1 Introduction
2 Mathematical Model of Pilot-operated Relief Valve
2.1 Description of Pilot-operated Relief Valve
2.2 Static Characteristics
2.3 Dynamic Mathematical Model
2.4 Linearization Analysis
Sign | Physical meaning | Expressions |
---|---|---|
KA | Pressure-flow coefficient of main exit port. | \(K_{{\text{A}}} = C_{{{\text{d}}1}} {\uppi }d_{1} \sin \alpha y_{{\text{x}}} \sqrt {\frac{1}{{2\rho p_{{{\text{sx}}}} }}}\) |
KB | Flow gain of main exit port. | \(K_{{\text{B}}} = C_{{{\text{d1}}}} {\uppi }d_{1} \sin \alpha \sqrt {\frac{{2p_{{{\text{sx}}}} }}{\rho }}\) |
KC | Hydraulic conductivity of orifice R1. | \(K_{{\text{C}}} = \frac{{{\uppi }d_{{{\text{r1}}}}^{2} }}{4}C_{{{\text{r1}}}} \sqrt {\frac{1}{{2\rho \left( {p_{{{\text{sx}}}} - p_{{{\text{cx}}}} } \right)}}}\) |
KD | Equivalent stiffness of steady hydrodynamic force of the main valve. | \(K_{{\text{D}}} = C_{{{\text{d1}}}} C_{{{\text{v1}}}} {\uppi }d_{{1}} \sin \left( {2\alpha } \right)\) |
KE | Pressure-flow coefficient of pilot port. | \(K_{{\text{E}}} = C_{{{\text{d2}}}} {\uppi }d_{2} x_{{\text{x}}} \sin \beta \sqrt {\frac{1}{{2\rho p_{{{\text{cx}}}} }}}\) |
KF | Flow gain of pilot port. | \(K_{{\text{F}}} = C_{{{\text{d2}}}} {\uppi }d_{2} \sin \beta \sqrt {\frac{{2p_{{{\text{cx}}}} }}{\rho }}\) |
KG | Equivalent stiffness of Steady hydrodynamic force of the pilot valve. | \(K_{{\text{G}}} = C_{{{\text{d2}}}} C_{{{\text{v2}}}} {\uppi }d_{{2}} \sin \left( {2\beta } \right)\) |
Gr | Hydraulic conductivity of orifice R2. | \(G_{{\text{r}}} = \frac{{{\uppi }d_{{{\text{r2}}}}^{4} }}{{128\mu l_{{{\text{r}}2}} }}\) |
3 Contrast Model
4 Theoretical Analysis
4.1 System Block Diagram
Sign | Physical meaning | Expressions |
---|---|---|
ω1 | Natural frequency of main mass-spring vibration system. | \(\omega_{1} = \sqrt {\frac{{k_{{1}} + K_{{\text{D}}} p_{{{\text{sx}}}} }}{{m_{{1}} }}}\) |
ω2 | Natural frequency of pilot mass-spring vibration system. | \(\omega_{2} = \sqrt {\frac{{k_{{2}} + K_{{\text{G}}} p_{{{\text{cx}}}} }}{{m_{{2}} }}}\) |
ω3 | Break-frequency of chamber A. | \(\omega_{3} = \frac{{E\left( {K_{{\text{A}}} + K_{{\text{C}}} } \right)}}{{V_{{\text{A}}} }}\) |
ω4 | Break-frequency of chamber C(original model). | \(\omega_{4} = \frac{{E\left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}{{V_{{\text{C}}} }}\) |
ω5 | Break-frequency of the main port differential element. | \(\omega_{5} = \frac{{K_{{\text{B}}} }}{{A_{{1}} }}\) |
ω6 | Break-frequency of the pilot port differential element. | \(\omega_{6} = \frac{{K_{{\text{F}}} }}{{A_{{2}} }}\) |
ω7 | Break-frequency of chamber B(original model). | \(\omega_{7} = \frac{{EG_{{\text{r}}} }}{{V_{{\text{B}}} }}\) |
ω8 | Break-frequency produced by the orifice R2. | \(\omega_{8} = \frac{{G_{{\text{r}}} }}{{A_{1}^{2} }}\) |
ω9 | Break-frequency of integration element corresponding to chamber B. | \(\omega_{9} = \frac{E}{{V_{{\text{B}}} }}\) |
ω10 | Break-frequency of chamber B(contrast model). | \(\omega_{10} = \frac{{E\left( {K_{{\text{C}}} + K_{{\text{E}}} } \right)}}{{V_{{\text{B}}} }}\) |
Parameter | Value | Parameter | Value |
---|---|---|---|
m1 (g) | 30.0 | δ1(μm) | 10.0 |
m2 (g) | 3.5 | δ2(μm) | 20.0 |
k1 (N/mm) | 20 | VA (mL) | 1500 |
k2 (N/mm) | 40 | VB (mL) | 1.905 |
x0 (mm) | 3.4 | VC (mL) | 0.388 |
y0 (mm) | 5.8 | Cd1、Cd2 | 0.65 |
d1 (mm) | 14.0 | Cv1、Cv2 | 0.98 |
d2 (mm) | 3.0 | E (MPa) | 800 |
α (°) | 30 | ρ (kg/m3) | 833 |
β(°) | 15 | μ(Pa·s) | 1.96e-3 |
Am(mm2) | 220 | l1(mm) | 25.0 |
Ap(mm2) | 94.2 | l2(mm) | 10.0 |
dr1(mm) | 1.0 | dr2(mm) | 0.8 |
Cr1 | 0.61 | lr2(mm) | 10.0 |