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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.