## Nomenclature
  
\begin{tabular}{l l} \(f_{\text{n3,nof,mad}}\) & natural frequency of the actuator mover with added mass \\  & when testing in the non-fluid state (Hz) \\ \(f_{\text{p,3,nof}}\) & peak frequency of \(\left|\bar{Y}_{3}/I\right|\) (Hz) \\ \(f_{\text{p,5,nof}}\) & peak frequency of \(\left|F_{5}/I\right|\) (Hz) \\ \(f_{\text{R}}\) & frequency of the fixed-point \(R\) of frequency- and \\ \(j\) & amplitude-variant passive dynamic stiffness in-phase (Hz) \\ \(k^{\prime}\), \(k,k^{\prime}\), \(k^{\prime}\) & dynamic stiffness and its modulus, dynamic stiffness in-phase and out-of-phase (N-mm\({}^{-1}\)), respectively, \\ \(k^{*}=k(\cos\varphi+j\sin\varphi)=k^{\prime}+jk^{*}=k^{\prime}+j\omega c\) \\ \(k_{1}\) & dynamic stiffness in-phase of the main rubber spring (N-mm\({}^{-1}\)) \\ \(K_{1}\) & dynamic bulk stiffness of the main rubber spring (GN-m\({}^{-5}\)) \\  & dynamic bulk stiffness of the lower fluid chamber (GN-m\({}^{-5}\)), \\ \(K_{2}\) & \(K_{2}=0\) \\  & dynamic stiffness in-phase of the decoupler membrane (N-mm\({}^{-1}\)), \\ \(k_{3}\) & \(k_{3}=A_{3}^{2}K_{3}\) \\ \(K_{3}\) & dynamic bulk stiffness of the decoupler membrane (GN-m\({}^{-5}\)) \\ \(k_{\text{M}}\) & voice coil constant (T-m = kg\cdot m\cdot A^{-1}\cdot s^{-2}\)=N-A\({}^{-1}\)),\(k_{\text{M}}=Bl\) \\ \(K_{\text{u1}}\) & dynamic bulk stiffness of the upper fluid chamber (GN-m\({}^{-5}\)) \\ \(l\) & coil length (m) \\ \(l_{2}\) & length of the inertia track (mm) \\ \(L_{2}\) & wet perimeter of the cross-sectional area of the inertia track(mm) \\ \(m_{1}\) & mass at the engine side (kg) \\ \(m_{2}\) & mass of fluid in the inertia track (g), \(m_{2}=\rho l_{2}A_{2}\) \\ \(m_{3}\) & total mass of the actuator mover including attached fluid in \\  & the fluid-filled state (g) \\ \(m_{\text{3,nof}}\) & net mass of the actuator mover in the non-fluid state (g) \\ \(m_{\text{3,nof,mad}}\) & mass of the actuator mover including added mass \(m_{\text{ad}}\) when \\  & testing in the non-fluid state (g), \(m_{\text{3,nof,mad}}=m_{\text{3,nof}}+m_{\text{ad}}\) \\ \(m_{\text{ad}}\) & added mass to the actuator mover when testing in the \\ \(p_{1}\) & dynamic fluid pressure in the upper chamber (Pa) \\ \(p_{2}\) & dynamic fluid pressure in the lower chamber (Pa), \(p_{2}=0\) \\ \(R\) & fixed point of frequency- and amplitude-variant \\  & dynamic stiffness in-phase \\ \(y_{1}\) & excitation displacement applied at the engine side of the \\  & mount (mm) \\ \(Y_{1}\) & amplitude of \(y_{1}\) (mm) \\ \(y_{2}\) & reaction displacement of inertia fluid (mm) \\ \(Y_{2}\) & amplitude of \(y_{2}\) (mm) \\ \(y_{3}\) & reaction displacement of the mover (mm) \\ \(y_{5}\) & displacement at the body/chassis side (mm) \\  & 1 st horizontal segment in drive-point/cross-point passive \\ \(k^{\prime}_{\text{oo,1}}\) & dynamic stiffness in-phase in the mid-frequency band \\  & (N-mm\({}^{-1}\)), \(k^{\prime}_{\text{oo,1}}=k_{1}+A_{1}^{2}K_{\text{u1}}\) \\  & 2 nd horizontal segment in drive-point passive dynamic \\ \(k^{\prime}_{\text{oo,2}}\) & stiffness in-phase in the high-frequency band (N-mm\({}^{-1}\)), \\  & \(k^{\prime}_{\text{oo,2}}=k_{1}+A_{1}^{2}K_{\text{l}}\) \\  & 3 rd horizontal segment in cross-point passive dynamic \\ \(k^{\prime}_{\text{oo,3}}\) & stiffness in-phase in the high-frequency band (N-mm\({}^{-1}\)), \\  & \(k^{\prime}_{\text{oo,3}}=k_{1}+A_{1}^{2}K_{\text{l}}-A_{1}A_{3}K_{\text{l}}\) \\  & 4 th horizontal segment in active FRFs in the mid-frequency band in the fluid-filled state (N-A\({}^{-1}\)), \\ \(F_{i,\text{oo,4}}\) & \(-\left.\frac{F_{i}}{I}\right|_{\lambda_{2}\rightarrow\infty}=-\left.\frac{F_{i} }{I}\right|_{\lambda_{2}\rightarrow\infty}=\left.\frac{A_{i}K_{\text{u1}}}{A_{3} K_{\text{l}}}\text{M}=F_{i,\text{oo,4}}\) \\  & \(-\left.\frac{F_{i}}{I}\right|_{\lambda_{3}\rightarrow\infty}=-\left.\frac{F_{i} }{I}\right|_{\lambda_{3}\rightarrow\infty}=\left.\frac{A_{i}K_{\text{u1}}}{A_{3} K_{\text{l}}}\text{M}=F_{i,\text{oo,4}}\) \\ \(F_{i,\text{oo,5}}\) & 5 th horizontal segment in active FRF in the high-frequency band in the fluid-filled state (N-A\({}^{-1}\)), \(-\left.\frac{F_{i}}{I}\right|_{\lambda_{2}\rightarrow\infty}=k_{\text{M}}=F_{i, \text{oo,5}}\) \\ \(F_{i,\text{oo,5}}\) & 5 th horizontal segment in active FRFs in the mid-frequency band in the fluid-filled state (N-A\({}^{-1}\)), \\ \(F_{i,\text{oo,5}}\) & \(-\left.\frac{F_{i}}{I}\right|_{\lambda_{2}\rightarrow\infty}=-\left.\frac{F_{i} }{I}\right|_{\lambda_{2}\rightarrow\infty}=\left.\frac{A_{i}K_{\text{u1}}}{A_{3} K_{\text{l}}}\text{M}=F_{i,\text{oo,4}}\) \\ \(F_{i,\text{oo,6}}\) & 5 th horizontal segment in active FRFs in the mid-frequency band in the fluid-filled state (N-A\({}^{-1}\)), \\ \(F_{i,\text{oo,6}}\) & \(6 th horizontal segment in active FRF in the high-frequency band in the non-fluid state \(\left(\mathrm{g\cdot A^{-1}}\right)\), \(\left|\left.\frac{\tilde{Y}_{3}}{I}\right|\right|_{\lambda_{\mathrm{3,not}} \rightarrow\infty}=\left.\frac{k_{\mathrm{M}}}{m_{\mathrm{3,not}}}=\tilde{Y}_{i \
eq 0,6}\right.\)  
7 th horizontal segment in active FRF in the high-frequency band in the non-fluid state \(\left(\mathrm{N\cdot A^{-1}}\right)\), \(\left|\left.\frac{\tilde{F}_{3}}{I}\right|\right|_{\lambda_{\mathrm{3,not}} \rightarrow\infty}=\left.k_{\mathrm{M}}=F_{i\
eq 0,7}\right.\)  
peak value of active FRF \(\left|F_{1}/I\right|\) in the mid-high-frequency bands in the fluid-filled state \(\left(\mathrm{N\cdot A^{-1}}\right)\) and the corresponding frequency (Hz) and frequency ratio  
peak value of active FRF \(\left|\tilde{Y}_{3}/I\right|\) in the mid-high-frequency bands in the non-fluid state \(\left(\mathrm{g\cdot A^{-1}}\right)\) and the corresponding frequency (Hz) and frequency ratio  
peak value of active FRF \(\left|F_{5}/I\right|\) in the mid-high-frequency bands in the non-fluid state \(\left(\mathrm{N\cdot A^{-1}}\right)\) and the corresponding frequency (Hz) and frequency ratio  
loss angle  
frequency ratio, \(\lambda=\omega/\omega_{n}=f/f_{n}\)  
frequency ratio, \(\lambda_{2}=\omega/\omega_{n2}=f/f_{n2}\)  
frequency ratio in the fluid-filled state,  
\(\lambda_{3}=\omega/\omega_{n3}=f/f_{n3}\)  
frequency ratio in the non-fluid state,  
\(\lambda_{3,\mathrm{not}}=\omega/\omega_{n3,\mathrm{not}}=f/f_{n3,\mathrm{ not}}\)  
fluid viscosity(Pa\(\cdot\)s)  
fluid density (kg\(\cdot\)m\({}^{-3}\))  
excitation angular frequency (rad\(\cdot\)s\({}^{-1}\))  
nature angular frequency of inertia track (rad\(\cdot\)s\({}^{-1}\))  
nature angular frequency of the actuator mover in the fluid-filled state (rad\(\cdot\)s\({}^{-1}\))  
nature angular frequency of the actuator mover without added mass in the non-fluid state (rad\(\cdot\)s\({}^{-1}\))  
damping ratio of the decoupler/mover system  
in the fluid-filled state  
damping ratio of the decoupler/mover system without added mass in the non-fluid state