# Automatic Identification of Bridge Vortex-Induced Vibration Using Random Decrement Method
## 2 Presentation of Method
### Random Decrement Technique  
In this section, the background of RDT was briefly introduced. The random decrement technique is a time-domain method that uses the ensemble average of ambient data to approximate the free vibration responses when structures are exposed to linear, Gaussian white noise excitation. It was initially proposed by [PERSON] in 1973 [28; 29]. Later in 1982, [PERSON] provided a strict mathematical proof for a single-degree-of-freedom system under the assumption of input to be a zero-mean, stationary, and Gaussian random process [27]. The technique can also be extended to multi-degrees-of-freedom systems naturally, and is identified as the multimode random decrement technique (MRDT) [30; 31].  
For a linear dynamical model, the forced vibration response, \(y(t)\), at a measurement point of a structure under any linear, Gaussian white noise excitation, \(f(t)\), can be expressed as:  
\[y(t)=y(0)D(t)+\dot{y}(0)V(t)+\int_{0}^{t}h(t-\tau)f(\tau)d\tau, \tag{1}\]  
\[D(t)=e^{-\zeta_{d}\omega t}\bigg{[}\cos\omega_{d}t+\frac{\zeta_{d}}{\omega_{d} }\sin\omega_{d}t\bigg{]}, \tag{2}\]  
\[V(t)=\frac{1}{\omega_{d}}\sin\omega_{d}t, \tag{3}\]  
where \(D(t)\) is free vibration response with an initial displacement of 1 and an initial velocity of 0; \(V(t)\) is free vibration response with an initial displacement of 0 and an initial velocity of 1; \(y(0)\) and \(\dot{y}(0)\) are initial displacement and velocity of the vibration system, respectively; \(h(t)\) is unit impulse response function of a viscously damped single-degree-of-freedom system; \(f(t)\) is dynamic loading; and \(\zeta\), \(\omega\), and \(\omega_{d}\) are modal damping ratio, natural frequency, and damped frequency, respectively.  
As shown in Equation (1), the response \(y(t)\) in each time instant \(t\) can be decomposed as a linear superposition of three parts: The first two parts in the right hand side of Equation (1) represent the free vibration response due to initial displacement and initial velocity, and the third part is the forced vibration response due to dynamic loading, \(f(t)\). When \(f(t)\) follows a Gaussian white noise process, the responses, \(y(t)\), will also obey a Gaussian white noise process.  
Figure 3 shows the overview of random decrement technique. By selecting an appropriate constant \(A\) to intercept a measured random vibration response signal, \(y(t)\), a series of sub-response process, \(y(t-t_{i})\), can be obtained as follow:  
\[\begin{split}& y(t-t_{i})\\ &=y(t_{i})D(t-t_{i})+\dot{y}(t_{i})V(t-t_{i})+\int_{t_{i}}^{t}h(t -\tau)f(\tau)d\tau\end{split} \tag{4}\]  
Figure 3: Overview of the random decrement technique (RDT).  
Since the \(f(t)\) is a zero-mean stationary Gaussian process, the starting point of time does not affect its random characteristics. By moving a series of time starting points \(t_{i}\) of \(y(t-t_{i})\) to the origin of coordinate, a series of sample function, \(x_{i}(t)\) (\(I=1,\,2,\,\ldots\,,\,N\)), of random processes can be obtained:  
\[x_{i}(t)=AD(t)+\dot{y}(t_{i})V(t)+\int_{0}^{t}h(t-\tau)f(\tau)d\tau \tag{5}\]  
Because \(f(t)\) and \(\dot{y}(t)\) are zero-mean stationary Gaussian processes, the expectation of a sub-excitation process, \(E[f(t)]\), and the expectation of the sub-response process, \(E[\dot{y}(t)]\), are zero. The mathematical expectation of \(x_{i}(t)\) (\(I=1,\,2,\,\ldots\,,\,N\)) is obtained as follows:  
\[\begin{split} E[x(t)]&=E\Big{[}AD(t)+\dot{y}(t_{i} )V(t)+\int_{0}^{t}h(t-\tau)f(\tau)d\tau\Big{]}\\ &=E[A]D(t)+E\Big{[}\dot{y}(t_{i})\Big{]}V(t)+\int_{0}^{t}h(t- \tau)E[f(\tau)]d\tau\\ &=AD(t)\end{split} \tag{6}\]  
In practice, due to the limited measurement time and data, the average of \(x(t)\) is used to replace the mathematical expectation:  
\[x(t)=\frac{1}{N}\sum_{i=1}^{N}x_{i}(t)\approx E[x(t)]=AD(t) \tag{7}\]  
Therefore, a damped free vibration response with an initial displacement of \(A\) and an initial velocity of 0 is obtained.