# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 3 Force Transmission and Power Flow Analysis  
### Force Transmissibility  
Equation (25) may be rearranged to obtain the nonlinear receptance function of the system:  
\[\frac{r}{F_{0}}=\frac{1}{\sqrt{\left(2\xi\Omega\right)^{2}+\left\{\Omega^{2}-1+ \beta-\frac{4\beta(1+\alpha)}{\pi r^{2}}[K(r)-E(r)]\right\}^{2}}}, \tag{26}\]  
which represents the response amplitude of the system per unit excitation force. Note that this function depends on the response amplitude \(r\) when \(\alpha\
eq-1\), different from the receptance functions of linear systems. To evaluate the performance of vibration isolators, a conventional index is force transmissibility. For the current system, the non-dimensional transmitted force from the mass to base point D is  
\[f_{t1}=2\xi X^{\prime}+X. \tag{27}\]  
When a first-order approximation of response \(X=r\cos(\Omega T+\phi)\) is assumed, the amplitude of this transmitted force is  
\[\left|f_{t1}\right|\ =\ r\sqrt{\left(2\xi\Omega\right)^{2}+1}. \tag{28}\]  
The force transmissibility of the system can be approximated by  
\[TR=\frac{r}{F_{0}}\sqrt{\left(2\xi\Omega\right)^{2}+1}, \tag{29}\]  
which is the \(\sqrt{\left(2\xi\Omega\right)^{2}+1}\) times of the receptance function shown by Equation (26). Figure 8 provides the plots of the defined receptance function \(r/F_{0}\) and the transmissibility \(TR\) against the excitation frequency \(\Omega\). For five sets of parameters \(\alpha=-1.2,\ \beta=0.7\) (Case one), \(\alpha=-1.6,\ \beta=0.4\) (Case two), \(\alpha=-1,\ \beta=0.36\) (Case three), \(\alpha=0.2,\ \beta=2.2\) (Case four), and \(\alpha=0.5,\ \beta=2.5\) (Case five), the corresponding natural frequencies of the linearized system are 0.4, 0.6, 0.8, 1.2, and 1.5, respectively. The damping ratio \(\xi\) is fixed as 0.01. Cases one and two refer to softening stiffness systems, while Cases four and five correspond to hardening stiffness systems. On each surface, there is a peak curve, which can be described mathematically by examining the minimum condition of the denominator of the expression on the right-hand side of Equation (26) [14]. For Cases one and two, the surface shifts and bends to the low-frequency range. This is beneficial for vibration isolation, as there is a larger frequency range of effective isola  
Figure 7: Possible chaotic motions of the double–well potential system \(\xi=0.02,\ \alpha=-0.55\), \(\beta=2,\ F_{0}=0.05\). (**a**) Sweeping from low to high frequencies and (**b**) sweeping from high to low frequencies. Period–1 (black), period–2 (red), period–3 (cyan), period–4 (green), and chaotic motion (blue). frequencies, the surfaces merge with each other, and \(r/F_{0}\) and \(TR\) become insensitive to the nonlinear parameters \(\alpha\) and \(\beta\). Figure 9 shows the force transmission behaviour of the proposed SBM design (Case three: \(\alpha=-1\) and \(\beta=0.36\)), the analogical NSM design [14] (\(\alpha=2P/kl=0.02\), damping and external force are the same as Case three), and the linear spring-damper isolator (i.e., \(\beta=0\)). For the conventional NSM, negative stiffness is obtained by applying constant compression forces at the ends of the bars, while variable spring forces are used for the same purpose in the current SBM. Figure 9 presents that the resonant peak of the \(TR\) curve shifts to lower frequencies compared to the conventional linear and the NSM cases, thereby expanding the effective isolation range. In the high-frequency range, three different isolators have similar force transmission performance. Figure 9 shows the time history of the steady-state transmitted force \(f_{\rm t1}\) within four periodic cycles. When the excitation frequency \(\Omega=1\), the force amplitudes are approximately equal to 0.33, 0.050, and 0.029 for the linear, the referenced NSM and the proposed SBM cases, respectively. This indicates that the SBM in the proposed isolator can effectively reduce the force transmission level near the original resonance.