### Coil Output Power and Electromagnetic Damping Force  
The fundamental frequency of the actual bridge structure vibration is generally a few hertz to a dozen hertz, which belongs to low-frequency vibration. Therefore, the frequency of the vibration signal transmitted to the permanent magnet is mainly low-frequency. The inductance of the coil is far less than the internal resistance of the coil, and the inductance can be ignored. Assuming that the coil resistance is \(R_{0}\) and the external resistance is \(R_{1}\), regardless of the inductance and capacitive reactance of the external load, the current and output power of the energy acquisition circuit can be expressed as  
\[P=I^{2}R_{1}=\left(\frac{\varepsilon}{R_{0}+R_{1}}\right)^{2}R_{1} \tag{4}\]  
Figure 2: Schematic diagram of magnetic field and coil.  
If we input the electromotive force \(\varepsilon\) into Equation (4), then the output power \(P\) can be expressed as  
\[P=\frac{R_{1}}{\left(R_{0}+R_{1}\right)^{2}}\cdot{k_{u}}^{2}\cdot{v}^{2}(t) \tag{5}\]  
Therefore, when the coil internal resistance and external resistance are fixed, the greater the electromechanical coupling coefficient, and the greater the output power.  
The magnitude of the electromagnetic damping force \(F_{uv}\) provided by a single group of coils is equal to the Ampere force \(F_{I}\) in the opposite direction, which can be represented as follows:  
\[F_{uv}=F_{I}=\oint_{L}\left(\stackrel{{\rightarrow}}{{I}} \times\stackrel{{\rightarrow}}{{B}}\right)\cdot d\stackrel{{ \rightarrow}}{{I}}=k_{u}I=k_{u}\cdot\frac{\varepsilon}{R_{0}+R_{1}}=k_{u} ^{2}\cdot\frac{v(t)}{R_{0}+R_{1}} \tag{6}\]  
where \(\stackrel{{\rightarrow}}{{I}}\) is the current vector, \(\stackrel{{\rightarrow}}{{B}}\) is the Magnetic field vector and \(I\) is the current scalar.  
In general, the coil internal resistance \(R_{0}\) is a fixed value. From Equations (5) and (6), it can be seen that as the load resistance increases, the output power and electromagnetic damping cannot reach the maximum at the same time.  
The output power and the electromagnetic damping force were normalized and expressed as  
\[P_{N}=\frac{4R_{0}R_{1}}{\left(R_{0}+R_{1}\right)^{2}}=\frac{4\cdot\frac{R_{1 }}{R_{0}}}{\left(1+\frac{R_{1}}{R_{0}}\right)^{2}} \tag{7}\]  
\[F_{uvN}=\frac{R_{0}}{R_{0}+R_{1}}=\frac{1}{1+\frac{R_{1}}{R_{0}}} \tag{8}\]  
As shown in Figure 3, the energy output and electromagnetic damping could not reach the maximum at the same time. The calculation of the electromagnetic damping force can provide a reference for the finite element analysis of structural vibration. In order to consider both the output power and the electromagnetic damping force, the optimum ratio of the external resistance to the internal resistance of the coil was \(1/3\). The value of each parameter was \(75\%\) of the maximum. If the electromagnetic damping force is maximized, the resistance value of the external load is zero, i.e., the coil is closed:  
\[F_{uv}=\frac{\varepsilon^{2}}{R_{0}\cdot v(t)}=\frac{k_{u}^{2}v(t)}{R_{0}} \tag{9}\]  
If the output power was maximized, the electromagnetic damping force was only \(50\%\) of the maximum value, as  
\[F_{uv}=\frac{\varepsilon^{2}}{2R_{0}\cdot v(t)}=\frac{k_{u}^{2}v(t)}{2R_{0}} \tag{10}\]  
Figure 3: Change in normalized value with load.