Identification Methods for Structural Systems
Prof. Dr. Eleni Chatzi
DEMO 2 - 09 March, 2015
Institute of Structural Engineering
Identification Methods for Structural Systems
2DOF system
Two Degree of Freedom system (2dof)
x2(t)
F(t)
m2
EIb
H
EIc
EIc
c
x1(t)
m1
EIb
H
EIc
EIc
c
L
Institute of Structural Engineering
Identification Methods for Structural Systems
2DOF System –
State Space representation
2DOF system
F1 ( t )
k1
c1
k2 ( x2 − x1 )
m1
m2
c2 ( x2 − x1 )
c1 x1
c1
x1 ( t )
F2 ( t )
k2 ( x2 − x1 )
k1 x1
m2
m1
F1 ( t )
FBD
F2 ( t )
k2
c2 ( x2 − x1 )
(Lumped Mass System)
x2 ( t )
The equations of motion can be written as
m1 ẍ1 + c1 ẋ1 + k1 x1 + c2 (ẋ1 − ẋ2 ) + k2 (x1 − x2 )
= F1
m2 ẍ2 + c2 (ẋ2 − ẋ1 ) + k2 (x2 − x1 )
We introduce the augmented state vector:
= F2

 
z1
x1
 z2   x2
 
z=
 z3  =  ẋ1
z4
ẋ2


 =⇒


.
 
ẋ1
z3
 ẋ2  
z4
=
=⇒ ż = 
 ẍ1   −(k1 + k2 )z1 /m1 + k2 z2 /m1 − (c1 + c2 )z3 /m1 + c2 z4 /m1
k2 z1 /m2 − k2 z2 /m2 + c2 z3 /m2 − c2 z4 /m2
ẍ2
Institute of Structural Engineering
Identification Methods for Structural Systems




2DOF System –
State Space representation
Then,

State eq.
0
0
0
0
1
0
0
1
k2
m2
k2
m1
− mk22
− c1m+c1 2
c2
m2
c2
m1
− mc22

ż = 
 − k1m+k2
1
|




}|
{z
A
x1
x2
ẋ1
ẋ2
{z
z


0
0
  0
0
+ 1
  m
0
1
1
0
m
} |
{z 2
B
=⇒ ż = Az + Bu

Output eq.
y=
|
1
0
0
{z
C
0


}
|
x1
x2
ẋ1
ẋ2
{z
z
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
 + 0 0
 | {z }
D
F1
= Cz + Du
F2
| {z }
u
}
Identification Methods for Structural Systems




F1
F2
| {z }
u
}
2DOF System –
State Space representation
or,

State eq.
0
0
0
0
1
0
0
1
k2
m2
k2
m1
− mk22
− c1m+c1 2
c2
m2
c2
m1
− mc22

ż = 
 − k1m+k2
1

}|
{z
|



A
x1
x2
ẋ1
ẋ2
{z
z


0
0
  0
0
+ 1
  m
0
1
1
0
m
} |
{z 2
B
=⇒ ż = Az + Bu

Output eq.
y=
|
0
0
0
{z
C
1


}
|
x1
x2
ẋ1
ẋ2
{z
z
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
 + 0 0
 | {z }
D
F1
= Cz + Du
F2
| {z }
u
}
Identification Methods for Structural Systems




F1
F2
| {z }
u
}
2DOF System –
State Space representation
or,

State eq.
0
0
0
0
1
0
0
1
k2
m2
k2
m1
− mk22
− c1m+c1 2
c2
m2
c2
m1
− mc22

ż = 
 − k1m+k2
1
|




}|
{z
A
x1
x2
ẋ1
ẋ2
{z
z


0
0
  0
0
+ 1
  m
0
1
1
0
m
} |
{z 2
B




F1
F2
| {z }
u
}
=⇒ ż = Az + Bu

Output eq.
y=
0
k2
m2
|
0
− mk22
0
c2
m2
1
− mc22
{z
C



}
|
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x1
x2
ẋ1
ẋ2
{z
z


+

|
0
0
0
1
m2
{z
D
F1
= Cz + Du
F2
|
{z
}
}
u
}
Identification Methods for Structural Systems
2DOF System –
Transfer Function representation (Laplace transform)
2DOF system
F1 ( t )
k1
k1 x1
c1
c1 x1
c1
x1 ( t )
F2 ( t )
k2 ( x2 − x1 )
k2 ( x2 − x1 )
c2 ( x2 − x1 )
c2 ( x2 − x1 )
m1
m2
m1
F1 ( t )
FBD
F2 ( t )
k2
m2
(Lumped Mass System)
x2 ( t )
The equations of motion can be written as
m1 ẍ1 + c1 ẋ1 + k1 x1 + c2 (ẋ1 − ẋ2 ) + k2 (x1 − x2 ) =F1
m2 ẍ2 + c2 (ẋ2 − ẋ1 ) + k2 (x2 − x1 ) =F2
)
L(·) with zero i.c.
=⇒
)
{m1 s 2 + (c1 + c2 )s + (k1 + k2 )}X1 (s) − (c2 s + k2 )X2 (s) =F1 (s)
(m2 s 2 + c2 s + k2 )X2 (s) − (c2 s + k2 )X1 (s) =F2 (s)
=⇒

{m1 s 2 + (c1 + c2 )s + (k1 + k2 )}X1 (s) − (c2 s + k2 )X2 (s) =F1 (s) 
=⇒
1
m2 s 2 + c2 s + k2
· X2 (s) −
· F2 (s) =X1 (s)
c2 s + k2
c2 s + k2
Institute of Structural Engineering
Identification Methods for Structural Systems
2DOF System –
Transfer Function representation (Laplace transform)
2s 4 + 0.25s 3 + 500.0075s 2 + 30s + 30000
0.05s + 100
· X2 (s) −
2s 2 + 0.15s + 300
0.05s + 100
s 2 + 0.05s + 100
0.05s + 100
2s 4 + 0.25s 3 + 500.005s 2 + 20s + 20000
0.05s + 100
· X2 (s) −
s 2 + 0.05s + 100
0.05s + 100
2s 2 + 0.15s + 300
2s 4 + 0.25s 3 + 500.005s 2 + 20s + 20000
· F2 (s) +
2s 2 + 0.15s + 300
0.05s + 100
· X2 (s) −
· F1 (s) +
0.5s 2 + 0.025s + 50
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
F1 (s) +
0.05s + 100
· X2 (s) −
· X2 (s) −
1
0.05s + 100



· F2 (s) =F1 (s) 






· F2 (s) =X1 (s)
=⇒
1
0.05s + 100



· F1 (s) = X2 (s)


· F2 (s) =X1 (s)
s 2 + 0.075s + 150
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
0.025s + 50
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
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


X2 (s) =F1 (s) 






· F2 (s) =X1 (s)
2s 4 + 0.25s 3 + 500.005s 2 + 20s + 20000
0.05s + 100
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
1
0.05s + 100
0.05s + 100
s 2 + 0.05s + 100
0.025s + 50
0.0025s 2 + 10s + 10000
· F2 (s) −
=⇒








· F2 (s) = X2 (s)


F2 (s) =X1 (s)
Identification Methods for Structural Systems





=⇒
2DOF System –
Transfer Function representation (Laplace transform)
Displacement
X1 (s)
F1 (s)
X2 (s)
F1 (s)
0.5s 2 + 0.025s + 50
=
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
0.025s + 50
=
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
X1 (s)
,
F2 (s)
X2 (s)
,
F2 (s)
=
=
0.025s + 50
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
s 2 + 0.075s + 150
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
Velocity
sX1 (s)
F1 (s)
sX2 (s)
F1 (s)
=
=
s(0.5s 2 + 0.025s + 50)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
s(0.025s + 50)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
sX1 (s)
,
F2 (s)
sX2 (s)
,
F2 (s)
=
=
s(0.025s + 50)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
s(s 2 + 0.075s + 150)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
Acceleration
s 2 X1 (s)
F1 (s)
s 2 X2 (s)
F1 (s)
=
=
s 2 (0.5s 2 + 0.025s + 50)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
s 2 (0.025s + 50)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
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,
,
s 2 X1 (s)
F2 (s)
s 2 X2 (s)
F2 (s)
=
=
s 2 (0.025s + 50)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
s 2 (s 2 + 0.075s + 150)
s 4 + 0.125s 3 + 250.0025s 2 + 10s + 10000
Identification Methods for Structural Systems
2DOF System –
Frequency Response Function (Bode diagrams)
Driving example: transfer function between the first output (X1 (s)) and the
first input (F1 (s)).
0.5s 2 + 0.025s + 50
X1 (s)
= 4
F1 (s)
s + 0.125s 3 + 250.0025s 2 + 10s + 10000
√
Assume sinusoidal excitation and set s = jω, j = −1, ω frequency of
excitation (rad/s).
0.5(jω)2 + 0.025(jω) + 50
X1 (jω)
=
4
F1 (jω)
(jω) + 0.125(jω)3 + 250.0025(jω)2 + 10(jω) + 10000
Manipulate
X1 (jω)
(−0.5ω 2 + 50) + (0.025ω)j
= 4
F1 (jω)
(ω − 250.0025ω 2 + 10000) + (10ω − 0.125ω 3 )j
Institute of Structural Engineering
Identification Methods for Structural Systems
2DOF System –
Frequency Response Function (Bode diagrams)
Multiply by the conjugate of the denominator
X1 (jω)
=
F1 (jω)
(−0.5ω 6 + 175ω 4 − 17500ω 2 + 500000) + (0.0875ω 5 + 5ω 3 − 250ω)j
(ω 4 − 250.0025ω 2 + 10000)2 + (10ω − 0.125ω 3 )2
Define H11 (jω): the steady–state FRF of the first output to the first
input, when the latter is sinusoidal. H11 (jω) is a complex number
H11 (jω) =
−0.5ω 6 + 175ω 4 − 17500ω 2 + 500000
(ω 4 − 250.0025ω 2 + 10000)2 + (10ω − 0.125ω 3 )2
|
{z
}
+
0.0875ω 5 + 5ω 3 − 250ω
(ω 4 − 250.0025ω 2 + 10000)2 + (10ω − 0.125ω 3 )2
|
{z
}
real part
complex part
Representation
phase
z }| {
H11 (jω) = |H(jω)| e φ(jω)
| {z }
magnitude
Bode diagram: plot of the magnitude and the phase of an FRF against
frequency.
Institute of Structural Engineering
Identification Methods for Structural Systems
j