# Development of a Self-Powered Magnetorheological Damper System for Cable Vibration Control
## 2 Description of the Self-Powered MR Damper System for Cable Vibration Control
### Model and Test of a Rotary DC Generator  
The equivalent circuit model of a rotary DC generator is depicted in Figure 2. The back EMF of the generator \(V_{\mathrm{g}}\) is expressed as  
\[V_{\mathrm{g}}=K_{\mathrm{e}}\eta\dot{\theta}=K_{\mathrm{e}}\dot{u}/r \tag{3}\]  
where \(K_{\mathrm{e}}\) denotes the back EMF constant. According to [PERSON]'s law, the voltage in the circuit should be in balance as follows:  
\[V_{\mathrm{g}}=L_{\mathrm{g}}\dot{I}+R_{\mathrm{g}}I+R_{\mathrm{L}}I \tag{4}\]  
where \(I\) denotes the current flow in the circuit; \(L_{\mathrm{g}}\) and \(R_{\mathrm{g}}\) represent the inductance and resistance of the generator, respectively; and \(R_{\mathrm{L}}\) s the load resistance in the circuit. Applying the Laplace transform to Equations (3) and (4), the transfer function of the current \(I\) can be obtained using  
\[I(s)=\frac{K_{\mathrm{e}}sU(s)/r}{L_{\mathrm{g}}s+R_{\mathrm{g}}+R_{L}} \tag{5}\]  
where \(U(s)\) and \(I(s)\) are the Laplace transform of \(u(t)\) and \(I(t)\), respectively. Equations (3)-(5) indicate that the amplitude of the current flow \(I\) is proportional to the back EMF coefficient \(K_{\mathrm{e}}\) and the cable velocity at the generator location (i.e., displacement amplitude and frequency of the cable), while it is inversely proportional to the effective transmission radius of the sprocket. Consequently, the output voltage \(U_{\mathrm{output}}\) and the output power \(P_{\mathrm{output}}\) of the generator can be respectively given as  
\[U_{\mathrm{output}}=R_{\mathrm{L}}I,\ P_{\mathrm{output}}=R_{\mathrm{L}}I^{2}. \tag{6}\]  
Equation (6) indicates that the output power of the generator is proportional to the square of the current flow in the circuit, which mainly relates to the square of the cable's displacement amplitude and frequency at the location of the generator. Experimental study on the rotary DC generator was conducted to illustrate its potential to provide the power supply for an MR damper. The rotary DC generator adopted in the test is a commercial CFX-04 speed-measuring permanent magnet DC motor, as shown in Figure 3a. The mass of the generator is 3.7 kg, and the maximum angular velocity can reach as much as 2800\(\pi\)/min. The internal resistance of the generator is 6.4 \(\Omega\). To acquire the back EMF constant of the generator, the shaft of the generator was forced to rotate synchronously with a servomotor at different rotational speeds, as shown in Figure 3b. Figure 4 shows the relationship between the back EMF and the rotational speed of the generator. Finally, the back EMF constant \(K_{\mathrm{e}}\) of the generator was identified as 0.0594 v/(r/min). Figure 2: The equivalent circuit model for a rotary DC generator. Next, the chain in the linear-to-rotational conversion mechanism was subjected to sinusoidal excitations with ten different frequencies ranging from 0.5 Hz to 5 Hz at amplitudes of 5 mm and 10 mm, respectively. The external load in the circuit was 15 \(\Omega\). The output voltage of the generator was measured. A laser vibrometer was used to monitor the linear displacement of the excitation. The corresponding experimental setup is shown in Figure 5, and all the data were recorded using the Donghua Data Acquisition System. Figure 4: The relationship between the back electromotive force and the rotational speed of the generator. Figure 5: Experimental setup for the energy harvesting capability of the generator. Figure 3: Experimental setup for identifying the back electromotive force constant of the generator. Figure 6a,b illustrate the relationship between the output voltage and the vibration frequency, and the relationship between the harvested electrical power and the vibration frequency, respectively. It is shown that both the output voltage and the output power are quite dependent on the frequency and the amplitude of the vibration source. Hence, it can be inferred that energy harvesting in civil engineering structures vibrating with low amplitudes and frequencies is quite challenging.