## 3 Results: Numerical Analysis
### Numerical Simulation Results for Bilinear Spring Design
\begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Definition** & **SI Units** \\ \hline \(K_{r}^{\prime}\) & Real component of rubber stiffness & 325,000 N/m \\ \hline \(tan\delta\) & Tangent delta, \(K_{r}^{\prime}\) / \(K_{r}^{\prime}\) & 0.0414 \\ \hline \(K_{r}^{\prime}\) & Imaginary component of rubber stiffness & 13,455 N/m \\ \hline \(R_{2}\) & The damping in Figure 7 = \(K_{r}^{\prime}\)/ \(\omega\) & 13,455/\(\omega\) \\ \hline \(A_{p}\) & Effective pumping or piston area & 0.0019635 m\({}^{2}\) \\ \hline \(d_{t}\) & Inertia track diameter & 1.65 cm \\ \hline \(A_{t}\) & Track cross-sectional area & 2.1382 \(\times\) 10\({}^{-4}\) m\({}^{2}\) \\ \hline \(A_{p}/A_{t}\) & Piston area to inertia track area ratio & 9.2 \\ \hline \(A_{m}\) & Cross-sectional area of the bottom piston & 0.0019635 m\({}^{2}\) \\ \hline \(L_{t}\) & Inertia track length & 10.0 cm \\ \hline \(\rho\) & Fluid density & 1720 kg/m\({}^{3}\) \\ \hline \(\
u\) & Fluid kinematic viscosity & 1.8 Cst \\ \hline \(I_{f}\) or \(I_{10}\) & Inertia track fluid inertia & \(8.044\times 10^{5}\) N-s\({}^{2}\)/m\({}^{5}\) \\ \hline \(R_{a}\), or \(R_{8}\) & Inertia track flow losses & \(6.4\times 10^{6}\) N-s/m\({}^{5}\) \\ \hline \(K_{\text{eff}}\) & Top chamber volumetric stiffness & \(4.51\times 10^{10}\) N/m\({}^{5}\) \\ \hline \(K_{\text{eb}}\) & Bottom chamber volumetric stiffness & \(1.0186\times 10^{13}\) N/m\({}^{5}\) \\ \hline \(M_{p}\) or \(I_{14}\) & Bottom piston mass (made of steel) & 0.4624 kg \\ \hline \(R_{16}\) & Damping of spring below the piston & 9.54 N-s/m \\ \hline \(K_{1}\) & Stiffness of the bilinear spring before \(S\) & 33,183 N/m \\ \hline \(K_{2}\) & Stiffness of the bilinear spring after \(S\) & 230,000 N/m \\ \hline \(X_{in}\) & Input amplitude to mount & \(\pm 0.1\) to \(\pm 0.4\) mm \\ \hline \(f_{n notch}\) & Notch frequency, in Hz & 32 Hz (linear run) \\ \hline \(f_{p\text{eak}}\) & Peak frequency, in Hz & 38 Hz (linear run) \\ \hline \end{tabular}
\end{table} 
Table 1: Hydraulic engine mount parameters used in MATLAB simulations. shocks or high-amplitude engine motions are present, the new hydraulic engine mount design stiffens up just like the hydraulic engine mount design with a decoupler, but the new design still acts like a vibration isolator. In addition, the peak frequency, which is undesirable, is highly attenuated without impacting the notch frequency. To make sure the designs of Figures 6a and 7 are practical, it is necessary to check for chamber pressures and bottom piston displacements. It is advisable to keep the chamber pressures below 0.82 MPa (<100 to 120 psi); otherwise, repeatedly high chamber pressures can result in the rubber bursting and the fluid leaking. Figure 12 shows the top chamber pressure versus frequency at different engine displacement amplitudes. The maximum pressure occurs at the peak frequency, and it can be clearly seen that the top chamber pressure is highly attenuated at the peak frequency when a bilinear spring is used. The top chamber pressures are well below 0.82 MPa for the new design. Figure 12 clearly shows that when linear springs are used, the top chamber pressures are worse than in the case where a bilinear spring is used. Figure 11: Dynamic stiffness at the notch at different displacement inputs, linear vs. bi-linear springs. Figure 10: Dynamic stiffness at different displacement inputs, linear vs. bi-linear springs. Figure 13 shows the bottom chamber pressure versus frequency at different engine displacement amplitudes. The maximum pressure occurs at the peak frequency, but unlike the top chamber pressures, when a bilinear spring is used, the bottom chamber pressure rises as the engine displacement amplitude rises. This is expected, since the higher the input engine displacement amplitudes are, the stiffer the bilinear spring will be. Thus, the bottom piston moves less, and bottom chamber pressures rise. Even though the bottom chamber pressures have risen, they are well below 0.82 MPa for the new hydraulic engine mount design.