )

2 2

31 2a b K CC C

F s Ks s a b s a b s s a b s a b

− = = + + − − − + − − − +

2 1

n

n

a

b

ζω

ω ζ

= −

= −

MechatronicsMechatronics Second order system

Question: Obtain the complete solution for equation:

With ζ > 1 using Laplace transformation. Assume all initial

conditions are zero.

2

2 2

22

n n n

d f dff K

dt dtςω ω ω+ + =

( ) 1 1 1a b a bF s K

− + = + −

Taking inverse Laplace transform,

( )( ) ( )

1 1 1

2 2

a b a bF s K

s b s a b b s a b

− + = + − − + − −

1 1 atL e

s a

− = −

( ){ } ( ) ( ){ } ( )1 / 2 / 2

a b t a b tf K a b b e a b b e

+ − = + − − +

MuPAD: ilaplace(wn*wn*K/(s*(s^2+2*z*wn*s+wn^2)), s, t)

MechatronicsMechatronics Problem

For the system shown below do the following:

33 /

15 /

3

k N m

c Ns m

m kg

=

=

=m

kx

i. Find the transfer function G(s) = X(s) / F(s).

ii. Find the system equation for transient response subjected

to a unit step input.

iii. Find ωn, ζ, %OS, Ts, Tp, and Tr.

3m kg=

c

MechatronicsMechatronics Problem

For the system shown below do the following:

33 /

15 /

3

k N m

c Ns m

m kg

=

=

=m

kx

i. Find the transfer function G(s) = X(s) / F(s).

=

c

( ) ( )2

2 2 2

/ 1

2 3 5 11

n

n n

kG s

s s s s

ωζω ω

= =+ + + +

11n

k

mω = =

5

2 2 11c

c c

c mkζ = = =

MechatronicsMechatronics Problem

For the system shown below do the following:

33 /

15 /

3

k N m

c Ns m

m kg

=

=

=m

kx

ii. Find the system specifications for transient response

subjected to a unit step input.

=

c

( ) ( )2

2 2

1

2

n

n n

X sk s s s

ωζω ω

=+ +

2

5 / 2

1 19 / 2

n

d n

a ω ζ

ω ω ζ

= − = −

= − =( ) ( ){ }1

1 cos / sinat

d d dx t e t a t

kω ω ω = − −

MechatronicsMechatronics Problem

For the system shown below do the following:

33 /

15 /

3

k N m

c Ns m

m kg

=

=

=m

kx

ii. Find the system specifications for transient response

subjected to a unit step input.

=

c

2

5 / 2

1 19 / 2

n

d n

a ω ζ

ω ω ζ

= − = −

= − =

( ) ( ) ( )5 /21 51 cos 19 / 2 sin 19 / 2

33 19

tx t e t t− = − +

MechatronicsMechatronics Problem

For the system shown below do the following:

33 /

15 /

3

k N m

c Ns m

m kg

=

=

=m

kx

iii. Find ωn, ζ, %OS, Ts, Tp, and Tr.

=

c2/ 1

% 100 2.72

41.6 sec

s

n

OS e

T

ζπ ζ

ζω

− −= × =

= =2

1.44 sec1

1.11 sec

p

n

r

d

T

T

π

ω ζ

π βω

= =−

−= = 1tan 0 .7 2 raddω

βσ

− = =

MechatronicsMechatronics Problem

For the system shown below do the following:

33 /

15 /

3

k N m

c Ns m

m kg

=

=

=m

kx

=

c

( ) ( )2

2 2 2

/ 1

2 3 5 11

n

n n

kG s

s s s s

ωζω ω

= =+ + + +

>> g = tf([1],[3 15 33])

>> damp(g)

>> stepinfo(g,'Risetimelimits',[0 1])

>> step(g)

RiseTime: 1.1133

SettlingTime: 1.7204

SettlingMin: 0.0303

SettlingMax: 0.0311

Overshoot: 2.7224

Undershoot: 0

Peak: 0.0311

PeakTime: 1.4411

MechatronicsMechatronics

Question: Consider the transfer function for a feedback system

given below

Calculate the transient response specifications for the system,

when the system subjected to a unit step input. Here, ωd = ωn √(1-

ζ2) and σ = ζωn.

2

25( )

6 25G s

s s=

+ +

Example: Transfer Function

ζ2) and σ = ζωn.

2

5, 0 .6 1

1 4, 3

n

d n n

ω ζ

ω ω ζ σ ζ ω

= = <

= − = = =

Underdamped system

2/ 1

2

% 100 9.5

41.33 sec

0.785 sec1

0.55 sec

s

n

p

n

r

d

OS e

T

T

T

ζπ ζ

ξω

π

ω ζ

π βω

− −= × =

= =

= =−

−= =

1tan 0 .9 3 raddωβ

σ− = =

MechatronicsMechatronics Home Work

For the system shown below do the following:

a) Find the transfer function G(s) = X(s) / F(s).

b) Find ωn, ζ, %OS, Ts, Tp, and Tr.