# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 3 Force Transmission and Power Flow Analysis
### Time-Averaged Input Power and Kinetic Energy  
The time-averaged dissipated and input powers in the system over the time span \(\tau_{p}\) are  
\[\overline{p}_{\rm d}=\frac{1}{\tau_{p}}\int_{\tau_{0}}^{\tau_{0}+\tau_{p}}p_{\rm d }{\rm d}\tau,\ \overline{p}_{\rm in}=\frac{1}{\tau_{p}}\int_{\tau_{0}}^{\tau_{0}+\tau_{p}}p_{ \rm in}{\rm d}\tau, \tag{30}\]  
Respectively, where \(p_{d}=2\xi X^{\prime}X^{\prime}\) is the dissipated power and \(p_{\rm in}=X^{\prime}F_{0}\cos\Omega\tau\) is the input power, \(\tau_{0}\) is starting time of the integral, and \(\tau_{p}\) is the integral time period, which equals one excitation cycle, i.e., \(\tau_{p}=2\pi/\Omega\). When a first-order approximation of the response is used, the velocity of the mass is \(Y=-r\Omega\sin(\Omega T+\phi)\). Consequently, the time-averaged input power over a cycle of oscillation \(2\pi/\Omega\) is  
\[\overline{p}_{\rm in}=-\frac{r\Omega F_{0}}{2}\sin\phi. \tag{31}\]  
Replacing \(\sin\phi\) in Equation (31) using Equation (23) and further simplifying, the time-averaged input power is rewritten as  
\[\overline{p}_{\rm in}=\xi r^{2}\Omega^{2}=\overline{p}_{\rm d}. \tag{32}\]  
Equation (32) shows that the time-averaged input power is equal to the time-averaged dissipated power within one excitation cycle, indicating that total external power input is consumed by the viscous damping. The maximum kinetic energy of the system in the steady-state motion is encountered when the mass has the maximum speed of \(|V_{max}|=\ r\Omega\), so that  
\[K_{\rm max}=\frac{1}{2}r^{2}\Omega^{2}=\frac{\overline{p}_{\rm in}}{2\xi}. \tag{33}\]  
Note that from Equations (25), (32), and (33), \(\overline{p}_{\rm in}\) and \(K_{\rm max}\) are re-expressed as  
\[\overline{p}_{\rm in}=\frac{\xi F_{0}^{2}}{4\xi^{2}+\left\{\frac{\Omega^{2}-1 +\beta}{\Omega}-\frac{4\beta(1+\alpha)}{\pi r^{2}\Omega}[K(r)-E(r)]\right\}^{ 2}}\,, \tag{34}\]  
\[K_{\rm max}=\frac{0.5F_{0}^{2}}{4\xi^{2}+\left\{\frac{\Omega^{2}-1+\beta}{ \Omega}-\frac{4\beta(1+\alpha)}{\pi r^{2}\Omega}[K(r)-E(r)]\right\}^{2}}\,, \tag{35}\]  
respectively, from which we have  
\[\overline{p}_{\rm in}\leq\frac{F_{0}^{2}}{4\xi_{\rm e}^{2}},\ K_{\rm max} \leq\frac{F_{0}^{2}}{8\xi^{2}}. \tag{36}\]  
These two expressions show the existence of an upper bound power flow values, which only depend on the damping ratio \(\xi\) and the excitation amplitude \(F_{0}\). The upper bound is only reached when the terms in the curly brackets of Equations (34) and (35) vanish:  
\[\Omega^{2}-1+\beta-\frac{4\beta(1+\alpha)[K(r)-E(r)]}{\pi r^{2}}=0. \tag{37}\]  
With reference to the frequency-response relationship in Equation (25) to replace \(\Omega\), a nonlinear equation of response amplitude \(r\) can be established:  
\[F_{0}^{2}=4\xi^{2}r^{2}\bigg{(}1-\beta+\frac{4\beta(1+\alpha)[K(r)-E(r)]}{\pi r ^{2}}\bigg{)}, \tag{38}\]  
It is the feasible solutions of which that provide the response amplitude \(r\) corresponding to the upper bound value. Together with Equation (38), responses with maximum time-averaged power flow can be identified. The upper bound value cannot be achieved if there is no solution to the equation. Additionally, it can be seen from Equations (34) and (35) that in the high-frequency range where \(\Omega\) is large, the following approximations may be made:  
\[\overline{p}_{\rm in}\approx\frac{\xi F_{0}^{2}}{4\xi^{2}+\Omega^{2}},\quad K _{\rm max}\approx\frac{0.5F_{0}^{2}}{4\xi^{2}+\Omega^{2}} \tag{39}\]Thus, the time-averaged input power and kinetic energy become independent of the stiffness parameters \(\alpha\) and \(\beta\) at large excitation frequencies. To obtain the transient response and power flow variables, Equation (8) is first transformed into two first-order differential equations. Then, the Runge-Kutta method based numerical integrations is used for solutions.