Free Vibration Analysis of Hydraulic Quick Couplings Considering Fluid–Structure Interaction Characteristics

Abstract

As an important component of the automatic hydraulic quick coupling device (AHQCD) on rescue vehicles, the hydraulic quick couplings (HQCs) are used to rapidly dock hydraulic lines and transport fluid while changing and operating hydraulic working tools. However, during tool operation at rescue sites, pressure pulsations at multiple frequencies in the hydraulic lines can coincide with the natural frequencies of the HQCs, potentially causing resonance that severely affects the stability of fluid conveying and damages the connection of hydraulic lines accidentally. To investigate the natural frequencies of the HQCs with upstream and downstream lines, the characteristics of fluid–structure interaction were considered between the poppets and the fluid in this study, and an equivalent stiffness model of the fluid domain was derived based on the fluid compressibility. A dynamic model, along with 6-DOF equations for the system, was established, and the natural frequencies and mode vectors were determined by free vibration analysis. In addition, the effects of working pressure, air content, and stiffness of the springs on the natural frequency of the HQC system were analyzed. The results show the natural frequency increases with a higher working pressure and lower air content, while the effect of spring stiffness on natural frequencies varies with different modes. Furthermore, the proposed model is validated by experimental pressure signals, showing good agreement, with an average error of 2.7% for the first-order natural frequency. This paper presents a theoretical method for improving the stability of fluid transport when operating various hydraulic tools under complex rescue conditions.

1. Introduction

China is among the countries most severely affected by geohazards globally, characterized by diverse types, a widespread geographic distribution, a high occurrence frequency, significant destructiveness, and challenges in effective prediction. Therefore, the ability to immediately and efficiently carry out emergency rescue operations during disasters is essential [1,2]. Rescue vehicles currently play a significant role at disaster sites, including demolishing concrete structures, clearing landslide rocks, and repairing damaged roads. However, these rescue vehicles are typically equipped with only one type of hydraulic working tool, providing a single operational function, which limits their efficiency and performance during rescue operations.

To address this issue, automatic hydraulic quick coupling devices (AHQCDs) have been designed and attempted to be installed on rescue vehicles, enabling quick tool changes within minutes and allowing a single vehicle to perform multiple tasks [3,4]. During the tool-changing process, the hydraulic lines must be docked with the mechanical frame simultaneously, presenting a technical challenge that has been extensively studied. Liebherr and Oilquick have developed automatic hydraulic quick coupling systems (AHQCSs), including the Likufix and OQ series, respectively [5,6,7]. Specialized hydraulic quick couplings (HQCs) have been used to simultaneously dock hydraulic lines during tool changes, with an average time of 60 s [8]. The K100 boom change system has been developed by Komatsu for replacing booms while simultaneously connecting the hydraulic lines in high-reach demolition machines [9]. Yun Chao [10,11] provided a comprehensive review of specialized HQCs, examining their applicability in hydraulic quick coupling systems. He proposed a solution for automatic docking of HQCs that balances cost and manufacturing complexity.

However, the proposed docking systems are rarely applied in practical rescue scenarios or heavy-load conditions. The HQCs simultaneously endure upstream pressure pulsations from a pump [12,13], downstream hydraulic impulses generated by the tools [14,15], and internal vortexes caused by complex geometric structures during fluid transport [16]. Resonance may occur in the docking system when the frequency of an excitation is close to the natural frequency of the HQCs, significantly amplifying hydraulic pulsation and potentially damaging the entire docking lines [17,18]. Additionally, resonance can increase the amplitude of vibrations in the internal poppets and exacerbate wear of the seals, leading to system leakage [19,20]. Therefore, to ensure a reliable connection of hydraulic lines, it is essential to first determine the natural frequency of the HQC system to prevent resonance.

The research on hydraulic pipeline modeling is abundant, including fluid network theory, the finite element method, the semi-analytical method, and others [21,22,23]. Pan et al. [21] developed a natural frequency model of a self-resonating water jet based on fluid network theory and experimentally determined the natural frequency ranges of low-frequency pulsation and high-frequency oscillations in the fluid network and nozzle. Cao [22] investigated a hydraulic-line system with connected hoses, determining the equivalent support stiffness of the hoses using a frequency response function, and established a dynamic model of the system based on the finite element method, analyzing the natural frequencies and mode shapes of the system under different pressures. Ding et al. [23] studied the free vibration characteristics of high-speed pipelines using a combination of analytical and numerical methods, analyzing the impact of nonlinear stiffness on the natural frequencies.

Different from a simple pipeline, the movable poppet element was contained in a docking system, which complicates the analysis of the natural frequency of the hydraulic line. Specifically, the HQCs have a very compact configuration with complex and narrow valve channels. As hydraulic fluid flows through the narrow channels, the poppets inside are gradually accelerated by significantly varying pressure caused by the strong geometric constraint, leading to vibration of poppets. This vibration of the poppets, in turn, affects the velocity and pressure distribution within the flow field, further intensifying the poppet vibration and resulting in a fluid–structure interaction effect. Therefore, the dynamic behavior of the poppets should be considered primarily when studying the natural characteristics of a docking system in hydraulic lines. Yuan et al. [24,25] considered the fluid–structure interaction characteristics between the spray core and the fluid in the fire-fighting jet system, and they established a 7-DOF free vibration equation based on the equivalent stiffness model of the upstream fluid and using the lumped parameter method. Liao et al. [26,27,28] analyzed the phenomenon of vortex-induced vibration caused by fluid flowing through a narrow chamber and investigated the causes of abnormal pressure fluctuations in the system based on a fluid–structure interaction dynamic model. Buscarino et al. [29] analyzed nonlinear jump resonance caused by a cubic term in stiffness, implying that a simple model of a hydraulic line exhibits very complicated dynamic behaviors due to nonlinear factors. Wu [30] analyzed the abnormal dynamic behavior of the relief valve’s main spool caused by alternating pressure in the downstream pipeline, which was induced by working tools.

The studies mentioned above have effectively highlighted the significance of modeling both the poppet and the pipeline together when analyzing natural characteristics. However, in previous analyses of natural characteristics, the movable poppet and the connecting pipeline have been considered separately, and fluid–structure interaction in the docking system has rarely been incorporated. In this paper, the effects of fluid–structure interaction are taken into consideration, which are shown to have a significant impact. The pressure force exerted by the fluid is equivalent to a linear spring force based on the bulk modulus theory. The fluids in upstream and downstream hydraulic lines, along with the movable poppets, are incorporated innovatively to establish the free vibration equations using a lumped parameter method. The study is organized as follows: In Section 2, a comprehensive model of the HQCs with hoses is developed based on the lumped parameter method. In Section 3, the equivalent stiffness of the fluid is derived, and a 6-DOF free vibration equation is developed. In Section 4, the natural frequency and mode shapes are determined, and the effects of working pressure, air content, and structural parameters on the natural frequency of the HQCs are also analyzed. In Section 5, the validity of the proposed model is verified through experiments.

2. Dynamic Model of the Hydraulic Quick Coupling System

2.1. Introduction to the Automatic Hydraulic Quick Coupling Devices and Hydraulic Quick Coupling System

The hydraulic quick couplings (HQCs) are a critical component of the automatic hydraulic quick coupling device (AHQCD) system on rescue vehicles, designed for quickly docking hydraulic lines and facilitating fluid transport during working tool changes, as shown in Figure 1. The AHQCD consists of two parts: A₁, mounted at the boom side, and A₂, mounted at the tool side. These parts can be connected at the positioning point M and secured together with a pair of spaced latching axles, O and F, when the boom is moved to hitch with the tool. The HQCs also consist of two parts: the female coupling C₁ and the male coupling C₂, which are embedded in mounting plates B₁ and B₂, respectively, with B₁ and B₂ arranged within A₁ and A₂. When A₁ and A₂ are connected together, C₁ and C₂ are automatically docked through three steps: approaching, aligning, and securely connecting [31].

The detailed internal structure of the HQCs is illustrated in Figure 2. In the disconnected state, the poppet (3) of the female coupling and the poppet (14) of the male coupling are each closed by the supporting forces of springs k₁ and k₂, respectively, thereby disconnecting and sealing their corresponding hydraulic lines, as illustrated in Figure 2a. In the working state, the two couplings push against each other’s poppets, thereby connecting the oil chambers at both ends, as illustrated in Figure 2b.

2.2. Dynamic Model of the Hydraulic Quick Couplings Considering Fluid–Structure Interaction

As shown in Figure 2, the HQCs have a very compact configuration with a complex and narrow valve channel. In the working state, as hydraulic fluid flows through this narrow channel, and the internal poppets are accelerated due to pressure pulsations resulting from the strong geometric constraints. This vibration of poppets, in turn, affects the pressure distribution within the flow field, further intensifying the vibration of the poppets. Therefore, fluid–structure interaction between the poppets and the fluid must be considered. Moreover, the fluid–structure interaction at the interface involves the fluid unit, which includes the upstream lines, the shared coupling area, downstream lines, as well as the solid unit, which consists of the two poppets. The influence of the fluid unit on the system’s dynamics primarily involves inertia and compressibility, while the influence of poppets on dynamics includes their inertia and the stiffness of the mechanical spring. The interaction between these units within the coupling system is illustrated in Figure 3.

In Figure 3, Fluid Unit 1 refers to the fluid within the upstream hose and the volume enclosed by the inner chamber of the female poppet. Fluid Unit 3 represents the fluid inside the downstream hose and the volume enclosed between the outer side of the male poppet and the body. Fluid Unit 2 refers to the fluid in the shared coupling chamber. Fluid Unit 4 corresponds to the fluid within the inner chamber of the male poppet, as shown in Figure 2c.

To support theoretical analyses, the dynamic model of the docking system is based on the following assumptions:

(1)

The HQC system is modeled based on the lumped parameter method. It is assumed that the density, stiffness, pressure, and other attribute parameters of each fluid unit in the HQC system are uniformly distributed across the control volume.

(2)

Except for the fluid units and springs, it is assumed that components such as the poppets, body enclosure, and hydraulic hoses are rigid, and their deformation under pressure is neglected.

(3)

Only the axial motion of the poppets is considered, and the forces exerted by the fluid units on the poppets are represented as linear spring forces in the axial direction.

(4)

The damping between the fluids and the poppets is reduced to linear damping in the axial direction.

(5)

Mechanical tolerances and installation errors, such as eccentricity, are neglected.

Building on the aforementioned analysis and assumptions, the dynamic model of the HQCs with connecting hoses is developed, as shown in Figure 4. In this model, m₁ and m₂ are the masses of the female and male poppets, respectively. k₁ and k₂ denote the mechanical stiffnesses of the springs associated with the female and male poppets, respectively. c₁ and c₂ denote the mechanical damping between the poppets and the valve seats. Here, m_fi (where i ∈ {1,2,3,4}) represents the mass of each fluid unit, while k_f1i, k_f2i, k_f3i, and k_f4i (where i ∈ {1,2}) represent the equivalent stiffness of each fluid unit at a contact boundary with the solid unit, respectively. c_f1i, c_f2i, c_f3i, and c_f4i (where i ∈ {1,2}) represent the damping characteristics of each fluid unit. x_i (where i ∈ {1,2,3,4,5,6}) denotes the displacement of the equivalent mass blocks of the fluid and solid units.

3. Equations of the Hydraulic Quick Coupling System

3.1. Equivalent Stiffness Model of the Fluid Unit

The concept of equivalent stiffness for the fluid unit in contact with the solid unit was introduced in the previous section. In this subsection, the equivalent stiffness is quantitatively calculated. According to the bulk modulus of elasticity theory for a fluid, as described in references [32,33], we let the bulk modulus of elasticity be E_v, defined as follows:

$$E_v=-{\displaystyle \frac{\Delta p}{\Delta V}}V,$$

(1)

where V represents the total volume of the fluid unit, ∆V denotes the volume variation, and ∆p is the pressure variation.

The volume change in the fluid unit, determined by the average cross-sectional area S_a and the axial length change ΔL of the fluid unit, is

$$\Delta V=S_{\mathrm{a}}\Delta L,$$

(2)

According to the definition of mechanical stiffness, there is

$$k_{\mathrm{f}}=-{\displaystyle \frac{\Delta F}{\Delta L}}=-{\displaystyle \frac{S_{\mathrm{a}}\Delta p}{\Delta L}},$$

(3)

where ΔF represents the change in load applied to the fluid unit. Based on Equations (1)–(3), the equivalent stiffness of fluid can be expressed by the bulk modulus of elasticity:

$$k_{\mathrm{f}}={\displaystyle \frac{E_v{S}_{\mathrm{a}}^2}{V}}.$$

(4)

The fluid–structure interaction is concerned with the coupled dynamics of structures in contact with a fluid at their interface. Therefore, the equivalent stiffness of the fluid unit in contact with the solid unit must be calculated separately, based on their contact area, as shown in Figure 5.

Considering Equation (4), the equivalent stiffness k_f′ of the fluid unit in contact with the left-side solid unit can be expressed as

$${k_{\mathrm{f}}}^{\prime }={\displaystyle \frac{E_v{{S_{\mathrm{a}}}^{\prime }}^2}{V}},$$

(5)

while the equivalent stiffness k_f″ of the fluid unit in contact with the right-side solid unit can be expressed as

$${k_{\mathrm{f}}}^{″}={\displaystyle \frac{E_v{{S_{\mathrm{a}}}^{″}}^2}{V}},$$

(6)

where S_a′ and S_a″ denote the contact areas between the fluid unit and the solid unit on the left-hand side and the right-hand side, respectively.

As shown in Figure 2, each fluid unit has a relatively complex geometrical feature. To enhance the calculation accuracy of the fluid equivalent stiffness, Fluid Unit 1 is segmented into six parts, while Fluid Unit 2 is divided into four parts based on the structural shape. L₁ and L₁₃ represent the upstream and downstream connecting hydraulic lines of the HQCs, respectively. The volumes of the four fluid units are V₁, V₂, V₃, and V₄, respectively, as indicated in Equations (7)–(10). The dimensional parameters of each segment of the fluid units are presented in

Table 1. Dimensional parameters of each segment of the fluid units.

Fluid Unit	Segment	Axial Length of the Fluid Domain Li/(mm)	Average Area of Flow Cross-Section Sai/(mm2)
Fluid Unit 1	I–II	L1 = 1500	Sa1 = 401.1
	II–III	L2 = 55.5	Sa2 = 201.1
	III–IV	L3 = 7.5	Sa3 = 471.4
	IV–V	L4 = 13.5	Sa4 = 283.5
	V–VI	L5 = 8	Sa5 = 57.7
	VI–I	L6 = 34.5	Sa6 = 99.5
Fluid Unit 2	II–VII	L8 = 26.1	Sa8 = 171.1
Fluid Unit 3	VII–VIII	L10 = 26.5	Sa10 = 35.3
	VIII–IX	L11 = 5.2	Sa11 = 283.5
	IX–X	L12 = 40	Sa12 = 153.9
	X–XI	L13 = 1500	Sa13 = 201.1
Fluid Unit 4	III–VIII	L9 = 36.75	Sa9 = 63.6

, while the cross-sectional area parameters of the two poppets are detailed in

Table 2. Cross-sectional parameters of the two poppets.

Poppet	Location of the End Face 1	End Face Area Api/mm2
Female poppet	VI	Ap11 = 68.33
	I	Ap12 = 37.70
Male poppet	VII	Ap21 = 168.73
	III	Ap22 = 63.62

¹ Location of the end face of two poppets labeled in Figure 2.

.

$$V_1={\displaystyle \sum _{i=1}^6{S}_{\mathrm{a}i}{L}_i},$$

(7)

$$V_2=S_{\mathrm{a}8}{L}_8,$$

(8)

$$V_3={\displaystyle \sum _{i=10}^{13}{S}_{\mathrm{a}i}{L}_i},$$

(9)

$$V_4=S_{\mathrm{a}9}{L}_9.$$

(10)

Based on Equations (5) and (6), and utilizing the parameters from Table 1 and Table 2, the equivalent stiffness between the fluid unit and the poppet can be calculated under the assumption that the bulk modulus of elasticity E_v for hydraulic oil, which contains a certain amount of air, is 700 MPa. The results are presented in

Table 3. Equivalent stiffness from the interaction between the fluid unit and the poppet.

Fluid Unit	Equivalent Stiffness (N/mm)	Value
Fluid Unit 1	kf11	199.7
	kf12	7.6
Fluid Unit 2	kf21	1761.8
	kf22	8460.1
Fluid Unit 3	kf31	62.2
	kf32	88.4
Fluid Unit 4	kf41	8.8
	kf42	8.8

.

3.2. Free Vibration Equation of the Hydraulic Quick Coupling System

According to the dynamic model of the HQCs with connecting hoses, the equation of free vibration of the docking system is developed as

$$\mathit{m}\ddot{\mathit{x}}+kx=0,$$

(11)

where the mass matrix of the docking system is

$$m=\mathrm{diag}(m_{\mathrm{f}1},m_1,m_{\mathrm{f}2},m_2,m_{\mathrm{f}3},m_{\mathrm{f}4}),$$

(12)

where the stiffness matrix of the system is

$$k=\left[\begin{array}{cccccc}{k}_{\mathrm{f}11}+k_{\mathrm{f}12}& -k_{\mathrm{f}12}& 0& 0& 0& 0\\ -k_{\mathrm{f}12}& k_{\mathrm{f}12}+k_{\mathrm{f}21}+k_1& -k_{\mathrm{f}21}& 0& 0& 0\\ 0& -k_{\mathrm{f}21}& k_{\mathrm{f}21}+k_{\mathrm{f}22}& -k_{\mathrm{f}22}& 0& 0\\ 0& 0& -k_{\mathrm{f}22}& k_{\mathrm{f}22}+k_{\mathrm{f}31}+k_{\mathrm{f}41}+k_2& -k_{\mathrm{f}31}& -k_{\mathrm{f}41}\\ 0& 0& 0& -k_{\mathrm{f}31}& k_{\mathrm{f}31}+k_{\mathrm{f}32}& 0\\ 0& 0& 0& -k_{\mathrm{f}41}& 0& k_{\mathrm{f}41}+k_{\mathrm{f}42}\end{array}\right],$$

(13)

the displacement vector x of the system can be expressed as

$$x={\left(\begin{array}{cccccc}{x}_1& x_2& x_3& x_4& x_5& x_6\end{array}\right)}^{\mathrm{T}}.$$

(14)

4. Modal Analysis of the Hydraulic Quick Coupling System

4.1. Natural Frequencies and Mode Shapes

Additional physical parameters of the docking system are listed in

Table 4. Physical parameters of the docking system.

Parameter Name	Value
Mass of female poppet m1 (kg)	0.029
Mass of male poppet m3 (kg)	0.022
Mass of Fluid Unit 1 mf1 (kg)	0.4934
Mass of Fluid Unit 2 mf2 (kg)	0.0041
Mass of Fluid Unit 3 mf3 (kg)	0.2802
Mass of Fluid Unit 4 mf4 (kg)	0.0020
Stiffness of the spring for female poppet k1 (N/mm)	1.23
Stiffness of the spring for male poppet k2 (N/mm)	3.85
Density of hydraulic oil ρ (kg·m−3)	875
Initial gas content of hydraulic oil α (%)	5
Bulk elastic modulus of hydraulic oil Ev (MPa)	700
Temperature T (°C)	25

, with hydraulic oil containing a certain amount of air at room temperature. Modal analysis of the docking system was conducted by substituting the parameters from Table 3 and Table 4 into the free vibration Equation (11). The natural frequencies and corresponding mode shapes of the docking system are presented in

Table 5. Natural frequencies and modal shapes of the HQC system.

Order		First-Order	Second-Order	Third-Order	Fourth-Order	Fifth-Order	Sixth-Order
Natural frequency fni/Hz		fn1	fn2	fn3	fn4	fn5	fn6
		88.7	103.4	203.4	469.4	1691	8505
Modal shapes uni	Fluid Unit 1	0.1408	1	−0.0127	0	0.0001	0
	Female poppet	1	−0.1146	1	−0.0239	−0.8379	−0.0217
	Fluid Unit 2	0.9993	−0.1187	0.9782	−0.0206	0.7153	1
	Male poppet	0.9990	−0.1195	0.9729	−0.0198	1	−0.1558
	Fluid Unit 3	0.9786	−0.2296	−0.1973	0.0005	−0.0020	0
	Fluid Unit 4	0.5182	−0.0628	0.6002	1	−0.0413	0.0002

.

Table 5 shows that the first four natural frequencies of the docking system are below 500 Hz, rendering it susceptible to resonance with the pressure pulsations from the pump, the pressure impulses generated by hydraulic tools, or internal fluid vortices within the system. This can further intensify the fluid pulsations and poppet oscillation, potentially destroying the docking system. Given that the frequency ranges of the aforementioned excitations are typically below 500 Hz, only the first four natural frequencies will be considered in the subsequent analysis of influence factors.

The first-order natural frequency f_n1 of the docking system is 88.7 Hz. The corresponding mode shape reveals that when the excitation frequency approaches f_n1, resonance occurs, leading to noticeable displacement of both poppets and Fluid Unit 3. Based on the dynamic model described in Figure 4, and Equations (4) and (5), it can be concluded that the equivalent displacement x_i of the fluid unit characterizes the magnitude of the pressure fluctuation. The mode shape u_n1 indicates that the large amplitudes observed in both poppets and Fluid Unit 3 suggest that, under the influence of fluid–structure interaction, the poppets may experience significant vibrations. This results in noticeable pressure pulsations in the downstream hydraulic line, potentially impacting the stability of fluid transmission and reducing the operational efficiency of the working tools.

When the excitation frequency reaches the second natural frequency f_n2, the vibration amplitude of both poppets is relatively small, but the upstream pressure pulsation is quite intense. When the excitation frequency reaches the third natural frequency f_n3, the vibration amplitude of both poppets increases significantly, while the pressure pulsations in upstream and downstream hydraulic lines become considerably smaller. When the excitation frequency reaches the fourth natural frequency f_n4, only the pulsation in Fluid Unit 4 is relatively intense. Furthermore, as indicated in Table 5, the effect of fluid–structure interaction causes the originally independent movements of both poppets to become coupled, which exacerbates the wear of the sealing components.

4.2. Influence of Important Parameters on the Natural Frequency

When keeping the docking configuration and the dimensions of the HQCs unchanged, as constrained by the structure of the AHQCD, the equivalent stiffness of the fluid unit and the mechanical spring stiffness have a significant impact on the natural frequencies and mode shapes of the system. The equivalent stiffness of the fluid unit is directly related to both the working pressure and the air content in the hydraulic oil. Therefore, to identify the variation range of natural frequencies where resonance occurs, it is crucial to assess the impact of these factors—namely, working pressure, air content, and spring stiffness—on the natural frequencies of multiple modes within the system.

4.2.1. Effect of Working Pressure on the Natural Frequency

Based on the Nykanen model of bulk modulus of elasticity presented in [32], E_v is fundamentally influenced by the pressure and air content for a specified fluid. Therefore, keeping the air content of the oil constant, the effect of working pressure on the first four natural frequencies of the docking system is illustrated in Figure 6.

Figure 6 demonstrates that the first four natural frequencies of the docking system monotonically increase with a rising working pressure. As the working pressure increases, small, entrained air bubbles in the oil (initially undissolved) gradually dissolve into the fluid. This process enhances the bulk modulus of elasticity E_v, thereby reducing the compressibility of fluid. Consequently, the equivalent stiffness of the fluid increases, leading to a rise in the first four natural frequencies with an increasing working pressure.

4.2.2. Effect of Air Content on the Natural Frequency

In rescue scenarios, rescue vehicles frequently switch between hydraulic tools, leading to the repeated disconnection and reconnection of HQCs, as illustrated in Figure 2. This process inevitably introduces a certain amount of air into the hydraulic system. Keeping the working pressure unchanged, the effect of the air content on the first four natural frequencies of the system is obtained, as shown in Figure 7.

It is indicated in Figure 7 that the first four natural frequencies of the system decrease gradually with an increase in air content. When the working pressure remains constant, the bulk modulus of elasticity E_v of the oil decreases with a rising air content, leading to a reduction in the equivalent stiffness of the fluid unit. Meanwhile, the equivalent mass of the fluid unit continuously decreases. However, since the decrease in equivalent mass is much smaller than the decrease in equivalent stiffness, the first four natural frequencies of the system gradually decline as the air content in the fluid increases. This implies that an increase in the number of connection and disconnection cycles will gradually alter the natural frequencies of the docking system, which could negatively impact performance.

4.2.3. Effect of Spring Stiffness on the Natural Frequency

The stiffness of the springs within two poppets is an important initial design parameter that significantly affects the natural frequencies of the docking system. When the stiffness k₁ of the spring in the female poppet ranges from 2.5 to 20 N/mm, and k₂ ranges from 10 to 70 N/mm, the variation curves of the first four natural frequencies are as shown in Figure 8 and Figure 9. It can be observed that the sensitivities of each mode’s natural frequency to the spring stiffness are different. The first-order and third-order natural frequencies f_n1 and f_n3 gradually increases with the rise in spring stiffness k₁ and k₂, and the growth rate of third-order natural frequency f_n3 is significantly larger. In contrast, spring stiffness has little effect on the second-order and fourth-order natural frequencies.

5. Experimental Verification

5.1. Experimental Set-Up

To validate the accuracy of the modal analysis of the docking system, a platform for testing the HQCs with connecting lines was constructed. A schematic of the platform is shown in Figure 10a, while Figure 10b shows a picture of the hydraulic test platform. A three-plunger pump, driven by an electric motor, is used to provide a constant flow of 60 L/min. A pressure relief valve is used to regulate the pressure of the hydraulic system in the range of 2 to 6 MPa. Pressure sensors PS1, PS2, and PS3 are installed to measure the pressure at the pump outlet, upstream line pressure, and output pressure of the testing HQCs, respectively. A reservoir is mounted on the test platform to reduce pressure pulsation and mechanical vibration from the pump, thereby facilitating the measurement of small-amplitude resonance characteristics. It should be noted that, due to constraints of leakage and environmental protection during the docking process, pure water is used as the working fluid in the experiment. The primary differences in physical parameters between pure water and hydraulic oil are presented in

Table 6. The primary different physical parameters of pure water with hydraulic oil.

Parameter	Value
Bulk modulus of elasticity Ev (MPa)	220
Viscosity μ (N·s/m2)	1 × 10−3
Mass density ρ (kg·m−3)	998

, and all other parameters in the test are set to be identical to those shown in Table 4.

5.2. Experimental Results

The fast Fourier transform was applied to analyze the pressure signal output p₃ from the test system in both the time and frequency domains. The amplitude–frequency curves of the pressure signal of the docking system, influenced by fluid self-pulsation excitation, are shown in Figure 11.

It can be observed from Figure 11 that five relatively prominent peaks are present in the frequency domain, while the amplitude at other positions is small. Here, 6 Hz corresponds to the frequency of the three-plunger pump in the experimental system (at a speed of 621 r/min with a gearbox reduction ratio of 4.45), while 50 Hz is the disturbance frequency of the induced current. The peaks labeled f_n1, f_n2, and f_n3 represent the first-order, second-order, and third-order natural frequencies of the docking system, respectively. However, weak resonance amplitudes of other natural frequencies, combined with noise disturbances caused by fluid turbulence in the HQC, complicate frequency spectrum recognition. Consequently, the natural frequencies of other modes cannot currently be identified experimentally.

The first three natural frequencies of the docking system, as determined from the experimental tests, are compared with the theoretical model in Figure 12. As illustrated in Figure 12, the natural frequencies calculated by the proposed theoretical model are basically consistent with the experimental results, and these natural frequencies increase with the increase in pressure p₃.

Errors compared with the experimental results can be calculated, as shown in

Table 7. Comparison of experimental data with theoretical values.

Order	Pressure Conditions	Experimental Data (Hz)	Proposed Model (Hz)	Error (%)
fn1	2.51	61	62.5	2.4	Avg. 2.7
	4.23	80	82.2	2.8
	6.29	92	94.7	2.9
fn2	2.51	67	70.0	4.5	Avg. 5.8
	4.23	90	94.8	5.3
	6.29	103	110.9	7.6
fn3	2.51	130	136.2	4.8	Avg. 6.7
	4.23	178	190.4	6.9
	6.29	205	222.4	8.4

. The average errors of the first-order f_n1, second-order f_n2, and third-order f_n3 are 2.7%, 5.8%, and 6.7%, respectively. Due to the neglect of elastic deformation, pressure loss, and fluid viscosity in the hydraulic lines in the proposed theoretical model, the experimental data are consistently lower than the theoretical results. However, both the magnitudes and trends exhibit good agreement, thereby verifying the accuracy of the proposed model.

6. Conclusions

When rescue vehicles utilize an AHQCD to connect and operate working tools, as shown in Appendix A, pressure pulsations of multiple frequencies occur in the hydraulic lines. These pulsations may coincide with the natural frequency of the HQCs, potentially resulting in resonance and ultimately leading to failures in fluid transport. To avoid resonance issues, a comprehensive dynamic model of the HQCs with connecting hoses was developed in this study, which can be used to identify the natural frequency of the system. The proposed model was verified through experiments. Additionally, the impacts of working pressure, air content, and spring stiffness on natural frequencies were discussed. The following conclusions can be drawn:

• Considering the fluid–structure interaction within the HQC system, the dynamic model and 6-DOF equations were developed based on the lumped parameter method. The equivalent stiffness of the fluid units was derived and quantitatively calculated based on the compressibility of the fluid and the configuration of the HQC system.
• A free vibration analysis of the docking system was conducted. The natural frequencies for each mode, along with their corresponding mode shapes under given conditions, were obtained. The first four natural frequencies below 500 Hz were primarily analyzed, and the characteristics of poppet oscillation and pressure pulsation were thoroughly discussed under resonance conditions based on the corresponding mode shapes. Additionally, it was found that the natural frequencies increase with an increase in working pressure and a decrease in air content, while the spring stiffness only affects the first- and third-order natural frequencies.
• An experimental test platform for the dynamic model of the HQC system was constructed, and the theoretical model results were compared with experimental data based on the frequency spectrum of the pressure signal. It was found that the error in first-order natural frequency was only 2.7%, while the errors in the second-order and third-order natural frequencies were 5.8% and 6.7%, respectively. These errors may be attributed to the neglect of elastic deformation in hydraulic lines. However, both results exhibit good agreement in terms of magnitudes and trends, confirming the accuracy of the proposed model.

The limitations of this study include its inability to account for the time-varying equivalent stiffness of the fluid due to pressure pulsations. More accurate analytical models will be adopted to describe the fluid–structure interaction in complex hydraulic pipelines. The added coupled coefficients of the fluids will be determined to further investigate the dynamic behaviors in the docking system. Future research will consider the effects of fluid pressure pulsations in the HQC system and develop a comprehensive parametric vibration model based on the time-varying fluid stiffness, which may lead to main resonances and combined resonances that seriously deteriorate the dynamic behavior of the system. Further investigation will optimize the design parameters under various working conditions to improve the stability of fluid transmission.