# Experimental and Analytical Study of Secondary Path Transfer Function in Active Hydraulic Mount with Solenoid Actuator

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## Abstract

**:**

## 1. Introduction

## 2. Working Mechanism of the Active Hydraulic Mount with Solenoid Actuator

#### 2.1. Structure of Active Hydraulic Mount

#### 2.2. Structure and Working Principle of Solenoid Actuator

## 3. Active Characteristics of Active Hydraulic Mount

#### 3.1. Model of Active Characteristic

_{1}and the force is f

_{1}; the displacement of the inertial liquid column relative to the inertial track wall is y

_{2}, the displacement of the decoupling membrane is y

_{3}, the displacement of the mover is y

_{4}, and the displacement of the frame side is y

_{5}; The pressure fluctuation of the upper liquid chamber relative to the static state is p

_{1}; the active power of the actuator is f

_{a}. The above are the variables in the lumped parameter model. Considering that the dynamic bulk stiffness K

_{2}of the lower liquid chamber surrounded by the wrinkled rubber bellow is almost zero, the pressure fluctuation p

_{2}of the lower liquid chamber is not considered in the model [11,12,13] and p

_{2}= 0.

_{n2}is generally in the range of 12~15 Hz. For the low frequency, large amplitude vibrations of the powertrain at rigid body modal frequency (about 8~15 Hz) are induced by shock conditions such as starting, stopping, and crossing speed bumps. Effective attenuation can be achieved by means of fluid damping, no active control is required, and the actuator does not work. The frequency band that needs to be actively controlled is above 25 Hz. At this time, the inertial track is almost closed due to the increase of fluid resistance. The degree of freedom of the inertial liquid column can be ignored, and y

_{2}= 0. When studying the transfer function of the secondary path of the force from the actuator to the frame side of the active mount, that is, the active characteristics, set m

_{1}= 0, k

_{2}= 0, y

_{1}= 0, y

_{5}= 0, and combine K

_{2}= 0, p

_{2}= 0, the mathematical model of the active characteristics of the active hydraulic mount with solenoid actuator in the filling state is obtained as Equation (1), and the relevant parameters and their meanings are shown in Table 1.

_{a}during the actuator operation is transferred partly to both the engine side and the frame side with the component f

_{1}, and partly to the frame side only with the component $-\left({m}_{3}{\ddot{y}}_{3}+{m}_{4}{\ddot{y}}_{4}\right)$ being the sum of the inertial forces of the mover mass m

_{4}and the ejector rod, decoupling membrane and its attached liquid mass m

_{3}. The purpose of the active control of the actuator is to adjust the magnitude and phase of the active force f

_{a}in real time, to offset the force transferred from the engine to the frame side through the primary path to the greatest extent, so as to achieve the purpose of vibration and noise reduction.

#### 3.2. Analysis of Active Characteristic

_{3}can be regarded as the equivalent dynamic bulk stiffness of the decoupling membrane line stiffness k

_{3}, so there is:

_{n4}of the mover m

_{4}is much higher than the modal frequency f

_{n3}of the decoupling membrane m

_{3}. Avoid introducing unnecessary modal in the active control frequency band. Therefore, in the mechanical model shown in Figure 4, m

_{3}and m

_{4}can be considered as a rigid connection, i.e., y

_{3}= y

_{4}, at which point Equation (3) can be simplified as:

_{1}and the dynamic bulk stiffness of the decoupling membrane K

_{3}are in a parallel relationship in terms of the stiffness contribution of the decoupling membrane/mover vibration.

_{3}, f

_{1}, and f

_{5}are obtained as follows.

_{1}and K

_{3}; $\lambda =\omega /{\omega}_{\mathrm{n}3}$ is the frequency ratio; $\xi ={c}_{3}/2\sqrt{\left({m}_{3}+{m}_{4}\right){A}_{3}^{2}\left({K}_{1}+{K}_{3}\right)}$ is the damping ratio.

#### 3.2.1. Frequency Response Characteristics of Engine Side Restraining Reaction Force f_{1} in the Mid-High-Frequency Band

_{n3}of the decoupling membrane m

_{3}is much larger than the natural frequency f

_{n2}of the inertial track, considering that the mathematical model shown in Equation (1) is suitable for the frequency range of the excitation frequency much larger than f

_{n2}, it is called the mid-high-frequency band. Accordingly, the frequency range much lower than f

_{n3}is called the mid-low-frequency band. The division of low-frequency, mid-frequency, and high-frequency band can be shown in Figure 5.

_{n2}<< f << f

_{n3})

_{a}to f

_{1}is a constant, and let it be G, because the parameters on the right side of the equation are constant in the case of the hydraulic mount with inertia track and decoupling membrane determined as active mounts carriers. Therefore, the active characteristic of the engine side is equivalent to a proportional link in the mid-frequency band, and the amplitude-frequency characteristic curve appears as a horizontal line.

_{n3})

_{n3}) where the excitation frequency is much larger than the nature frequency of the decoupling membrane, the active power is not transferred to the engine side.

#### 3.2.2. Frequency Response Characteristics of the Force f_{5} Transferred to the Frame Side in the Mid-High-Frequency Band

_{n2}<< f << f

_{n3})

_{1}and f

_{5}of the mount due to the inertial forces of m

_{3}and m

_{4}, so the inertial forces can be neglected; this conclusion is consistent with the mechanical vibration theory, and experimental verification has been done for this purpose.

_{n3})

_{n3}) where the excitation frequency is much larger than the natural frequency of the decoupling membrane, the active power is reproduced at the frame/chassis side in reverse phase 1:1, and is not transferred to the engine side. This provides convenience for the control of the active power.

_{a}= k

_{M}⋅i(t), for the actuation force f

_{a}as the input frequency response curve and the current i as the input frequency response curve only differs by a constant k

_{M}. As shown in Figure 6, the frequency response curves at both ends of the mount almost overlap within 100 Hz, indicating that the difference in frequency response characteristics between the engine and frame sides forces f

_{1}and f

_{5}of the mount caused by the inertial force in the mid-low-frequency band is negligible. Above 100 Hz, the influence of the decoupling membrane/mover mass inertial force gradually increases, and the influence of inertial force must be considered.

## 4. Actuating Force of Solenoid Actuator

#### 4.1. Alternating Suction Force of Electromagnet

_{m}is the amplitude of magnetic induction intensity B

_{0}in the solenoid air gap, unit [T]. Then the alternating electromagnetic suction force is also known as the solenoid actuator active force instantaneous value f

_{a}(t) and its amplitude F

_{a}are:

_{0}is the cross-sectional area of the cone air gap, 379.5 mm

^{2}; Φ

_{m}is the maximum value of the magnetic flux in the core magnetic circuit, that is, the amplitude, the unit is [Wb], 1 Wb = 1 T·m

^{2}=1 V·s.

#### 4.2. Alternating Suction Force of the Cone Air Gap Solenoid

_{m}is the amplitude of the alternating current $i\left(t\right)={I}_{\mathrm{m}}\mathrm{sin}\omega t$, unit [A]; N is the number of turns of the coil, 150; R is the total reluctance of the magnetic flux loop, R

_{m}is the reluctance of the core, R

_{δ}is the reluctance of the air gap, unit [H

^{−1}].

_{a}~I

_{m}is independent of the excitation current frequency f, but is squared with the reluctance of air gap as well as the current amplitude I

_{m}, which has nonlinear characteristics.

_{c}is the outer diameter of mover, 26.5 mm; d

_{i}is the inner diameter of mover cone, 14.5 mm; ${\mu}_{0}$ is the magnetic permeability of the vacuum, 4π × 10

^{−7}H·m

^{−1}; $\delta $ is the length of air gap, unit [m].

#### 4.3. Frequency Response Characteristics of Full-Wave-Rectified Current Excitation

_{m,a}is the amplitude of the alternating current before rectification;

_{m,0}is the direct current component, ${I}_{\mathrm{m},0}=\frac{2{I}_{\mathrm{m},\mathrm{a}}}{\pi}$;

_{m,2}is the second harmonic amplitude, ${I}_{\mathrm{m},2}=\frac{4{I}_{\mathrm{m},\mathrm{a}}}{3\pi}$;

_{m,4}is the fourth harmonic amplitude, ${I}_{\mathrm{m},4}=\frac{4{I}_{\mathrm{m},\mathrm{a}}}{15\pi}$, which is only 20% of the second harmonic amplitude I

_{m,2}.

_{m,a}to the second harmonic amplitude I

_{m,2}, then to the amplitude of the alternating active power of the solenoid actuator F

_{a}, and then to the frequency response function of force F

_{5}at the frame side, by Equations (14), (24) and (25), there are:

_{1}, K

_{u1}, A

_{3}, K

_{3}, δ and the errors caused by it can be eliminated, and the implementation of active control algorithms is facilitated.

## 5. Experimental of Secondary Path Transfer Function in Active Hydraulic Mount with Solenoid Actuator

_{m,a}between 0 and 4 A, and get several amplitude data pairs for different currents at that frequency F

_{5}~I

_{m,a}, forming a curve. Multiple curves can be obtained under multiple experiment frequencies, and the experiment results are shown in Figure 9. The curves of different frequencies in the figure basically overlap and conform to the shape of a quadratic curve. The fitting can be performed based on all the experimental data and according to Equation (26). The fitting result is K = 2.12.

_{5}of f

_{5}at each frequency, and get the curve of ${F}_{5}/{I}_{\mathrm{m}}^{2}$ changing with frequency. The experiment result is shown in Figure 10, and the curve appears as a horizontal line and coincides with the curve calculated by the fitting constant K = 2.12.

## 6. Conclusions

- (1)
- In the active hydraulic mount with the inertial track-decoupling membrane hydraulic mount as the carrier, the transfer function of the active power of the actuator to the force on the frame side is constant in the mid-frequency band, which is equivalent to the proportional link.
- (2)
- Under the action of harmonic current of solenoid actuator, the frequency of its alternating suction force is twice the frequency of current, and the amplitude of alternating suction force is independent of frequency. Under the action of full-wave-rectified current, the frequency of the current is the same as the second harmonic component of the main component of the alternating suction force, which is convenient to calculate and test the frequency response function in the approximate sense.
- (3)
- The analysis and experiment verify that the transfer function of the solenoid actuator from the full-wave-rectified current to the force on the frame side is constant, which not only provides convenience for active control, but also eliminates the tedious testing of the parameters related to the traditional hydraulic mount, reduces the error links, and improves the accuracy of the transfer function.
- (4)
- The next step will be to study a full-wave-rectified current shaping technology to eliminate its high-order harmonic components, so as to obtain higher-precision transfer function, which is conducive to active control.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Comparison of experimental results of active characteristics at engine and frame sides of AHM.

Parameter | Name | Value |
---|---|---|

c_{1} | Viscous damping of main rubber spring in vertical direction/N·s·m^{−1} | N/A |

c_{2} | Fluid damping/N·s·m^{−1} | N/A |

c_{3} | Viscous damping of decoupling membrane/N·s·m^{−1} | 19.5 |

c_{4} | Mover damping/N·s·m^{−1} | 16 |

k_{1} | Dynamic stiffness in-phase of main rubber spring in vertical direction/N·m^{−1} | N/A |

k_{2} | Fluid stiffness/N·m^{−1} | N/A |

k_{3} | Dynamic stiffness of decoupling membrane/N·m^{−1} | 8.025 × 10^{3} |

k_{4} | Mover stiffness/N·m^{−1} | 1.547 × 10^{6} |

m_{1} | Mass at engine side/kg | N/A |

m_{2} | Mass of fluid in inertia track/kg | N/A |

m_{3} | Mass of ejector rod, decoupling membrane and attached liquid/kg | 0.266 |

m_{4} | Mass of mover/kg | 7.1 × 10^{−2} |

p_{1} | Pressure fluctuation of upper liquid chamber/Pa | N/A |

p_{2} | Pressure fluctuation of lower liquid chamber/Pa | N/A |

A_{1} | Equivalent piston area of main rubber spring/mm^{2} | N/A |

A_{2} | Cross-sectional area of inertia track/mm^{2} | N/A |

A_{3} | Decoupling membrane pump liquid piston area/mm^{2} | 1.3872 × 10^{3} |

K_{1} | Dynamic bulk stiffness of Main rubber spring/GN·m^{−5} | 22.67 |

K_{2} | Dynamic bulk stiffness of lower liquid chamber/GN·m^{−5} | 0 |

K_{3} | Dynamic bulk stiffness of decoupling membrane/GN·m^{−5} | N/A |

A_{0} | Cross-sectional area of cone air gap/mm^{2} | 379.5 |

N | Number of turns of the coil | 150 |

R | Total reluctance of the magnetic flux loop/H^{−1} | N/A |

R_{m} | Reluctance of core/H^{−1} | N/A |

R_{δ} | Reluctance of air gap/H^{−1} | N/A |

α | Cone angle/deg | 45 |

d_{c} | Outer diameter of mover/mm | 26.5 |

d_{i} | Inner diameter of mover cone/mm | 14.5 |

μ_{0} | Magnetic permeability of vacuum/H·m^{−1} | 4π × 10^{−7} |

δ | Length of air gap/m | N/A |

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**MDPI and ACS Style**

Fan, R.-L.; Dou, Y.-F.; Yao, F.-H.; Qi, S.-Q.; Han, C.
Experimental and Analytical Study of Secondary Path Transfer Function in Active Hydraulic Mount with Solenoid Actuator. *Actuators* **2021**, *10*, 150.
https://doi.org/10.3390/act10070150

**AMA Style**

Fan R-L, Dou Y-F, Yao F-H, Qi S-Q, Han C.
Experimental and Analytical Study of Secondary Path Transfer Function in Active Hydraulic Mount with Solenoid Actuator. *Actuators*. 2021; 10(7):150.
https://doi.org/10.3390/act10070150

**Chicago/Turabian Style**

Fan, Rang-Lin, Yu-Fei Dou, Fang-Hua Yao, Song-Qiang Qi, and Chen Han.
2021. "Experimental and Analytical Study of Secondary Path Transfer Function in Active Hydraulic Mount with Solenoid Actuator" *Actuators* 10, no. 7: 150.
https://doi.org/10.3390/act10070150