## 3 Establishment of a Single-Degree-of-Freedom Eddy Current Damping Vibration Suppression Model
### Establishment of Eddy Current Damping Vibration Suppression Model
Taking the derivative of Equation (5) as follows:  
\[\dot{x_{1}}=i\omega A_{1}e^{\omega t}e^{-\phi_{1}i} \tag{6}\]  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]  
\[\ddot{x_{0}}=i\omega A_{0}e^{\omega t}e^{-\phi_{0}i} \tag{8}\]  
Substituting Equations (5)-(8) into Equation (4), it can obtain the following:  
\[\left[i\omega(c+c_{0})-m\omega^{2}+k\right]A_{1}e^{-\phi_{1}i}-[k+i\omega(c+ c_{0})]A_{0}e^{-\phi_{0}i}=0 \tag{9}\]  
Let:  
\[a=i\omega(c+c_{0})-m\omega^{2}+k \tag{10}\]  
\[b=i\omega(c+c_{0})+k \tag{11}\]  
So, the rearranged form of Equation (9) is as follows:  
\[A_{1}=\frac{bA_{0}}{a}e^{(\phi_{1}-\phi_{0})i} \tag{12}\]  
Take the modulo of Equation (12) as follows:  
\[A_{1}=A_{0}\sqrt{\frac{k^{2}+\omega^{2}(c+c_{0})^{2}}{\left(k-m\omega^{2} \right)^{2}+\omega^{2}(c+c_{0})^{2}}} \tag{13}\]  
The calculation of natural angular frequency and equivalent damping ratio is as follows:  
\[\omega_{n}=\sqrt{\frac{k}{m}}\sqrt{(1-\xi^{2})} \tag{14}\]  
\[\xi=\frac{c+c_{0}}{2m\omega_{n}} \tag{15}\]  
To preclude the overdamped scenarios (i.e., \(\xi>1\)) and to ensure the validity of Equations (5) and (14), the constraint \(c+c_{0}<2m\omega_{n}\) is imposed. This condition guarantees the oscillatory nature of the system response under critical damping. Substitute Equations (14) and (15) into Equation (13) as follows:  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]0Define the frequency ratio as follows:  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]1  
Substituting Equation (17) into Equation (16), simplify it as follows:  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]2  
Amplitude transfer coefficient is set to \(r=\frac{A_{1}}{A_{0}}\),  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]3  
From Equation (19), it can be observed that the amplitude transfer coefficient depends on the system's equivalent damping ratio and the frequency ratio. Under uncontrollable frequency ratio conditions, controlling the equivalent damping ratio allows us to reduce the amplitude transfer coefficient, achieving vibration suppression. \[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]4  
where:  
**S** is the projected area of the electromagnet on the conductor plate. **T** is the thickness of the conductor plate. **B** is the magnetic flux density. **\(\sigma\)** is the electrical resistivity (\(\Omega\cdot\)m). Refer to \[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]5  
When calculating the magnetic flux density \(B\), due to the small air gap, it can be regarded as a uniform magnetic field, and the magnetic resistance is as follows:  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]6  
where, **L** is magnetic path length, **S** is magnetic path area, \(\mu\) is magnetic permeability. \[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]7 Schematic diagram of eddy current damping. Magnetic flux is calculated as follows:  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]8  
where **N** is the number of turns in the electromagnet coil, **I** is the current flowing through a single turn of the coil, **R1** is the magnetic reluctance of the electromagnet, **R2** is the magnetic reluctance of the conductor plate, **R\({}_{\delta}\)** is the magnetic reluctance of the air gap. So, the magnetic flux density is as follows:  
\[\ddot{x_{1}}=-\omega^{2}A_{1}e^{\omega t}i^{e}e^{-\phi_{1}i} \tag{7}\]9  
From Equations (20)-(23), it can be seen that, with fixed structural and magnetic material properties, the magnetic potential (**NI**) of the coil is related to its equivalent damping coefficient. By controlling the magnetic potential of the coil, we can adjust the system's damping, as in Equation (24).