Amortiguamiento Estructural: Teoría, Identificación y Limitantes Dionisio Bernal Northeastern University, Civil and Environmental Engineering, Center for Digital Signal Processing, Boston MA 02115 Resumen: Amortiguamiento es el término que se utiliza para hacer referencia a una colección de mecanismos que en su conjunto extraen energía de un sistema vibrante. En ingeniería estructural, por conveniencia en la manipulación matemática, es usual utilizar un modelo donde las fuerzas de amortiguamiento son proporcionales a la velocidad, o sea un modelo viscoso. El modelo viscoso puede clasificarse como clásico, en caso de que su presencia no impide vibración libre en fase o no clásico, en caso contrario. La especificación del amortiguamiento equivalente de modelos clásicos y los aspectos especiales que aparecen en estructuras donde es necesario considerar modelos no clásicos se presentan en la primera parte de este seminario. La segunda parte de la presentación enfoca el problema de identificación del amortiguamiento en estructuras existentes. En esta parte se esboza la teoría donde se apoya la solución del problema inverso y se describen resultados recientemente obtenidos en el análisis de más de 100 edificios reales. La tercera y última parte de la charla examina las limitantes que impone la teoría de la información de Fisher en la identificación del amortiguamiento a partir de mediciones tomadas durante sismos. The identification of damping ratios from input-output measurements is a standard problem in identification and exact results are obtained by all theoretically consistent algorithms when the data generating system is viscously damped, linear, time invariant and the input-output records are noise free. In practice, however, these assumptions are never satisfied and, as a consequence, identified damping ratios are random variables. In the particular case of seismic analysis it is well known that identified damping values have relatively high variance and it is shown here that this is a consequence of the low Fisher information contained in the response; a significant contributor being the relatively short durations of seismic signals. Notwithstanding the difficulties, values for damping are needed to formulate predictive models and expressions to estimate the expected value for buildings have been proposed through the years. Although not explicitly stated in most cases, these expressions are based on analyses that reflect dissipation within the structure, as well as energy loss through the soil structure interface. This paper summarizes recent work on characterizing the uncertainty in the estimation of damping, discusses the issue associated with isolating structural characteristics from those of the structure-soil system, and presents some new statistical expressions for expected value of the first mode damping ratio derived from analysis of a large collection of seismic responses. 1.0 Introduction
Transcript
Amortiguamiento Estructural: Teoría, Identificación y Limitantes
Dionisio Bernal
Northeastern University, Civil and Environmental Engineering, Center for Digital Signal Processing, Boston MA 02115
Resumen:
Amortiguamiento es el término que se utiliza para hacer referencia a una colección de
mecanismos que en su conjunto extraen energía de un sistema vibrante. En ingeniería
estructural, por conveniencia en la manipulación matemática, es usual utilizar un modelo
donde las fuerzas de amortiguamiento son proporcionales a la velocidad, o sea un modelo
viscoso. El modelo viscoso puede clasificarse como clásico, en caso de que su presencia no
impide vibración libre en fase o no clásico, en caso contrario. La especificación del
amortiguamiento equivalente de modelos clásicos y los aspectos especiales que aparecen en
estructuras donde es necesario considerar modelos no clásicos se presentan en la primera
parte de este seminario. La segunda parte de la presentación enfoca el problema de
identificación del amortiguamiento en estructuras existentes. En esta parte se esboza la teoría
donde se apoya la solución del problema inverso y se describen resultados recientemente
obtenidos en el análisis de más de 100 edificios reales. La tercera y última parte de la charla
examina las limitantes que impone la teoría de la información de Fisher en la identificación
del amortiguamiento a partir de mediciones tomadas durante sismos.
The identification of damping ratios from input-output measurements is a standard problem
in identification and exact results are obtained by all theoretically consistent algorithms when
the data generating system is viscously damped, linear, time invariant and the input-output
records are noise free. In practice, however, these assumptions are never satisfied and, as a
consequence, identified damping ratios are random variables. In the particular case of seismic
analysis it is well known that identified damping values have relatively high variance and it is
shown here that this is a consequence of the low Fisher information contained in the
response; a significant contributor being the relatively short durations of seismic signals.
Notwithstanding the difficulties, values for damping are needed to formulate predictive
models and expressions to estimate the expected value for buildings have been proposed
through the years. Although not explicitly stated in most cases, these expressions are based
on analyses that reflect dissipation within the structure, as well as energy loss through the soil
structure interface. This paper summarizes recent work on characterizing the uncertainty in
the estimation of damping, discusses the issue associated with isolating structural
characteristics from those of the structure-soil system, and presents some new statistical
expressions for expected value of the first mode damping ratio derived from analysis of a
large collection of seismic responses.
1.0 Introduction
The energy input from an earthquake is the work done by the forces acting at the soil-structure
interface. Energy balance shows that this work is at any time equal to the sum of the kinetic and the
strain energies plus the running integral of the work of the non-conservative forces. It is customary to
separate the non-conservative work into the work done by hysteresis plus the work of a collection of
unspecified mechanisms that are treated as an aggregate an referred to as “the damping”. This
aggregate is not usually viscous but viscosity is commonly assumed on grounds of mathematical
convenience.
Characterization of equivalent viscous damping has not been a central issue in earthquake engineering
because the force-ductility pairs needed to ensure adequate structural performance for the design level
earthquake are relatively insensitive to the specification of the dissipation, especially for wide band
excitations. The definition of what constitutes adequate performance, however, has evolved and at
present includes minimization of economic losses from non-structural damage for earthquakes
associated with relatively short recurrence periods. In these cases the earthquake intensity is
moderate, the anticipated response is linear (or quasi-linear), and the values of damping assigned to
the predictive models take increased importance.
This paper presents a review of some theoretical issues regarding the conventional specification of
equivalent damping, discusses the issue that arise in identification from seismic records, in particular,
it notes that the estimates of damping have large uncertainty due to the low Fisher information
contained in the transient response. The paper also comments on the effects of soli-structure
interaction and reviews some recent results on formulas derived from the statistical analysis of a
relatively large data set of identified first mode damping ratios.
2.1 The Viscous Model
If equivalent viscosity is accepted the damping model, either explicitly or implicitly, is described by a
damping matrix which we shall designate as C nxnR , where n is the number of degrees of freedom.
The damping matrix is typically specified as classical, in which case C is diagonalized by a congruent
transformation using the un-damped mode shape matrix .
Classical-Damping
In the classical model one has
1 1
2 2
2
2
.
2
T
n n
C
(1)
where the mode shapes are normalized to the mass matrix and j jand are the un-damped
frequency and damping ratio of the jth mode. A necessary and sufficient condition for the damping
matrix to be classical is that , the eigenvector matrix of M-1
K , be also the eigenvector matrix of M-1
C. This can be shown as follows: multiplying the homogeneous equations of motion by the inverse of
the mass matrix one has 1 1 0q M Cq M Kq (2)
If the matrices in the second and third terms on the lhs share the eigenvectors one has
1 1 1 0c kq q q (3)
So taking 1Y q and pre-multiplying by 1 gives
0c kY Y Y (4)
which is a diagonal system. Since matrices that have the same eigenvectors commute, classical
damping matrices are those for which the matrices M-1
K and M-1
C commute. A widely used classical
damping matrix is
C M K (5)
which is a special form known as Rayleigh or proportional damping. An issue discussed in the
literature of earthquake engineering is whether the stiffness in eq.5 should be the original or the
tangent stiffness (Priestly 2002). From a fundamental perspective the question is whether a model of
viscous dissipation with constant coefficients is reasonable for representing the non-hysteretic
dissipation when the response is nonlinear. The perception of this writer is that the consensus at the
time of writing is that there is no good reason for changing the equivalent viscosity when nonlinear
behavior ensues.
A situation where the constant coefficient model can lead to poor results is when there are massless
degrees of freedom. Indeed, in this case equilibrium at the massless coordinates, satisfied entirely by
the stiffness terms when the damping is classical, receives damping contributions during nonlinear
excursions that can lead to physically meaningless results, e.g., large moments at the free ends of a
cantilever, or to the observation that columns that are much stronger than the connecting beams yield
in a simulation. The noted behavior was clarified in Bernal (1993), where a solution that removes the
spurious effects while retaining a constant damping matrix, was put forth.
The fact that M-1
C shares the eigenvectors with M-1
K when the damping is classical implies that any
classical damping matrix can be written in terms of mode shapes as
1
nT
j j jj
C M M
(6)
where the j ’are arbitrary coefficients. It is also possible, without computing the mode shapes to
specify classical damping matrices using the Caughey series (Caughey 1960, Caughey and O’Kelly
1965)
1b
bb
C M a M K (7)
where b are arbitrary integers and the coefficients ab are related to the damping ratios by
21
2
bj b j
j b
a
(8)
Non-Classical Damping
Support for the classical premise is found on the fact that, except for cases where the distance
between two eigenvalues is small, errors in response predictions due to deviations from the classical
model are small in lightly damped structures. This result is not always appreciated so we present a
derivation that clarifies it. For simplicity consider a 2-DOF system (which can also be viewed as two
adjacent modes of an n-dof system). Transferring the equations of motion to the coordinates of the
undamped modes, one gets
21 1 1 1 1 21 1
22 2 22 2 2 2 2
2( )
2
T
T
Y bY Yf t
YY Y b
(9)
or
222 ( )T
j j j j j j kY Y Y b f t Y (10)
with j=1 k=2 or j=2 k=1. Taking a Fourier transform of eq.10 and solving for the generalized
amplitude gives
2
2 2
( ) ( ) ( )( )
( )( ) 2
Tj k j k
jjj j j
b f Y a YY
di
(11)
or
2
1 2 1 2
( )1 ( )
j kj
j
a aY
d d d d d
(12)
if 0 damping is classical and the solution for the modal amplitude is the first term of the rhs of
eq.12. The potential for error from deviations from classical depend on how large the second term in
the parenthesis of the lhs is, compared to one, and on how large the second term in the rhs is
compared to the first term. Consider first the term in the parenthesis. This term is
2 2
2 2 2 21 2 1 1 1 2 2 2
( ) ( )
( ) 2 ( ) 2d d i i
(13)
or
2
2
2 21 2 1 1 1 2 2 2
( )
( 1) 2 ( 1) 2d d i i
(14)
where jj
. To make this ratio large one needs to make the numerator large or the
denominator small (or both). Accepting that the damping ratios are small it is reasonable to inspect
this values at resonance. Say we take 1 1 , one gets
2 2
21 1
221 2 2 12 1 2 1 2
( )
2(1 )4 2 ( 1)
i
d d i
(15)
Taking the value of as 50% of the first entry in the diagonal (relatively large) and taking the
damping ratios as equal one has
2
21 2
( )
2(1 )
i
d d
(16)
Since we used the first frequency to arrive at the previous expression it follows that >1. Fig.1 plots,
for 2% and 5% damping, the value of eq.16 vs.. As can be seen the ratio is small, except in cases
where the frequency of the second mode is very close to the first. For 2% and 5% the coefficient
defined by eq.16 is less than 0.1 in absolute value for >1.04 and 1.12 respectively.
1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5-0.25
-0.2
-0.15
-0.1
-0.05
0
Eq.
16
Fig.1 Eq.16 vs the ratio of undamped frequencies of adjacent modes.
Examination of the importance of the second term in eq.12, compared to the first, can be carried out
in the same fashion as before, namely, taking the ratio of the second term to the first (on the rhs of
eq.12), assuming that a1 and a2 are equal, that the damping ratios are small, and taking as stated
previously one gets that this ratio is
2( 1)
(17)
which, at any is simply twice the value plotted in Fig.1. It follows then that the term is no larger
than 0.1 if > 1.1 and 1.25 for 2% and 5% damping respectively. It’s appropriate to note that the
closeness of the frequencies as an important factor for the relevance of the deviations of damping
from the classical model was pointed out early on by Rayleigh, who showed that the first order
approximation of the complex mode shape, zj, is given by
2 2
1 ( )
jj kj
j j kk j kk j
Cz x i x
(18)
where xj is the real mode shape and Ckj is the off-diagonal coefficient of the damping matrix.
Damping Ratio Definition
The concept of critical damping ratio is connected with the classical damping model and does not
have a counterpart in the non-classical situation. To be precise, in the classical model one can say that
if the damping matrix of a system is divided by a scalar, p, then all the damping ratios are divided by
p. If the damping in all the modes is 5%, for example, multiplication of the damping matric by 20
makes every mode critically damped. This interpretation does not hold in the arbitrary damping case.
To illustrate we note that the homogeneous equation of motion in the Laplace domain is
2( ) y(s) 0Ms Cs K (19)
The system poles are the roots of the polynomial matrix in the parenthesis and, given that M,C and K
are real, they must be real or appear in complex conjugate pairs. In typical earthquake engineering
applications the poles come in complex conjugates. The rate of decay of the unforced response is
governed by the real part of the pole and the frequency of vibration by the imaginary. The pole can
be expressed as
21i (20)
where case and coincide with un-damped natural frequency and the fraction of critical damping
when the damping distribution is classical. From eq.20 it is a simple matter to show that
R
(21)
The definition of eq.21, although universally adopted, cannot be literally interpreted as “the damping
ratio” in the arbitrary case. To illustrate consider a 2-DOF shear building with m={1,1} k={50,50}
c={a,0}. The smallest value of the dashpot constant “a” for which one of the modes is critically
damped is a = 17.678 and for this case one finds, from eq.21, that the “damping ratios” are {0.25,1}.
Assume now that one reduces “a” to “0.05a”. Since the damping matrix has uniformly decreased to
5% of the original one may expect that the “damping ratios” would be 5% of the previous ones,
namely {0.0125, 0.05}. What is obtained from eq.12, instead, is {0.028, 0.028}. Five percent of the
matrix that leads to a critically damped mode does not, therefore, lead to 5% critical damping in that
mode because the distribution is not classical.
2. Identification Methods that apply only to SDOF systems like the logarithmic decrement of the half-power
bandwidth approach can be found in any structural dynamics textbook and are not repeated here. We
limit the presentation to the basic time domain identification of a system from input-output signals
because this is the situation that prevails in the earthquake case, given that the base motion is always
measured.
Time domain algorithms are typically indirect, namely, a model that maps the sampled input and the
sampled output is obtained from the digital data and this model is then converted to continuous time.
The postulated model in sampled time has the form
1k k kx Ax Bu (19)
and the output equation is
k k ky Cx Du (20)
where uk is the measured input, ky is the output and the matrices {A,B,C,D} are determined from the
measurements.
2. STATISTICAL ACCURACY LIMITS ON DAMPING ESTIMATION
2.1 ON THE IDENTIFIED SYSTEM
System identification is the process of extracting, from measurements, information on the
properties of a system. In the case of buildings, which are invariably connected to the ground,
the question arises as to what exactly is the system to which the identified properties apply.
One expects that they correspond to the building with a certain set of boundary conditions
and some commentary on what these are is opportune. The matter is more clearly discussed
by replacing the structure with a discretized model. We designate the domain of the building
as , the interface with the ground as and the degrees of freedom (DOF) on (exclusive
of those in ) as y1 and those on as y2.
We begin by recognizing that either the load or the displacement has to be prescribed at each
coordinate of a model and that not all the DOF on are prescribed. It follows, therefore, that
some of the coordinates on the interface are treated in the identification as un-prescribed and
that their connection to the ground must be reflected by some connection impedance.
Accepting that the impedances can be represented by masses, springs and dashpots, the
equations of motion can be written without making reference to frequency dependent terms
as
11 12 1 11 12 1 11 12 1
21 22 2 21 22 2 21 22 2
0
e
M M y C C y K K y
RM M y C C y K K y (2.1)
where Re are the reactions at the prescribed coordinates. The displacements that are not
prescribed can be expressed as a linear combination of the prescribed ones plus a residual and
one can thus write
1 2y ry u (2.2)
which when substituted in the top partition of eq.2.1 leads to
11 11 11 12 11 2 12 11 2 12 11 2( ) ( ) ( )M u C u K u M M r y C C r y K K r y (2.3)
Taking the matrix r as
1
11 12r K K (2.4)
neglecting the damping contribution to the rhs in eq.2.2, and recognizing that for lumped
mass models M12 = 0 one gets
11 11 11 11 2M u C u K u M r y (2.5)
which is the conventional expression used to represent earthquake excitation. The point to
note is that the properties of the systems on the lhs of eq.2.5 are those of the building with
restraints at the prescribed coordinates only. Therefore, in the common case where the input
is horizontal base motion the identified properties are those of the building with flexibility
and dissipation at all DOF in other than horizontal translation. In the subsequent treatment
we drop the subscripts in eq.2.5 and replace 2y by the more commonly used gx , namely, we
use
gMu Cu Ku Mr x (2.6)
An outline of the identification approach used to extract the modal properties of this system
from the measured data is presented in Appendix A.
Damping Ratio
Let the rhs of eq.2.6 equal zero, namely
0Mu Cu Ku (2.7)
the solution to eq.2.7 is of the form
( ) it
i iu t e and one finds, by substitution that
2 0i i iM C K (2.8)
where i’s are scalars. The values of i ’s that satisfy eq.2.8 are the complex poles. Writing
the solution in terms of its real and its imaginary part and calling on Euler’s identity one finds
that
( ) (cos( ) sin( ))iRt
i i iI iIu t e t i t (2.9)
which shows that the rate of decay of the free vibration is determined by the real part of the
pole and the vibration frequency by the imaginary. By analogy with the solution for free
vibration of a SDOF system, where the exponent of the rate of decay is the product of the
undamped frequency times the damping ratio one has
R
(2.10)
where we have used the fact that the magnitude of the pole in the single DOF system is equal
to the undamped frequency. 2.2 UNCERTAINTY IN DAMPING ESTIMATION
All system identification results are afflicted with inherent uncertainty because measured
signals are always noise corrupted and because it is often difficult to fully characterize the
input. All identified values, therefore, are realizations of a stochastic process and are subject
to variance errors. Eq.2.10 allows for a simple appreciation of why it is difficult to identify
damping ratios with low variance. Namely, let the true pole for a given mode be a point in the
complex plane and let there be a region around the pole where, due to noise, the identification
algorithm places the pole. Assume the region of uncertainty around the pole is a circle of
radius R, where R is a fraction of the pole magnitude, say R . Recalling that the
magnitude of the pole is an estimate of the undamped frequency (exact for classical damping)
and recognizing that is small, one concludes that the variability in frequency is small. The
estimation of damping, however, which is given by eq.2.10, is subject to much larger
variations. In fact, examination of the geometry shows that the percent error in the frequency
is essentially equal to while the damping ratio varies from the true value to plus or minus .
If is 0.02, for example, the frequency error is no more than 2% but the damping ratio can be
over or under estimated by 0.02. Namely, if the true damping is 5% one gets values as large
as 7% and as low as 3%. To determine if the circular assumption for the uncertainty region is
reasonable we performed a Monte Carlo simulation study where a system was identified for a
ground motion using 1000 different random realizations of the noise. As can be seen from
fig.2.1, which shows results for the first and the second pole, the circular premise is not
unreasonable.
Fig.2.1 Uncertainty of the real part vs. the imaginary part of the 1st pole and 2
nd pole in a 6-DOF
system identified using white excitation and 5% additive noise.
Fisher Information and the Crámer-Rao Lower Bound
The accuracy with which any parameter can be estimated from noisy data is limited by the
amount of information on the parameter that is contained in the data. For any distribution of
the noise affecting the input and the output the information on a set of parameters, , is
quantified by the Fisher Information matrix, )(I . The lower bound of the covariance of
any estimator of these parameters is given by its inverse, known as the Cramér-Rao Lower
Bound (CRLB) (Casella & Berger 2001). The Fisher information matrix and, as a
consequence, the CLRB depend only on the statistical distribution of the noise and on the
sensitivity of the data to the parameter but not on the estimator. The FI can be defined in
terms of the gradient or the hessian of the probability distribution with respect to the
parameters. For the gradient the expression is
2
) log ( | )( f YI
E (2.11)
Where ( | )f Y is the likelihood function of the observed data Y given the parameter . If the
sensitivity of the likelihood to the parameter is high the derivative in eq.2.11 is large and so
is )(I . In practice the likelihood function ( | )f Y is in general unknown so other quantities
derived from the data are typically used to estimate )(I . For example, if the data Y can be
used to generate a vector X whose distribution is a member of the linear exponential family
having a mean ( ) and a covariance (that is the CRLB of X contained in Y), then the FI
of the parameter contained in X can be obtained as (van den Bos 2007)
1) ( ) ( )( TI J J where ( )
J . (2.12)
To illustrate the significance of eq.2.12 in a simple setting let the “true” value of Y be
deterministically dependent on as depicted schematically in fig.2.2. Assume one wants to
know the value of based on noisy values of Y. From the sketch in the figure it is evident
that the statistical accuracy of depends on the slope of the functional relation at the location
of the estimate and it is not difficult to see that the variances are related by the square of the
local slope. Eq.2.12 is the generalization of this concept to the multivariate situation.
Fig.2.2 Schematic illustration of eq.2.12 in the scalar case.
The Pole as a Feature
Denoting as the CRLB of the real and imaginary parts ( ) and ( ) of a pole , the
FI of the frequency and the damping follows from eq.12 as
1
, ,( , ) f f
TI f J J (2.13)
where the sensitivity of the pole with respect to damping ratio and frequency is given by
1,2 22
( ( ), ( ))2
( , ) (1 ) 1f
f
f f
J . (2.14)
Due to the relation between the FI and the CRLB, an analytical relationship between the
coefficients of variation (COV) of damping and frequency can be obtained from eq.2.14 and
is detailed in Appendix B. This relation shows that the ratio depends only on the damping
ratio. Assuming that the uncertainty region around the complex poles is circular, as depicted
in the Monte-Carlo simulation in fig.1, the ratios between the COVs are shown in fig.2.3. As
can be seen, the uncertainty on the damping ratios is around 50 times higher than that for the
frequencies at 2% , and the ratio is near 25 for 5% . These results are consistent with
the findings in (Gersch 1974), where maximum likelihood estimation of modal parameters
from ARMA models was considered.
Fig.2.3 Range of the ratio of the coefficient of variation of damping and frequency when the
uncertainty region around the pole is circular.
The Frequency Response Function (FRF) as a Feature
Let the modal parameters be estimated from experimental FRF’s. Accepting classical
damping the FRF from horizontal ground motion to absolute acceleration at any level can be
derived as follows: the modal amplitude is
22j j j j j j j j gY Y Y x (2.15)
where 2j jf . Taking a Fourier transform and solving for the modal acceleration gives
2
2 2
( )( )
( ) 2
j g
j
j j j j
xY
i
(2.16)
Specifying the location of the output sensor with the superscript k and adding the ground
acceleration to convert the results to absolute acceleration gives
2 ( )
2 21
( ) 1( ) 2
kNj j
j j j j j
frfi
(2.17)
Differentiating eq.2.17 with respect to damping and frequency one gets
3 ( )
2
2 k
j j i
D
J (2.18)
2 ( )
2
2
j
k
j j j ji
D
J (2.19)
where
2 2( ) 2j j j jD i (2.20)
Eqs.2.18 and 2.19 can be evaluated at any number of desired frequencies and the results
placed in two vectors to form the sensitivity matrix. Eq.2.12 provides the Fisher information,
where is the CRLB of the FRF, and by inversion one obtains the CRLB of damping and
frequency. In the case of multiple channels the FRFs are stacked (one under the other) and
the computations carried out identically.
The reason for showing the FRF alternative is because the results in this case make clear that
the sensitivity is high only near resonance and that, as a consequence, low variance in the
identification of damping requires that the CRLB of the feature be small in this region. This,
of course, points towards the well-known fact that accurate damping estimation is possible if
one can excite the system harmonically.
Numerical Illustration
To show some numerical results consider a uniform 5 DOF shear building with unit masses, a
fundamental period of one second and 2% damping in each mode. We estimate the
covariance of the 5th
mode contribution to the FRF connecting the ground to each output
sensor using 1000 simulations of the noise and we assume that the result is a reasonable
approximation of the CRLB. For the noise considered the COV of the damping ratios proved
to be: {0.30, 0.11, 0.14, 0.17, 0.46}, with the first value corresponding to the sensor in level
#1 and the last to the roof. To test these results we performed 20 identifications using a
record from the Northridge earthquake and different realizations of the noise. In ten cases the
output is taken as the response at level#2 and in the other ten at level #5. The COV of the
results from the simulations were 0.085 and 0.46 for the sensors at levels #2 and #5
respectively. These results, albeit from a small number of simulations, are in general
agreement with the CRLB predictions. 2.3 INCREASE IN THE DAMPING OF HIGHER MODES
The observation that damping ratios in higher modes increase relative to that of the first
mode have been made, for example, by Satake et al. (2003), who based his observation on the
analysis of data from the first 4 modes of a number of buildings. Here we propose a simple
mechanistic explanation for this observation. Namely, we contend that the relation between
the damping of various modes is likely dependent on the effectiveness that the mode shape
has in activating the dissipation mechanism. To illustrate, we formulate a 6-story one bay
model where the damping is assumed to come from dashpots of equal magnitude located at
each of the connections between beams and columns and compute the equivalent modal
damping for the six modes. Results for the case where the behavior is dominated by frame
action (relatively rigid beams) and where flexure dominates (relatively flexible beams) are
depicted in fig.2.4. As can be seen, the damping increases in the early modes but eventually
decreases, as the joint rotations for sufficiently high modes (due to the wavy nature of the
mode) are small. It is interesting that the results for the shear type behavior are, in this case at
least, in qualitative agreement with the empirical result obtained by Satake for increases from
the 1st to 2
nd and the 2
nd to 3
rd mode.
Fig.2.4 Ratio of damping between various modes in a 6-story model with dissipation simulated with
dashpots at the beam-column joints.
3. NON-LINEARITY DETECTION AND OTHER ISSUES
3.1 NON-LINEARITY DETECTION
An implementation of a classifier that can automatically identify which responses from
database are linear (or quasi-linear) and which are not, is investigated. None of the
procedures tried proved sufficiently robust when operating with seismic data from real
structures. The procedures tried were: test of Kramers-Kronig Relations (KKR) (Tomlinson
and Ahmed, 1987) and Coherence test (Heylen et al. 1997).
Kramers–Kronig Relations
It is known that The KKR test is based on the fact that when a system is linear and causal the
real and the imaginary parts of the transfer functions (FRF) are related by Hilbert transforms,
namely
( ( ( ))) ( ( ))
( ( ( )) ( ( ))
H G G
H G G
(3.1)
where the H(.) stands for the Hilbert transform operator which defined as
1 ( )
( )u
H u dt
(3.2)
and the transfer function G(ω) is defined as
( )
( )( )
F YG
F X (3.3)
Where F(Y) and F(X) are the Fourier transforms of the output and input respectively. Studies
have shown that if the ratio of output to input spectra is treated as a transfer function but the
output comes from a nonlinear response, then FRF found to be non-causal. Therefore, the
KKR fails to be satisfied.
To confirm the validity of this test, the simulation is carried out for a 5-story shear building
and white noise is added to the output acceleration. Fig.3.1 shows the result of examining
KKR to check how they react to the linear and non-linear responses. The results are shown
for the first equality in eq.3.1 in the neighborhood of the first and second frequency. As can
be seen in the linear case the two sides of the equality are very close, and in the non-linear
case the two terms deviate in most frequencies. This test also performed on the real responses
of station #24322 due to Northridge (0.46g) and Chino Hills (0.049g) ground motions and the
result for the first equality in eq.3.1 is shown in fig.3.2.
2 4 6 8 10 12 14 16 18 20
-5
0
5
10
Hz
2 4 6 8 10 12 14 16 18 20
-5
0
5
10
15
Hz
(a)
(b)
( ( ))G
( ( ( )))H G
( ( ))G
( ( ( )))H G
Fig.3.1 KKR for the linear (a) and non-linear (b) response of a 5-story shear building
Fig.3.2 KKR for the response of station# 24322 due to Northridge (0.46g) (a) and Chino Hills (0.049)
(b)
2 3 4 5 6 7 8 9-2
0
2
4
Hz
-5 0 5
-500
0
500
drift
forc
e
2 3 4 5 6 7 8 9
-4
-2
0
2
Hz
-5 0 5-400
-200
0
200
400
drift
forc
e
( ( ))G
( ( ( )))H G
( ( ))G
( ( ( )))H G
(a)
(b)
To evaluate how the two terms deviate, we defined an index I as
2
2
( )A BI
B
(3.4)
where ( ( ( )))A H G and ( ( ))B G . This index is 111 for (a) and is 26 for (b) in
fig.3.2, which seems promising since Northridge earthquake has the ground motion 10 times
stronger than the Chino Hills and the non-linearity in the response expected to be more for
the Northridge case. However experience shows this index is not always low for weaker
ground motions compare to Northridge. Fig.3.3 shows the KKR for station# 57536 response
in Loma Prieta (0.09g) earthquake where I=94.
Fig.3.2 KKR for the response of station# 57536 due to Loma Prieta (0.09g)
Therefore, it seems hard to define a threshold for index I to be able to distinct linear and non-
linear responses. We should mention that the results for the second equality in eq.3.1 are
quite similar. Thus, in practice the KKR relations cannot provide reliable results.
2 4 6 8 10 12 14 16 18 20-10
-5
0
5
10
15
20
25
Hz
( ( ))G
( ( ( )))H G
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hz
ϒ2
(a)
Coherence Function
Coherence is defined as the ratio of the H₁ to the H₂ estimators of the transfer function
between an output and an input. The H₁ and H₂ estimators are defined as
*
11
*
1
1( ) ( )
( )( )
1 ( )( ) ( )
N
i iuyi
N
yyi i
i
u yGN
HG
y yN
(3.4)
*
12
*
1
1( ) ( )
( )( )
1 ( )( ) ( )
N
i i
i uu
N
yui i
i
u uGN
HG
y uN
(3.5)
Therefore coherence (squared) is defined as
2
2 1
2
| ( ) |
( ) ( )
uy
uu yy
GH
H G G
(3.6)
where Gjk are the transfer functions from the input j to the output k. The coherence is a
function of frequency and, being a correlation coefficient, varies between 0 and 1.The idea is
that if the system is linear and the output is a filtered version of the input, then the coherence
should be near to the identity. When the system is nonlinear, the coherence is expected to
decrease.
In the seismic case, an important difficulty comes from the fact that the signals are of
relatively short duration and this is aggravated by the fact that one does not have multiple
tests so it is necessary to divide the signal in pieces to compute the averages, making the
duration issue even more severe. In any case, the approach used was to divide the signal in 8
equal pieces.
Unfortunately, the coherence deviates from unity for many reasons other than nonlinearity,
i.e., because the noise is not white, because the input is not just one motion (that is the output
is the result of multi-component input) and, of course, because the Fourier transforms have to
be taken over short duration segments. An illustration of typical results for the case of station
#24322 is shown in fig.3.4 for 2 earthquakes (the first three identified frequencies are 0.32,
1.08, and 1.96 Hertz). In one case, the response is expected to display significant nonlinearity
while in the other the intensity of the motion is small and one anticipates linear response. As
can be seen, Northridge earthquake with the strong ground motion acceleration of 0.46g and
Chino hills with a weak one (0.049g) show very different values for coherence, and it is
difficult to decide based on the coherence value weather the response is linear.
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Hz
ϒ2
Fig.3.4 Coherence function for Northridge (a) and Chino Hills (b) ground motions – station # 24322
To summarize, none of the two methods tried proved sufficiently robust for the response of
the real structures.
3.2 SELECTING BETWEEN IDENTIFICATION ALGORITHMS
In the absence of noise all the rigorous identification techniques yield identical results so the
selection issue is one of bias and variance for the operating conditions. We carried out a
study to determine which algorithm has the smallest confidence interval for damping
estimation. The algorithms considered were:
1. Eigensystem realization algorithm (ERA-OKID)
2. Subspace based (SubID)
3. Subspace based (DS-R)
Examination shows that selection of a model order that is optimal without individual
examination of identification results is not feasible. The approach that must be followed to
optimize accuracy is to perform identification for various model orders and to take the
eigenporoperties as the values that stabilize over a range of model orders. In this regard a
pole is treated as stable when fluctuations in the system order lead to changes that are less
than:
1) 1% in frequencies
2) 5% in damping values
3) 2% in eigenvector entries
In the experimental modal analysis field the approach described previously is known as
model order selection by means of stabilization diagrams. To investigate the difference
between identification algorithms we implemented them on simulated and real data.
Simulations carried out with 2 shear buildings of 6 and 25 stories for 10 different ground
motions did not show any of the ID algorithms to be clearly superior regarding the estimation
of damping, all of them performed well, even when substantial additive noise was included.
Table 3.1 shows a sample of the results obtained.
Table 3.1 Results for 6-story using Earthquake Ferndale from Station 99261 (true damping is 5%)
Since the correct answer for damping is not known for real data, the best that can be done is
to look for signs of consistency. We examined how the results varied when the analysis was
carried out with the sensors in only one direction (basically symmetric buildings) and using
all the available channels. The results for station #24288 suggests that SubID and DS-R are
the most consistent algorithms, in the sense that the results considering all sensors, or a
partial set of sensors (in a single direction in nominally symmetric buildings) differ by the
smallest percentage compare to the OKID algorithm. The distribution and labeling of the
measured coordinates at this station are depicted in fig.3.5.
Fig.3.5 Station 24288 We performed identification in two ways: in case A channel 3 is treated as the input
excitation and channels {6 9 12 15} as outputs (one directional response) while in case B the
input channels are {2 3 4} and channels {5-15} are outputs (3-D response). The results for
damping are summarized in Table 3.2. For interest we also present the results for frequency
in Table 3.3 As can be seen, the error differences in frequencies are small for all algorithms,
but for damping ratios the error in OKID is more compare to SubID and DS-R. DS-R is the
method selected in this project as identification algorithm. Table 3.2 Identified damping values (%) for the modes in N-S direction
OKID SubID DS-R
Case
A
Case
B
Error
(%)
Case
A
Case
B
Error
(%)
Case
A
Case
B
Error
(%)
1st
mode 6.38 3.10 51 2.82 2.77 2 2.82 2.88 2
2nd
mode 2.97 1.30 56 3.10 3.21 3 3.06 3.21 5
3rd
mode 3.20 2.33 27 3.70 3.51 5 3.64 3.54 3
Table 3.3 Identified frequencies (Hz) in N-S direction
OKID SubID DS-R
Case
A
Case
B
Error
(%)
Case
A
Case
B
Error
(%)
Case
A
Case
B
Error
(%)
1st
mode 0.32 0.30 6 0.31 0.31 1 0.31 0.31 1
2nd
mode 0.85 0.86 1 0.85 0.86 0.7 0.85 0.86 0.7
3rd
mode 1.48 1.48 0 1.47 1.50 2 1.47 1.50 2
3.3 SENSITIVITY OF DAMPING TO NOISE AND SAMPLING RATE
We examined the effect of the Signal to Noise Ratio (SNR) in the identified damping ratios.
First, damping ratios for a 25-story shear building were computed for three different SNRs
for the noise just in the output. Each case is run with 20 random noises. Table 3.4 shows the
mean (µξ )and standard deviation (σξ) of the damping ratios for 20 runs in each case. As can
be seen identified damping ratios are not strongly affected by the SNR when the noise is just
added to the output.
Table 3.4 First mode identified damping ratios for the Chinohills earthquake in a 25-story model for
different SNR’s (noise just in output and true damping is 5%).
SNR 100 50 25
µξ 5 4.96 4.99
σξ 0.029 0.067 0.108
Next, we investigated the effect the noise of input and output in identification of the
damping. As can be seen in Table, 3.5 the difficulty in identifying damping increases by the
increase in the noise level. Table 3.5 First mode identified damping ratios for the Chinohills earthquake in a 25-story model for
different SNR’s (noise in input and output and true damping is 5%).
SNR 100 50 25 17
µξ 5.02 5.31 5.49 6.22
σξ 0.07 0.16 0.33 0.54
To investigate the effect of sampling rate, identified damping ratios for the same simulation
example were computed for two different sampling rates. Results for the case of the
Chatsworth earthquake with SNR of 45 are shown in Table 3.6.
Table 3.6 First mode identified damping ratios for the 25-story model for different sampling rates
(true damping is 5%).
dt(s) 0.005 0.01 0.02
µξ 4.94 4.98 5.17
σξ 0.69 0.48 0.60
These results are representative and illustrated that the sampling rate had no noticeable effect
on the computed damping.
3.4 FREQUENCY CONTENT OF THE MOTION
It is well known that the viscous model is widely used due to its mathematical convenience
but that dissipation in structural systems is more nearly independent of frequency that
linearly related to frequency, as the viscous model implies. In the event of a pure harmonic
input the equivalent viscous damping for a structure that has frequency independent
dissipation would depend on the frequency of the excitation. The previous observation lead
us to wonder whether in the case of earthquake engineering the frequency content of the
motion could have a role to play on the identified damping. If the issue is going to be
practically relevant it would become apparent in the case of narrow band excitation. Namely,
one would find that the equivalent damping ration would prove frequency dependent. What
was found is that for practical excitations and low values of damping the equivalent viscosity
is the one that matches the viscous and the hysteretic transfer functions at the resonant
frequency. In the numerical investigation we used the SCT record of the 1985 Mexico City
earthquake as representative of a narrow band motion.
Variability in the damping does not come from changes in the frequency content from one
motion to the next (in a given structure).
4. REGRESSION ANALYSIS
Regression is the process by which one attempts to estimate the conditional expectation of a
dependent variable given the independent ones. With as the vector of regressors and as
the model parameters one has, for damping
( , )g (4.1)
where g is the postulated functional relationship. The functional relationship g is not
suggested by theory so it must be based on inspection of the data. The regressors considered
here are the peak ground motion parameters (PGA, PGV and PGD), the spectral ordinates
(SA, SV and SD), the building height, H, the frequency of the mode, f, and the effective
duration t0.9. This last entry defined as the time interval between the attainment of 5% and
95% of the total integral of the acceleration squared (Arias 1970). Not included due to lack of
information, but potentially important, are parameters related to the soil and the foundation
and to the type and density of partitions. In this study concrete and steel buildings are treated
in different data sets so building material is not a parameter in the regression list. For a given
form of the regression expression the model parameters are typically obtained as
2
2min ( ( , ) )i i iw g
(4.2)
where wi are weights. The estimate of the regression parameter coincides with the
maximum likelihood solution if the error on the measurement is Gaussian and the weights are
inversely proportional to the standard deviation of the error. From the results in Appendix B
one finds that the asymptotic standard deviation of the damping ratio is
2
1 1s
(4.3)
where s1 is the variance of the real part of the pole. Assuming that the coefficient of variation
of frequency is constant one concludes that the standard deviation of the damping is also
constant, a conclusion that coincides with that by Gersch (1974). Examination of the effect of
duration on the variance of identified damping presented in Appendix C shows that the
standard deviation varies inversely with the square root of the duration normalized by the
modal period. Combining this observation with the result in eq.4.3 one concludes that the
standard deviation of the damping can be taken as
0.9
1
t
(4.4)
where we have assumed that that the record duration can be taken as proportional to t0.9. In the
estimation of the regression model parameters the weights wi in eq.4.2 are taken as the reciprocals of
eq.4.4.
Goodness of Fit
When measurements have significant inherent variance the goodness of fit cannot be judged
from a simple inspection of the scatter. The objective is to obtain an expression such that the
differences between predictions and measurements (the residuals) agree with the distribution
that is anticipated for these errors. The standard goodness of fit test is the F-test for lack of fit
(FTLF). In this test the residuals are assumed to come from realizations of equal variance.
Since the identified damping ratios are not of equal variance we divide the residuals by the
result in eq.4.4. Namely, the adjusted residuals are
, 0.9,( ) i i p i i ir t (4.5)
Taking the adjusted deviations of the data as
0.9,( )i i i i id t (4.6)
the F statistic is given by
2
,
1 1
2
,
1 1
1
( )
1
( )
i
i
nni j
j i i
nn
i j
j i
r
n pnF
dN n
(4.7)
where n = number of bins, ni = number of samples in the ith
bin and N = total number of
observations, i = data point, p,i = predicted value and = mean of the data points in a bin.
Bins are such that the prediction varies little within the bin (5% of the average in our
numerical results). The F statistic has a Fisher-Snedecor distribution with (n-p) and (N-n)
degrees of freedom for the numerator and the denominator and a low value indicates a good
fit. Results on the goodness of fit, however, are most easily interpreted in terms of the p-
value of the F statistic. The p-value is such that if it is smaller than the acceptable Type I
error rate the proposed fit is rejected. The Type I error, in this case, consists in rejecting the
proposed fit when it is valid one. The typical p-value threshold is 0.05.
Coefficient of Determination
The coefficient of determination, R2, is defined as
2
,2
2
( )
1( )
i p i
i
i
i
y y
Ry y
(4.8)
This coefficient is essentially a measure of how much the regression line reduces the scatter
relative to the mean. It must be emphasized, however, that low values of R2 do not invalidate
regression results when the intrinsic variance of the data is large.
Functional Form
Inspection of the trend in the “local mean” of the data plotted vs. each regressor provides
guidance for selection of the functional form. We considered a number of different forms and
settled on two exponential ones: the first for cases where the damping decreases and the other
for cases where the damping increases with the regressor, namely
2
0 1
aa a e
(4.9)
and
2
0
11
b
b
b e
(4.10)
4.1 RESULTS
The regression was carried out for the first mode damping ratio for steel, concrete, masonry,
and wood buildings (The numerical values and the regressors for each considered case are
presented in Appendix D). When the mode considered is dominated by translation in one
direction the ground motion in this direction was used to compute the ground motion
parameters. When the mode is strongly coupled, or torsional, the average of the ground
motion parameters for the two directions was used. The best results for the expected value of
the first mode damping ratios are:
0.0131.22 4.26 H
s e (steel) (4.11)
0.0182.91 3.54 H
c e (concrete) (4.12)
8.84
1
0.11 0.23
A
m Se
(masonry) (4.13)
3.37
1
0.09 0.17
A
w Se
(wood) (4.14)
where H is in meters and SA is the 5% pseudo-spectral acceleration in g’s. Plots of the
regression, the 95% confidence intervals, and the data, are presented in figs.4.1 to 4.4.
Fig.4.1 Regression result (steel buildings) and 95% confidence limits, R2=0.37, F-test p-value
= 0.85
0 50 100 150 200 2500
1
2
3
4
5
6
7
8
9
10
H(m)
(%)
H(m)
0 10 20 30 40 50 60 702
3
4
5
6
7
8
9
10
(%)
H(m)
Fig.4.2 Regression result (concrete buildings) and 95% confidence limits, R2=0.11, F-test p-
value=0.72
Fig.4.3 Regression result (masonry buildings) and 95% confidence limits, R2=0.15, F-test p-value
=0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8
10
12
14
(%)
SA (g’s)SA (g’s)
0 0.2 0.4 0.6 0.8 1 1.2 1.42
4
6
8
10
12
14
SA (g’s)
(%)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
1
2
3
4
5
6
7
8
9
SA (g’s)
(%)
Fig.4.4 Regression result (wood buildings) and 95% confidence limits, R2=0.64, F-test p-value =0.97
Discussion
For steel buildings H provided the best correlation with damping ratio by a significant
margin. In concrete buildings the expression based on SA produced results that are only
slightly less correlated than those for H. For masonry and wood buildings the correlation with
SA was clearly the superior choice. These results are along the line of what one expects from
qualitative reasoning. Namely, in steel most of the intensity related increase in damping (in
the linear range) is related to non-structural components while in the other structural types
there is also a lateral load resisting mechanism dependence. For completeness, figs. 4.5 and
4.6 show the correlation of damping ratio with SA for steel and concrete buildings. The fact
that dependence on SA saturates very quickly in steel and less so in concrete is evident from
the plots and from the coefficient in front of SA in the best fit expressions.
A question that comes to mind is whether a multivariate regression using both H and SA
could lead to notable improvements but the answer to this proved negative because the two
parameters happen to be correlated. It is not difficult to see that this is so because as the
height increases the period lengthens and the spectral accelerations, except for short
buildings, decrease. For the buildings considered here the correlation coefficients between H
and SA are -0.57 and -0.39 for steel and concrete, respectively.
SA (g’s)
Fig.4.5 Regression with 5% damped pseudo spectral acceleration (steel buildings) and 95%
confidence limits, R2=0.15, F-test p-value =0.25
Fig.4.6 Regression with 5% pseudo-spectral acceleration (concrete buildings) and 95% confidence
limits, R2=0.64, F-test p-value =0.97
A final item worthy of attention is whether or not there is consistency in the results of the
regression for damping as a function of H and SA. Examination shows that the results are
indeed consistent. To clarify consider these two situations: a) say a steel building has H = 50
m. From the plot in fig.4.1 one finds that the 95% confidence for the regression is from 2.8%
to 4.05%. Looking at fig.4.5 one finds that this could be the solution for any value of SA, b)
consider H = 200 m. In this case the 95% confidence interval is from 0 to 3%. Looking at
fig.4.5 one concludes that consistency requires that SA for this building be no larger than
0.047 g. The data shows that there are 28 records for buildings whose heights are 200 m or
more and that, of all these, there are only two 2 cases where SA exceeded this limit. This
small violation is, of course, reasonable since a 95% confidence interval does not provide
complete certainty.
5. SUMMARY AND CONCLUSIONS
0 0.2 0.4 0.6 0.8 1 1.2 1.42
3
4
5
6
7
8
9
10
(%)
SA (g’s)SA (g’s)
APPENDICES
APPENDIX A – IDENTIFICATION
Time domain algorithms are typically based on an indirect approach. Namely, a model
mapping the sampled input and the sampled output is obtained and then it is converted to
continuous time. The postulated model in sampled time has the form
1k d k d k k
d kk k
A x B u
y
x v
wC x
(A.1)
where the measured inputs ( )k gu x k of dimension m and the measured acceleration outputs
ky of dimension r at time instants k are given. The sequences ( )kv and ( )kw are the
unknown process and measurement noise. From the measurements, the system matrices are
obtained from a system identification procedure. We have chosen subspace identification
methods (Van Overschee & De Moor, 1996) for their robustness, good numerical and
statistical properties as well as ease of implementation. In the following, the DSR method (Di
Ruscio, 1996) is described.
From the measurements, the data matrices
1 2 1 2
2 3 1 2 3 1
| |
1 1
,
k k k N k k k N
k k k N k k k N
k L k L
k L k L k N L k L k L k N L
y y y u u u
y y y u u uY U
y y y u u u
(A.2)
are filled, where L is a user defined parameter. Define the matrices
0| 0|
| | | 1 1| 1| | 1
0| 0|
,
T T
L L
k L k L k L k L k L k L
L L
Y YZ Y U Z Y U
U U
(A.3)
with projection 1( )T TU I U UU U . Note that matrices |k LZ and 1|k LZ are not computed
directly, but instead numerically stable QR decompositions of the data are used as described
in detail in (Di Ruscio, 1996). Matrix |k LZ possesses a factorization property |k L LZ O X
with the observability matrix
1
d
d d
L
Ld d
C
C AO
C A
(A.4)
and some other matrix X, and matrix 1|k LZ possesses the factorization property
1|k L L dZ O A X . The observability matrix is obtained from a Singular Value Decomposition
1
| 1 2
2
1
2
TL
T
k
S VZ U U
S V
(A.5)
which is truncated at the desired model order, as 1LO U . Then, 1 1
TX S V and matrix dA is
obtained from the factorization property of 1|k LZ as
| 1
1
1 11d k L
TA U Z V S
(A.6)
Matrix dC is found in the first r rows of LO . As the task is modal identification, only the
system matrices dA and dC are necessary. Once these matrices from the sampled time model
are available their conversion to continuous time follows as (Bernal 2007)
1ln( )c dA A
t
(A.7)
c dC C (A.8)
The damping ratios are obtained as the real part of the eigenvalues of Ac divided by their
magnitude, as indicated by eq.10.
APPENDIX B – RELATIONSHIP BETWEEN COEFFICIENTS OF VARIATION
It follows from eq.2.13 that the CRLB of damping and frequency is
1
, , ,
1( , ) f
T
f fI f J J (B.1)
Plugging in eq.2.14, a straightforward computation shows
2
, 2
/ /1
(2 )f
a a af
af f ff
(B.2)
where 2 1/2(1 )a . Accepting that the uncertainty of the pole has circular shape, the variance of
its real part equals the variance of its imaginary part, and is of the shape
1 2
2 1
s s
s s
(B.3)
where 1 0s , 2 1s s and 2 1s s . Then,
2
1 11 1 21
2
2 2
, 21 1
2
2
1 11
1
2
( ) ( )0 0 1
(2 )0 0
2
( ) 2
aaf
a
as a s
ff f a
s a s
ass as s
(B.4)
from where the ratio of the coefficients of variation follows as
2
2 1
2
2
2
1
2
1 1
COV( ) 1·
COV
2
2( )
s
f a s
as a s
as a s
(B.5)
This expression takes the minimum at 2 1s s and the maximum at 2 1s s , respectively, and is
shown for different damping values in Fig.2.3.
APPENDIX C – RELATIONSHIP BETWEEN VARIANCE AND NUMBER OF CYCLES
The estimation of the modal parameters varies as the number of samples increases. Most
parametric system identification methods, such as maximum likelihood, prediction error or
subspace methods, ensure asymptotic normality of the estimates, i.e. for any parameter
whose estimate ˆN is computed on N data samples, the Central Limit Theorem (CLT)
2ˆ( ) (0, )d
NN N (C.1)
holds, where 2 is the asymptotic variance. It follows that the COV of the estimate is
inversely proportional to N , and the estimation of the parameters gets more accurate as N
gets larger. To illustrate this effect, the COV of damping values from stochastic subspace
identification of a 6 DOF mass spring chain system with 2% damping and time step 0.02
were obtained from a Monte Carlo simulation for different numbers of samples in Table C1.
Due to the limited data length during an earthquake, modes with low frequencies are difficult
to estimate as they have only few cycles in the measured data. For example, a mode at 0.5 Hz
has only 30 cycles in an earthquake record of 60 seconds, while a mode at 5 Hz shows
already 300 cycles in the measured data. Thus the accuracy of the estimate of a mode with
frequency f depends on the number of cycles, defined as
,cn N f (C.2)
leading to a CLT for the damping estimates of the form
2,ˆ( (0 )) d
cn N (C.3)
We contend that equal accuracy of estimation is only possible when the respective modes
have an equal number of cycles, i.e. damping of a mode at 0.5 Hz can be estimated equally
well from a record of 600 seconds as the damping of a mode at 5 Hz estimated from 60
seconds of data. This contention is supported by the data obtained in Table C1. Multiplying
its rows with N leads to very similar (asymptotic) COVs for each frequency predicted by
(C.1), while multiplying in addition the columns with f
leads also to very similar
(asymptotic) COVs of damping ratios of modes with different periods, supporting the
previous contention and CLT (C.3). These results are presented in Table C2. Table C1. COVs of damping values for different number of samples and frequencies.
N 1.97 Hz 5.88 Hz 9.43 Hz 16.1 Hz 17.6 Hz 19.0 Hz
2000 0.53 0.21 0.19 0.15 0.14 0.14
5000 0.31 0.14 0.11 0.09 0.09 0.09
10000 0.21 0.10 0.08 0.07 0.07 0.06
15000 0.17 0.08 0.07 0.06 0.05 0.05
20000 0.15 0.07 0.05 0.05 0.05 0.04
Table C2. COVs of damping values normalized by N f .
