# Theoretical Design of a Novel Vibration Energy Absorbing Mechanism for Cables
## 3 Theoretical Modeling of the VEAM
### Nonlinear Stiffness Isolator Model  
In the VEAM sub-system, there is a set of mutually exclusive PMs at each end of the cable fixing hex, these two sets of PMs form a magnetic spring. The force exerted on the cable fixing hex by the repulsion of the magnetic spring can be expressed as the following Duffing type cubic polynomial equation [40; 41].  
\[\rm{F_{ms}}=k_{b}y+by^{3} \tag{11}\]  
where, \(\rm{F_{ms}}\) represents the resultant force provided by the magnetic spring and \(\rm{k_{b}}\) and \(\rm{b}\) represent the equivalent linear and non-linear stiffness coefficients, respectively.  
When \(\rm{F_{m}}\leq\rm{F_{f}}\), the hydraulic cylinder does not work, and the PM at the end of the hydraulic piston rod can be assumed as grounded. The theoretical model of the non-linear stiffness isolator module of the VEAM can be established as a system composed of; (1) the mass (m), (2) the linear stiffness of the coil spring (\(\rm{k_{S1}}\), \(\rm{k_{S2}}\)), (3) the structural damping from the two coil springs on each side of the sub-system (which is simplified as Rayleigh damping in this study) (\(\rm{c_{1}}\), \(\rm{c_{2}}\)), and (4) the equivalent linear and non-linear stiffness due to the magnetic spring (\(\rm{k_{b}}\)), as shown in Figure 5b. The linear stiffness coefficient of the VEAM subsystem can be integrated as \(\rm{k_{I}}=\rm{k_{S1}}+\rm{k_{S2}}+\rm{k_{b}}\).  
To simplify the calculations, the vibration excitation from the stay cable is calculated as the following sine function:  
\[\rm{y_{c}}=\rm{Y_{c}}\sin(\omega t+\varphi) \tag{12}\]  
where, \(\rm{Y_{c}}\) is the vibration amplitude, \(\rm{\varpi}\) is the angular frequency, \(\rm{t}\) is the time, and \(\varphi\) is the phase shift. Thus, the displacement of the VEAM subsystem (y), which is the relative displacement between the stay cable and the deck, can be expressed as:  
\[\rm{y}=\rm{Ysin}(\omega t) \tag{13}\]  
When the VEAM sub-system receives the vibration transmitted from the stay cable and enters a steady state, the equation of motion of the sub-system according to the HBM can be written as the following Duffing type equation [53; 54]:  
\[\rm{m\ddot{y}}+2c\dot{y}+k_{l}y+by^{3}=-m\ddot{y}_{c} \tag{14}\]  
Because the same springs are used on both sides of the sub-system, we assume that their structural damping coefficients are equal, i.e., \(\rm{c}=\rm{c_{1}}=\rm{c_{2}}\).  
Substituting Equations (12) and (13) into Equation (14) and deriving the time, the following equation is obtained.  
\[-mY\omega^{2}\sin(\omega t)+2 cY\omega\cos(\omega t)+k_{l}Ysin(\omega t)+bY^{3} \sin^{3}(\omega t)=mY_{c}\omega^{2}[\sin(\omega t)\cos\varphi+\cos(\omega t) \sin\varphi] \tag{15}\]The cubic term in this equation can be broken down as: \(\sin^{3}(\omega t)=[3\sin(\omega t)-\sin(3\omega t)]/4\). The non-harmonic high frequency term that, in general, does not simply decay with time can be omitted and the constant terms with \(\sin(\omega t)\) and \(\cos(\omega t)\) can be integrated as:  
\[-\rm{mY}\omega^{2}+k_{\rm{I}}Y+\frac{3}{4}bY^{3}=mY_{c}\omega^{2}cos\varphi \tag{16}\]  
and  
\[2 cY\omega=mY_{c}\omega^{2}sin\varphi \tag{17}\]  
The absolute displacement of the deck is the sum of the vibration excitation of the stay cable and the relative displacement between the stay cable and the deck:  
\[Y_{\rm{g}}=y+y_{\rm{c}}=Y\sin(\omega t)+Y_{\rm{c}}\sin(\omega t+\varphi) \tag{18}\]  
Therefore, the displacement transmissibility of the VEAM sub-system can be sorted as:  
\[T=\frac{Y_{\rm{g}}}{y_{\rm{c}}}=\sqrt{\frac{\left(Y+Y_{\rm{c}}\rm{cos}\, \varphi\right)^{2}+\left(Y_{\rm{c}}\rm{sin}\,\varphi\right)^{2}}{Y_{\rm{c}}^{ 2}}}=\sqrt{\frac{Y^{2}+2 YY_{\rm{c}}\rm{cos}\,\varphi+Y_{\rm{c}}^{2}}{Y_{\rm{ c}}^{2}}} \tag{19}\]  
Substituting Equation (16) into Equation (19), we get:  
\[T=\sqrt{\frac{Y^{2}}{Y_{\rm{c}}^{2}}+1+\frac{2Y}{Y_{\rm{c}}}\left(\frac{-\rm{ mY}\omega^{2}+k_{\rm{I}}Y+\frac{3}{4}bY^{3}}{\rm{mY}_{\rm{c}}\omega^{2}}\right)} \tag{20}\]