## 3 Results: Numerical Analysis  
The bond graph [19] model of Figure 5 is shown in Figure 8. The same bond graph model can be used for the designs of Figures 6 and 7. In the bond graph model of Figure 8, the input to the hydraulic engine mount is velocity, labeled as \(V_{\text{in}}\) (input velocity).  
The effective piston (or pumping) area of the thick elastomer pumping fluid between the fluid chambers is represented by \(A_{p}\) in the bond graph model of Figure 8, while C\({}_{3}\) and \(R_{2}\) represent the vertical compliance and damping of the thick elastomer (labeled as flexible element). C\({}_{6}\) represents the top chamber volumetric compliance, \(R_{8}\) and \(I_{10}\) represent the flow losses and fluid inertia of the inertia track, respectively, \(C_{11}\) represents the bottom chamber volumetric compliance, \(A_{m}\), represents the area of the bottom piston, \(I_{14}\) represents the mass of the bottom piston, and \(C_{15}\) and \(R_{16}\) represent the compliance and damping of the nonlinear spring, respectively. Since low friction seals and bearings are to be used, the friction between the bottom piston and the walls is considered negligible. The bond graph model shown in Figure 8 yields the following state space equations:  
\[\dot{q}_{3}=V_{in} \tag{1}\]  
\[\dot{q}_{6}=A_{p}V_{in}-\frac{P_{10}}{I_{10}} \tag{2}\]  
\[\dot{p}_{10}=\frac{q_{6}}{C_{6}}-\frac{q_{11}}{C_{11}}-R_{8}\frac{P_{10}}{I_{1 0}} \tag{3}\]  
\[\dot{q}_{11}=\frac{p_{10}}{I_{10}}-A_{m}\frac{p_{14}}{I_{14}} \tag{4}\]  
\[\dot{p}_{14}=A_{m}\frac{q_{11}}{C_{11}}-e_{15}-R_{16}\frac{p_{14}}{I_{14}} \tag{5}\]  
\[\dot{q}_{15}=\frac{p_{14}}{I_{14}} \tag{6}\]  
Please notice that the effort \(e_{15}\) in Equation (5) is a nonlinear relationship, relating spring force \(F_{k}\left(e_{15}\right)\) to its relative deflection (\(\Delta x=q_{15}\)). In the above state space equations,  
Figure 8: Bond graph model of Figure 5.  
Figure 7: Variation in design Figure 6a, bottom piston (**a**) with a rubber tube form or (**b**) on metal springs.  
\(q_{3}\), \(q_{6}\), \(q_{11}\), and \(q_{15}\) are the generalized displacement variables, and \(p_{10}\) and \(p_{14}\) are the momentum variables. The state-space variables are defined as follows:  
\(q_{3}\)--Relative motion across the engine mount  
\(q_{6}\)--Top chamber change in volume  
\(p_{10}\)--Integral of the pressure drop in the inertia track  
\(q_{11}\)--Bottom chamber change in volume  
\(p_{14}\)--Linear momentum of the bottom piston  
\(q_{15}\)--Motion of the piston or the nonlinear spring  
The output equations, namely the input force, \(F_{in}\), applied to the engine mount; top chamber pressure, \(P_{t}\); bottom chamber pressure, \(P_{b}\); and bottom piston displacement, \(X_{p}\), are determined as follows:  
\[F_{in}=\frac{q_{3}}{C_{3}}+R_{2}V_{in}+A_{p}\frac{q_{6}}{C_{6}} \tag{7}\]  
\[P_{t}=\frac{q_{6}}{C_{6}} \tag{8}\]  
\[P_{b}=\frac{q_{11}}{C_{11}} \tag{9}\]  
\[X_{p}=q_{15} \tag{10}\]  
The dynamic stiffness of the engine mount can be found from the ratio of the input force to the input displacement, K* = \(F_{in}/X_{in}\). The reason we are interested in plotting chamber pressures versus frequency is to make sure the chamber pressures are not excessively high or else the rubber can potentially burst. As for the bottom piston, its dynamic displacement needs to be reasonable or else our proposed design becomes impractical. With the state space equations of (1) through (6) and output equations of (7) through (10), numerical simulations can now be conducted.