## 3 Results: Numerical Analysis
### MATLAB Numerical Results for the Cubic Spring  
To see the difference in the dynamic stiffness between an engine mount with a linear spring (\(F_{spring}=Kx\)) versus a cubic spring (\(F_{spring}=33,183x+1.88\times 10^{10}\)\(x^{3}\)), a comparison was made using MATLAB. Figures 20 and 21 show a comparison of the dynamic stiffness of the hydraulic engine mount design of Figure 6b with a linear spring (\(F_{\text{spring}}=Kx\)) versus a cubic spring of Figure 15 at different vibration amplitudes. From Figure 21, we can clearly see that at vibration amplitudes of \(\pm 0.1\) mm, the notch frequency is the same as the notch frequency of the hydraulic engine mount with a linear spring, but the peak frequency is considerably attenuated (See Figure 20). The resonant peak frequency is considerably attenuated by simply using a cubic spring with no added damping, and this is extremely desirable. Figure 19: Best scenario—beam on 5 mm tall columns, welded to a rigid substrate. Figure 18: Load–deflection plot of the beam as the column height and material change. Figure 21 is zoomed to the notch frequency. If the engine is operating near the notch frequency (at 32 Hz), if the engine vibration amplitude is suddenly increased from \(\pm\)0.1 mm to \(\pm\)0.4 mm due to shock, the dynamic stiffness increases from 67 \(\times\) 10\({}^{3}\) N/m to 103 \(\times\) 10\({}^{3}\) N/m by a factor of 1.53 times without using any decoupler. Figure 21 clearly shows that at the notch frequency, when 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, just like the hydraulic engine mount with a bilinear spring, 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) or else repeated high chamber pressures can result in the rubber bursting and fluid leaking. Figure 22 shows the top chamber pressures versus the frequency at different engine displacement amplitudes. The maximum pressure occurs at the peak frequency, and it can be clearly seen that the top chamber pressures are highly attenuated at the peak frequency when a cubic spring is used. The top chamber pressures are well below 0.82 MPa for the new design. Figure 22 clearly shows that when linear springs are used, the top chamber pressures are worse than the case where a cubic spring is used. Figure 21: Dynamic stiffness at the notch at different displacement inputs, linear vs. cubic springs. Figure 20: Dynamic stiffness at different displacement inputs, linear vs. cubic springs. Figure 23 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 cubic spring is used, the bottom chamber pressures rise as the engine displacement amplitudes rise. Just like the bilinear case, this is expected, since the higher the input engine displacement amplitudes are, stiffer the cubic 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 with a cubic spring. Figure 24 clearly indicates that the bottom piston displacements of the new hydraulic engine mount design with a cubic spring are smaller than for a mount with a linear spring. A hydraulic engine mount with a linear spring has a piston displacement of almost \(\pm\)9 mm at the peak frequency at \(\pm\)0.4 mm input displacement, but using a cubic spring resulted in the bottom piston displacements being less than \(\pm\)3 mm at the peak frequency at \(\pm\)0.4 mm.