## 3 Collapse Control Mechanism
### Fluid Viscous Damper Shock Absorption and Collapse Control  
A fluid viscous damper is a kind of damper. Its working principle is to drive the piston to reciprocate in the cylinder by the piston rod, thus liquid (silicone oil) flows through a specially shaped piston head and provides different damping forces by different flow rates of liquid. Figure 1 shows the specific construction. Through the energy consumption of the damper, the input energy by the seismic wave is absorbed, and the damage of the bridge structure is reduced by the damper so that the risk of the structure collapse is reduced. Therefore, the collapse control of the damper is mainly related to its own energy-absorption mechanism and capacity. The energy consumption principle of the FVD is described below. The FVD is a velocity-dependent damper whose damping force can be expressed as:  
\[F=Cv^{\alpha}\] (11a) where \[C\] is the damping coefficient, \[v\] is the velocity, and \[\alpha\] is the velocity index. When the seismic input is certain, then \[v\] is the definite value, so the damping force is related to the values of \[C\] and \[\alpha\]. When \[\alpha\] is certain, the larger the \[C\] is, the greater the \[F\] is. When \[C\]0 is certain, \[C\]1 varies with the velocity index \[C\]2. When \[C\]3 is greater than 1 m/s, the curve of F with \[C\]4 greater than 1 rises faster than that with \[C\]5 less than 1, and the change speed of curve F with \[C\]6 equal to 1 is between the two. When the speed is less than 1 m/s, the curve \[C\]7 with \[C\]8 less than 1 is faster than that with \[C\]9 more than 1. If \[v\]0 is less than 1 m/s, the law is the opposite. See Figure 2 for details. The increasing speed of the damping force F is not the most important factor of the velocity index \(\alpha\). The change of \(\alpha\) directly influences the energy consumption level of the damper in the case of assumed \(v\) and \(\alpha\). Assuming that the motion of the damper is the simplest sinusoidal curve \(u=u_{0}\sin\omega_{0}t\), thus the damper completes one cycle of the displacement-consumption energy \(W_{d}\) is  
\[v\]1  
where \(T\), \(u_{0}\), \(\omega\), \(F\) are sine wave vibration period (2\(\pi\)/\(\omega_{0}\)), amplitude, circular frequency, and output force of the damper, respectively. \[v\]2 Schematic Diagram of Fluid Viscous Damper. \[v\]3 Damping Force and Velocity Curve with different Velocity Indices. According to Equation (12), the integral can be obtained:  
\[v\]4  
where \(\Gamma\) is the Gamma function. The maximum \(F_{\max}\) of the output force of the damper is \(C\upsilon_{0}^{\alpha}\omega_{0}^{\alpha}\), so the Equation (13) can be simplified as follows:  
\[v\]5  
If we assume \(\psi=4\cdot 2^{\alpha}.\frac{\Gamma^{2}(1+\frac{\alpha}{2})}{\Gamma(2+\alpha)}\), Equation (14) can be abbreviated as:  
\[v\]6  
If the maximum output \(F_{\max}\) and the maximum displacement \(\upsilon_{0}\) of the damper are fixed design parameters, the energy consumption level is linearly related to the parameter \(\psi\). The value of \(\psi\) is essentially the value of the velocity index \(\alpha\), and the smaller \(\alpha\) is, the stronger the energy dissipation capacity is. For example, when \(\alpha=0.4\), \(\psi=3.58\), while \(\alpha=0.8\), \(\psi=3.27\). However, when the value of \(\alpha\) is close to \(0\), \(\upsilon^{\alpha}\) is close to \(1\) and the damping force \(F\) is basically only related to \(C\), which will lose the basic characteristics of the velocity correlation of viscous dampers. This is contrary to the original intention of the design, so the value of velocity index \(\alpha\) is not needed to be too small, usually between \(0.2\) and \(1\). The velocity index of the FVD used for the anti-seismic designed bridge has not been unified, and the appropriate \(\alpha\) can only be analyzed according to the specific conditions. When \(\alpha\) is \(1\), the damper is linear, and when \(\alpha\) is less than \(1\), it is nonlinear.