Extraction of contact-point response in indirect bridge health monitoring using an input estimation approach

Abstract

Identification of bridge dynamic properties from moving vehicle responses presents several practical benefits. However, a problem that arises when working with vehicle responses for indirect bridge health monitoring is that the bridge dynamics may get low-pass filtered by the vehicle suspension dynamics, rendering the identification of higher bridge modes difficult. Instead, the contact-point (CP) response—response at the contact point of the vehicle with the bridge surface—is a superior alternative to the vehicle response for identifying the bridge modal features. In the $\text {CP}$ response, the vehicle dynamics is suppressed and the higher bridge modes are significantly enhanced, thus making it better suited for modal identification. Extracting the $\text {CP}$ response from vehicle response is, however, not straightforward for a multiple degrees of freedom (MDoF) vehicle model. In this study, a novel methodology is proposed to extract $\text {CP}$ acceleration from the measured vehicle acceleration using the knowledge of the $\text {MDoF}$ vehicle dynamics. The $\text {CP}$ acceleration is shown to act as a base-excited input to the test vehicle and is extracted via a joint input-state estimation procedure employing a Gaussian process latent force model (GPLFM). Numerical case studies are considered to assess the quality of the $\text {CP}$ acceleration estimated with the proposed approach. It is found that the proposed method performs well and the extracted $\text {CP}$ acceleration response is able to reduce the effect of vehicle dynamics and improve the prominence of higher bridge modes.

Appendices

Appendix 1: Vehicle-bridge interaction equations

The $\text {VBI}$ equations formulated here follow closely with that provided in [13].

1.1 Equation of motion of bridge

The governing partial differential equation (PDE) of the simply supported bridge deck under the influence of a moving vehicle is written as follows:

$$\begin{aligned}& {EI} \frac{\partial ^4 d_\mathrm{b}(x,t)}{\partial x^4} + \mu _\mathrm{b} \frac{\partial d_\mathrm{b}(x,t)}{\partial t} + m_\mathrm{b} \frac{\partial ^2 d_\mathrm{b}(x,t)}{\partial t^2}\nonumber \\&\quad = f(x,t) \delta (x-vt) \end{aligned}$$

(24)

with f(x, t) being the interaction force between the bridge and the vehicle

$$\begin{aligned} f(x,t) = -(m_\mathrm{u}+m_\mathrm{s}) g + k_\mathrm{t} \left( d_\mathrm{u}(t) - d_\mathrm{b}(x,t) - r(x)\right) , \end{aligned}$$

(25)

g is the acceleration due to gravity, $\delta \left( \cdot \right)$ is the Dirac’s delta function, and $\mu _\mathrm{b}$ is the bridge damping per unit length. The material damping $\mu _\mathrm{b}$ is ignored, instead modal damping is assumed.

A modal solution is employed for the bridge PDE, that is, the bridge response is expressed using a set of $n_m$ participating modes:

$$\begin{aligned} d_\mathrm{b}(x,t) = \sum _{j=1}^{n_m} \phi _j(x) \; \eta _j(t). \end{aligned}$$

(26)

$\phi _j(x) = \sqrt{\frac{2}{m_\mathrm{b} L}} \sin \left( \frac{j \pi x}{L}\right)$ denotes the jth mass-normalized bridge mode shape and $\eta _j(t)$ denotes the jth modal response. Substituting Eq. (26) in Eq. (24) and applying modal orthogonality conditions yields an ordinary differential equation (ODE) for the jth mode:

$$\begin{aligned}&\ddot{\eta }_j(t) + 2 \zeta _j \omega _j {\dot{\eta }}_j(t) + \left( \omega _j^2 + k_\mathrm{t} \phi _j^2\left( vt\right) \right) {\eta }_j(t) \nonumber \\&\quad - k_\mathrm{t} d_\mathrm{u} \phi _j(vt) = F(t) \phi _j(vt), \end{aligned}$$

(27)

where

$$\begin{aligned} F(t) = -(m_\mathrm{u}+m_\mathrm{s}) g -k_\mathrm{t} r(vt), \end{aligned}$$

(28)

and $\omega _j$ and $\zeta _j$ are the jth bridge modal frequency and damping ratio, respectively; $\omega _j^2 = \frac{{EI}}{m_\mathrm{b}} \left( \frac{j \pi }{L}\right) ^4$ and $\zeta _j = \frac{\mu _\mathrm{b}}{2 m_\mathrm{b} \omega _j}$.

Ambient excitation, in the form of bandlimited Gaussian white noise, is applied to the left and right supports as support excitations. The support accelerations at the left and right bridge supports are denoted as $\ddot{d}_{\mathrm{{ls}}}(t)$ and $\ddot{d}_{\mathrm{rs}}(t)$, respectively. Adding the support excitations, the equation of motion for the jth vibration mode of the deck becomes

$$\begin{aligned} \begin{aligned}&\ddot{\eta }_j(t) + 2 \zeta _j \omega _j {\dot{\eta }}_j(t) + \left( \omega _j^2 + k_\mathrm{t} \phi _j^2\left( vt\right) \right) {\eta }_j(t) \;-\; k_\mathrm{t} d_\mathrm{u}\phi _j(vt) \\&\quad = \;F(t) \phi _j(vt) - P_j(t), \end{aligned} \end{aligned}$$

(29)

where

$$\begin{aligned} P_j(t)= & {} \ddot{d}_{\mathrm{{ls}}}(t) \int _{0}^{L} m_\mathrm{b} \left( 1-\frac{x}{L}\right) \phi _j(x) \mathrm{d}x \nonumber \\&+ \ddot{d}_{\mathrm{rs}}(t) \int _{0}^{L} \rho \frac{x}{L} \phi _j(x) \mathrm{d}x, \end{aligned}$$

(30)

for $j=1,\ldots ,n_m$.

1.2 Vehicle-bridge interaction equation

Combining Eqs. (2) and (29), the equation of motion of the $\text {VBI}$ system can be written as a matrix ODE:

$$\begin{aligned} \mathbf {{M}}_{a}(t) \varvec{{\ddot{d}}}_a(t) + \mathbf {{C}}_a(t) \varvec{{{\dot{d}}}}_a(t) + \mathbf {{K}}_a(t) \varvec{{d}}_a(t) = \varvec{{F}}_a(t) \end{aligned}$$

(31)

where

$$\begin{aligned} \begin{aligned} \mathbf {{M}}_a(t)&= \begin{bmatrix} 1 &{} 0 &{} \ldots &{} 0 &{} \mathbf {{0}}\\ 0 &{} 1 &{} \ldots &{} 0 &{} \mathbf {{0}}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \ldots &{} 1 &{} \mathbf {{0}}\\ \mathbf {{0}}&{} \mathbf {{0}}&{} \ldots &{} \mathbf {{0}}&{} \mathbf {{M}}_\mathrm{v} \end{bmatrix} \\ \mathbf {{C}}_a(t)&= \begin{bmatrix} 2 \zeta _1 \omega _1 &{} 0 &{} \ldots &{} 0 &{} \mathbf {{0}}\\ 0 &{} 2 \zeta _2 \omega _2 &{} \ldots &{} 0 &{} \mathbf {{0}}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \ldots &{} 2 \zeta _{n_m} \omega _{n_m} &{} \mathbf {{0}}\\ \mathbf {{0}}&{} \mathbf {{0}}&{} \ldots &{} \mathbf {{0}}&{} \mathbf {{C}}_\mathrm{v} \end{bmatrix}\\ \mathbf {{K}}_a(t)&= \begin{bmatrix} \omega _1^2 + k_\mathrm{t} \phi _1^2(vt) &{} 0 &{} \ldots &{} 0 &{} - k_\mathrm{t} \phi _1(vt) \mathbf {{L}}^\mathrm{T}\\ 0 &{} \omega _2^2 + k_\mathrm{t} \phi _2^2(vt) &{} \ldots &{} 0 &{} - k_\mathrm{t} \phi _2(vt) \mathbf {{L}}^\mathrm{T} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \ldots &{} \omega _{n_m}^2 + k_\mathrm{t} \phi _{n_m}^2(vt) &{} - k_\mathrm{t} \phi _{n_m}(vt) \mathbf {{L}}^\mathrm{T}\\ - \mathbf {{L}} k_\mathrm{t} \phi _1(vt) &{} - \mathbf {{L}} k_\mathrm{t} \phi _2(vt) &{} \ldots &{} -\mathbf {{L}} k_\mathrm{t} \phi _{n_m}(vt) &{} \mathbf {{K}}_\mathrm{v} \end{bmatrix} \\ \varvec{{d}}_a(t)&= \begin{bmatrix} \eta _1(t)\\ \eta _2(t) \\ \vdots \\ \eta _{n_m}(t) \\ \varvec{{d}}_\mathrm{v}(t) \end{bmatrix} \quad \varvec{{F}}_a(t) = \begin{bmatrix} \phi _1(vt) F(t) - P_1(t)\\ \phi _2(vt) F(t) - P_2(t)\\ \vdots \\ \phi _{n_m}(vt) F(t) - P_{n_m}(t)\\ \mathbf {{L}} k_\mathrm{t} r(vt) \end{bmatrix}, \quad \mathbf {{L}} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \end{aligned} \end{aligned}$$

Equation (31) represents a linear ODE with time-varying coefficient matrices. To solve this equation, the Newmark-Beta average acceleration method [7] is employed. On solving, one obtains the vehicle responses as well as the bridge modal responses.

Appendix 2: Kalman filter and RTS smoother

The Kalman filter and $\text {RTS}$ smoother equations for obtaining the smoothed state estimates $\hat{\varvec{{x}}}^a_{k|N}$ and covariance estimates $\hat{\mathbf {{V}}}^a_{k|N}, \hat{\mathbf {{V}}}^a_{k+1,k|N}$ of the discrete-time $\text {LTI}$ system described by Eq. (20) are given below:

Kalman filter: Do for $k = 1,\ldots , N$

$$\begin{aligned} {e}_k&= y_k - \mathbf {{G}}_{ad} \hat{\varvec{{x}}}_{k|k-1} \end{aligned}$$

(32a)

$$\begin{aligned} \varSigma _k&= \mathbf {{G}}_{ad} \hat{\mathbf {{V}}}^a_{k|k-1} \mathbf {{G}}_{ad}^\mathrm{T} + R \end{aligned}$$

(32b)

$$\begin{aligned} \mathbf {{K}}_k&= \hat{\mathbf {{V}}}^a_{k|k-1} \mathbf {{G}}_{ad}^\mathrm{T} \;/\; \varSigma _k \end{aligned}$$

(32c)

$$\begin{aligned} \hat{\varvec{{x}}}^a_{k|k}&= \hat{\varvec{{x}}}^a_{k|k-1} + \mathbf {{K}}_k {e}_k \end{aligned}$$

(32d)

$$\begin{aligned} \hat{\mathbf {{V}}}^a_{k|k}&= \hat{\mathbf {{V}}}^a_{k|k-1} - \mathbf {{K}}_k \varSigma _k \mathbf {{K}}_k^\mathrm{T} \end{aligned}$$

(32e)

$$\begin{aligned} \hat{\varvec{{x}}}^a_{k+1|k}&= \mathbf {{F}}_{ad} \hat{\varvec{{x}}}^a_{k|k} \end{aligned}$$

(32f)

$$\begin{aligned} \hat{\mathbf {{V}}}^a_{k+1|k}&= \mathbf {{F}}_{ad} \hat{\mathbf {{V}}}^a_{k|k} \mathbf {{F}}_{ad}^\mathrm{T} + \mathbf {{Q}}_d. \end{aligned}$$

(32g)

Here $\hat{\varvec{{x}}}^a_{k|k-1}$ and $\hat{\varvec{{x}}}^a_{k|k}$ represent the kth predicted and filtered state estimate respectively, and, $\hat{\mathbf {{V}}}^a_{k|k-1}$ and $\hat{\mathbf {{V}}}_{k|k}$ denote the kth predicted and filtered state error covariance matrices, respectively. The Kalman filter recursion is started from an initial state $\hat{\varvec{{x}}}_{1|0}$ and an initial covariance $\hat{\mathbf {{V}}}_{1|0}$. ${e}_k$ and $\varSigma _k$ represent the innovation and the innovation variance at the kth time step; note they are scalar-valued in this study.

Following the filtering step, the (fixed interval) smoothing recursions given by the $\text {RTS}$ smoother are computed as follows:

Kalman smoother: Do for $k = N,\ldots , 1$

$$\begin{aligned} \mathbf {{N}}_k&= \hat{\mathbf {{V}}}^a_{k|k} \mathbf {{F}}_{ad}^\mathrm{T} \left( \hat{\mathbf {{V}}}^a_{k+1|k}\right) ^{-1} \end{aligned}$$

(33a)

$$\begin{aligned} \hat{\varvec{{x}}}^a_{k|N}&= \hat{\varvec{{x}}}^a_{k|k} + \mathbf {{N}}_k \left( \hat{\varvec{{x}}}^a_{k+1|N} - \hat{\varvec{{x}}}^a_{k+1|k} \right) \end{aligned}$$

(33b)

$$\begin{aligned} \hat{\mathbf {{V}}}^a_{k|N}&= \hat{\mathbf {{V}}}^a_{k|k} + \mathbf {{N}}_k \left( \hat{\mathbf {{V}}}^a_{k+1|N} - \hat{\mathbf {{V}}}^a_{k+1|k} \right) \mathbf {{N}}_k^\mathrm{T} \end{aligned}$$

(33c)

$$\begin{aligned} \hat{\mathbf {{V}}}^a_{k+1,k|N}&= \hat{\mathbf {{V}}}^a_{k+1|N} \mathbf {{N}}_k^\mathrm{T}. \end{aligned}$$

(33d)