DESCRIPTION OF SELECTED SYSTEM IDENTIFICATION TECHNIQUES
2.1. INTRODUCTION
2.3.2 Ibrahim Time Domain Identification Technique
crossing times tjare determined from one designated measurement. Ueng et al. [2000]
suggested for utilizing lower floor measurements for determination of crossing times as because it contains greater weight for higher modes. Then all the measurements are processed using step (3) with starting time tj to obtain their free decay signatures.
Fig. 2.6 Flow chart for extraction of RANDOMDEC signature
parameters. Here, the procedure of obtaining modal parameters of a single degree of freedom system is explained first and formulation is extended for a multi degree of freedom system.
A classical damped single degree of freedom system governed by free vibration can be expressed as:
=0 + +cx kx x
m&& & (2.24)
The characteristic equation is given by
2m+λc+k=0
λ (2.25) and solution of the above equation is given as:
et
t
u()=ψ λ (2.26) for an over damped system, ψ and λ are both real-valued; for an under damped system, they are complex and occur in a conjugate pairs.
In the more common under damped case, the roots of the characteristic equation are
iωd
σ
λ= + (2.27)
where ωdis the damped natural frequency in rad/sec and the corresponding undamped natural frequency is ωn = σ2+ωd2and damping factor or fraction of critical damping is
ωn
ξ = σ .
For a multi-degree-of-freedom linear-dynamic system with ‘p’excited modes, the free response of the structure at any sensor location ‘i’ and instant of time ‘tj’can be expressed by summation of the individual response of each mode as:
∑
=
=
=
p
k
t ik ij
j i
j
e k
x t x
2
1
)
( ψ λ (2.28)
where, ψikand λkare both complex numbers. The free response values for location i, xiis available from extended random decrement method.
With help of Eq. 2.28 free-response values for ‘2p’locations and ‘s’instants of time can be represented as:
ps p
p
s s
x x
x
x x
x
x x
x
2 2
2 1 2
2 22
21
1 12
11
...
. . .
..
...
..
...
=
ps p
p
s s
2 2
2 1 2
2 22
21
1 12
11
...
. . .
..
...
..
...
ψ ψ
ψ
ψ ψ
ψ
ψ ψ
ψ
X
s p p
p
s s
t t
t
t t
t t t t
e e
e
e e
e
e e
e
1 2 2
2 1 2
2 2
2 1 2
1 2
1 1 1
..
...
. . .
. ...
..
...
λ λ
λ
λ λ
λ
λ λ
λ
(2.29)
or simply in matrix form
s p p p s
px 2 x2 2 x
2 ΨΨΨΨ xΛΛΛΛ Φ
Φ Φ
Φ = (2.30) Similarly, free response value after a time interval of (∆t)1than those in Eq. 2.28 at the same stations
[ ] ∑
[ ]=
∆
= +
∆ +
p
k
t t ik j
i
j
e k
t t x
2
1
) ( 1
) 1
( ψ λ
=
∑ [ ]
=
∆ p
k
t t ik
j
k e k
e
2
1
)
( 1 λ
ψ λ
=
∑ [ ]
= p
k
t ik
j
e k
2
1
ˆ λ
ψ (2.31) or in matrix form
s p p p s
px 2 x2 2 x
2 ˆ x
ˆ ΨΨΨΨ ΛΛΛΛ Φ
Φ Φ
Φ = (2.32)
Eq. 2.30 and 2.32 can be related by eliminating Λ.
p p p p p
px2 2 x2 2 x2
2 xΨΨΨΨ ΨΨΨΨˆ AA
AA = (2.33)
where, Ts p
p T p s p
T x2 xAAAA 2 x2 ΦΦΦΦˆ x2
Φ Φ Φ
Φ = (2.34) From Eq. 2.31 we have
k t k
e k ψψψψ ψ
ψψ
ψˆ = λ (∆)1
The system can be defined with single eigenvalue problem
k t k
ek ψψψψ A
A A
Aψψψψ = λ (∆)1 (2.35) The matrix AAAA is referred to as system matrix and contains information of modal parameters of the system. The desired damped natural frequencies and damping factors are determined from the eigenvalues of AAAA , k k
t a ib
eλk(∆)1 = + as follows:
The damped natural frequency is
) ( ) tan ( 2 1 )
( 1
1 k
k k
k
d a
b
f t −
= ∆
= π
ω (2.36) and damping factor or fraction of critical damping is
2 2 ( d)k
k k
c k
c c
ω σ
σ +
=
(2.37)
where, ln( )
) ( 2
1 2 2
1
k k
k a b
t +
= ∆
σ .
The eigenvectors of AAAA are desired structural mode shapes (complex), ψψψψk. AT
A A
A can be obtained from Eq. 2.34 and 2.35. First the free response functions are placed into the rows of ΦΦΦΦ and the free response functions from the same locations after time shift
)1
(∆t are placed into the rows ofΦΦΦΦˆ . Matrices ΦΦΦΦ and ΦΦΦΦˆ are termed as response matrices whose time shift is(∆t)1. The least square solution of Eq. 2.35 gives the complex
eigenvalues and eigenvectors ofAAAA. It may be clearly seen that the size of the matrix ΦΦΦΦ is s
x
2p . Thus, pseudo-inverse of this matrix is computed in MATLAB using the command
“pinv” and the solution corresponds to a least squares estimate of eigen parameters. The system modal parameters can be evaluated directly from the complex eigenvalues ofAAAA using Eq. 2.36 and 2.37. The various steps mentioned in the flow chart [Fig. 2.7] have been executed in MATLAB to obtain modal parameters. The modes identified through ITD are in complex form. Procedures adopted through Eq. 2.18 to 2.21 are used to process complex modes for computing the corresponding normal modes.
Fig. 2.7 Flow chart for identification of eigenpairs and damping using ITD
Construction of response matricesΦΦΦΦandΦΦΦΦˆ of free decay responses of measured responses:
Ibrahim [1981] explained the method of construction of response matrices ΦΦΦΦ and ΦΦΦΦˆ . The fundamental time increment between all data placed into ΦΦΦΦ and ΦΦΦΦˆ is(∆t) . Two other 1 user-selectable time shifts are used in positioning overlapping segments of the measured free decay responses into the rows of the response matrices. The selection of time shifts denoted by(∆t) , 1 (∆t) and 2 (∆t) is very critical. The eigenvalues of the system matrix 3 A depends on the fundamental time shift(∆t) . The dimension ‘p’ is referred as the “number 1 of allowed (computational) degrees-of-freedom” or NDOF. The number of rows in the response matrices is kept twice the degree of freedom desired in the identification process.
When the number of free decay responses (i.e. number of response measurements) p0is less than the number of desired degrees-of-freedom, fewer than half the rows ofΦΦΦΦare filled up by the original unshifted free decay responses. Under these circumstances,
‘assumed’ or ‘transformed’ stations are created for the additional rows of both response matrices by shifting the original free decay responses placed in the first p0by multiples of a second user-selectable time shift, (∆t) :2 (∆t) ,22 (∆t) , 32 (∆t) etc., until the upper half of 2 the matrix ΦΦΦΦare filled. On the other hand, if the NDOF is smaller than p0, only NDOF of the available free decay responses are used in the analysis. The bottom half of the response matrix ΦΦΦΦare formed by duplicating the upper rows, but delaying an additional user- selectable time shift,(∆t) . Pappa and Ibrahim [1981] established a relationship between 3
(∆t) and3 (∆t) or1 (∆t) . There they refrained from using the time shift 2 (∆t) equal to either 3 (∆t) or1 (∆t) . They used to get satisfactory results in most of the cases by putting 2 (∆t) 3 equal to one-half of the value of(∆t) . Again if all the data of lower half are obtained by 2 delaying the data in the upper half by(∆t) , frequencies,3 fr =n/2(∆t)3, for integer values
of ‘n’ will not be identified. They got acceptable results by considering(∆t) < (∆t) . It 3 1 essential to know the Nyquist frequency frof the structure before selecting(∆t) . The 1 value of fris simply 1/2(∆t) . Hence, pre-filtering is required to ignore identification of 1 frequency larger than fr. Thus, (∆t) should be kept lower than1 1/2fr.
2.4 STRUCTURAL PARAMETER IDENTIFICATION OF MULTI-STOREY SYMMETRIC–PLAN SHEAR BUILDING
The stiffness which is directly related to moment of inertia or Young’s modulus of the structure is considered to be the most important structural parameter of a structure.
Presences of damage affect the moment of inertia or Young’s modulus of structure and in turn reduction of stiffness is observed. The change in these parameters will allow us to identify, locate and quantify the damage of a structure. In this section a method is proposed to identify structural parameters from the modal parameters obtained from N4SID identification techniques.