Time domain response analysis of multi-flow channel hydraulic mount

2.1. Hydraulic mount structure

Fig. 1 shows a schematic diagram of the hydraulic mount, where Fig. 1(a) shows the structure of the passive hydraulic mount, and Fig. 1(b) shows the lumped parameter model of the passive hydraulic mount. In this case, the hydraulic mount structure includes an inertial channel and an orifice flow channel. The flow rates in the inertial and orifice flow channels are denoted by $q_i$ and $q_o$, respectively. The rubber unit is modeled using the rubber stiffness $K_r$and the viscous damping coefficient $B_r$. The compliances of the upper and lower chambers of the hydraulic mount are $C_1$ and $C_2$ respectively, and the pressure of the upper chamber #1 is $P_1$, and the pressure of the lower chamber #2 is $P_2$. where the inertial channel and the orifice flow channel are represented by the fluid inertia $I_i$ and $I_o$ and the fluid resistance $R_i$ and $R_o$, respectively.

Fig. 1Hydraulic mount: a) structure diagram, b) lumped parameter model

a)

b)

2.2. Hydraulic mount flow channel design with multiple configurations

From the literature [12, 32], it is known that increasing the number of hydraulic mount inertia channels can improve the peak frequency of the dynamic stiffness and loss angle of the mount, and the combination of different flow channels can broaden the range of hydraulic mount vibration isolation frequency. However, these literature studies are all from the mount frequency characteristics and rarely involve the analysis of multi-flow channel hydraulic mount time domain characteristics. To investigate common multi-flow hydraulic mounts, six prototype configurations as shown in Table 1 and Fig. 2 were analyzed.

Fig. 2Multiple configurations of the hydraulic mount prototype a) single long inertia channel, configuration Z1, (L1= 212 mm, D1= 8.53 mm), configuration Z2 (L2=4L1, D2=D1); b) parallel combination of long inertia channel and orifice flow channel, configuration Z3 (L3=4L1, D3=D1, Lo1= 2 mm, Do1= 0.2 mm); configuration Z4 (L4=4L1, D4=D1, Lo2=Lo1, Do2=5Do1); c) parallel combination of three long inertia channels, configuration Z5 (L51=L52=L1, L53=4L1, D51=D52=D53=D1), configuration Z6 (L51=L52=L53=L1, D51=D52=D53=D1)

a)

b)

c)

The first configuration (named $Z_1$, $Z_2$) hydraulic mount upper and lower chambers are connected only by long inertia channels, this structure is the most common at present. Where, the length of $Z_1$ inertia channel $L_1=$ 212 mm, the length of $Z_2$ inertia channel $L_2=4L_1$, the diameter of $Z_1$ and $Z_2$ inertia channels are equal $D_1=D_2=$ 8.53 mm. In the second configuration (named $Z_3$ and $Z_4$), the upper and lower chambers of the hydraulic mount are connected in parallel by long inertia channels and orifice flow channel. Where $Z_3$ and $Z_4$ inertia channels have equal lengths and diameters, $L_3=L_4=4L_1$, $D_3=D_4=$8.53 mm, respectively; The length of the $Z_3$ configuration orifice flow channel and $Z_4$ are equal to $L_{o1}=L_{o2}=$ 2 mm, the diameter of the $Z_3$ orifice flow channel is $D_{o1}=$ 0.2 mm, and the diameter of the $Z_4$ orifice flow channel is $D_{o2}=$ 1 mm. The third configuration (named $Z_5$, $Z_6$) is a parallel combination of the number of long inertia channels $n=$3. Where the diameters of the $Z_5$ and $Z_6$ inertial channels are equal $D_5=D_6=$ 8.53 mm, $L_{51}=L_{52}=L_1$, $L_{53}=4L_1$ in the $Z_5$ configuration; the lengths of the three inertial channels in the $Z_6$ configuration are equal $L_{61}=L_{62}=L_{63}=L_1$.

Fig. 3 shows the effect of variation in inertial channel length and cross-sectional area on the flow rate of the decoupler membrane channel. The decoupler membrane channel flow increases slightly with increasing inertial channel length. The flow rate of the decoupler membrane channel decreases slightly as the cross-sectional area of the inertial channel increases.

Fig. 3Effect of hydraulic mount inertia channel length and cross-sectional area variation on decoupling membrane channel flow: a) length variation; b) cross-sectional area variation

a)

b)

2.3. AMEsim-based hydraulic mount study

Fig. 4 shows the hydraulic mounts models of six configurations built using AMEsim software.

Fig. 4(a) shows the hydraulic mounts of $Z_1$ and $Z_2$ configurations, Fig. 4(b) shows the hydraulic mounts of $Z_3$ and $Z_4$ configurations, and Fig. 4(c) shows the hydraulic mounts of $Z_5$ and $Z_6$ configurations.

Build 6 hydraulic mounts configurations using AMEsim, and the dynamic characteristics of the combined inertial channel and orifice flow channel hydraulic mounts were simulated for the above six configurations. The test uses a sinusoidal displacement excitation $x_e\left(t\right)=A\mathrm{s}\mathrm{i}\mathrm{n}\omega t$, where $A$ is the amplitude of the harmonic input and $f=\omega /2\pi$ is the frequency of the excitation, where the unit is Hz. The transfer forces of the hydraulic mounts and the dynamic pressures $P_1\left(t\right)$ and $P_2\left(t\right)$ within the upper and lower chambers are calculated in 1 Hz increases between 1 and 30 Hz in $X_e=$0.3 mm, $X_e=$ 0.9 mm and, where $X_e=2A$ is the peak-to-peak (P-P) value. The calculated dynamic stiffness and loss angle for a given excitation amplitude $X_e=$ 0.3 mm and $X_e=$ 0.9 mm hydraulic mounts can be obtained as shown in Fig. 4. It is worth noting that configurations $Z_5$ and $Z_6$ exhibit greater dynamic stiffness and loss angle peak frequency. While configurations $Z_3$ and $Z_4$ show the smallest peak frequencies for dynamic stiffness and hysteresis angle, this is due to the inclusion of orifice flow channels in both configurations. Finally, the dynamic stiffness and loss angle of $Z_1$ and $Z_2$ are relatively small relative to the peak, due to the number of flow channels of one in the configuration. When the excitation amplitude $X_e$ is increased from 0.3 mm to 0.9 mm, the dynamic stiffness and hysteresis angle of the hydraulic mounts decrease, which is more obvious for $Z_1$, $Z_2$, $Z_5$ and $Z_6$.

Fig. 4AMEsim hydraulic mount model: a) single inertia channel; b) combination of inertia channel and orifice flow channel; c) combination of 3 inertia channels

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b)

c)

Fig. 5Dynamic stiffness and loss angle of six configurations of hydraulic mount: a) Xe= 0.3 mm; b) Xe= 0.9 mm

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b)

3.1. Model I single inertia channel hydraulic mount

When the hydraulic mount is excited by $x_e\left(t\right)$, applying the continuity equation to the fluid control hydraulic unit in combination with Fig. 1 and Fig. 1(a), the following Eq. (1)-(5) are obtained. Where the fluid inertia $I_i$ and the fluid resistance $R_i$ are expressed:

where $\rho$ is the density of the fluid, $L$ is the length of the flow channel, $A_i$ is the cross-sectional area of the flow channel, and $v$ is the viscosity of the fluid, $D_h$ is the equivalent “hydraulic diameter” of the flow channel. The first term on the right side of Eq. (1) is the pressure drop due to the fluid in the inertial channel due to inertia; the second term on the right side is the pressure drop due to the viscous damping of the fluid in the inertial channel.

Therefore, the hydraulic mount transfer force $F_T$ under external excitation $X_e$ is defined as follows:

Then, by Laplace transformation of Eq. (6), the expressions for the variation of the transfer complex stiffness and the upper chamber pressure with the excitation amplitude for a single-flow channel hydraulic mount are obtained as follows:

Since $1/C_1=K_1$, $1/C_2=K_2$, and $K_1\gg K_2$. So, Eqs. (7) and (8) can be further simplified as follows:

Eq. (9) and Eq. (10) can be simplified to a standard second-order system:

Thus, Eq. (11) can be transformed into the following standard form, where $\lambda =A_p^2{K}_1$, $\tau =R_i/I_i$:

where ${\varsigma }_2=\sqrt{R_i^2/4K_1{I}_i}$, ${\omega }_{n2}=\sqrt{K_1/I_i}$.

3.2. Model II combined inertial channel and orifice flow channel hydraulic mount

According to Fig. 1 and Fig. 2(b), the fluid equation of the hydraulic mount subjected to displacement excitation $x_e\left(t\right)$ for the parallel combination of inertial channel and orifice flow channel can be obtained as shown below:

In the equation, $R_0$ is the fluid resistance of the orifice flow channel. Since the inertia in the orifice flow channel is negligible, Eq. (16) only considers the viscous damping term resulting in the hydraulic mounts differential pressure change. Reference [33] shows that when the fluid is assumed to be in the form of laminar flow in the orifice flow channel, the fluid resistance of the orifice flow channel is solved for by the following expression:

where, $D_o$ is the diameter of the orifice flow channel.

Thus, the hydraulic mounts parallel to the inertial channel and the orifice flow channel transfer the complex stiffness $K_{d2}\left(s\right)$ and the variation of the upper chamber pressure with the excitation amplitude $G\left(s\right)$ defined as follows:

Since $1/C_1=K_1$, $1/C_2=K_2$, and $K_1\gg K_2$ can be simplified by Eqs. (20) and (21) as follows:

So, reduced to the standard second-order system form shown:

where the expressions for ${\omega }_{n2}$, ${\varsigma }_2$, $\kappa$ and $F$ are shown:

3.3. Model III parallel combination of 3 flow channel hydraulic mount

To easily solve for the multi-inertia channel hydraulic mount transfer complex stiffness $K_{d3}\left(s\right)$ and the upper chamber pressure variation $G\left(s\right)$ with excitation amplitude for the $Z_5$ and $Z_6$ configurations. The lumped parameter model of the multi-inertia channel hydraulic mounts with $Z_5$ and $Z_6$ configurations of Fig. 6(a) is equated to the equivalent mechanical model of Fig. 6(b). Because the cross-sectional area of the hydraulic mount inertia channel is much smaller than the equivalent cross-sectional area of the rubber main spring, when the fluid flows through the inertia channel, the fluid velocity is amplified to play the role of vibration attenuation, the literature [6] referred to it as a velocity amplification of large absorbing array.

Fig. 6Multi-inertia channel hydraulic mount: a) lumped parameter model, b) equivalent mechanical model

a)

b)

According to Fig. 6(a), the mathematical model equation of the lumped parameters of the multi-inertia channel hydraulic mounts can be obtained as follows:

where, $q_{i1}$, $q_{i2}$, $q_{i3}$ are the flow rate through the three inertia channels of configuration $Z_5$ and $Z_6$ respectively, $I_{i1}$, $I_{i2}$, $I_{i3}$ are the inertia flow through the three inertia channels respectively, $R_{i1}$, $R_{i2}$, $R_{i3}$ are the liquid resistance flow through the three inertia channels respectively.

In Fig. 6(b), $k_1=A_p^2/C_1$ is the equivalent linear stiffness of the upper chamber of the hydraulic mount, $k_2=A_p^2/C_2$ is the equivalent linear stiffness of the lower chamber of the hydraulic mount, $b_{in}$ ($n=$1, 2, 3) is the equivalent damping of the fluid in the inertia channel of the hydraulic mount, and $m_{in}$ ($n=$1, 2, 3) is the equivalent mass of the fluid in the inertia channel of the hydraulic mount [34]. It is known that the flow rate $q=A_i{\dot{x}}_i$, where $A_i$ is the cross-sectional area of the inertial channel and ${\dot{x}}_i$ is the average flow velocity of the fluid in the inertial channel. Therefore, Eqs. (32-35) can be written as shown in the following equation:

where, $A_{i1}$, $A_{i2}$, $A_{i3}$ are the cross-sectional areas of the three inertial channels in configurations $Z_5$ and $Z_6$, and $x_{i1}$, $x_{i2}$, $x_{i3}$ are the displacements of the fluid flowing through the three inertial channels in configurations $Z_5$ and $Z_6$, respectively.

Combining the equivalent mechanical model of the multi-inertia channel hydraulic mount in Fig. 6(b) and the literature [2] the Eqs. (33-36) can be written as follows:

Because $C_1\ll C_2$, The equivalent masses of the three inertia channels of the hydraulic mounts of the $Z_5$ and $Z_6$ structures are $m_{i1}=I_{i1}{A}_p^2$, $m_{i2}=I_{i2}{A}_p^2$, $m_{i3}=I_{i3}{A}_p^2$, and the equivalent damping of the three inertial channels of the hydraulic mounts of the $Z_5$ and $Z_6$ are $b_{i1}=R_{i1}{A}_p^2$, $b_{i2}=R_{i2}{A}_p^2$, $b_{i3}=R_{i3}{A}_p^2$ respectively. Therefore, Eqs. (37)-(39) can be further simplified as shown below:

Referring to the literature [32], the equivalent mass and equivalent damping of the inertial channel can be expressed as follows:

Therefore, the Eqs. (40-43) can be reduced to the following expression:

As a result, the transfer function expressions for the transfer complex stiffness and the upper chamber pressure variation with excitation for the multi-inertia channel hydraulic mounts of $Z_5$ and $Z_6$ structures are as follows:

Therefore, the expressions for Eqs. (48) and (49) reduced to a standard second-order system are shown below:

where the expressions for $\kappa$, ${\omega }_{n2}$ and ${\varsigma }_2$ are as follows:

The key parameters of the hydraulic mount can be obtained by combining model I, model II and mode III as shown in Table 2.

4.1. Time domain steady-state harmonic response

To analyze the hydraulic mounts in Models I, II, and III configurations for steady-state harmonic response, the hydraulic mounts transfer the force $F_T\left(t\right)$ and the pressure $P_1\left(t\right)$ in the upper chamber, the specific derivation process was referred to the reference [15, 16]. According to the convolution of the bishop function:

where $x\left(t\right)*y\left(t\right)$ is the mathematical expression of the convolution.

Therefore, the analytical equations of the transfer force $F_T\left(t\right)$ and the pressure $P_1\left(t\right)$ in the upper chamber under the steady-state harmonic response in the time domain for models I, II and III of the hydraulic mount configurations can be solved according to Eq. (52), which is expressed as the following equation: