An analytical method for programming piston displacements for constant flow rate piston pumps is presented. A total of two trigonometric transition functions are introduced to express the piston velocities during the transition processes, which can guarantee both constant flow rates and the continuity of piston accelerations. A kind of displacement function of pistons, for two-piston pumps, and two other kinds, for three-piston pumps, are presented, and the physical meaning of their parameters is also discussed. The results show that, with the given transition functions, cam profiles can be designed analytically with parameterized forms, and the maximum accelerations of the pistons are determined by the width of the transition domain and the rotational velocities of the cams, which will affect contact forces between cams and followers.

The piston pump is a type of positive displacement pump in which the high-pressure seal reciprocates with the piston, which is a vital component in hydraulic fluid power systems. There are four types of displacement pumps commonly used in the industry, i.e. gear pumps, vane pumps, radial piston pumps, the axial piston pumps (Ye et al., 2019). Piston pumps are widely used in many fields, such as oil exploitation, hydraulic circuits, and control systems, owing to their high specific power, efficiency, and reliability. However, piston pumps also have some shortcomings when compared to when compared to equivalent pumps.

Piston pumps usually contain several cylinders. Multiphase pumps can efficiently increase oil and gas production in crude oil drilling owing to their good internal compression and anti-gas resistance performance (Deng et al., 2018; Dogru et al., 2004; Falcimaigne and Decarre, 2008; Hua et al.,
2011). However, oscillations in the pump flow rate may give rise to pressure pulsations which can lead to vibrations, noise, and harmful impacts on
pipelines (Karassik et al., 2001). Therefore, the flow rate of the piston pump must be kept constant. The invariability in the pump flow rate is also
essential for controlling systems, and one can obtain the desired control only through adjusting the rotational speed of the pump cams. Pre-pressure airbags
and infusion pumps are generally used to fulfil the constant flow rate requirement. Still, there are some patents aiming at solving this problem
(Couillard and Garnier, 1998; Sipin, 2002). They provide the methods and apparatus for supplying constant liquids. The system, according to the
invention, comprises several primary pumping units in parallel on a single mixing head. By adjusting the phases, the piston strokes, and their
velocities, each piston output flows intermittently in order to obtain a substantially constant discharge rate. Furthermore, a novel variable displacement pump architecture (Foss et al., 2017) was designed for displacement control circuits that uses the concept of alternating flow (AF) between piston pairs that share a common cylinder. The AF pump was constructed from two inline triplex pumps, which is efficient across a wide operating range. Wilhelm and Van De Ven (2014a) presented the optimization and machine design of an 8.5

Structural diagram of the piston pumps.

To make the piston pump adapt to the developmental needs of the modern industrial sector, high efficiency and energy saving, small flow, and pressure pulsation are the goals of piston pump design (Gandhi et al., 2014; Huang et al., 2017; Wilhelm and Van De Ven, 2014b). Initially, most of the piston pumps were driven by the crank-connecting rod mechanism. Its remarkable feature was the transient flow rate, which caused a pressure pulsation in the discharge and suction system and had a direct impact on the working effect (Berezovskii and Nakorneeva, 1969). Therefore, the low shear, constant flow, non-fluctuating piston pump with a cam transmission mechanism as the power solved the problem of flow and pressure fluctuations. Dong et al. (2002) compared and analysed the performances of the crank-connecting rod piston pump and the cam piston pump. In the end, they pointed out that, under the same conditions, the cam-driven piston pump had a smaller flow pulsation and inertial load, which was conducive to extending the service life. Therefore, in this paper, the pistons are driven by rotational cams to pump liquid out in turns. The structural diagram of the piston pump is shown in Fig. 1. The piston pumps contain three cylinders driven by cam systems with a flat-faced follower.

By changing the speed of the drive motor to adjust the flow, it is necessary to add a corresponding control and detection system, which increases the design difficulty and the design cost. Therefore, the improvement of the cam profile is a fundamental improvement. With the development of technology, the cam profile has been deeply studied, which makes the cam system drive more flexible, and it has less impact vibration (Gatti and Mundo, 2010; Hsieh, 2010). The proposed piston pump motion law determines the shape of the cam profile curve. In turn, the cam profile surface parameters affect the motion and dynamic characteristics of the plunger and even the whole machine. The cam profile includes the transition section and the working section. The transition section can eliminate the impact of the pressure pulsations of the piston pump, and the working section directly affects the work efficiency of the piston pump (Li et al., 2013). To this end, we must first determine the movement law of the follower, namely the plunger.

The simplest way to keep constant flow rates is to give proper design schemes of piston displacements that are finally determined by the profiles of
cams. Traditionally, the design of cams was based on the desired movement function by specifying translated displacements of the follower in terms of
a cam rotational angle

In the existing literature, the acceleration of this kind of cam was not continuous, and the transition function was difficult to use when planning frequency and avoiding resonance. Moreover, these researchers are devoted to optimizing the structure of piston pumps and lack a systematic method for the design of piston pumps driven by the cam mechanism. Therefore, this paper presents the study on an innovative design method for high-speed cams using trigonometric transition function, selects the basis and law of intermediate parameter, and discusses the physical meaning of their parameters.

In order to meet the constant flow rates of the pumps and

For piston pumps with a constant flow rate and a good performance, the piston velocities should meet the requirements of constant flow rates and

Piston velocities in transition processes.

The velocities of the forward and backward strokes, corresponding with the output and input processes of the cylinder, have to be changed between the
two processes in some transition domains. Suppose that when one piston moves forward with a speed-up velocity

According to the

In order to simplify the task, a couple of standard transition functions,

According to conditions (1) and (2),

Transition function

In order to meet condition (1),

Transition functions.

In Fig. 3,

Using boundary conditions (4) and (5), unknown parameters

According to boundary condition (6),

Obviously,

The acceleration of the piston is another important factor for piston pumps, which affects the contact forces between cams and the
followers. According to Eq. (12), the accelerations in the transition domain are as follows:

Ignoring the directions, the maximum accelerations are

From Eq. (14), we can see that the maximum accelerations in the transition domain are determined by angular velocities of the cam, the width of the
transition domain, and

It should be noted that spline functions might be workable, but they are more complex and could cause more difficulties in the computation of displacement functions.

By lending the previous transition functions to the design of cam profiles, the goal of achieving the constant flow rate and the continuity of piston displacement through the second derivative can be achieved easily.

For a two-piston pump, the phase between the two pistons is usually 180

Imaginary piston velocity of the two-piston pump.

Piston velocity of the two-piston pump.

However, the velocity is not continuous, as shown in Fig. 4. Given the fact that the acceleration goes to infinity sometimes, serious collisions
between cams and pistons will occur. Therefore, transition processes are needed to guarantee the

The velocity of the pistons in the backward stroke is not constant since it has no effect on the constant flow rates, and it can obtain the minimum
acceleration, so contact forces are also reduced. It easy to verify that the velocity function, given by Eq. (15), is

For a two-piston pump, Eq. (15) describes the velocity of piston 1. The same function applies to the velocity of piston 2, provided (

The displacement function

Therefore,

The velocity

When the cams run with the angular velocity

From Eq. (16), the minimum and maximum displacements of the pistons can be obtained as follows:

For knife-edge and flat-faced follower systems,

The stroke of the pistons

According to Eqs. (20) and (21),

From Eq. (15), the accelerations of pistons are derived in the following form:

Therefore, the maximum accelerations in the forward and backward strokes,

Piston acceleration of the two-piston pump per cycle.

In Eq. (25), the relationship between

Piston maximum accelerations respect to transition angles (

Figure 7 shows the maximum accelerations corresponding to different transition angles

The three-piston pumps are used in many places, such as polymer injection in the exploitation of oil and control systems. In general, the difference in
phase among the three pistons is 120

Piston velocity of the three-piston pump using cam A.

Using a similar method to that of two-piston pumps, the piston velocity

For a three-piston pump, Eq. (26) gives the velocity of piston 1. The same equations apply to the velocity of piston 2 and piston 3, provided
(

Thus, the flow rate of the three-piston pump is constant, and the flow rate per minute has the following form:

By integrating the velocity Eq. (26) over time, we can obtain the following displacement function:

By means of the following continuity conditions:

As

After the values of

According to Eq. (31), the minimum and maximum displacements of the pistons are as follows:

Similarly, the acceleration is as follows:

Piston maximum accelerations with respect to transition angles

Figure 9 shows the accelerations corresponding to different transition angles

Piston velocity of the three-piston pump using cam B.

For two-piston pumps, one piston must be in the forward stroke with constant velocity to provide a constant flow rate when the other piston is in the
backward stroke. For three-piston pumps, there can be two pistons in the forward stroke to provide the total constant flow rate when the third piston is in the backward stroke. Thus, each cycle can be divided into three 120

The profile of cam A in three sizes.

The profile of cam B in three sizes.

The velocities of the piston pump in cam A and cam B.

Cam systems with profiles with flat-faced and roller followers, respectively.

According to Eqs. (9) and (11),

Therefore, Eq. (38) can be rewritten as follows:

For a three-piston pump, Eq. (40) gives the velocity of piston 1. The same equations apply to the velocity of piston 2 and piston 3, provided (

By integrating the velocity function over time, the displacement curve

Since the cam profiles are continuous, the height function must obey the following:

From Eq. (43), we have the following:

At last, Eq. (42) can be rewritten as follows:

Noting Eq. (42), the minimum and maximum displacements of the pistons are as follows:

Similarly, the acceleration is as follows:

As it provides a greater width of transition domains than cam A in both the forward and backward processes, cam B can run with lower accelerations of pistons, which can also be seen through the comparison among Eqs. (34), (35), (48), and (49).

The displacements corresponding to the cams are designed in the previous sections, and the cam profiles corresponding to different followers can be determined by rational theories (Casciola and Morigi, 1996; González-Palacios and Angeles, 2012; Jensen, 2020). The computational formulas of the cam profiles corresponding to these followers are given in the following.

The cam profiles corresponding to knife-edge followers can be prescribed by the following functions:

Take the cams driven by hydraulic fluid power systems as a specific calculation example. According to the flow rate and the application requirements, the system parameters were fixed as

Flat-faced and roller followers have usually been used in high-speed cams rather than knife-edge followers because of the rapid rate of wear. Flat-faced followers can only work with the cam profile of all external curves with a small pressure angle, high efficiency, and good lubrication. The roller followers improved the contact condition between the follower and the cam profile, which can greatly reduce frictional losses and bear a high load.

Flat-faced followers (Fig. 14) are the most common in piston pumps, and the cam profiles, described by the coordinates

Corresponding to the roller followers (Fig. 14), the cam profile, coordinating with

Trigonometric splines are employed in the design of various cam mechanisms as transition functions, which can to help to plan the frequency and avoid resonance. With the given transition functions, cam profiles, required to provide a constant flow rate, can be designed analytically with parameterized forms, and the parameters can be determined according to application requirements. The piston acceleration continuity can also be guaranteed simultaneously.

Defined in a standard domain, the transition functions are uniformly described, with very simple forms, and can be used conveniently by lending coordinate transformations.

Two types of piston velocities among the three pistons and two types of cam systems with different followers are presented to provide more choices for the design of the piston pumps.

The maximum accelerations of the pistons are determined by the width of the transition domain and the rotational velocities of the cams, which will affect contact forces between cams and followers. The two parameters should be noted during the designation of the cams.

All codes generated or used during the study are available from the corresponding author upon request.

All data sets used in the paper can be requested from the corresponding author.

All work related to this paper has been accomplished by the efforts of both authors. ZT proposed the design of the piston pumps, analysed the numerical results, and wrote the paper. XL provided guidance on theoretical methods, edited the paper, and prepared the figures.

The authors declare that they have no conflict of interest.

Zhongxu Tian and Xingxing Lin greatly acknowledge the financial support from the National Key Research and Development Program of China and the Shanghai Engineering Research Center of Marine Renewable Energy, which made this research possible.

This research has been supported by the National Basic Research Program of China (973 Program; grant no. 2019YFD0900800) and the Shanghai Engineering Research Center of Marine Renewable Energy (grant no. 19DZ2254800).

This paper was edited by Daniel Condurache and reviewed by three anonymous referees.