## 2 Method to Evaluate the Lateral Vibration of Footbridges Including the Postlock-in Behaviour
### Footbridge Response Due to the Auto-Induced Component of the Pedestrian Load  
As described before, when walking on a moving surface, people perceive the floor vibration and modify their gait to maintain balance. This interaction with the motion of the structure produces the auto-induced components of the pedestrian load. In the case of lightly damped footbridges, the structural response is amplified only in a very narrow frequency interval, and the resonant harmonic of the auto-induced force induces the maximum dynamic response in the footbridge's natural frequency. Experiments show that this resonant harmonic is independent of the pedestrian frequency. Even if its magnitude is small for one individual, the passage of a crowd can trigger a multiplying effect. Hence, the proposed method assumes that the auto-induced response is controlled by the resonant harmonic, omitting other components such as that proportional to acceleration, which slightly modifies the natural frequency but affects little the response magnitude. In the frequency domain, the spectral value of the auto-induced response can be directly obtained once the spectral value of the auto-induced resonant harmonic is known. Considering that the steady-state velocity response of a damped system due to a resonant harmonic vibration (_f = fb_) is in-phase with the load, the resonant force can be expressed through the velocity proportional component of the experimentally measured actions. Hence, the auto-induced component \(\alpha\)(f) \(x\) Y(f) in Figure 2 is defined from the dynamic tests performed by [PERSON] et al. in 2009 at the University of Florence in Prato, Italy [31]. Seventy-one volunteers participated by walking on a swaying treadmill with lateral harmonic displacement expressed by \(y(t)=y_{0}\ sin(2\pi\ell_{b}t)\) (m), where \(y_{0}\) (m) is the amplitude of the lateral movement and \(f_{b}\) (Hz) is the frequency. [PERSON] et al. measured both the treadmill motion and the lateral forces that were exerted by the participants. The auto-induced components, which were determined through the cross-covariance between the measured pedestrian force and the velocity of the treadmill, were expressed as proportional to the treadmill velocity in the following way:  
\[c_{p}\left(\frac{f_{b}}{f_{p}},y_{0}\right)y^{\prime}(t)\ \ \stackrel{{ Fourier Transform}}{{\longrightarrow}}\ \ c_{p}\left(\frac{f_{b}}{f_{p}},y_{0}\right)2\pi f\ Y(f) \tag{2}\]  
in which the coefficient of proportionality \(c_{p}\)(_f_/_f_,\(y\)0) (Nsm\({}^{-1}\)) that depends on the frequency ratio \(f_{b}\)(_f_/_f_p and on the amplitude of the movement \(y_{0}\) is defined by fitting an exponential function to the data [17], and thus considering the large randomness of the measured load. Therefore, the feedback function \(\alpha\)(f) (Nm\({}^{-1}\)) can be obtained from Equation (2) and expressed as follows:  
\[\alpha(f)=c_{p}\left(\frac{f_{b}}{f_{p}},y_{0}\right)2\pi f \tag{3}\]  
In order to present an easy formulation, the coefficient of proportionality is defined by using the simplest expression proposed by [PERSON] [42]. This expression omits the dependency with the amplitude \(y_{0}\) and is expressed as follows:  
\[c_{p}\left(\frac{f_{b}}{f_{p}}\right)=\ -794\left(\frac{f_{b}}{f_{p}}\right)^{2} +1558\frac{f_{b}}{f_{p}}-580\ ;\ \ \ 0.4\leq\frac{f_{b}}{f_{p}}\leq 1.2 \tag{4}\]  
Figure 3 compares the value of the coefficient of proportionality \(c_{p}\) obtained using the amplitude dependent stochastic model [17] at four different vibration amplitudes (4.5, 10, 19.4 and 31 mm), with the value obtained using Equation (4) [42]. It can be observed that the value of \(c_{p}\) decreases as the amplitude increases, resulting in the second order polynomial (Equation (4)) being an upper bound of the experimental results. Furthermore, the dependence of the coefficient \(c_{p}\)(_f_/_f_/_p_) on the step frequency \(f_{p}\) shows the inter-subject variability in the auto-induced loads.