**2. Modal parameter estimation methods**

materials, and be able to carry light loads. But, nowadays, these conventional bridges have been

There are various bridge types constructed during the last century according to the carrier system type, span lengths, and material properties such as masonry arch bridges, long span concrete/steel/composite highway bridges, base isolated bridges, footbridges, steel bridges, suspension bridges, cable‐stayed bridges, and wooden/timber bridges. Masonry bridges have been built worldwide for social, economic, and strategic purposes. Originally intended to carry only pedestrian and horse‐drawn vehicles, many of these historical bridges currently serve as critical components of transportation systems and, thus, must withstand significantly larger loads. Among various types of civil engineering structures, long span highway bridges, which are commonly used for passing large rivers, dam reservoirs, and deep valleys, attract the greatest interest for study particularly in terms of structural performance. Footbridges are generally situated to allow pedestrians to cross water or railways in areas where there are no nearby roads to necessitate a road bridge, and also across busy roads to let pedestrians cross safely without slowing down the traffic. Steel offers many advantages to the bridge builder, not only the material itself, but also its broad architectural possibilities such as high strength‐ to‐weight ratio, high‐quality material, speed of construction, versatility, modifications, recycling, durability, and aesthetics. Suspension and cable‐stayed bridges are widely used across long spans (>550 m) and give rise to the usage of domains under the bridge. For this reason, the uses of suspension and cable‐stayed bridges have increased recently. Wood is one of the most used and common materials for bridge constructions from the ancient times when

humans first started finding ways on how to cross rivers and hard terrains.

Determination of dynamic response of bridges under static and dynamic loads, such as wind, earthquake, or traffic, is very complex and requires special studies. Finite element method has been widely used in civil engineering application since 1950s. Static, dynamic, linear, and nonlinear behavior can be obtained and illustrated using this method. It is generally expected that finite element models (FEMs) based on technical design data and engineering judgments can yield reliable simulation. However, because of modeling uncertainties, these models often cannot predict dynamic characteristics with the required level of accuracy. This raises the need for verification of finite element models using nondestructive experimental measurement tests. There are two basically different methods available to experimentally identify the dynamic system parameters of a structure: experimental modal analysis (EMA) and operational modal analysis (OMA). In the EMA, the structure is excited by known input forces and the structural behavior is evaluated. In the OMA, the ambient vibrations such as vehicle load, wind, or wave loads have been used to actuate the structures. Heavy forced excitations may become expensive and sometimes may cause damage to the structure. Ambient excitations and their combination are environmental or natural excitations. Structural identification using this method gains the major importance. In this case, only response data of ambient vibrations are measurable while actual loading conditions are unknown. A system identification procedure will therefore need

It is well accepted that the finite element model updating is used to minimize the differences between analytically and experimentally determined dynamic characteristics by changing

replaced to steel and reinforced concrete.

166 Structural Bridge Engineering

to base itself on output‐only data.

#### **2.1. Enhanced frequency domain decomposition (EFDD) method**

Enhanced frequency domain decomposition (EFDD) method is an extension of frequency domain decomposition (FDD) method which is a basic and easy‐to‐use method. In this method, modes are simply picked locating the peaks in singular value decomposition plots [1, 2].

In EFDD, the single degree of freedom (SDOF) power spectral density (PSD) function, identified around a peak of resonance, is taken back to the time domain using the Inverse Discrete Fourier Transform. In EFDD method, the relationship between unknown input and measured responses can be expressed as [2, 3]:

$$
\left[\left\{G\_{\mathcal{W}}\left(j\mathbf{w}\right)\right\}\right] = \left[H\left\{j\mathbf{w}\right\}\right] \left[\left\{G\_{\text{xx}}\left\{j\mathbf{w}\right\}\right\}\right] H \left(j\mathbf{w}\right) \right]^r \tag{1}
$$

where *Gxx*{*jw*} is the PSD matrix of the input, *Gyy*{*jw*} is the PSD matrix of the responses, *H*{*jw*} is the frequency response function (FRF) matrix, and \* and superscript *T* denote complex conjugate and transpose, respectively. The FRF can be written in partial fraction, i.e., pole/ residue form as [4]

$$H\left\{j\nu\right\} = \sum\_{k=1}^{n} \frac{R\_k}{j\nu - \lambda\_k} + \frac{R\_k^\*}{j\nu - \lambda\_k^\*} \tag{2}$$

where *n* is the number of modes, *λk* is the pole, and, *Rk* is the residue. Substituting Eq. (2) into (1), we have

$$G\_{\rm{yy}}\left(j\text{w}\right) = \sum\_{k=1}^{n} \sum\_{s=1}^{n} \left[ \frac{R\_{k}}{j\text{w} - \lambda\_{k}} + \frac{R\_{k}^{\ast}}{j\text{w} - \lambda\_{k}^{\ast}} \right] \left[ G\_{\rm{xx}}\left\{ j\text{w}\right\} \right] \left[ \frac{R\_{s}}{j\text{w} - \lambda\_{s}} + \frac{R\_{s}^{\ast}}{j\text{w} - \lambda\_{s}^{\ast}} \right]^{H} \tag{3}$$

where *s* is the singular value, superscript *H* denotes complex conjugate and transpose. Multiplying the two partial fraction factors and making use of the Heaviside partial fraction theorem, the output PSD can be reduced to a pole/residue form as follows:

$$G\_{\rm yr} \left( j\nu \right) = \sum\_{k=1}^{n} \frac{A\_k}{j\nu - \lambda\_k} + \frac{A\_k^\*}{j\nu - \lambda\_k^\*} + \frac{B\_k}{-j\nu - \lambda\_k} + \frac{B\_k^\*}{-j\nu - \lambda\_k^\*} \tag{4}$$

where *Ak* is the *k*th residue matrix of the output PSD. In the EFDD identification, the first step is to estimate the PSD matrix. The estimation of the output PSD, *Gyy*(*jw*) known at discrete frequencies *w* = *wi* is then decomposed by taking the SVD of the matrix

$$\left(G\_{\mathcal{Y}}\left(j\mathbf{w}\_{i}\right) = U\_{i}S\_{i}U\_{i}^{H}\right) \tag{5}$$

where the matrix <sup>=</sup> 1, 2,…., is a unitary matrix holding the singular vectors, *uij*, and *Si* is a diagonal matrix holding the scalar singular values *sij* [4–6].

#### **2.2. Stochastic subspace identification (SSI) method**

Stochastic subspace identification (SSI) method is an output‐only time domain method that directly works with time data, without the need to convert them to correlations. The model of structural vibrations can be defined by a set of linear, constant coefficient and second‐order differential equations [7]:

$$
\dot{V}M\ddot{U}\left(t\right) + C\_2\dot{U}\left(t\right) + KU\left(t\right) = F\left(t\right) = B\_2u\left(t\right)\tag{6}
$$

where *M, C*2, and *K* are the mass, damping, and stiffness matrices, *F*(*t*) is the excitation force, and *U*(*t*) is the displacement vector depending on time *t*. Note that the force vector *F*(*t*) is factorized into a matrix *B*2 describing the inputs in space and a vector *u*(*t*). The equation of dynamic equilibrium (6) will be converted to a more suitable form: the discrete‐time stochastic state‐space model [7, 8]. With the following definitions

$$\mathbf{x}(t) = \begin{pmatrix} U(t) \\ \dot{U}(t) \end{pmatrix}, A = \begin{pmatrix} 0 & I\_{n\_2} \\ -M^{-1}K & -M^{-1}C\_2 \end{pmatrix}, B = \begin{pmatrix} 0 \\ M^{-1}B\_2 \end{pmatrix} \tag{7}$$

Eq. (6) can be transformed into the state equation

$$
\dot{X}(t) = A x(t) + B u(t) \tag{8}
$$

where *A* is the state matrix, *B* is the input matrix, and *x*(*t*) is the state vector. If it is assumed that the measurements are evaluated at only one sensor location, and that this sensor can be accelerometer, velocity, or displacement transducer, the observation equation is [9]:

where *s* is the singular value, superscript *H* denotes complex conjugate and transpose. Multiplying the two partial fraction factors and making use of the Heaviside partial fraction

*kk k k*

*k kk k k*

<sup>=</sup> *jw jw jw jw*

where *Ak* is the *k*th residue matrix of the output PSD. In the EFDD identification, the first step is to estimate the PSD matrix. The estimation of the output PSD, *Gyy*(*jw*) known at discrete

where the matrix <sup>=</sup> 1, 2,…., is a unitary matrix holding the singular vectors, *uij*, and

Stochastic subspace identification (SSI) method is an output‐only time domain method that directly works with time data, without the need to convert them to correlations. The model of structural vibrations can be defined by a set of linear, constant coefficient and second‐order

where *M, C*2, and *K* are the mass, damping, and stiffness matrices, *F*(*t*) is the excitation force, and *U*(*t*) is the displacement vector depending on time *t*. Note that the force vector *F*(*t*) is factorized into a matrix *B*2 describing the inputs in space and a vector *u*(*t*). The equation of dynamic equilibrium (6) will be converted to a more suitable form: the discrete‐time stochastic

( ) <sup>0</sup> <sup>0</sup> , , ( )

*U t MK MC M B* - - -

1 1 1

2 2

*x t Ax t Bu t* &() () = + ( ) (8)

æ ö æ ö æ ö = = ç ÷ ç ÷ <sup>=</sup> ç ÷ è ø è ø - - è ø & (7)

is then decomposed by taking the SVD of the matrix

\* \*

= ++ + - - -- -- å (4)

 l

( ) *<sup>H</sup> G jw U S U yy i i i i* = (5)

*MU t C U t KU t F t B u t* () () () () () + + == 2 2 && & (6)

 l

( ) \* \*

*AA B B G jw*

ll

theorem, the output PSD can be reduced to a pole/residue form as follows:

1 *n*

is a diagonal matrix holding the scalar singular values *sij* [4–6].

**2.2. Stochastic subspace identification (SSI) method**

state‐space model [7, 8]. With the following definitions

Eq. (6) can be transformed into the state equation

( ) <sup>2</sup>

*<sup>n</sup> U t I xt A B*

*yy*

frequencies *w* = *wi*

168 Structural Bridge Engineering

differential equations [7]:

*Si*

$$\mathbf{x}\left(\mathbf{y}\left(t\right) = \mathbf{C}\mathbf{x}\left(t\right) + Du\left(t\right)\right) \tag{9}$$

where *C* is the output matrix and *D* is the direct transmission matrix. Eqs. (8) and (9) constitute a continuous‐time deterministic state‐space model. This is not realistic: measurements are available at discrete time instants *k*Δ*t, k* ∈ *N* with Δ*t*, the sample time and noise is always influencing the data. After sampling, the state‐space model looks like [6]:

$$\begin{aligned} \mathbf{x}\_{k+1} &= A\mathbf{x}\_k + Bu\_k\\ \mathbf{y}\_k &= C\mathbf{x}\_k + Du\_k \end{aligned} \tag{10}$$

where *xk* = *x*(*k*Δ*t*) is the discrete‐time state vector. The stochastic components are included and obtained discrete‐time combined deterministic‐stochastic state‐space model:

$$\begin{aligned} \mathbf{x}\_{k+1} &= A\mathbf{x}\_k + Bu\_k + \mathbf{w}\_k\\ \mathbf{y}\_k &= C\mathbf{x}\_k + Du\_k + \mathbf{v}\_k \end{aligned} \tag{11}$$

where *wk* is the process noise due to disturbances and modeling inaccuracies and *vk* is the measurement noise due to the sensor inaccuracy. They are both immeasurable vector signals but we assume that they are zero mean, white, and covariance matrices [7]:

$$E\left[\begin{pmatrix}\boldsymbol{\nu}\_{\boldsymbol{\rho}}\\\boldsymbol{\nu}\_{\boldsymbol{\rho}}\end{pmatrix}\begin{pmatrix}\boldsymbol{\nu}\_{q}^{\boldsymbol{r}}&\boldsymbol{\nu}\_{q}^{\boldsymbol{r}}\end{pmatrix}\right] = \begin{pmatrix}\boldsymbol{\mathcal{Q}} & \boldsymbol{S}\\\boldsymbol{\mathcal{S}}^{\boldsymbol{r}}&\boldsymbol{R}\end{pmatrix}\boldsymbol{\delta}\_{pq} \tag{12}$$

where *E* is the expected value operator and *δpq* is the Kronecker delta. This is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. It can be written as the symbol *δpq*, and treated as a notational shorthand rather than a function:

$$\delta\_{\rho q} = \begin{cases} 1, \text{if } p = q \\ 0, \text{if } p \neq q \end{cases} \tag{13}$$

The vibration information that is available in structural health monitoring is usually the responses of a structure excited by the operational inputs that are some immeasurable inputs. It is impossible to distinguish deterministic input *uk* from the noise terms *wk*, *vk* in Eq. (11). If the deterministic input term *uk* is modeled by the noise terms *wk*, *vk* the discrete‐time purely stochastic state‐space model is obtained:

$$\begin{aligned} \mathbf{x}\_{k+1} &= A\mathbf{x}\_k + \mathbf{w}\_k\\ \mathbf{y}\_k &= \mathbf{C}\mathbf{x}\_k + \mathbf{v}\_k \end{aligned} \tag{14}$$

Eq. (14) constitutes the basis for the time‐domain system identification through operational vibration measurements.

#### **2.3. Modal assurance criterion**

The modal assurance criterion (MAC) is defined as a scalar constant relating the degree of consistency (linearity) between one modal and another reference modal vector [10] as follows:

$$MAC = \frac{\left| \left< \mathcal{D}\_{\boldsymbol{\omega}\boldsymbol{i}} \right>^{\boldsymbol{r}} \left< \mathcal{D}\_{\boldsymbol{\omega}\boldsymbol{j}} \right> \right|^{2}}{\left< \mathcal{D}\_{\boldsymbol{\omega}\boldsymbol{i}} \right>^{\boldsymbol{r}} \left< \mathcal{D}\_{\boldsymbol{\omega}\boldsymbol{i}} \right> \left< \mathcal{D}\_{\boldsymbol{\omega}\boldsymbol{j}} \right>^{\boldsymbol{r}} \left< \mathcal{D}\_{\boldsymbol{\omega}\boldsymbol{j}} \right>} \tag{15}$$

where ∅ and ∅ are the modal vectors of *i*th and *j*th for different methods, respectively.
