Dynamic characteristics of coupling model of valve-controlled cylinder parallel accumulator

. To solve the electro-hydraulic control problems caused by the accumulator ’ s passive involvement, the internal principle of the valve-controlled luf ﬁ ng cylinder was explored based on one ship crane; comparison models with and without the accumulator were established, and experiments were performed to verify the dynamic characteristics. By comparing simulations with experimental data, it was found that in the outstretch stage of the cylinder, the accumulator releases energy, accelerates the extension of the cylinder, and increases the response velocity. However, the accumulator also reduced the precision of location control. Therefore, to effectively solve the problem of ﬂ uctuation rate in the outstretch stage, experiments were conducted to adjust the valve-core opening according to the energy-release characteristics of the accumulator. In the retracting stage of the cylinder, the accumulator absorbed oil and inhibited the pressure ﬂ uctuation of the cylinder effectively. Finally, the feasibility of the compensation solution was veri ﬁ ed through the experiments.


Introduction
In the ship crane, the valve-controlled luffing cylinder can be used for wave compensation to keep the hoist stable. However, the active compensation control method often has problems such as large power consumption and sudden hit. The accumulator is an important hydraulic element, and it is widely applied when designing energy-saving and anti-damping systems [1][2][3]. Thus, the accumulator is applied in the valve-controlled luffing cylinder to reduce the power consumption. In order to get a precise position track, especially in harsh sea conditions, electro-hydraulic proportion control requires that the hydraulic system should have a timely velocity-controlled feature. Due to the energy's passive involvement in the accumulator, the accumulator causes some difficulties to electro-hydraulic proportion control, as shown in Figure 1. Therefore, it is of great significance to study the influence of the accumulator's passive involvement in electro-hydraulic proportion control.
Experts have conducted many basic studies on accumulators, mainly focusing on noise reduction and energy-saving aspects. To achieve noise reduction, a bladder-type accumulator in coal mine machinery is used to reduce high flow capacity (2000 L/min) and load impact [4]. Simulations show that the peak pressure fluctuation (5 MPa) can be eliminated. Additionally, Yang and coworkers [5] show that the accumulator can suppress the water hammer phenomena of hydraulic steering systems and prove that the suppression effect on the water hammer is optimized when the inherent frequency of the water accumulator is equal to or close to the frequency of the water hammer. In references [6][7][8][9][10][11][12], hydraulic potential energy regenerative vehicles are designed to reduce the vibration amplitude of vehicles on rough roads and decrease the structural noise. The motor location, the hole area of the valve, the inner diameter of pipelines, and the accumulator pressure are key factors contributing to suspension performance. By combining the accumulator's pressure signal and PID algorithm, there is a better effect in the aspect of reducing vibration and pitching movement. The accumulator plays an important role of adjusting rigidity in vehicle suspension. The whole system can have a higher modal stiffness and damping coefficient by changing the accumulator's inflation pressure. If the mode is applied to a valve-controlled motor system, the motor can achieve constant velocity [8]. To address energy conservation, the accumulator is integrated to develop a five-valve flow control and is combined with a new algorithm to keep accurate motion-tracking performance of the hydraulic cylinder and reduce energy consumption [13]. According to modeling and simulation, the accumulator's highest energy conservation rate of potential energy recovery in an electric hydraulic forklift is 36% [14,15]. As a source of auxiliary power, the accumulator improves the complex nonlinear response and control accuracy of valve-controlled motors; AMESIM and Simulink models are adopted to make dynamic simulations [16]. A kilowatt-level hydraulic buck converter is designed, and it is proved by experiments that the accumulator has an obvious role in guaranteeing system linearity and reducing energy consumption [17]; wave energy is transformed into potential energy and stored in the accumulator, and the theoretical peak of transformation rate can be up to 27%. However, the overall transformation efficiency of equipment in model prototype tests is low, with a mean value of about 6% [18,19]. In general, the model is effective, and the overall efficiency is low. Cylinders and accumulators are wildly used in excavators [20][21][22][23]. The accumulator plays a key role in hydraulic design. Through coupling power output, the overall energy conservation effect can be improved to 10.1%. Related designs are applied in rock drills [24], scrapers [25], fast-forging presses [26], and other engineering machinery. Some studies replace the throttle valve, unloading valve, and fuel tank with an accumulator; in this way, the energy regeneration efficiency of the system is 44% or even higher [27][28][29][30][31][32][33]. Another study uses a variable area gas piston with a fixed cylinder area, and the results reveal that the constant pressure accumulator provides a 16% improvement in energy density over a conventional accumulator [34].
Most literature regarding accumulators focuses on the energy-saving and vibration-reduction characteristics of accumulators. The effect of passive intervention of accumulators on the valve-controlled cylinder oil is rarely reported. Accumulators are often used in applications where the cylinder moves slowly, but wave compensation requires that the cylinder can conduct a fast response. Furthermore, there are several problems with wave compensation. First, the accumulator will face an inertia delay caused by the cylinder's outstretch and frequent retraction. Second, the hydraulic components' nonlinearity and oil pressure hysteresis can also lead to the delayed response of the accumulator. Third, the components' parameters are uncertain. As a result, the angle error cannot be compensated in time. The rapid increase of compensation signal can easily lead to another overshoot during the next time period.
Our contribution is to establish the coupling dynamic model of the valve-controlled cylinder with an accumulator. Comparative analyses, simulations, and experiments are conducted to study the accumulator's influence on electro-hydraulic proportional control. Finally, the compensation method is presented and verified.

Dynamic model
The crane is fixed on the ship and the telescopic arm outstretches from the ship. The wave influences the ship, leading to rotation in three directions, and the luffing mechanism compensates the impact of the rotation in the Z direction. The hydraulic system is shown in Figure 2.
The accumulator and valve provide oil together into the rodless cavity at the outstretch stage, and the oil from the rod cavity returns to the tank through the balance valve. The maximum work pressure of the accumulator can be adjusted by changing the opening pressure of the sequential valve. The oil from the rodless cavity flows into the accumulator at the retracting stage.

Valve's flow equation
In the hydraulic system, the flow (Q L ) is related to the pressure (P L ) and the spool displacement (X V ): Q L = f(x v , P L ). The following assumptions are made: (1) the valve is an ideal positive-opening four-way slide valve and the four orifices are matching and symmetrical, (2) the valve is always working close to its working position, and (3) the fluid cannot be compressed. The valve's flow equation is shown in equation (1) [35]: where q 1 denotes the oil flow from the valve to the rodless cavity, m 3 /s; C d denotes the flow coefficient of the valve; w denotes the valve area gradient, m 2 /m; U denotes the open size of the valve at zero position, m; x v denotes the displacement of the valve core, m; r denotes the oil density, kg/m 3 ; p s denotes the system pressure, N/m 2 ; and p 1 denotes the rodless cavity's pressure, N/m 2 . Then, the linearization method can be used to linearize the equation Q L = f(x v , P L ) to obtain equation (2): where K q1 denotes the flow-displacement gain coefficient of the rodless cavity, m 2 /s; and K c1 denotes the flow-pressure gain coefficient of the rodless cavity, m 5 /(s · N). Equations (3) and (4) can be deduced by taking the derivative of equation (1).
where P 10 denotes the initial pressure of the rodless cavity, N/m 2 .
The flow equation of the rod cavity is shown in equation (5).
where q 2 denotes the oil flow from the valve to the rod cavity, m 3 /s; p 2 denotes the pressure of the rod cavity, N/m 2 ; p t denotes the return oil pressure, N/m 2 ; and p t ≈ 0. By applying linearization on equation (5), equation (6) can be obtained: By taking the derivative of equation (5), equations (7) and (8) can be deduced.

Flow continuity equation of valve-controlled cylinder
The flow continuity equation of the rodless cavity is shown in equation (9), and its Laplace transformation is shown in equation (10).
where Q L denotes the total flow of the rodless cavity of the cylinder, m 3 /s; Q X denotes the oil flow from the accumulator, m 3 /s; X P denotes the displacement of the pistol, m; C i denotes the internal leakage coefficient, m 5 /(N · s); V 1 denotes the total volume of the rodless chamber (containing the connecting pipe and the inside of the cylinder), m 3 ; and b e denotes the liquid Bulk Elastic Modulus, N/m 2 . By combining equations (2) and (10), equation (11) can be obtained.
The flow continuity equation of the rod cavity is shown in equation (12), and its Laplace transformation is shown in equation (13).
where C e denotes the external leakage coefficient, m 5 /(N · s); and V 2 denotes the total volume of the rod chamber (containing the connecting pipe and the inside of the cylinder), m 3 . By combining equations (6) and (13), equation (14) can be obtained.

Dynamic equation of the cylinder's piston rod
The force equation of the cylinder's piston rod is shown in equation (15), and its Laplace transformation is shown in equation (16).
where m t denotes the mass of the piston, kg; B P denotes the viscosity damping coefficient, (N · s)/m; K z denotes the spring stiffness, N/m; and F L denotes the external load, N.

Dynamic equation of the accumulator
The bladder accumulator's structure [36] and simplified force model are shown in Figure 3.
Equation (17) is derived from the ideal gas equation where k denotes the gas variable index, k = 1.4; P 0a denotes the initial pressure of the accumulator, N/m 2 ; and V 0a denotes the initial volume of the accumulator, m 3 . By the definition a = 1.4P 0a ⋅ V 0a À1 ⋅ A a , equation (18) can be simplified as equation (19) The force equation of the oil in the accumulator is shown in equation (20), and its Laplace transformation is shown in equation (21).
where A n denotes the cross-section area of the pipeline, m 2 ; A a denotes the cross-section area of the accumulator, m 2 ; V a denotes the gas cavity volume of the accumulator, m 3 ; P a denotes the gas cavity pressure of the accumulator, N/m 2 ; m e denotes the quality of oil in the accumulator, kg; B e denotes the equivalent damping coefficient of oil, (N · s)/m; and K a denotes the stiffness coefficient of gas, N/m.

Flow continuity equation of the accumulator
The output flow of the accumulator is equal to the volume change rate, so equation (22) is obtained. Its Laplace transformation is shown in equation (23): Equation (24) can be obtained by combining equations (19), (21), and (23).
A a ⋅A n ⋅P 1 ¼ 1 s ⋅Q X ⋅ðm e ⋅s 2 þ B e ⋅s þ k a þ a⋅A n Þ þ P 0a ⋅A a ⋅A n :

Simulation analysis
Equations (11), (14), (16), and (24) are utilized to establish the block diagram of power elements. Figure 4 shows the working process from the control signal to the cylinder's piston rod displacement in Simulink software. The simulation parameters are shown in Table 1.
The stiffness coefficient of gas (K a ) is relative to the current cavity volume of the accumulator: If V is 4.4 L at the stable working position, then K a = 4.03 Â 10 5 N/m.

Analysis of step signal response
A positive step signal is assigned to simulate the protruding condition. The simulation results from the oscilloscope are shown in Figure 5. At the beginning, the cylinder outstretches rapidly due to the electro-hydraulic valve and the accumulator's powered oil. The consumed time with the accumulator is 0.25 s, which is less than the consumed time without the accumulator (2 s). When the cylinder outstretches to the end without the accumulator, the pressure of the rodless cavity rises quickly to the threshold value. In contrast, the pressure with the accumulator rises slowly.
Conclusion: In the outstretch phase, the accumulator's oil flow is nonlinear. The boom angle changes quickly in the beginning and then tends to be stable. The slowly changing pressure with the accumulator leads to the velocity fluctuation.
A negative step signal is assigned to simulate the retracting movement. The simulation results from the oscilloscope are shown in Figure 6.
In the step retraction movement, the accumulator adsorbs the oil due to the rising oil pressure. Because of the same powered oil, the boom angle curves in the two situations are same. In the condition without the accumulator, the pressure of the rodless cavity reaches the threshold value of the sequence valve, and then the oil flows into the tank. Conclusion: In the cylinder retraction phase, the accumulator has no obvious effect on the cylinder movement. The pressure of the rodless cavity rises slowly, and the accumulator adsorbs the oil until the pressure reaches the sequence valve's threshold value.

Analysis of PID control response
The process of the valve-controlled cylinder system with the PID algorithm is shown in Figure 7, where Gp(s) is the algorithm, which is used to get the control signal, and Dx is the deviation between the cylinder's actual displacement (x m ) and given displacement ( The step signal is set, and the cylinder's displacement is shown in Figure 8.
The cylinder fluctuates around the target location and tends to be stable eventually. By comparison, the deviation and consumed time in the condition with the accumulator is larger than in the condition without the accumulator.

Analysis of wave compensation response in one period
The ship sways periodically with the character of the sinecosine rule. In order to keep the boom angle stable, the cylinder's trajectory also needs to adhere to the sine-cosine rule to compensate for the ship's swaying. In the 0-0.5 period, the cylinder outstretches with the sine-cosine character; in the 0.5-1 period, the cylinder extracts with the sine-cosine character. The wave compensation curves in one period under the PID algorithm are shown in Figures 9 and 10.
In the outstretch stage, the wave compensation movement with the accumulator shows higher deviation fluctuation, but the vibration frequency is lower.    The crane, as shown in Figure 11, includes the luffing mechanism, the telescopic mechanism, the rotary mechanism, and the hoist mechanism.
The PLC outputs a 4-20 mA current signal, which is used to control the valve through a proportional amplifier. The angle and displacement sensors are applied to reflect the cylinder's movements. The parameters of the equipment are shown in Table 2.
The designed control interface is shown in Figure 12. It can display the current angle and the set angle and has the function of reading and writing parameters. The data are recorded in real time at a rate of 5 data points per second.

Experimental strategy and data analysis
The experimental strategy is divided into three steps. First, the threshold value of the sequence valve is set to make the accumulator work in a proper condition. Second, the step signal is applied, and the responses are compared in these two conditions to verify the influence of the accumulator. Third, the swing table is wobbled to simulate harsh sea conditions, the compensation experiments are conducted, and the influence of the accumulator is studied in wave compensation movement.

The external conditions of the accumulator
The working pressure range of the accumulator is related to multiple factors, such as charging pressure, accumulator volume, pipe diameter, and piston position. We adjust the valves' working pressure and record the process of oil absorption and release, as shown in Figure 13. As the cylinder retracts from 25°to À8°, the accumulator absorbs energy. The pressure of the rodless cavity rises from  Stiffness coefficient of gas N/m 7.56 Â 10 6 Pa to 9.28 Â 10 6 Pa. The accumulator releases the oil in a nonlinear way when the cylinder outstretches. At first, the oil is released quickly, so the curve is very steep, and then the angle-change tends to be smooth. The consumed time is about 6 s. The color block diagram in Figure 13 shows the relationship among the energy releasing value, external pressure, and cylinder's velocity. The left color block diagram shows that the energy's released velocity is significantly related to the cylinder's outstretch velocity and has no obvious relationship with pressure. The color block diagram on the right shows that the energyabsorbing velocity is significantly related to both the pressure and the velocity of the cylinder.
Conclusion: The working pressure of accumulator is related to the volume of the accumulator and the precharge pressure. The precharge pressure should be 80% of the minimum working pressure. The sequence valve's threshold value should be higher than the largest working pressure of the accumulator. The volume of the accumulator should be larger than the volume of the cylinder rodless cavity.

Periodic step control analysis
The following steps are conducted in two conditions. (1) Apply a negative step signal to make the cylinder conduct retraction movement. (2) Apply a zero signal when the boom angle drops down to À8°, and then wait for 2 s. (3) Apply a positive signal to make the cylinder conduct outstretch movement. (4) Apply a zero signal when the boom angle rises up to 25°, and then wait for 2 s. The response curves are recorded in Figures 14 and 15. The step response appears to have a periodic character. Hence, one period is divided into four stages, and each stage is studied separately.

Retraction stage
The retraction movement of the cylinder is shown in Figure 16. When the accumulator is shut down, the pressure increases rapidly to the stable state within 1 s, and the curve below indicates that the boom angle changes smoothly. By comparison, when the   Step control without accumulator.
accumulator is started, the pressure increases slowly to the stable state within 5 s, and the angle changes smoothly as shown in the right curve below.
Conclusion: In the retraction stage, the pressure curve changes from step state to smooth state after the accumulator is added. The accumulator reduces the  pressure fluctuation when absorbing the oil. The angle changes smoothly in the whole stage.

Waiting stage
In the waiting stage, the valve-core goes back to the zero position, and the pressure is shown in Figure 17. The pressure decreases because of the oil leakage. In particular, the pressure drop is obvious. When the accumulator is added, the pressure drops slowly and the fluctuation is small.

Conclusion:
In the waiting stage, the accumulator plays a stabilizing role, and the pressure-changing range is less than in the condition without the accumulator.

Outstretch stage
The cylinder's movement is shown in Figure 18. When the accumulator is connected, the pump and accumulator supply the oil together. The cylinder outstretches rapidly at the beginning, and then tends to be smooth. The experimental process is in accordance with the simulation results in Figure 5. The boom angle's changing rate with the accumulator shows that the angle changes violently, and the cylinder's movement linearity is poor.
Conclusion: In the cylinder's outstretch stage, the accumulator releases energy to assist the outstretch movement, and at the same time, the velocity changes unsteadily with the phenomenon of velocity fluctuation.

Waiting stage
As shown in Figure 19, the pressure decreases obviously without the accumulator. By comparison, the pressure's change is small with the accumulator, and due to the disappearance of inertia force, the pressure increases to keep the balance state.
Conclusion: In the cylinder's waiting phase, the cylinder's pressure is more stable with the accumulator. The accumulator shows its shock-absorption function.

Influence of wave compensation in several periods
In this test, the swing platform is used to simulate the ship's movement with a frequency of 0.11 Hz and an amplitude of ±7°. To keep the arm stable, the PID algorithm is used to conduct the wave compensation experiment. The simulation in Figures 9 and 10 is conducted in one cycle, and the following experiments are carried out in multiple cycles.

Wave compensation without the accumulator
The experimental curves without the accumulator are shown in Figure 20. The cylinder's pressure is unstable. The pressure's fluctuation scope rises up to 1.0 × 10 6 Pa, and the rod cavity's pressure is about 3.0 × 10 6 Pa. The cylinder's movement is fast, and the compensation effect is obvious and good. The wave compensation's angle error is about ±0.6°. Conclusion: The wave compensation without the accumulator has good stability and easy control characters, but it has high pressure fluctuation.

Wave compensation experiment with the accumulator
The experimental curves with the accumulator are shown in Figure 21. The pressure fluctuation is small, but the dynamic compensation effect is poor, and the angle error is about ±3°. By analyzing the control signal, it is found that the signal switches between positive and negative frequently, and the phenomenon leads to the poor performance of wave compensation. The conclusion is that the accumulator is likely to cause the cylinder to overshoot. Figure 22 shows the accumulator's oil flow, which is irregular. 3. Solution scheme to improve wave compensation with the accumulator The optimized method of reducing the control signal is adopted to limit the valve's oil supply, which makes the system's total power decrease according to the output flow from the accumulator. The optimized algorithm makes the cylinder extend smoothly, reduces the cylinder's jitter, and improves the effect of the wave compensation. The angle compensation deviation is stable in the range of ±1°, as shown in Figure 23. Figure 24 shows that the accumulator's largest output flow happens exactly at the highest angle-changing rate. The accumulator plays the role of oil supplementation source in a timely manner.
Conclusion: At the outstretch stage with the accumulator, the two oil sources cause the oil cylinder to outstretch rapidly, which is likely to lead to overshoot. The nonlinear oil supply of the accumulator causes the fluctuation of the cylinder, which results in the velocity inconsistency. To smooth the movement, the opening of the valve core should be smaller according to the accumulator's released energy, and then it can solve the fluctuation effectively.

Conclusion
The comparative analysis method is used to study the working characteristics of the accumulator, and the electro-hydraulic control law with accumulator under passive intervention is obtained.
The working pressure range of the accumulator is related to factors such as inflation pressure, accumulator volume, pipe diameter, and piston rod position. The accumulator inflation pressure (6.0 × 10 6 Pa) should be at or slightly above 80% of the system operating minimum pressure (7.56 × 10 6 Pa). The opening pressure of the sequence valve should be higher than the maximum working pressure (9.28 × 10 6 Pa). The oil-absorbed velocity is significantly related to the cylinder retraction velocity and has no obvious relationship with pressure. The energyreleased velocity of the accumulator is significantly related to both the cylinder's pressure and the cylinder extension velocity.  During the entire step response process, the accumulator stabilizes the system, and the pressure's drop scope is less. In the cylinder retraction stage, the pressure curve changes from step response to stable state. When the cylinder is extended, the accumulator releases energy to accelerate the cylinder, and the velocity becomes unstable and the volatility is large.
In electro-hydraulic control of wave compensation, the accumulator reduces the compensation effect. At the outstretch stage, the oil cylinder responds rapidly but is likely to cause overshoot. The opening of the reversing valve should be correspondingly reduced according to the potential energy characteristic curve of the accumulator to effectively solve the problem of pressure fluctuation. From another point of view, the accumulator exerts an energy-saving function in the system and reduces the energy consumption.