Amplitude-Sensitive Single-Pumper Hydraulic Engine Mount Design without a Decoupler

Abstract

Engine mounts serve three primary purposes: (1) to support the weight of the engine, (2) to lessen the transmitted engine disturbance forces to the vehicle structure/chassis or airplane fuselage, and (3) to limit the engine motion brought on by shock excitations. The engine mount’s stiffness must be high to control large engine motions and low to control chassis or vehicle body vibration. When hydraulic engine mounts are used, a device called a decoupler creates the dual stiffness requirement. However, numerous investigations have shown that the decoupler has the potential to rotate within its cage bound and become stuck or sink and obstruct fluid flow between the fluid chambers due to a density mismatch between the decoupler and the working fluid. In addition, most hydraulic engine mounts with a decoupler no longer act as vibration isolators but as hydraulic dampers. This study suggests a new amplitude-sensitive hydraulic engine mount design without a decoupler, where the vibration isolation of the engine mount is retained and there is a 75% reduction in the peak frequency, which further enhances the engine mount’s capabilities in comparison to the current hydraulic engine mounts with a decoupler. The new design concept and its mathematical model and simulation results will be presented.

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 fluidlastic 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.

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. To have a very low dynamic stiffness at the notch frequency, thus, lower cabin noise and vibration, rubber damping, and inertia track flow losses must be low, and the fluid inertia (mass of the fluid in the inertia track) of the track must be high.

The limited amplitude sensitivity of the hydraulic engine mount design shown in Figure 2 is solely attributable to rubber’s strain sensitivity. To make the hydraulic or fluid-filled engine mount significantly more amplitude sensitive, decouplers [8,9,10,11,12] are employed, as shown in Figure 4.

As seen in Figure 4, hydraulic engine mounts with decouplers typically use two inertia tracks: the spiral-shaped inertia track and the center inertia track with a decoupler. The majority of hydraulic engine mount solutions with decouplers operate more like a fluid damper rather than a vibration isolator.

The center inertia track has a large diameter due to the decoupler and its length is short; thus, the fluid inertia, mentioned on page 2, is very low. When the engine amplitudes are low, the fluid flows around the decoupler, moves the decoupler up and down, and only a little amount of fluid passes through the spiral-shaped inertia track; thus, the flow losses are significant, and the fluid inertia is very low. Thus, the hydraulic engine mount will exhibit a very shallow notch or no notch frequency at all, act like a fluid damper rather than a vibration isolator, and its dynamic stiffness will be, in general, low.

When the engine motions are high due to shock, the decoupler, which is floating, blocks the center fluid inertia track, and fluid flow will be through the spiral-shaped inertia track. When the spiral-shaped inertia track is active, there is fluid inertia, but since the engine displacement amplitudes are high, the spiral-shaped inertia track flow losses will be also high. Therefore, a shallow notch or a hardly present notch will be observed in the dynamic stiffness curve. At high engine displacements due to shock, the engine mount feels stiff; thus, the engine mounts are able to control engine motions. The decoupler will again float once the shock has subsided and the engine motions have returned to being minimal.

The existence of the decoupler, in most cases, eliminates the notch frequency; thus, often, the hydraulic or fluid-filled engine mounts with decouplers no longer act as a vibration isolators but as a hydraulic dampers. The experimental dynamic stiffness plot shown in reference [1] clearly shows that at low engine motions, ±0.1 mm, the dynamic stiffness is flat with no notch at all, and at ±1 mm engine motions, a shallow notch is observed. A similar phenomenon is seen in the dynamic stiffness plot of Figure 5a of reference [13]. In the dynamic stiffness of Figure 3 in reference [12], at small motions, a shallow notch is seen, and at large engine motions, no notch is seen.

In reference [14], a rigid decoupler plate is replaced with a flexible membrane and the tension of the membrane is adjusted, thus adjusting the hydraulic engine mount dynamic stiffness. In the proposed design of reference [14], the decoupler no longer floats. The paper mentions that the flexible decoupler can be made from a thin metal membrane, and its stiffness/tension can be adjusted by using PZT patches. The work of reference [14] is very interesting but requires PZT patches, a power supply, sensors, a controller, and a control policy to detect when to increase or decrease decoupler tension. The work is mostly theoretical, and the proposed design has not been verified experimentally. In this paper, it is shown that the dynamic stiffness can be adjusted as membrane tension is adjusted, but this design is unable to adjust dynamic stiffness passively and automatically as engine displacement amplitudes are increased.

In reference [15], a nonlinear mathematical model of a hydraulic engine mount with a decoupler is developed through the theory of fluid dynamics. This engine mount is experimentally tested, and just like previous documented work of hydraulic engine mounts with decouplers, again it can be seen that the notch frequency is very shallow, indicating that the hydraulic engine mount does not act like an isolator as it should.

In reference [16], the low-frequency performance of a hydraulic engine mount with a decoupler was investigated. A mathematical model of the engine mount with a decoupler was developed, and its performance was then investigated in a quarter car model. The engine mount model was then compared with the experimental results of reference [17]. The experimental results of reference [17] used by reference [16] again show no notch frequency, meaning the engine mount with a decoupler does not act like a vibration isolator, unlike our proposed design in this paper.

In reference [18], the work of reference [16] was used with different inertia track lengths, diameters, and multiple inertia tracks. The effect of track length inertia, diameter, and multiple tracks were studied while having a decoupler.

As explained earlier, decouplers have the drawback of rotating and becoming trapped or stuck in their cage bound [9]. The second issue is that decoupler may stop floating. The decoupler is usually made of plastic or rubber, and it is meant to float. But due to temperature and aging, its density can change, and in time, it stops floating, thus blocking the center inertia track. The third issue is that most hydraulic engine mounts with a decoupler do not act like vibration isolators. So, the engine mount is only used to control engine motion, and the reduction in transmitted vibration from the engine to the car body or the fuselage is minimal.

In this study, a new amplitude-sensitive hydraulic engine mount design without a decoupler is proposed that not only can control engine motions due to shock but also acts like a vibration isolator. In addition, the peak frequency, which is often undesirable, is greatly minimized.

2. New Hydraulic or Fluid-Filled Engine Mount Design

In the new hydraulic engine mount design depicted in Figure 5, a piston, a spring- energized Teflon-coated seal, a Teflon bearing with extremely low surface friction, and a nonlinear spring with low hysteretic damping replace the soft rubber diaphragm shown in Figure 4. In addition, the inertia track that housed the decoupler is removed along with the decoupler. The nonlinear spring will be soft when the engine displacement amplitudes are low but stiff when the engine displacement amplitudes are high. At large engine motions, since the spring stiffness is high, the piston dynamic motions will be small, which keeps fluid trapped in the chambers. This causes the hydraulic engine mount to hydraulically lock up or experience an increase in the dynamic stiffness. In contrast to the design shown in Figure 5, when engine motions are small, the spring beneath the piston will have low spring rates, leaving the piston free to vibrate. As a result, the hydraulic engine mount behaves more like a vibration isolator than a fluid damper, unlike the engine mount design shown in Figure 4.

The nonlinear spring, shown in Figure 5 as $K_s$, can be a bilinear spring (see Figure 6a) or a cubic spring ($F_k=K_{1}x+K_2{x}^3$). Since the bottom piston can move up or down, the nonlinear spring needs to work in both the up and down directions, in compression as well as in tension. The below designs are considered in this paper. Figure 6b shows a possible cubic spring using two separate cantilever beams or one beam supported on two bocks with the bottom piston shaft attached to its center. Figure 7 shows other design variations.

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 Figure 6 and Figure 7. In the bond graph model of Figure 8, the input to the hydraulic engine mount is velocity, labeled as V_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 C₃ and R₂ represent the vertical compliance and damping of the thick elastomer (labeled as flexible element). C₆ represents the top chamber volumetric compliance, R₈ and I₁₀ represent the flow losses and fluid inertia of the inertia track, respectively, C₁₁ represents the bottom chamber volumetric compliance, A_m, represents the area of the bottom piston, I₁₄ represents the mass of the bottom piston, and C₁₅ and R₁₆ 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:

$${\dot{q}}_3=V_{in}$$

(1)

$${\dot{q}}_6=A_p{V}_{in}-\frac{P_{10}}{I_{10}}$$

(2)

$${\dot{p}}_{10}=\frac{q_6}{C_6}-\frac{q_{11}}{C_{11}}-R_8\frac{P_{10}}{I_{10}}$$

(3)

$${\dot{q}}_{11}=\frac{p_{10}}{I_{10}}-A_m\frac{p_{14}}{I_{14}}$$

(4)

$${\dot{p}}_{14}=A_m\frac{q_{11}}{C_{11}}-e_{15}-R_{16}\frac{p_{14}}{I_{14}}$$

(5)

$${\dot{q}}_{15}=\frac{p_{14}}{I_{14}}$$

(6)

Please notice that the effort $e_{15}$ in Equation (5) is a nonlinear relationship, relating spring force $F_k$ ($e_{15}$) to its relative deflection ($∆x=q_{15}$). In the above state space equations, q₃, q₆, q₁₁, and q₁₅ are the generalized displacement variables, and p₁₀ and p₁₄ are the momentum variables. The state-space variables are defined as follows:

• q₃—Relative motion across the engine mount
• q₆—Top chamber change in volume
• p₁₀—Integral of the pressure drop in the inertia track
• q₁₁—Bottom chamber change in volume
• p₁₄—Linear momentum of the bottom piston
• q₁₅—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:

$$F_{in}=\frac{q_3}{C_3}+R_2{V}_{in}+A_p\frac{q_6}{C_6}$$

(7)

$$P_t=\frac{q_6}{C_6}$$

(8)

$$P_b=\frac{q_{11}}{C_{11}}$$

(9)

$$X_p=q_{15}$$

(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.

3.1. Frequency Response Function (FRF) of a Nonlinear System

Dynamic stiffness versus frequency, chamber pressures versus frequency, and bottom piston displacement versus frequency are to be plotted. Conducting a frequency response analysis of a nonlinear system is somewhat problematic. To develop the dynamic stiffness of a hydraulic engine mount with a nonlinear spring rate, the mount is given a sinusoidal input displacement, and a numerical simulation is run long enough to see the input force $F_{in}$ (Equation (7)) reach steady state. Then, the amplitude of the steady-state $F_{in}$ is divided by the amplitude of the input displacement $X_{in}$ to obtain the dynamic stiffness at each input frequency as well as chamber pressures and bottom piston displacement.

3.2. Nonlinear System

Since in this paper we are dealing with nonlinear systems, it will be good to review peculiar behaviors of nonlinear systems, described below:

• Nonlinear systems respond differently to step inputs of different magnitudes.
• The period of oscillation can change as the amplitude of the vibration changes.
• Jump resonances may be observed in the frequency response functions.
• Subharmonic oscillations may be observed.
• Self-excited oscillations or limit cycles can occur.
• These systems may have no, one, or multiple equilibrium points.

3.3. Engine Displacement Amplitudes

The engine vibration amplitudes depend on the application and how well the engine has been balanced by the engine manufacturer; therefore, there is no standard vibration amplitude range. In references [1,3], dealing with automotive applications, the engine vibration amplitudes were assumed to be in the range of ±0.1 mm to ±1 mm. In reference [2], again dealing with automotive applications, the hydraulic engine mount with a decoupler was tested from near zero (±0 mm) to ±0.32 mm displacement amplitudes.

For fixed wing applications, the engine vibration amplitudes are in the general range of ±0.0025 mm to ±0.025 mm [5], and for rotary wing applications, the vibration amplitudes can be in the range of ±0.5 mm to ±4 mm. In this paper, the engine vibration amplitudes have been assumed to be from ±0.1 mm to ±0.4 mm. When different engine vibration amplitudes than the ones assumed in this paper are encountered, the mathematical model of this paper can be used to simulate and find the dynamic stiffness, chamber pressures, and bottom chamber piston displacements at those vibration amplitudes.

3.4. Comparison of Bilinear Spring versus Cubic Spring

There are many choices regarding which bilinear spring or cubic spring to use below the bottom piston. It is important to mention that the displacement of the bottom piston will be many times larger than the input displacement given to the engine mount. Figure 9 compares the bilinear spring versus the cubic spring used in this paper. As Figure 9 indicates, for the bilinear spring, there is a change in stiffness at 0.5 mm (let us refer to this point as the switch point S, S = 0.5 mm). To find the optimum switch point for a given application, one will need to conduct many numerical simulations at different switch points to determine the optimum location of S. For the bilinear spring, in this paper, for piston deflections less than 0.5 mm, $K_1$ was chosen as 33,183 N/m, matching the nonlinear spring stiffness at small motions, and $K_2$ was chosen as 230,000 N/m. For the bilinear spring, the $K_2$ value of 230,000 N/m was chosen to match the force value of the cubic nonlinear spring at a piston deflection of ±3 mm, namely 600 N. Notice that both bilinear and cubic springs share the same slope for deflections less than ±0.5 mm.

For the bilinear spring, choosing $K_1$ is straightforward. $K_1$ is found by the desired location of the notch frequency. But, $K_2$ requires simulations. Just like the optimum location of the switch point S for the bilinear spring, to find the optimum value of $K_2$, many simulations are necessary for a given application. Our mathematical model can be used to determine the optimum value for $K_2$. For the cubic spring, we have $F_{cubic}=\mathrm{33,183} x+1.88\times {10}^{10} x^3$. The details of the cubic spring design will be provided later.

3.5. Numerical Simulation Results for Bilinear Spring Design

To simulate the design of Figure 6a and Figure 7, the following engine mount parameters, see

Table 1. Hydraulic engine mount parameters used in MATLAB simulations.

Parameters	Definition	SI Units
$K_r^{\prime }$	Real component of rubber stiffness	325,000 N/m
tanδ	Tangent delta, $K_r^{″}/K_r^{\prime }$	0.0414
$K_r^{″}$	Imaginary component of rubber stiffness	13,455 N/m
R2	The damping in Figure 7 = $K_r^{″}$/ω	13,455/ω
Ap	Effective pumping or piston area	0.0019635 m2
dt	Inertia track diameter	1.65 cm
At	Track cross-sectional area	2.1382 × 10−4 m2
Ap/At	Piston area to inertia track area ratio	9.2
Am	Cross-sectional area of the bottom piston	0.0019635 m2
Lt	Inertia track length	10.0 cm
ρ	Fluid density	1720 kg/m3
ν	Fluid kinematic viscosity	1.8 Cst
If or I10	Inertia track fluid inertia	8.044 × 105 N-s2/m5
Ro, or R8	Inertia track flow losses	6.4 × 106 N-s/m5
Kvt	Top chamber volumetric stiffness	4.51 × 1010 N/m5
Kvb	Bottom chamber volumetric stiffness	1.0186 × 1013 N/m5
Mp or I14	Bottom piston mass (made of steel)	0.4624 kg
R16	Damping of spring below the piston	9.54 N-s/m
K1	Stiffness of the bilinear spring before S	33,183 N/m
K2	Stiffness of the bilinear spring after S	230,000 N/m
Xin	Input amplitude to mount	±0.1 to ±0.4 mm
fnotch	Notch frequency, in Hz	32 Hz (linear run)
fpeak	Peak frequency, in Hz	38 Hz (linear run)

, were used for the MATLAB simulations.

To see the difference in dynamic stiffness between an engine mount with a linear spring (F_spring = Kx) versus a bilinear spring, a comparison was made using MATLAB R2023a. Figure 10 and Figure 11 show a comparison of the dynamic stiffness of the hydraulic engine mount design in Figure 6a with a linear spring (F_spring = Kx) versus a bilinear spring in Figure 9 at different vibration amplitudes. From Figure 11, we can clearly see that at vibration amplitudes of ±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 has been considerably attenuated (See Figure 10) by simply using a bilinear spring with no added damping. Elimination of the peak frequency is extremely desirable.

The green color graph in Figure 10, near 35 Hz, may appear to show a sudden change in slopes. If one zooms to 35 Hz (see a smaller graph of Figure 10), the slope of the green-colored graph is continuous and not abrupt. It is important to mention that the frequency increments (∆f) chosen for the simulation can also make some of the graphs appear to have a sudden change in slopes. Since in this paper, we were dealing with nonlinear simulations and the run times were high, efforts were made to avoid using very small frequency increments (∆f). Because of ∆f, some of the plots may appear to show sudden changes in slopes, but this is not the case. Figure 11 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 ±0.1 mm to ±0.4 mm due to shock, the dynamic stiffness increases from 66,795 N/m to 192,851 N/m by a factor of 2.88 times without using any decoupler. Figure 11 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 Figure 6a and Figure 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 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.

Figure 14 clearly indicates that the bottom piston displacements of the new hydraulic engine mount design with a bilinear 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 ±9 mm at the peak frequency at ±0.4 mm input displacement. However, using a bilinear spring resulted in the bottom piston displacements being less than ±4 mm at the peak frequency at ±0.4 mm.

3.6. Numerical Simulation Results for a Cubic Spring Design

In this section, the bilinear spring is replaced with a cubic spring with the following constitutive model: $F_{spring}=K_{1}x+K_2{x}^3$. In the cubic spring, the change in stiffness is gradual and not abrupt like the bilinear spring. To create a cubic spring, an extensive literature review was conducted. The best nonlinear spring solution for our application appears to be using either beams, air springs, or rubber mounts. Since the bottom piston moves in both the up and down directions, the cubic spring needs to be active in both directions. With air springs, the stiffness increases when they are compressed; therefore, two air springs will be required, or one air spring with a special design that can provide cubic stiffness in both directions. The same is true for rubber mounts.

Another approach to achieve cubic stiffness is to use beams. Using two cantilever beams or one clamped–clamped beam supported on two blocks and attached to the bottom side of the floating piston, as shown in Figure 15, can provide cubic stiffness. The two cantilever beams are shown in the grey color in Figure 15. The cantilever beams need to be attached to the piston shaft, either through welding or using bolts.

Reference [20] states that the nonlinear stiffness of just one clamped-guided beam is found using the following expressions:

$$F_k=K_{0}x+\frac{K_0}{2h^2}{x}^3$$

(11)

When two cantilever beams are attached to a piston or one clamped–clamped beam is attached to a piston, the stiffness felt by the floating piston will be as follows:

$$F_k={2K}_{0}x+\frac{K_0}{h^2}{x}^3$$

(12)

where $K_0=\frac{12EI}{L^3}$, and $E$, $I$, and $L$ are beam’s Young’s modulus, area moment of inertia, and length, respectively. Using a beam with a rectangular cross-section of height h and width w, will have an area moment of inertia equal to $I=\frac{1}{12}{h}^{3}w$. From Figure 15, we can clearly see that when the bottom piston moves up or down, the clamped–clamped beam provides nonlinear stiffness in both directions. This design is now investigated in the next section.

3.7. Finite Element (FE) Model of the Nonlinear Spring

In this section, a comparison is made between Equation (12) and the ANSYS finite element (FE) results. A beam made from aluminum with a thickness of 1 mm, a width of 16 mm, and a length of 90 mm is modeled using ANSYS, see Figure 16a. The 1 cm wide middle component, shown in Figure 15 and Figure 16a, represents the shaft of the bottom chamber piston. The plane stress with thickness (of 16 mm) option was used in ANSYS for this FE run. The FE model of Figure 15, shown in Figure 16a, was developed using the ANSYS 2023R1 finite element software program. The FE model of Figure 16a is given a 3 mm displacement in the negative y-direction with the beam held fixed at the edges. SI units were used in the FE model and the FE analysis. The ANSYS “Large Displacement Static” option was turned on (NLGEOM, ON). Thirty sub-steps were prescribed, meaning at each sub-step, a motion of 0.1 mm was applied until the FE model was deformed by 3 mm. The results of all the 30 sub-step runs were saved. Figure 16b shows the beam’s deformed shape at a 3 mm input displacement.

Figure 17 shows an excellent correlation between the ANSYS load-deflection results versus the results of using Equation (12), with the highest error being 8.1%.

Figure 17 shows that at small deformations, the slope (stiffness) is 33,183 N/m and at high deflections, near 3 mm, the slope (stiffness) is about 506,300 N/m. A cubic stiffness equation of $F=\mathrm{33,183}x+1.88\times {10}^{10} x^3$ matches the ANSYS results very well.

3.8. Effects of Block or Column Height and Material on Load–Deflection Curve

In practice, to create a cubic spring, the beam needs to be mounted on top of two columns or blocks, as shown in Figure 15. The height of the columns or blocks are shown as h in Figure 15. The rigidity of the columns or blocks is very important. A detailed “2D plane stress with thickness” finite element analyses in ANSYS was conducted to see the impact of the support structure (columns or blocks) height and rigidity on the nonlinear stiffness curve.

The columns of this paper were 16 mm by 16 mm, but the column height was varied from 0 to 40 mm, and the column material was changed from aluminum to steel to see the effects of column height and material on the nonlinear stiffness. As for the beam that sits on the columns, just like the FE analysis of Figure 16, it has a thickness of 1 mm, width of 16 mm, and length of 90 mm. It is made from aluminum, with the piston shaft attached to the center of the beam having a diameter of 10 mm.

Figure 18 shows the force–deflection curve for the beam at different column heights and materials. Figure 18 clearly shows that even though the columns are made of metal and may appear rigid, if the columns are too tall, the nonlinear stiffness behavior can significantly reduce. Figure 18 indicates that the beam should be mounted on steel columns or blocks and the height of the columns should be kept below 5 mm, as shown in Figure 19. The ANSYS results clearly show that the rigidity of the columns and the substrate to which the columns are to be attached are very important.

3.9. 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 ($F_{spring}=\mathrm{33,183}x+1.88\times {10}^{10} x^3$), a comparison was made using MATLAB. Figure 20 and Figure 21 show a comparison of the dynamic stiffness of the hydraulic engine mount design of Figure 6b with a linear spring (F_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 ±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 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 ±0.1 mm to ±0.4 mm due to shock, the dynamic stiffness increases from 67 × 10³ N/m to 103 × 10³ 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 Figure 6a and Figure 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 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 ±9 mm at the peak frequency at ±0.4 mm input displacement, but using a cubic spring resulted in the bottom piston displacements being less than ±3 mm at the peak frequency at ±0.4 mm.

4. Experimental Verification of the Cubic Spring

To experimentally verify the nonlinear load–deflection curve of Figure 18 for the design of Figure 15, Equation (12) was used to make a cubic spring using an aluminum beam. Figure 25 shows a 1 mm thick, 16 mm wide, and 122 mm long aluminum beam supported by two (16 mm by 16 mm) rectangular aluminum columns or blocks at its edges. The 1 mm aluminum beam was connected to the columns by 4 bolts, as shown in Figure 25. The length of the beam that hangs free from the columns is 90 mm. The center column (16 mm × 10 mm wide made from aluminum), shown in Figure 25, represents the piston shaft. Figure 25a shows the 1 mm thick beam supported on two 40 mm long columns, and Figure 25b shows the same beam on two 10 mm long columns.

The FEA results of Figure 18 indicated that to have a proper cubic stiffness from a beam, it is best to make the beam and the columns as one body and not to use bolts to attach the beam to the columns, as was done in our experiment. Not properly bolting the beam to the columns can result in the beam slipping relative to the columns when loaded, thus affecting the nonlinear stiffness and creating a large hysteresis loop in the load–deflection curve.

In the FE analysis, the substrate structure was assumed to be completely rigid. However, in the experiment, the substrate structure may not be necessarily rigid. To reach the black-colored nonlinear load–deflection plot of Figure 18, the FE results indicated that the substrate structure needed to be very rigid, the columns needed to be short and rigid, and it was preferable to weld the beam to the columns and the piston shaft and also have the columns welded to the substrate structure, as shown in Figure 26.

The ANSYS FE results indicated that when the columns were 40 mm tall (see Figure 25a), the load–deflection curve would be only slightly nonlinear, and our experimental results indicated exactly the same thing. So, to approach the black-colored load–deflection curve, all the bolts were over-tightened to ensure no slippage occurred. The weights and a dial indicator, as shown in Figure 26, were used to obtain the experimental load–deflection data for the beam. Since the applied forces to the beam were below 25 N and the beam deflections were very small, use of a MTS test machine to acquire the load–deflection data was not recommended since it would be susceptible to sensor noise.

Figure 27 shows the experimental load–deflection plot of the designed nonlinear stiffness beam, which is indeed nonlinear and is more strongly nonlinear than the ANSYS predictions.

The experimental test of Figure 26 was repeated three times, and the data were very consistent with no hysteresis.

5. Discussion

The new design is clearly amplitude-sensitive, and at large motions, is able to control engine motions. Yet, at small motions, it acts like a vibration isolator, unlike present hydraulic engine mount designs on the market employing a decoupler. Figure 10 and Figure 21 clearly indicate that at small motions (±0.1 mm), the addition of a bilinear or cubic spring underneath the bottom piston had no impact on the location and depth of the notch frequency compared to using a linear spring.

Table 2. Dynamic stiffness at the notch and at the peak frequencies for the bilinear and cubic cases.

Engine Mount Type	Dyn. Stiffness @Notch at ±0.1 mm	Dyn. Stiffness @Notch at ±0.4 mm	% Increase in Stiffness as Displ.	Dyn. Stiffness @Peak at ±0.1 mm NL Spring	Dyn. Stiffness @Peak at ±0.4 mm-NL Spring	Dyn. Stiffness @Peak (Linear)
Bilinear	0.071 × 106 N/m	0.19 × 106 N/m	165%	0.90 × 106 N/m	1.18 × 106 N/m	3.88 × 106 N/m
NL-Spring	0.067 × 106 N/m	0.103 × 106 N/m	52%	2.56 × 106 N/m	0.87 × 106 N/m	3.88 × 106 N/m

and

Table 3. Top and bottom chamber max. pressures, piston max. disp. for the bilinear, cubic, and linear spring cases.

Engine Mount Type	Top Chamber Max. Pressure at ±0.4 mm	Bottom Chamber Max. Pressure at ±0.4 mm	Maximum Bottom Piston Displacement at ±0.4 mm
Bilinear	0.288 MPa	0.310 MPa	±3.58 mm
Cubic-spring	0.221 MPa	0.221 MPa	±2.85 mm
Linear spring	0.776 MPa	0.031 MPa	±8.7 mm

provide summaries of the numerical simulations results.

Table 2 shows that when the engine operates near the notch frequency, if the engine displacement amplitudes increase from ±0.1 mm to ±0.4 mm, the new engine mount dynamic stiffness increases by 52% to 165%, thus controlling engine motion when shocks are present. The bilinear spring that was chosen in this paper appears to be more effective in controlling large engine motions. Table 2 also indicates that the peak frequency is greatly attenuated by 3.2 times for the bilinear spring and 4.45 times for the cubic spring. Table 3 below shows the chamber pressures and bottom piston displacements. Table 3 indicates that the top chamber pressures and the bottom piston displacements are higher in value by 2.7 to 3.5 times and 2.43 to 3 times, respectively, when a linear spring is used.

Table 3 indicates that even though the bottom chamber pressures are higher when bilinear and cubic springs are used, fortunately, the maximum pressures remain below 0.82 MPa (120 psi), which most rubbers can tolerate.

6. Conclusions

In this paper, a new hydraulic engine mount design is introduced that is amplitude-sensitive, but without a decoupler. The elimination of the decoupler in this new design will eliminate issues seen in the field with hydraulic engine mounts with a decoupler. Experimental data documented in the literature have shown that the majority of hydraulic engine mounts with a decoupler do not act as vibration isolators but as hydraulic dampers. The new hydraulic engine mount design proposed in this paper still acts as a vibration isolator, yet it is amplitude-sensitive and capable of limiting engine motions brought on by shock excitations. In addition, the stiffening effect of the engine mount at the resonant peak frequency, which is often undesirable, is considerably attenuated without impacting the depth of the notch frequency.

Our simulation and experimental results indicate that using a bilinear spring in our application might be easier than using a cubic spring, since engine motions are large and the possibility of the beam going through a plastic deformation is high. The cubic spring assumption, using beams, was realized at small deflections, but plastic deformation limited its use at large deflections.

Bilinear springs can be easily employed using metal springs or using a custom-designed rubber mount to create the bilinear spring effect, particularly given the fact that engine mount manufacturers have access to rubber since they are designing the hydraulic engine mounts.

If beams are used to create the cubic spring, attention needs to be given to the substrate rigidity, the support column rigidity and height, and the boundary conditions, since these greatly impact the cubic nonlinear stiffness properties of the beam. Plastic deformation of the beam is also of concern, and at large deformations, the beam may experience a plastic deformation if one is not careful. Another issue to also keep in mind is the static fluid pressure. To avoid vapor bubbles being created (avoid fluid cavitation) inside the chambers, hydraulic engine mount manufacturers pressurize the hydraulic engine mounts to some static pressures. This static pressure can cause static deformation of the beam-type cubic springs when no engine motions are present.

The beam-type cubic spring proposed in this paper suits fluid mount applications where the engine shock excitation motions are small and the static pressures are low.

There are other cubic spring designs that can be much more effective than our proposed cubic spring, made of beams. Metal conical springs or low-damped rubber conical or V-shaped springs can be a better solution than beams, since plastic deformation will not be an issue as it was for a beam-type cubic spring. The cubic spring technology should not be ruled out. We were unable to consider metal conical springs or rubber conical/V-shaped springs in our study, but we strongly feel hydraulic engine mount manufacturers should consider them.

Damping, in general, including the damping of the bilinear or the cubic springs, affects the notch frequency and thus the level of vibration isolation; therefore, it should be kept to a minimum.

Jump resonances were only observed in Figure 13, which is expected, given the fact that the spring underneath the piston is nonlinear. However, for the displacement range of our interest, these jump resonances were not significant enough to cause any problems. At larger input displacement amplitudes, these jump resonances can be problematic.