# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 3 Force Transmission and Power Flow Analysis
### Non-Periodic Response  
In the previous content, the power flow behaviour of the system with periodic responses was examined. It is noted that when stiffness nonlinearities are introduced by the SBM, the steady-state response may become non-periodic or even chaotic. The performance of the vibration isolation system exhibiting non-periodic motions is unclear and investigated herein. To examine the dynamic behaviour of the system over a large frequency range, numerical simulations based on the fourth-order Runge-Kutta method were conducted. For each interested value of excitation frequency, the steady-state displacement response was sampled over each cycle of excitation. In this way, by plotting the sampling point positions against the frequency, the response characteristics can be shown. Based on this, Figures 13 and 14 show the response behaviour and the associated time-averaged power flow of a system with the parameters set as \(\alpha=-0.7,\ \beta=1.5,\ \xi=0.015,\ F_{0}=0.01\). Figure 13a shows that non-periodic responses can be encountered in frequency bands located between 0.125 and 0.142. There are also period-2 motions in frequency range B from \(\Omega=0.645\) to \(\Omega=0.70\), with the period of the response being twice that of the excitation. The corresponding time-averaged input power to these responses is shown in Figure 13b. To reflect the long-time steady-state dynamic behaviour of the system, the starting time \(\tau_{0}\) and the averaging time \(\tau_{p}\) are set as 500 and 1000 cycles of the excitation, respectively. The results shown in Figure 13b indicate that the level of the time-averaged input power of the non-periodic responses in frequency band A is similar to that of the periodic responses near the band. A jump in the time-averaged input power occurs when \(\Omega=0.4\) with the response bifurcates from motions across two potential wells to a single-well motion. In the higher frequency range B with the system exhibiting period-2 response motions, the time-averaged input power is larger than the period-1 responses encountered close to the frequency band. Figure 13 shows that the bifurcation diagrams of the displacement response and input power can present different quantification outcomes; therefore, it is necessary to assess the level of vibration transmission and isolation performance from the two aspects of displacement and energy. In Figure 14, the system parameters and numerical settings remain the same as those used in Figure 14, but the excitation frequency varies from 10 to 0.1 to follow the response branches from high to low frequencies. Comparing this figure with Figure 13, it can be seen that there is a coexistence of the period-1 and period-2 responses from \(\Omega=0.57\) to \(\Omega=0.645\). This is mainly due to the different initial conditions in low-to-high and high-to-low sweeping. Additionally, non-periodic responses appear from \(\Omega=0.29\) to \(\Omega=0.35\), where period-1 responses were observed in the same region in Figure 13. Clearly, these multiple solutions of the system arise from the stiffness nonlinearity of the SBM. A jump up in time-averaged input power is encountered at \(\Omega=0.29\), with the system motion changing from being non-periodic to being periodic. In high-frequency range, Figures 13 and 14 show the same results, which means that the displacement and power flow behaviour are not sensitive to the initial conditions when the excitation frequency is large. Figure 14: Bifurcation diagrams of (**a**) displacement response and (**b**) time–averaged input power with high to low excitation frequencies. Figure 13: Bifurcation diagrams of (**a**) displacement response and (**b**) time–averaged input power with low to high excitation frequencies. To examine whether the non-periodic responses are chaotic, the corresponding Lyapunov exponent can be calculated [41]. This quantity has been widely used in the identification of chaotic responses by evaluating the rate of separation of nearby trajectories.