# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 3 Force Transmission and Power Flow Analysis
### Time-Averaged Input Power and Kinetic Energy
By using this method, Figure 10 shows the input power behaviour of the system with \(\alpha=-0.55,\ \beta=2,\xi=0.02,\ \text{and}\ F_{0}=0.05\), with the excitation frequency \(\Omega\) taken as 0.5 in Figure 10a-c and 0.8 in Figure 10d-f, respectively. It is shown in Figure 7 that at those two frequencies, the system will exhibit periodic and chaotic motion, respectively. Figure 10a shows that input power \(p_{\text{in}}\) is periodic and that its dominant frequency is located at the excitation frequency \(2\Omega\). This is due to the trigonometric product of the response velocity and the excitation. Figure 10d shows that when chaos occurs, the time series of \(p_{\text{in}}\) exhibits an irregular, random-like behaviour. Figure 10e and f further shows infinite frequency components in both \(p_{\text{in}}\) and the displacement \(X\). The dominant components in \(X\) are located at \(\Omega\) and \(\Omega/3\). Consequently, there are high spikes in the spectrum of \(p_{\text{in}}\) at \(2\Omega/3\), \(4\Omega/3\), and \(2\Omega\). Using the averaging formulation as well as numerical integrations, the effects of parameters \(\alpha\) and \(\beta\) on the power flows of the system are examined and shown in Figures 11 and 12. In the following content, the power flow variables are presented in decibel scales with a reference level of \(10^{-12}\). In Figure 11, the system changes from being of softening stiffness when \((\alpha,\ \beta)\ =\ (-1.5,\ 0.5)\) or \((-1.25,\ 0.6)\) to being linear when \((\alpha,\ \beta)\ =\ (-1,\ 0.75)\) and then to being of hardening stiffness when \((\alpha,\ \beta)\ =\ (-0.5,\ 1.5)\). The other parameters are set as \(\xi=0.01,\ F_{0}=0.01\). For all of these cases, the product of \(\alpha\) and \(\beta\) remains \(-0.75\) so that the natural frequency of the linearized system remains unchanged at 0.5. It can seen that the peaks in the power flow curves bend to the low-frequency range for softening systems and to the high-frequency range when the system is of hardening stiffness. The power flow curves for the \((\alpha,\ \beta)\ =\ (-1.5,\ 0.5)\) case are continuous with the peak values being lower than the case with \((\alpha,\ \beta)\ =\ (-1.25,\ 0.6)\). This indicates a potential benefit of using softening stiffness to reduce the peak time-averaged power flows. The  
\[\overline{p}_{\rm in}=-\frac{r\Omega F_{0}}{2}\sin\phi. \tag{31}\]0 Instantaneous input power. Parameters \(\alpha=-0.55,\ \beta=2,\xi=0.02,F_{0}=0.05\), and (**a**–c) \(\Omega=0.5\), period–1 motion; (**d**–f), \(\Omega=0.8\), chaotic motion. (**a**, **d**) are for transient input power; (**b**, **e**) are the frequency components in \(p_{\text{in}}\); (**c**, **f**) show the frequency components in displacement. time-averaged input power and the maximum kinetic energy for linear and the hardening cases are approximately of the same height. As \(\Omega\) moves away from the resonance region, the curves for different combinations of \(\alpha\) and \(\beta\) coincide, and the power flow levels are approximately the same. This results from the fact that the response amplitude at these frequencies is small, so the nonlinear stiffness effects are insignificant. Figure 12 presents the power flows of systems with different products of \(\alpha\) and \(\beta\). The system changes from being one with softening stiffness when \((\alpha,\ \beta)\ =\ (-1.2,\ 0.7)\) or \((-1.6,\ 0.4)\) to being linear when \((\alpha,\ \beta)\ =\ (-1,\ 0.36)\) and then having a hardening characteristic when \((\alpha,\ \beta)\ =\ (0.2,\ 2.2)\). The other parameters are set as \(\xi=0.015\), \(F_{0}=0.01\). The natural frequency \(\Omega_{n}\) of the linearized system varies from 0.4 to 0.6 to 0.8 and then to 1.2. The figure shows that with a small product of \(\alpha\) and \(\beta\), the peaks in the power flow curves shift to the low-frequency range. This is good for vibration isolation, as the amount of the input power of the system is low over a larger range of excitation frequencies.