Workflow overview
Data assembly
Decomposition
Matrix Realization
Eigenvalue problem solving
Extract system properties
Assemble the selected data sets into a Hankel
Matrix and a Shifted Hankel Matrix
Workflow overview
Data assembly
Decomposition
Matrix Realization
Eigenvalue problem solving
Extract system properties
Decompose the Hankel Matrix using Singular Value Decomposition
Workflow overview
Data assembly
Decomposition
Matrix Realization
Eigenvalue problem solving
Extract system properties
Extract the new controllability and
observability matrix; Calculate the system
realization matrix
Workflow overview
Data assembly
Decomposition
Matrix Realization
Eigenvalue problem solving
Extract system properties
Solve the eigenvalue problem for the system
realization matrix
Workflow overview
Data assembly
Decomposition
Matrix Realization
Eigenvalue problem solving
Extract system properties
Calculate natural frequencies and
damping factors using the obtained eigenvalues
Data acquisition
PC Acceleration signals
Note: The ERA is implemented for the case of free response data. Therefore Impact (Hammer, drop-weight) tests would be generally suitable.
Data assembly
y1 y2 y3 y4
y5 yn-1 yn
Hankel Matrix:
Shifted Hankel Matrix:
Channel 1
Channel 2
Channel 3
Channel 4 Channel i
The ERA works by exploiting the relationship of the series of outputs from different points (channels) of the structure to fundamental system properties (Markov Parameters)
1 ny y
Decomposition
Assume the state – space representation of a dynamic system
Assume an impulse force, at t = 0, and 0 Initial Conditions
Decomposition
Assume the state – space representation of a dynamic system
Assume an impulse force, at t = 0, and 0 Initial Conditions
Decomposition
By iterating system in time
Assume the state – space representation of a dynamic system
Decomposition
Iterate the system in time starting from I.C.
Markov Parameters
These constant parameters are termed & are system characteristics:
Decomposition
By constructing the Hankel matrix of the Markov Parameters yi :
Which is equivalent to the matrix product:
Decomposition
Controllability matrix
Observability matrix
By constructing the Hankel matrix of the Markov Parameters yi :
Decomposition
Controllability matrix
Observability matrix
By constructing the Hankel matrix of the Markov Parameters yi :
Decomposition
Controllability matrix
Observability matrix
In order to obtain these matrices we perform Singular Value Decomposition for H1:
Matrix Realization
Product of Singular Value Decomposition :
TIP:
New controllability
matrix
New observability
matrix
Matrix Realization
Note: The Decomposition 1H PQ= is not unique!
In fact by using a different number of shifts k, and total measurements n, different alternatives can occur. And this is due to the fact that if matrices (A, B, C) are a realization of the system:
Then matrices, 1 1, TB, CTTAT − − are also a realization through the system: 1
11
ì i i
i i i
x TAT x TBuy CT x Du
−+
−
= +
= +
1ì i i
i i i
x Ax Buy Cx Du+ = += +
Under the transformation: x Tx=
Therefore the state that occurs from the ERA is not necessarily the that corresponds to the structural dofs but some transformation of it.
x
x
Matrix Realization
By using the new observability and the new controllability matrices:
Realization of A
replaced by
replaced by
Then, using the Shifted Hankel Matrix :
Extract system properties
Damping factors;
Conversion for discrete time to continuous time
representation
Natural frequencies;
For obtaining the mode shapes:
Extension for Random Input
It has already been mentioned that the ERA operates using output measurements of impulse response data. However, it possible to appropriately extend the method so as to account for response to a measured input loading.
The ERA as an input-output Id method
Assuming measurements of the input f(t) and output of the system x(t) are available from m measurement locations. The Frequency Response Function (FRF) may be extracted as:
( ) ( )( )
, 1xfi
ff
S jH j i m
S jω
ωω
= =
Then by applying the Inverse Fourier Transfom, the Impulse Response Functions (IRF) per measurement channel (usually this implied per dof) are obtained. The ERA method, as described previously can then be implemented on the IRFs which essentially simulate the system’s response to impulse.
Extension for Random Input
As mentioned in Lecture 1, the system’s response to a random input can be obtained via discrete convolution with the IRF:
Proof of the FRF extraction formula:
[ ] [ ] [ ]xft
R x t f tτ τ∞
=−∞
= −∑
[ ] [ ] [ ]0x t h t f
ττ τ∞
== −∑
On the other hand, the cross-correlation of two discrete time signals is defined as:
(1)
(2)
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
( ) ( ) ( )
0
xf fft t
xf xx
x t f t h t f f t R R h
S j S j H j
τ
τ τ τ τ τ τ τ
ω ω ω
∞ ∞ ∞
=−∞ =−∞ =
− = − − ⇒ = ∗
=
∑ ∑ ∑However, convolution in the time domain is multiplication in the frequency domain. Thus, by taking the Fourier Transform we obtain:
Extension for White Noise (Ambient Data)
For the case of ambient (operational) loads, it may be assumed that the excitation and responses are each stationary random processes. Assuming that the structural parameter matrices are deterministic, postmultiplying the Eq. of motion by a reference scalar response process and taking the expected value of each side yields:
The Natural Excitation Technique (NExT)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2, , , ,i ii i
i i i i
XX FXXX XX
ME X t X t CE X t X t KE X t X t E F t X t
MR t t CR t t KR t t R t t
+ + = ⇒ + + =
where denote the displacement and excitation stochastic vector process respectively. Additionally, for weakly (or strongly) stationary processes, we know that:
( ) ( ), X t F t
( )1 2X t
( ) ( ) ( ) ( ) th2 1, , where denotes the m derivative.m
mABA B
R R t t mτ τ τ= = −
Recognizing that the responses of the system are uncorrelated to the disturbance for t>0, and assuming that the random vector processes are weakly stationary, we can write: , ,X X X
Thus, the vector of displacement process correlation functions, satisfies the homogeneous differential equation of motion. Using a similar approach it can be shown that the acceleration process correlation functions also satisfy this equation (Beck et al. 1994). We can therefore employ the ERA for the correlation signals!
( ) ( ) ( )1 2, 0i i iXX XX XXMR t t CR KRτ τ τ+ + =
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