1 Introduction

To guarantee a smooth and accurate performance of the actuator, hydraulic systems are typically designed to operate within a specified pressure range. Relief valves are frequently used in hydraulic systems to limit the maximum pressure in a system or to prevent hydraulic circuits from overloading [1]. In general, relief valves can be categorized into direct and pilot types. The pilot type is a two-stage structure invented by Vickers in 1931 [2]. Owing to their effective pressure control properties, pilot-operated relief valves are used extensively in hydraulic control systems featuring high pressures and large flows. However, during operation, relief valves occasionally vibrate, thus causing system pressure fluctuations and severely reducing the safety, stability, and reliability of the hydraulic system [3].

Several analyses from various perspectives have been performed to clarify the instability and vibration of relief valves. One approach is to conduct various studies to determine the causes of instability. Hayashi investigated the stability of single-stage poppet valves, and the results indicated that various factors, such as the effects of the valve poppet and seat, and the hysteresis of the transient hydrodynamic force, might cause the poppet valve to destabilize vibrations [4]. Additionally, some studies indicate that the stability of relief valves can be affected by the pre-compression shrinkage of springs [5], half cone angle [6], dam** orifice size [7], flow rate in valve chambers [8], spool dam** coefficient [9], valve orifice diameter [10], upstream and downstream pipelines [11, 12], cavitation [13,14,15,16], impact between the spool and seat [17], and valve chamber volume [18]. In addition, some scholars established a fluid–structure interaction model of the relief valve and investigated its stability [19, 20]. Meanwhile, some scholars investigated the effects of various parameters on the stability of the valve via theoretical analysis and simulations [21,22,23], whereas others have experimentally obtained the conditions for the steady operation of the relief valve [24].

Numerous strategies have been proposed to increase the stability of relief valves. Merrit placed a fixed office between a pressure-controlled chamber and valve port and discovered that reasonably matching the fixed office with the sensitive chamber volume can increase the relief valve stability [25]. Moreover, designing a buffer structure at the end of the valve poppet can enhance its stability [26].

Although the stability of relief valves has been investigated extensively, quantitative agreement between predicted and measured stabilities has not been achieved because parameters such as dam** and friction coefficients cannot be easily calculated accurately; therefore, the orifice size can only be confirmed through extensive experiments. However, because hydraulic technology is develo** toward miniaturization, intelligence, and high performance, the relevant theories must be supplemented urgently.

In this study, the stability of a cartridge-type pilot-operated relief valve is investigated via frequency-domain analysis. In addition, a Bode diagram is used to assess elements that are different between the original and contrast models. Finally, theoretical conclusions are presented based on numerical simulation and experimental results.

2 Mathematical Model of Pilot-operated Relief Valve

2.1 Description of Pilot-operated Relief Valve

Generally, a pilot-operated relief valve comprises a main valve and a pilot valve. Figure 1 shows the structural diagram of a cartridge-type antivibration pilot-operated relief valve, and its schematic diagram is illustrated in Figure 2. The system pressure ps is transmitted to the sensitive chamber C through the sharp-edged orifice R1 and then transmitted to the pressure-sensing chamber B through the dynamic pressure feedback orifice R2, which is sensed by the pilot valve. If ps does not exceed the cracking pressure, then the pilot valve is closed by the preload spring force. Based on Pascal’s law, ps = pb = pc; thus, the main poppet is hydraulically balanced. However, it is also maintained in the seat by a pre-loaded spring force. Any positive deviation from the reference value causes the pilot port to open and flow, thus causing the main poppet to be unbalanced due to the pressure difference Δp between chambers A and B. Consequently, the main poppet is lifted and relieves the system flow from chamber A to the tank.

Figure 1
figure 1

Structure diagram of pilot-operated relief valve

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Figure 2
figure 2

Schematic diagram of pilot-operated relief valve

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The following assumptions were introduced to derive a mathematical model for the relief valve:

  1. 1.

    The bulk modulus of the fluid is constant.

  2. 2.

    The valve outlet pressure is equal to the tank pressure.

  3. 3.

    Perturbations are minimal, which allows the mathematical model to be linearized at the rated operating point.

  4. 4.

    Leakage around the main poppet is negligible.

2.2 Static Characteristics

Under steady state, the physical model of the pilot-operated relief valve is described as follows:

The steady flow passing through the relief valve is expressed as

$$Q_{{\text{p}}} - Q_{{\text{L}}} = Q_{x} + Q_{y} ,$$
(1)

where Qp is the supply flow rate, QL the flow rate to the load, Qx the flow rate at the pilot port, and Qy the flow rate at the main port.

The mechanical equilibrium equation of the main poppet is

$$A_{1} \left( {p_{{\text{s}}} - p_{{\text{b}}} } \right) = k_{1} \left( {y + y_{0} } \right) + F_{{{\text{s1}}}} ,$$
(2)

where A1 is the cross-sectional area of the main poppet; ps is the pressure in chamber A; pb is the pressure in chamber B; k1 is the main spring stiffness; y is the main valve displacement; and y0 is the constant pre-compression of the main spring. Meanwhile, Fs1 is the steady flow force of the main valve, expressed as \(F_{{{\text{s}}1}} = \rho Q_{y} v_{y} \cos \alpha = C_{{{\text{d}}1}} C_{{{\text{v}}1}} {\uppi }d_{{1}} \sin \left( {2\alpha } \right)yp_{{\text{s}}}\), where ρ is the fluid density, vy the main port flow velocity, α the half-angle of the main valve, Cd1 and Cv1 are the discharge and velocity coefficients of the main exit port, respectively, and d1 the diameter of the main exit port.

The flow rate passing through the main exit port can be represented by the following well-established relationship:

$$Q_{y} = C_{{{\text{d1}}}} A_{y} \sqrt {\frac{{2p_{{\text{s}}} }}{\rho }} ,$$
(3)

where Ay denotes the cross-sectional area of the main port. Here, \(A_{y} = {\uppi }d_{1} y\sin \alpha \left( {1 - \frac{y}{{2d_{1} }}\sin 2\alpha } \right)\), and because y << d1, Ay can be approximated as \(A_{y} = {\uppi }d_{1} y\sin \alpha\).

The flow rate passing through the sharp-edged orifice R1 is expressed as

$$Q_{{{\text{r1}}}} = C_{{{\text{r1}}}} \frac{{{\uppi }d_{{{\text{r1}}}}^{2} }}{4}\sqrt {\frac{{2\left( {p_{{\text{s}}} - p_{{\text{c}}} } \right)}}{\rho }},$$
(4)

where Cr1 is the discharge coefficient of orifice R1, dr1 the diameter of orifice R1, and pc the pressure in chamber C.

The mechanical equilibrium equation of the pilot poppet is

$$A_{{2}} p_{{\text{c}}} = k_{{2}} (x + x_{0} ) + F_{{{\text{s2}}}},$$
(5)

where A2 is the cross-sectional area of the pilot poppet, k2 the pilot spring stiffness, x the pilot valve displacement, and x0 the pre-compression constant of the pilot spring. Fs2 is the steady flow force of pilot valve, expressed as \(F_{{{\text{s2}}}} = \rho Q_{x} v_{x} \cos \beta = C_{{{\text{d2}}}} C_{{{\text{v2}}}} {\uppi }d_{2} \sin \left( {2\beta } \right)xp_{{\text{b}}}\), where vx is the pilot port flow velocity, β the half-angle of the pilot poppet, Cd2 and Cv2 are the discharge coefficient and velocity coefficient of pilot port, respectively, and d2 is the diameter of the pilot port.

The flow rate passing through the dynamic pressure feedback orifice R2 is expressed as

$$Q_{{{\text{r}}2}} = \frac{{{\uppi }d_{{{\text{r}}2}}^{4} }}{{128\mu l_{{{\text{r2}}}} }}\left( {p_{{\text{b}}} - p_{{\text{c}}} } \right),$$
(6)

where dr2 and lr2 are the diameter and length of orifice R2, respectively; μ is the dynamic viscosity of the fluid; and pb = pc in steady state.

The flow rate passing through the pilot port is expressed as

$$Q_{x} = C_{{{\text{d2}}}} A_{x} \sqrt {\frac{{2p_{{\text{c}}} }}{\rho }},$$
(7)

where Ax is the cross-sectional area of the pilot port. Here, \(A_{x} = {\uppi }d_{2} x\sin \beta \left( {1 - \frac{x}{{2d_{2} }}\sin 2\beta } \right)\), and because x << d2, Ax can be approximated as \(A_{x} = {\uppi }d_{2} x\sin \beta\).

2.3 Dynamic Mathematical Model

In the transient state, the mathematical model of the pilot-operated relief valve is as follows:

As shown in Figure 2, the continuity equation applied to chamber A of volume VA yields

$$Q_{{\text{p}}} - Q_{{\text{L}}} - Q_{{{\text{r1}}}} - Q_{y} - \frac{{V_{{\text{A}}} }}{E}\frac{{{\text{d}}p_{{\text{s}}} }}{{{\text{d}}t}} - A_{1} \frac{{{\text{d}}y}}{{{\text{d}}t}} = 0,$$
(8)

where E is the bulk stiffness of the fluid.

The mechanical equilibrium equation of the main poppet is

$$m_{{1}} \frac{{{\text{d}}^{2} y}}{{{\text{d}}t^{2} }} = A_{{1}} (p_{{\text{s}}} - p_{{\text{b}}} ) - k_{{1}} (y + y_{0} ) - F_{{{\text{s}}1}} - f_{{1}} \frac{{{\text{d}}y}}{{{\text{d}}t}},$$
(9)

where m1 is the effective main poppet mass (including 1/3 of the spring mass); and f1 is the viscous dam** coefficient of the main poppet, which is expressed as

$$f_{1} = f_{{{\text{m1}}}} + f_{{{\text{t1}}}} ,$$
(10)

where, fm1 and ft1 are expressed as

$$f_{{{\text{m1}}}} = \frac{{\mu A_{{\text{m}}} }}{{\delta_{1} }},$$
(11)
$$f_{{{\text{t1}}}} = C_{{{\text{d1}}}} {\uppi }d_{1} l_{1} \sin (\alpha )\sqrt {2\rho p_{{\text{s}}} } ,$$
(12)

where, Am is the equivalent wetting area of the main poppet; δ1 the clearance between the main poppet and valve body; and l1 the dam** length of the main poppet.

The flow continuity equation applied to chamber B of volume VB can be written as

$$A_{1} \frac{{{\text{d}}y}}{{{\text{d}}t}} - Q_{{{\text{r2}}}} - \frac{{V_{{\text{B}}} }}{E}\frac{{{\text{d}}p_{{\text{b}}} }}{{{\text{d}}t}} = 0.$$
(13)

The continuity equation applied to chamber C of volume VC yields

$$Q_{{{\text{r1}}}} + Q_{{{\text{r2}}}} - Q_{x} - A_{2} \frac{{{\text{d}}x}}{{{\text{d}}t}} - \frac{{V_{{\text{C}}} }}{E}\frac{{{\text{d}}p_{{\text{c}}} }}{{{\text{d}}t}} = 0.$$
(14)

The mechanical equilibrium equation of the pilot poppet is

$$m_{{2}} \frac{{{\text{d}}^{2} x}}{{{\text{d}}t^{2} }} = A_{{2}} p_{{\text{c}}} - k_{{2}} \left( {x + x_{{0}} } \right) - F_{{{\text{s}}2}} - f_{2} \frac{{{\text{d}}x}}{{{\text{d}}t}},$$
(15)

where m2 is the effective pilot poppet mass (including 1/3 of the spring mass); and f2 is the viscous dam** coefficient of the pilot poppet, which is expressed as

$$f_{2} = f_{{{\text{m2}}}} + f_{{{\text{t2}}}} ,$$
(16)

where fm2 and ft2 are expressed as

$$f_{{{\text{m2}}}} = \mu \frac{{A_{{\text{p}}} }}{{\delta_{2} }},$$
(17)
$$f_{{{\text{t2}}}} = C_{{{\text{d2}}}} {\uppi }d_{2} l_{2} \sin (\beta )\sqrt {2\rho p_{{\text{c}}} } ,$$
(18)

where, Ap is the equivalent wetting area of the pilot poppet, δ2 the clearance between the pilot poppet and valve body, and l2 the dam** length of the pilot poppet.

2.4 Linearization Analysis

Directly analyzing a higher-order complex system using classical control theory is challenging. Linearization is a widely used mathematical approach in mechanical engineering. Consider a slight perturbation near the rated operating point. Subsequently, using the Laplace transform of Eqs. (3), (4), (6)–(9), and (13)–(15), the following mathematical model can be obtained:

$$Q_{y} = K_{{\text{A}}} p_{{\text{s}}} + K_{{\text{B}}} y,$$
(19)
$$Q_{{{\text{r}}1}} = K_{{\text{C}}} \left( {p_{{\text{s}}} - p_{{\text{c}}} } \right),$$
(20)
$$Q_{{{\text{r2}}}} = G_{{\text{r}}} \left( {p_{{\text{b}}} - p_{{\text{c}}} } \right),$$
(21)
$$Q_{x} = K_{{\text{E}}} p_{{\text{c}}} + K_{{\text{F}}} x,$$
(22)
$$Q_{{\text{p}}} - Q_{{\text{L}}} - Q_{y} - Q_{{{\text{r1}}}} - \frac{{V_{{\text{A}}} }}{E}p_{{\text{s}}} s - A_{1} ys = 0,$$
(23)
$$\left( {A_{{1}} - K_{{\text{D}}} y_{{\text{x}}} } \right)p_{{\text{s}}} - A_{{1}} p_{{\text{b}}} = \left( {m_{{1}} s^{2} + f_{1} s + k_{{1}} + K_{{\text{D}}} p_{{{\text{sx}}}} } \right)y,$$
(24)
$$A_{1} ys - Q_{{{\text{r2}}}} - \frac{{V_{{\text{B}}} }}{E}p_{{\text{b}}} s = 0,$$
(25)
$$Q_{{{\text{r1}}}} + Q_{{{\text{r2}}}} - Q_{x} - A_{2} xs - \frac{{V_{{\text{C}}} }}{E}p_{{\text{c}}} s = 0,$$
(26)
$$\left( {A_{{2}} - K_{{\text{G}}} x_{{\text{x}}} } \right)p_{{\text{c}}} = \left( {m_{{2}} s^{2} + f_{2} s + k_{2} + K_{{\text{G}}} p_{{{\text{cx}}}} } \right)x,$$
(27)

where, yx and xx are the main and pilot valve displacements under the rated operating point, respectively; and psx and pcx are the pressures in chambers A and C under the rated operating point, respectively.

The physical meanings and expressions of Kn are shown in Table 1.

Table 1 Physical meanings and expressions of Kn
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3 Contrast Model

To analyze the effect of the dynamic pressure feedback orifice R2, a pilot-operated relief valve without a dynamic pressure feedback orifice is proposed as a contrast model, as shown in Figures 3 and 4.

Figure 3
figure 3

Structure diagram of contrast model

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Figure 4
figure 4

Schematic diagram of contrast model

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4 Theoretical Analysis

4.1 System Block Diagram

In this section, the effect of the dynamic pressure feedback orifice is analyzed using the frequency-domain method. Using the flow rate at inlet Qp-QL as the input and the pressure of chamber A, ps, as the output, the system block diagrams of the original and contrast models are developed, as shown in Figures 5 and 6, respectively.

Figure 5
figure 5

System block diagram of original model

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Figure 6
figure 6

System block diagram of contrast model

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The physical meanings and expressions of ωn are presented in Table 2. A comparison of Figures 5 and 6 reveals the discrepancies between the original and contrast models. Circuits 1 and 3 contain the same components, which indicate the transfer function of the main valve mass–spring vibration system. Circuit 2 depicts the transfer function of pilot valve pressure control chamber C. Circuits 1, 2, and 3 of the original model have one more first-order inertial element than those of the contrast model. According to the classical control theory, for a system comprising a first-order inertial element and an oscillation element, the dynamic characteristics are primarily dominated by the first-order inertial element if the break frequency of the first-order inertial element is much lower than that of the oscillation element. The hysteresis of the first-order inertial element can counterbalance the lead effect of the oscillation element, thereby improving the stability of the subsystems. This may be beneficial to the global stability of the pilot-operated relief valve.

Table 2 Physical meanings and expressions of ωn
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The complete list of parameters used in the investigation is presented in Table 3.

Table 3 Main parameters used in current study
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Solving Eqs. (1)–(7) allows one to determine the steady-state operating point of the relief valve at a flow rate of 90 L/min, as listed in Table 4.

Table 4 Steady-state operating point
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4.2 Dynamic Characteristics of Pilot Valve Subsystem

For the original model, the system transfer function of the pilot valve subsystem can be expressed as

$$G_{{\text{pilot - o}}} \left( s \right) = \frac{{\frac{1}{{K_{{\text{C}}} + K_{{\text{E}}} }}}}{{\left( {\frac{s}{{\omega_{4} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right) + \frac{s}{{\omega_{9} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}}} \times \frac{{\frac{{K_{{\text{F}}} \left( {A_{{2}} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} }}\left( {\frac{s}{{\omega_{6} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} }}{{\omega_{2} }}s + 1}},$$
(28)

where Km2 is the equivalent mechanical spring stiffness of the main valve and Km2 = k2+KGpcx. Because VB >> VC, ω4 >> ω7, Eq. (28) can be approximated using Eq. (29).

$$\begin{gathered} G_{{\text{pilot - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{F}}} \left( {A_{2} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}\left( {\frac{s}{{\omega_{6} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left[ {\left( {\frac{s}{{\omega_{4} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right) + \frac{s}{{\omega_{9} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}} \right]\left( {\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} s}}{{\omega_{2} }} + 1} \right)}}, \\ \Downarrow \\ G_{{\text{pilot - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{F}}} \left( {A_{2} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}\left( {\frac{s}{{\omega_{6} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left[ {\left( {\frac{s}{{\omega_{7} }} + 1} \right) + \frac{s}{{\omega_{9} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}} \right]\left( {\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} s}}{{\omega_{2} }} + 1} \right)}}, \\ \end{gathered}$$
$$\begin{gathered} \Downarrow \\ G_{{\text{pilot - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{F}}} \left( {A_{2} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}\left( {\frac{s}{{\omega_{6} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left[ {\left( {\frac{{V_{{\text{B}}} }}{{EG_{{\text{r}}} }} + \frac{{V_{{\text{B}}} }}{{E\left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}} \right)s + 1} \right]\left( {\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} s}}{{\omega_{2} }} + 1} \right)}}, \\ \Downarrow \\ G_{{\text{pilot - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{F}}} \left( {A_{2} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}\left( {\frac{s}{{\omega_{6} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left[ {\left( {\frac{{V_{{\text{B}}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right) + G_{{\text{r}}} V_{{\text{B}}} }}{{EG_{{\text{r}}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}} \right)s + 1} \right]\left( {\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} s}}{{\omega_{2} }} + 1} \right)}}, \\ \end{gathered}$$
$$\Downarrow$$
$$G_{{\text{pilot - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{F}}} \left( {A_{2} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}\left( {\frac{s}{{\omega_{6} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left( {\frac{s}{{\omega_{{\text{a}}} }} + 1} \right)\left( {\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} s}}{{\omega_{2} }} + 1} \right)}},$$
(29)
$$\omega_{{\text{a}}} = \frac{{EG_{{\text{r}}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}{{V_{{\text{B}}} \left( {G_{{\text{r}}} + K_{{\text{C}}} + K_{{\text{E}}} } \right)}}$$
(30)

where ωa is the break frequency of the first-order inertial element related to orifice R2 as well as chambers B and C in the original model.

For the contrast model, the system transfer function of the pilot valve subsystem can be expressed as follows:

$$G_{{\text{pilot - c}}} \left( s \right) = \frac{{\frac{{K_{{\text{F}}} \left( {A_{2} - K_{{\text{G}}} x_{{\text{x}}} } \right)}}{{K_{{{\text{m}}2}} \left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}\left( {\frac{s}{{\omega_{6} }} + 1} \right)}}{{\left( {\frac{s}{{\omega_{4} }} + 1} \right)\left( {\frac{{s^{2} }}{{\omega_{2}^{2} }} + \frac{{2\zeta_{2} s}}{{\omega_{2} }} + 1} \right)}}$$
(31)

Based on comparison, Gpilot-o(s) has one more high-frequency first-order differential element than Gpilot-c(s). However, the value of ω7 is high; therefore, the effect of the first-order differential on the stability and rapidity of Gpilot-o(s) is negligible. The Bode diagrams of the pilot valve subsystem are shown in Figure 7.

Figure 7
figure 7

Bode diagrams of Gpilot-o(s) and Gpilot-c(s)

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4.2.1 Stability, Rapidity, and Immunity of Pilot Valve Subsystem

As shown in Figure 7, Gpilot-o(s) and Gpilot-c(s) do not differ significantly within the operating frequency range of the relief valve unless the value of ωa is sufficiently small that the delay of the first-order inertial element can significantly improve the stability of the pilot valve.

Similar to the stability analysis above, the rapidity of Gpilot-o(s) or Gpilot-c(s) does not differ significantly in the operating frequency range of the relief valve. However, if ωa is sufficiently small, then the delay in the first-order inertial element may reduce the rapidity of the pilot valve.

As shown in Figure 7, Gpilot-o(s) has one more high-frequency first-order differential element than Gpilot-c(s), which implies that the immunity of Gpilot-o(s) is lower than that of Gpilot-c(s). Additionally, similar to the observation that ω7 decreases with the size of orifice R2, the immunity of Gpilot-o(s) decreases with the size of R2 as well.

4.3 Dynamic Characteristics of Main Valve Subsystem

In the original model, the system transfer function of the main valve subsystem is expressed as follows:

$$G_{{\text{main - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{B}}} \left( {A_{1} - K_{{\text{D}}} y_{{\text{x}}} } \right)}}{{K_{{{\text{m1}}}} \left( {K_{{\text{A}}} + K_{{\text{C}}} } \right)}}\left( {\frac{s}{{\omega_{5} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left( {\frac{s}{{\omega_{3} }} + 1} \right)\left[ {\left( {\frac{{s^{2} }}{{\omega_{1}^{2} }} + \frac{{2\zeta_{1} }}{{\omega_{1} }}s + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right) + \frac{s}{{\omega_{{\text{c}}} }}} \right]}},$$
(32)
$$\omega_{{\text{c}}} = \frac{{G_{{\text{r}}} \left( {k_{1} + K_{{\text{D}}} p_{{{\text{sx}}}} } \right)}}{{A_{1}^{2} }},$$
(33)

where Km1 is the equivalent mechanical spring stiffness of the main valve, expressed as Km1 = k1+KDpsx; and ωc is the equivalent break frequency produced by the dynamic pressure feedback orifice R2 and main spring.

For the contrast model, the system transfer function of the main valve subsystem can be expressed as follows:

$$G_{{\text{main - c}}} \left( s \right) = \frac{{\frac{{K_{{\text{B}}} \left( {A_{1} - K_{{\text{D}}} y_{{\text{x}}} } \right)}}{{K_{{{\text{m1}}}} \left( {K_{{\text{A}}} + K_{{\text{C}}} } \right)}}\left( {\frac{s}{{\omega_{5} }} + 1} \right)}}{{\left( {\frac{s}{{\omega_{3} }} + 1} \right)\left( {\frac{{s^{2} }}{{\omega_{1}^{2} }} + \frac{{2\zeta_{1} }}{{\omega_{1} }}s + 1} \right)}}.$$
(34)

Figure 8 shows the Bode diagrams of the main valve subsystem. As shown in Figure 8, the dynamic characteristics of Gmain-o(s) differ from those of Gmain-c(s). Gmain-o(s) is more stable than Gmain-c(s). By contrast, Gmain-c(s) is more rapid than Gmain-o(s). In addition, the stability of Gmain-o(s) illustrates different states, depending on the orifice R2 aperture value. For further analysis, dimensional normalization was performed.

Figure 8
figure 8

Bode diagrams of Gmain-o(s) and Gmain-c(s)

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4.3.1 Normalization Analysis

By disregarding the minor effect of ζ1, the denominator term in Eq. (32) can be approximated using Eq. (35).

$$\left( {\frac{{s^{3} }}{{\omega_{1}^{2} \omega_{7} }} + \frac{{s^{2} }}{{\omega_{1}^{2} }} + \frac{s}{{\omega_{7} }} + \frac{s}{{\omega_{{\text{c}}} }} + 1} \right) \times \left( {\frac{s}{{\omega_{3} }} + 1} \right).$$
(35)

Subsequently, Eq. (35) is converted into the product of two linear terms and a quadratic term, as shown in Eq. (36).

$$\left( {\frac{s}{{\omega_{{\upalpha }} }} + 1} \right)\left( {\frac{{s^{2} }}{{\omega_{{\upbeta }}^{2} }} + \frac{{2\zeta_{{\upbeta }} }}{{\omega_{{\upbeta }} }}s + 1} \right) \times \left( {\frac{s}{{\omega_{3} }} + 1} \right),$$
(36)

where ωα is the break frequency of the first-order inertia element; and ωβ and ζβ are the resonance frequency and dam** ratio of the oscillation element, respectively. However, the values of ωα, ωβ, and ζβ cannot be obtained easily using a simple analytical method; hence, dimensional normalization was performed.

The assumptions introduced are as follows:

$$\omega_{{\upalpha }} \omega_{{\upbeta }}^{2} = \omega_{7} \omega_{1}^{2} ,$$
(37)
$$\frac{{2\zeta_{{\upbeta }} }}{{\omega_{{\upbeta }} \omega_{{\upalpha }} }} + \frac{1}{{\omega_{{\upbeta }}^{2} }} = \frac{1}{{\omega_{1}^{2} }},$$
(38)
$$\frac{1}{{\omega_{{\upalpha }} }} + \frac{{2\zeta_{{\upbeta }} }}{{\omega_{{\upbeta }} }} = \frac{1}{{\omega_{7} }} + \frac{1}{{\omega_{{\text{c}}} }}.$$
(39)

Subsequently, the dimensionless transformation of Eqs. (37) to (39) yields

$$\frac{{\omega_{{\text{c}}} }}{{\omega_{1} }} = \sqrt {\frac{{\left( {\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }} + 1} \right)\frac{{\omega_{{\upalpha }} }}{{\omega_{10} }} - \frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }}}}{{\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }}\left( {\frac{{\omega_{{\upalpha }} }}{{\omega_{10} }}} \right)^{2} - \left( {\frac{{\omega_{{\upalpha }} }}{{\omega_{10} }}} \right)^{3} }}} ,$$
(40)
$$\frac{{\omega_{{\text{c}}} }}{{\omega_{1} }} = \sqrt {\frac{{\left( {\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }} + 1} \right)\left( {\frac{{\omega_{{\upbeta }} }}{{\omega_{{{\text{h1}}}} }}} \right)^{4} - \frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }}\left( {\frac{{\omega_{{\upbeta }} }}{{\omega_{{{\text{h1}}}} }}} \right)^{6} }}{{\left( {\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }}\left( {\frac{{\omega_{{\upbeta }} }}{{\omega_{{{\text{h1}}}} }}} \right)^{2} - 1} \right)}}} ,$$
(41)

where Kh1 is the equivalent spring stiffness of the liquid in chamber B, which is expressed as \(K_{{{\text{h1}}}} = \frac{{EA_{1}^{2} }}{{V_{{\text{B}}} }}\). Because VB is small, Kh1 is large, and \(K_{{{\text{h1}}}} \gg K_{{{\text{m1}}}}\), ωh1 is the natural frequency of the mass–spring vibration system formed by the main poppet and the liquid spring of chamber B, which is expressed as \(\omega_{{{\text{h1}}}} = \frac{{K_{{{\text{h1}}}} }}{{m_{1} }}\).

The real root normalization curves of Eqs. (40) and (41) are shown in Figures 9 and 10, respectively. Because \(K_{{{\text{h1}}}} \gg K_{{{\text{m1}}}}\) and \(\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }} \gg 1\), the stability of the main valve subsystem exhibits different states as the ratio of \(\frac{{\omega_{{\text{c}}} }}{{\omega_{1} }}\) changes. Next, the following conditions are analyzed:

Figure 9
figure 9

Real root normalization curve of Eq. (40)

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Figure 10
figure 10

Real root normalization curve of Eq. (41)

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Condition I: If \(\frac{{\omega_{{\text{c}}} }}{{\omega_{1} }} \ge 0.5\), then

$$\frac{{\omega_{{\upalpha }} }}{{\omega_{{\text{c}}} }} = \frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }} = \frac{{\omega_{7} }}{{\omega_{{\text{c}}} }} \Rightarrow \omega_{{\upalpha }} = \omega_{{7}} ,$$
(42)
$$\omega_{{\upbeta }} = \omega_{1} \sqrt {\frac{{\omega_{{7}} }}{{\omega_{{\upalpha }} }}} \Rightarrow \omega_{{\upbeta }} = \omega_{1} ,$$
(43)
$$\frac{1}{{\omega_{{\upalpha }} }} + \frac{{2\zeta_{{\upbeta }} }}{{\omega_{{\upbeta }} }} = \frac{1}{{\omega_{7} }} + \frac{1}{{\omega_{{\text{c}}} }} \Rightarrow \zeta_{{\upbeta }} = \frac{{\omega_{1} }}{{2\omega_{{\text{c}}} }}.$$
(44)

In this case, Eq. (32) can be approximated as follows:

$$G_{{\text{main - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{B}}} \left( {A_{1} - K_{{\text{D}}} y_{{\text{x}}} } \right)}}{{\left( {K_{{\text{A}}} + K_{{\text{C}}} } \right)K_{{{\text{m1}}}} }}\left( {\frac{s}{{\omega_{5} }} + 1} \right)}}{{\left( {\frac{s}{{\omega_{3} }} + 1} \right)\left( {\frac{{s^{2} }}{{\omega_{1}^{2} }} + \frac{1}{{\omega_{{\text{c}}} }}s + 1} \right)}}.$$
(45)

Condition II: If \(\frac{{\omega_{{\text{c}}} }}{{\omega_{1} }} \le 2\sqrt {\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }}}\), then

$$\frac{{\omega_{{\upalpha }} }}{{\omega_{{\text{c}}} }} \to 1 \Rightarrow \omega_{{\upalpha }} = \omega_{{\text{c}}} ,$$
(46)
$$\omega_{{\upbeta }} = \omega_{1} \sqrt {\frac{{\omega_{{7}} }}{{\omega_{{\upalpha }} }}} = \omega_{1} \sqrt {\frac{{\omega_{7} }}{{\omega_{{\text{c}}} }}} = \omega_{1} \sqrt {\frac{{K_{{{\text{h1}}}} }}{{K_{{{\text{m1}}}} }}} \Rightarrow \omega_{{\upbeta }} = \omega_{{{\text{h1}}}} ,$$
(47)
$$\frac{{2\zeta_{{\upbeta }} }}{{\omega_{{\upbeta }} }} + \frac{1}{{\omega_{{\upalpha }} }} = \frac{1}{{\omega_{{7}} }} + \frac{1}{{\omega_{{\text{c}}} }} \Rightarrow \zeta_{{\upbeta }} = \frac{{\omega_{{{\text{h1}}}} }}{{2\omega_{7} }}.$$
(48)

In this case, Eq. (32) can be transformed into

$$G_{{\text{main - o}}} \left( s \right) = \frac{{\frac{{K_{{\text{B}}} \left( {A_{1} - K_{{\text{D}}} y_{{\text{x}}} } \right)}}{{\left( {K_{{\text{A}}} + K_{{\text{C}}} } \right)K_{{{\text{m1}}}} }}\left( {\frac{s}{{\omega_{5} }} + 1} \right)\left( {\frac{s}{{\omega_{7} }} + 1} \right)}}{{\left( {\frac{s}{{\omega_{3} }} + 1} \right)\left( {\frac{{s^{2} }}{{\omega_{{{\text{h1}}}}^{2} }} + \frac{1}{{\omega_{7} }}s + 1} \right)\left( {\frac{s}{{\omega_{{\text{c}}} }} + 1} \right)}}.$$
(49)

4.3.2 Stability, Rapidity, and Immunity of Main Valve Subsystem Under Condition I

Figure 11 shows the Bode diagrams of the main valve subsystem under Condition I. Gmain-o(s) and Gmain-c(s) did not differ significantly when the aperture of R2 was relatively large, except for the resonant peak of Gmain-c(s), which was slightly higher than that of Gmain-o(s). Therefore, the dam** ratio of Gmain-o(s) was slightly larger than that of Gmain-c(s). In this case, the stability of the main valve subsystem might not improve significantly. Similarly, the rapidity and immunity of Gpilot-o(s) and Gpilot-c(s) did not differ significantly.

Figure 11
figure 11

Bode diagrams of Gmain-o(s) and Gmain-c(s) under Condition I

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4.3.3 Stability, Rapidity, and Immunity of Main Valve Subsystem Under Condition II

Figure 12 shows the Bode diagrams of the main valve subsystem under Condition II. Some differences were indicated between the Gmain-o(s) and Gmain-c(s). First, the resonance frequency of the oscillation element of Gmain-o(s) was much higher than that of Gmain-c(s). Second, the dam** ratio of Gmain-o(s) was larger than that of Gmain-c(s). In addition, Gmain-o(s) had one more energy storage element (first-order inertial element) and one more high-frequency first-order differential element than Gmain-c(s). Similar to the previous discussion (Section 4.2), the value of ω7 was high; therefore, the effect of the first-order differential on the stability and rapidity of Gmain -o(s) is negligible. As shown in Figure 12, if the break frequency (ωc) of this additional energy-storage element is sufficiently small, then its delay can effectively improve the stability of the main valve. Contrary to the stability analysis above, if ωc is sufficiently small, then the delay of the energy-storage element may reduce the rapidity of the main valve. Notably, the immunity between Gmain-o(s) and Gmain-c(s) did not differ significantly because the delay of the energy-storage element effectively counteracted the lead of the first-order differential element. Therefore, Gmain-o(s) and Gmain-c(s) exhibited similar immunity.

Figure 12
figure 12

Bode diagrams of Gmain-o(s) and Gmain-c(s) under Condition II

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Based on the analysis above, one can conclude that if the diameter of R2 is sufficiently small, then a dynamic pressure feedback orifice designed between the main valve and pilot valve may effectively improve the local stability of the hydraulic pilot-operated relief valve, which contributes positively to the global stability of the entire valve. By contrast, the rapidity and immunity of the hydraulic pilot-operated relief valve may be weakened to some extent.

5 Simulation and Experimental Verification

The accuracy of the conclusions inferred the previous section was verified through numerical simulations and measurements.

5.1 Numerical Simulation Model

A dynamic numerical simulation model of a relief valve was developed using MATLAB, and the nonlinearities were considered using appropriate Simulink blocks. For an accurate computation, solver “ode45” was used as the nonlinear system dynamics simulation mode, where the Runge–Kutta method was used with a fixed time step (1 × 10-6 s). The parameters used in the simulation were consistent with those listed in Table 3. The supply flow (Qp-QL), which was gradually increased from 0 to 90 L/min, was used as the input signal, whereas the chamber A pressure ps, pilot valve displacement x, and main valve displacement y were the output signals.

5.2 Experimental Model and Test Conditions

5.2.1 Experimental Device

Schematic diagrams of the experimental system and device are shown in Figures 13 and 14, respectively.

Figure 13
figure 13

Schematic diagram of experimental system

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Figure 14
figure 14

Experimental device and tested model valve

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In the experimental system, RP-3 aerospace kerosene was used as the working medium, and a seamless steel pipe was used as the hydraulic pipeline. An external gear pump (1) driven by an electric motor was used to provide a constant flow of 140 L/min, and the pressure was measured using pressure sensors (13) and (15). The flow rate was measured using a flowmeter (16) (range: 16–160 L/min; accuracy: 0.5%FS). Data were acquired using a synchronous data acquisition instrument (17) at a sampling period of 1.0 ms. The parameters of the tested valve were consistent with those listed in Table 3.

5.2.2 Experimental Scheme

The operational process stability of the relief valve was analyzed. The experimental scheme was formulated as follows: the inlet flow was gradually increased from 0 L/min to the rated flow (90 L/min) while the pressure in chamber A, ps, was monitored. Subsequently, the pressure-flow characteristics of the relief valve were analyzed after all experimental data were obtained. During the experiment, the throttle valve (7) was opened to the maximum lift. Subsequently, the data acquisition device (17) was turned on, followed by the pump station, while the throttle valve (8) was adjusted to ensure that the flow rate increased gradually. Finally, the pump station was turned off, and data acquisition was terminated.

5.3 Simulation and Experimental Results

The pressure-flow characteristic curves of the original and contrast models obtained via simulation and experiment are shown in Figures 15 and 16, respectively. As shown, the simulation results agreed well with the experimental results.

Figure 15
figure 15

Pressure-flow diagrams of pilot-operated relief valve obtained via simulation

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Figure 16
figure 16

Pressure-flow diagrams of pilot-operated relief valve obtained via experiment

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As shown in Figures 15 and 16, during the entire process, the pressure-flow characteristic curves of the original model were smoother or more stable than those of the contrast model. By contrast, the pressure-flow characteristic curves of the contrast model exhibited significant fluctuations. The experimental results show that the pressure fluctuation amplitude of the original model near the rated operating point was 0.39 MPa, which was approximately 1.6% of the rated pressure. However, the pressure fluctuation amplitude of the contrast model was approximately 1.64 MPa, which was approximately 6.6% of the rated pressure.

Combining the above with the results presented in Section 4, one can conclude that the contrast model valve is an unstable control system. Additionally, as shown in Figures 15 and 16, the flow-pressure characteristics of the valve do not change significantly after the addition of a dam** orifice. Thus, designing an orifice between the main and pilot valves is beneficial for improving the stability of the hydraulic pilot-operated relief valve.

5.4 Effect on Valve Response Time

In general, stability and rapidity are contradictory. According to a previous study, an appropriate size of R2 may enhance the stability of the relief valve. However, its effect on the valve response time remains unclear. Hence, the pressure response curves of the valve were obtained via numerical simulation, and the supply flow (Qp-QL), which was stepped from 0 to 90 L/min at 50 ms, was used as the input signal, whereas chamber A pressure ps, pilot valve displacement x, and main valve displacement y were specified as the output signals. Figures 17 and 21 present the simulation results for the relief valve response time.

Figure 17
figure 17

Pressure curves of contrast model

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As illustrated in Figures 17, 18, 19, 20 and 21, the response speed of the contrast model was high under step-signal excitation, and the pressure of the controlled chamber can reach the set value within approximately 9.9 ms (Figure 17). However, the original model with the dynamic pressure feedback orifice required at least 12.5 ms to reach the set value (Figure 18). More importantly, the response time of the relief valve can reach 31.6 ms (Figure 21) when the diameter of orifice R2 does not exceed 0.8 mm for a steady pressure output.

Figure 18
figure 18

Pressure curves of original model (dr2 = 1.6 mm)

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Figure 19
figure 19

Pressure curves of original model (dr2 = 1.2 mm)

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Figure 20
figure 20

Pressure curves of original model (dr2 = 1.0 mm)

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Figure 21
figure 21

Pressure curves of original model (dr2 = 0.8 mm)

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The dynamic pressure feedback orifice results in a longer response time by the relief valve, although it improves the valve stability. Therefore, to satisfy the stability requirements, the diameter of R2 should be set as large as possible.

6 Conclusions

In this study, the dynamic characteristics of a cartridge-type hydraulic pilot-operated relief valve were investigated through theoretical analysis and experiments, and the following conclusions were obtained:

  1. (1)

    For the case involving a larger dynamic pressure feedback orifice, the stability of the pilot-operated relief valve did not change substantially, except when its dam** ratio was increased.

  2. (2)

    A small-diameter orifice resulted in high flow resistance, which was beneficial to the stability of the relief valve. In terms of control engineering, an energy-storage element with a lower break frequency was added to the pilot-operated relief valve system, which was beneficial for improving circuit stability. In addition, the oil inside the main spring chamber was equivalent to a liquid spring with an extraordinarily high stiffness, and the resonance frequency of the main valve subsystem increased significantly to a level far exceeding the regulating frequency of the relief valve.

  3. (3)

    The dynamic pressure feedback orifice resulted in a longer response time by the relief valve, although it improved the valve stability.

  4. (4)

    The simulation and experimental results showed that designing a dynamic pressure feedback orifice with an appropriate size between the main valve and pilot valve is beneficial to the stability of hydraulic pilot-operated relief valves.