## 1 Introduction  
The automotive and aerospace industries frequently use rubber-to-metal (RTM)-bonded engine mounts to lessen the vibration that is transmitted from the engine to the car body or airplane fuselage. The engine mounts, also known as motor mounts, serve several purposes. They stabilize and support the weight of the engine, lessen vibrations (small motions are referred to as vibrations) caused by the engine that are communicated to the vehicle body or the airplane fuselage, and control or restrict engine motion in the presence of shock excitations. The stiffness of the RTM-bonded engine mounts must be low to limit the vibration transmitted from the engine to the vehicle body, but if sudden significant engine motions (shock excitations) are present, the engine motion will be excessive since the stiffness of the engine mounts is low. As a result, the majority of RTM-bonded engine mount manufacturers create rubber mounts with dual stiffness properties, as seen in Figure 1. As seen in Figure 1, there are collapsible air gaps or voids between the inner and the outer members of the engine mount. When large engine motions caused by shock excitations are present, the air gaps can collapse to zero and the inner member contacts the outer member; thus, the engine or motor mount stiffness increases, thereby restricting the engine motions. There is another passive technology that provides an even greater reduction in the transmitted engine disturbance forces to the vehicle structure/chassis or the airplane fuselage, called fluid-filled mounts. Numerous publications have referred to fluid-filled mounts by a variety of names, including hydraulic mounts [1], hydraulic engine mounts [2], hydraulic motor mounts, hydro-mounts [3], hydroelastic mounts [4], and fluid or fluid-lastic mounts [5]. Single-pumper and double-pumper fluid mounts are the two different varieties of hydraulic engine mounts [6; 7] found in the literature. Since single-pumper fluid mounts, with and without decouplers, are often employed in the automobile and helicopter industries, they are the subject of this study. As seen in Figure 2, a passive single-pumper fluid (or hydraulic) engine mount comprises of a fluid that is housed in two elastomeric cavities (or fluid chambers) that are joined by an inertia track. The fluid in Figure 2 is shown in the blue color. The fluid will oscillate between the two fluid chambers when the fluid mount is subjected to a sinusoidal motion. The oscillating fluid enters resonance at a frequency known as the \"notch frequency\" after bouncing between the vertical (or axial) stiffness and the two-chamber volumetric stiffnesses. The hydraulic or fluid mount dynamic stiffness significantly decreases at this frequency, reducing cabin vibration and noise at this frequency. The notch frequency occurs because of low damping and high inertia track fluid inertia, \(I_{f}=\frac{\rho L_{t}}{A_{t}}\), where \(\rho\) is the fluid density and \(L_{t}\) and \(A_{t}\) are the length and the cross-sectional area of the inertia track, respectively. The hydraulic engine mount designer must employ the proper inertia track length, diameter, working fluid, and rubber stiffnesses to set the \"notch frequency\" at the ideal spot. A passive fluid mount's typical dynamic stiffness as a function of frequency is shown in Figure 3.  
Figure 1: Typical automotive RTM-bonded engine mounts with collapsible air gaps. Figure 2: A passive single-pumper hydraulic engine mount and its schematic physical model. As can be seen from Figure 3, at the notch frequency, the dynamic stiffness reduces considerably; thus, less vibration is transmitted from the engine to the cabin or to the airplane fuselage. But at the peak frequency, since the dynamic stiffness is very high, more vibration and noise are transmitted to the cabin or to the fuselage if the engine operates near the peak frequency.