Study of pressure shock characteristics of pump-controlled hydraulic steering system

3.1. Mathematic model of axial plunger pump

The flow equation of axial plunger pump is:

where $Q_p$ is outflow of pump, $D_p$ is distribution circle diameter of the plunger in cylinder, $d_p$ is diameter of plunger, ${\alpha }_p$ is pump’s swash plate angle, $n_p$ is pump revolving speed, $Z$ is plunger number in plunger pump, $\eta$ is the volume efficiency of plunger pump.

3.2. Mathematic model of the pump-controlled hydraulic cylinder

(1) The force equilibrium equation of hydraulic cylinder is:

where $p_1$ and $p_2$ are respectively the pressures of high-pressure pipe and low-pressure pipe in the hydraulic system, $m$is equivalent mass including the mass of hydraulic piston and moving parts, $B_p$ is the hydraulic oil damping of hydraulic cylinder, $F_L$ is load on piston, $A_p$ is effective area of the piston in hydraulic cylinder, $K$ is stiffness of load.

(2) The continuity equation of high pressure chamber flow in hydraulic cylinder is:

where $V_s=V+A_{p}x$, $C_{im}$ and $C_{em}$ are respectively the coefficients of leakage inside and outside hydraulic cylinder, $V_s$ and $V$ are respectively the total volume and the initial volume of working chamber of the pump and the hydraulic cylinder and the connecting pipe, ${\beta }_e$ is hydraulic oil elasticity modulus.

3.3. Mathematic model of hydraulic cylinder load

The normal force on the rudder plate at sea is:

where $\delta$ is rudder angle, $A_d$ is the lateral projection area of control vanes, $v$ is navigational speed.

3.4. Modeling valve-controlled servo mechanism

For the convenience of research, the valve-controlled servo cylinder is regarded as a symmetrical cylinder controlled by a three-position four-way valve [9]. And also define some assumptions as follows: (1) The oil-supply pressure $P_s$ is constant and the oil-return pressure is $P_0$; (2) The four throttle windows of the valve are matching and symmetry; the rectangular window is used; the inlet flow of the valve is turbulent flow; (3) The pipe loss and dynamic state is regardless; the temperature and density are constant.

$Q_1$ is the oil-inlet flow of the high-pressure chamber, $Q_2$ is the oil-return flow of the low-pressure chamber. And the equation [10]is:

where $C_d$ is flow coefficient of valve port, $W=\pi d_v$ is area gradient of spool, $d_v$ is diameter of spool, $P_s$ is oil-feed pressure, $P_1$ is the pressure of high-pressure chamber, $\rho$ is the hydraulic oil density, $x_v$ is the displacement of spool.

Regardless of hydraulic oil elasticity modulus, the flow continuity equation in the high-pressure chamber is:

where $C_i$ and $C_e$ are respectively the coefficients of leakage inside and outside hydraulic cylinder, $P_2$ is the pressure of low-pressure chamber, $A_1$ is piston area of servo cylinder, $x_p$ is piston displacement of servo cylinder.

3.5. SIMULINK simulation results and analysis

According to the principle block diagram, by combining the Eqs. (1) to (7), SIMULINK model of pump-controlled hydraulic steering system is obtained and its parameters are confirmed, and then the SIMULINK model is simulated.

The signal of the steering angle input is: $\theta =25\mathrm{s}\mathrm{i}\mathrm{n}\omega t$. Where $\omega =\text{0.06}\pi$, the speed of the steering angle input is 2°/s, the navigational speed is 6 knots. The change of rudder angle and speed of the hydraulic cylinder is shown in Fig. 2 and Fig. 3.

Fig. 2Speed of hydraulic cylinder

Fig. 3Rudder angle

5.1. Influence of speed change on pressure shock

In Fig. 8, it shows that the speed will change along with the actual complex conditions. Keeping the rudder angle input of 2°/s and setting the spring elasticity coefficient of the pressure spring in the hydraulic cylinder of the feedback mechanism to 10 N/mm, and then the high-pressure chamber’s pressure curves at the different navigational speed of 6 knots, 10 knots and 14 knots are obtained (speed curve is omitted).

Analysis shows that: when the navigational speed changes from 6 knots to 14 knots, the velocity jump of piston increases from 0.27 m/s to 1.2 m/s in the reverse; the working pressure’s peak value of chamber B increases from 65 bar to 230 bar, and the shake in the reverse strengthen, and the duration time changes from 2 s to 4 s, and the pressure jump changes from 18 bar to 110 bar, namely the percentage of working pressure by 28 % increased to 48 %. Therefore, in order to make the steering system stable, the appropriate navigational speed should be selected to reduce the hydraulic shock under the condition of satisfying the navigation requirements.

Fig. 8Pressure of cylinder chamber

Fig. 9Piston displacement-rubber angle

5.2. Effect of the feedback mechanism

Keeping the navigational speed of 14 knots and the rudder angle input of 2°/s, in the feedback mechanism, the hydraulic cylinder’s displacement-rudder angle input curves at the different spring elasticity coefficients of 10 N/mm, 15 N/mm, 20 N/mm of the pressure spring are shown in Fig. 9 (the pressure curve of the high-pressure chamber is omitted).

From the simulation results, when the signal input and the navigational speed are under the constant circumstances, with the increase of the spring’s elastic coefficient (namely feedback is weakened), the linear degree of the displacement and angular displacement will reduce, and the tracking error will increase, and the system accuracy will decrease. Due to the asymmetry of the hydraulic cylinder, the return curve does not coincide with the push curve. With the decrease of the spring elasticity, the pressure shake in the reverse enhances, and the reverse shock increases.

From the above analysis, enhancing system’s feedback will improve the linear relationship between the displacement and the angular displacement and the accuracy of the system. But at the same time it will enhance system shake, increase the reverse shock and reduce the system stability. Therefore, when the elastic coefficient of spring is set, the actual situation of the whole system should be taken into consideration to make the system optimal.

5.3. Effect of the input rudder angle velocity

Keeping navigational speed of 14 knots and setting spring elasticity coefficient of pressure spring in hydraulic cylinder of the feedback mechanism to 10 N/mm, the simulation curves with the different rudder angle input of 2°/s, 3°/s, 4°/s are shown in Fig. 10, Fig. 11 and Fig. 12. In Fig. 12, the curves 1-6 are respectively representative of the cylinder displacement signals of 2°/s input and output, 3°/s input and output, 4°/s input and output.

Fig. 10Pressure of cylinder chamber

Fig. 11Piston speed

In Fig. 10-13, it shows that when the input rudder angle speed from 2°/s to 4°/s, the system’s maximum response delay will change from about 0.6 s to 1.8 s, and the signal amplitude attenuation will increase from 2 % to 10 %. With the rudder speed increasing, the piston’s speed jump in the reverse will change from –0.4 m/s to +0.4 m/s to –0.65 m/s to +0.67 m/s and the jump peak and scope will enlarge; the maximum pressure pulsation increases from about 35 bar to 120 bar, and the shock pressure will enlarge with the increase of the rudder angle. So the selection of the rudder angle velocity should be appropriate, and the input rudder speed should be set reasonably in combination with the actual requirements, and the accuracy and stability of the system should be also considered.

Fig. 12Piston displacement

Fig. 13Pressure of cylinder chamber