Reviewers:

dr hab. inż. Andrzej Balawander prof. nadzw., Gdansk University of Science and Technology, Poland

dr hab. inż. Mariusz Olszewski prof. nadzw., Warsaw University of Science and Technology, Poland

English translation:

Piotr Storta

Project coordinator:

Izabela Jaźwińska

Editorial coordinator:

Mariola Grzywacka

Cover and title pages design:

Przemysław Spiechowski

English language editor:

Agnieszka Adamczyk-Karwowska

The publication was subsidised by the Faculty of Mechanical Engineering of Wrocław University of Science and Technology, Wrocław, Poland.

Copyright © by Polish Scientific Publishers PWN SA, Warszawa 2020.

eBook created for Polish Scientific Publishers PWN by (imię nazwisko/firma)

Monika Lipiec / Woblink

ISBN 978-83-01-21145-5

eBook based on printed edition from year 2020, (First edition)

Warszawa 2020

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Contents

Preface

1. Definition, systematics and methodology of designing hydraulic gear machines

1.1. Definition and systematics

1.2. Designing methodology

2. The process of energy transformation in hydraulic gear machines

2.1. General models of a pump and a motor

2.1.1. General model of a pump

2.1.2. General model of a motor

2.2. Ideal and real characteristics of a pump and a motor

2.2.1. Characteristics of a pump

2.2.2. Characteristics of a motor

2.3. Conclusions

3. Gears in the hydraulic gear machines

3.1. Teeth, gears and the external involute gear systems

3.1.1. The tooth, the external gear

3.1.2. The boundary number of teeth, correction of the external tooth

3.1.3. The external involute gear system

3.1.3.1. The uncorrected gear system, the principle of co-operation,the number of contact

3.1.3.2. The corrected gear system, the principle of co-operation, the number of contact

3.2. Teeth, gears and the internal involute gear systems

3.2.1. The tooth, the internal gear

3.2.2. The boundary number of teeth, the correction of the internal tooth

3.2.3. The internal involute gear system

3.2.3.1. The uncorrected gear system, the principle of co-operation,the number of contact

3.2.3.2. The corrected gear system, the principle of co-operation, the number of contact

3.2.3.3. Interference of the gear teeth

3.3. Teeth, gears and the internal cycloidal gear systems

3.3.1. Teeth, gears and the uncorrected internal epicycloidal gear system

3.3.2. Teeth, gears and the corrected internal epicycloidal gear system

3.3.3. Principle of co-operation in the epicycloidal gear system, the contact line, the number of contact.

3.3.4. Teeth, gears and the uncorrected internal hypocycloidal gear system

3.3.5. Teeth, gears and the corrected internal hypocycloidal gear system

3.3.6. Principle of co-operation in the hypocycloidal gear system, the line of contact, the number of contact

3.3.7. Principle of co-operation in the cycloidal gear system with moveable axes

3.3.8. Principle of co-operation in the cycloidal multi-gear systems

3.4. Conclusions

4. Channels and clearances in the fluid power gear machines

4.1. The system of fixed channels and clearances

4.2. The system of moveable channels and clearances

4.3. The system of channels and clearances in the machines of the first group

4.3.1. Inlet channel and inlet chamber

4.3.2. Inlet bridge

4.3.3. Outlet channel and outlet chamber

4.3.4. Outlet bridge

4.3.5. Axial and radial clearance

4.4. The system of channels and clearances in the machines of the second group

4.4.1. Inlet channel and inlet chamber

4.4.2. Inlet bridge

4.4.3. Outlet channel and outlet chamber

4.4.4. Outlet bridge

4.4.5. Axial and radial clearance

4.5. The system of channels and clearances in the machines of the third group

4.5.1. Inlet channel and inlet chamber

4.5.2. Inlet bridge

4.5.3. Outlet channel and outlet chamber

4.5.4. Outlet bridge

4.5.5. Axial and radial clearance

4.6. The system of channels and clearances in the machines of the fourth group

4.6.1. The system of channels and clearances in the multifunctional hydraulic gear machine (MHGM)

4.6.2. The system of channels and clearances in the ORBIT orbital motor

4.6.3. The system of channels and clearances in the MAX orbital motor with double cycloidal gearing

4.7. Conclusions

5. Delivery and delivery pulsation, capacity and capacity pulsation of the fluid power gear machines

5.1. General formulae

5.1.1. Instantaneous delivery

5.1.2. The proper, theoretical and average delivery

5.1.3. Delivery pulsation

5.2. Delivery and delivery pulsation in the machines of the first group

5.3. Delivery and delivery pulsation in the machines of the second group

5.4. Delivery and delivery pulsation in the machines of the third group

5.4.1. Epicycloidal gear machines

5.4.2. Hypocycloidal gear machines

5.5. Delivery of the machines of the fourth group

5.6. Conclusions

6. Pressure and pressure pulsation in the fluid power gear machines

6.1. Theoretical characteristics of changes of the pressure and pressure pulsation in intertooth displacement chamber T in machines of the first–third groups

6.2. Theoretical characteristics of changes of the pressure and pressure pulsation in the intertooth displacement chamber in machines of the fourth group

6.3. Conclusions

7. The visual study of the flow processes and phenomena in the channels and clearances of the gear machines

7.1. General comments

7.2. The subject and methodology of the visual study, the test stand

7.3. The visual study of the machines of the first group

7.3.1. Inlet channel and inlet chamber

7.3.2. Inlet bridge

7.3.3. Outlet channel and outlet chamber

7.3.4 Outlet bridge

7.3.5 Relief grooves in the outlet bridge zone

7.3.6. Axial clearance

7.4. The visual study of the machines of the second group

7.4.1. Inlet channel and inlet chamber

7.4.2. Inlet bridge

7.4.3. Outlet channel and outlet chamber

7.4.4. Outlet bridge

7.5. The visual study of the machines of the third group

7.5.1. Inlet channel and inlet chamber

7.5.2. Inlet bridge

7.5.3. Outlet channel and outlet chamber

7.5.4. Outlet bridge

7.6. Conclusions

8. Pressure study in the channels and clearances of the gear machines

8.1. The subject and methodology of the study, the test stand

8.2. Study of the pressure in the channels and clearances of the machines of the first group

8.2.1. Study of gear pump I without the axial clearance compensation

8.2.2. Study of gear pump II with the axial clearance compensation

8.3. Study of the pressure in the channels and clearances of the machines of the second and third group

8.4. Conclusions

9. Designing the housings of the fluid power gear machines

9.1. Methodology of designing the housings of the fluid power gear machines

9.2. Determining the design and technological requirements for the housings of the pumps

9.3. Determining the basic shape of the pumps

9.4. Strength analysis of the basic shape gear pump housings by means of FEM

9.4.1. Geometrical models, load and fixation

9.4.2. Numerical models, research programme

9.4.3. Results of the strength analysis of the basic shape pump housings

9.5. Global and local modification of the basic shape housing

9.6. Strength analysis of the housings of the modified shape gear pumps by means of FEM

9.7. Assumption of the final shape of the pumps

9.8. Conclusions

10. Designing of the axial clearance compensation system

10.1. Principle of axial clearance compensation. Methodology of designing the axial clearance compensation system

10.2. Designing the axial clearance compensation system in the machines of the first group

10.2.1. Assumption of a design solution of the compensation system

10.2.2. Defining Ap area of the pressure working in the axial clearance, and the division of the area into partial areas Api

10.2.3. Defining the distribution of partial pressure pi working on partial areas Api

10.2.4. Defining partial pressure repulsive forces Fpi and pressure repulsive forces Fp

10.2.5. Defining partial torques of the repulsive forces Mpi and total torque of the repulsive forces Mp

10.2.6. Defining points of application xpi, ypi of partial pressure repulsive forces Fpi and point of application xp, yp of total pressure repulsive forces Fp

10.2.7. Shaping total compensation area Ac and its division into partial areas Aci. Determining of compensation pressure pc

10.2.8. Defining partial compensation forces Fci and total compensation force Fc

10.2.9. Defining coordinates xci, yci of the points of application of partial compensation forces Fci and coordinates xc, yc of the points of application of total compensation force Fc

10.2.10. Comparison of total pressure repulsive forces Fp and its point of application xp, yp with total compensation force Fc and its point of application xc, yc.

10.3. Designing the axial clearance compensation system in the machines of the second group

10.3.1. Assumption of a design solution of the compensation system.

10.3.2. Defining partial pressure repulsive forces Fpi and total pressure repulsive force Fp

10.3.3. Defining points of application xpi, ypi of partial pressure repulsive forces Fpi and point of application xp, yp of total pressure repulsive force Fp

10.3.4. Shaping total compensation area Ac and its division into partial areas Aci. Determining of compensation pressure pc

10.3.5. Defining partial compensation forces Fci, coordinates xci, yci of the points of its application, total compensation force Fc and coordinates xc, yc of the point of its application

10.3.6. Comparison of total pressure repulsive force Fp and its point of application xp, yp with total compensation force Fc and its point of application xc, yc

10.4. Designing the axial clearance compensation system in the machines of the third group

10.5. Designing the axial clearance compensation system in the machines of the fourth group

10.6. Conclusions

11. Design solutions of fluid power gear machines

About Author

References

Preface

The concept of ‘hydraulic (fluid power) gear machines’ (HGM) refers to such a type of machines utilized in the fluid power drive and control systems, where gears constitute the basic system both in terms of the principle of design and of operation. HGM are employed to change mechanical energy into hydraulic energy or conversely. The former are pumps, and the latter – hydraulic motors.

The author has been dealing with HGM for over 40 years, first on his own, and for the last several years, with a group of collaborators within the Fluid Power Research Group who works at the Department of Fundamentals of Machine Design and Tribology at the Faculty of the Mechanical Engineering of Wroclaw University of Science and Technology. The original knowledge accumulated as a result of the author’s work has been ordered and coherently presented in this book. The knowledge has been put into a framework of the design process, and the book has been titled ‘Fundamentals of designing hydraulic gear machines’. The work contains both the theoretical knowledge (Chapters 1– 6 and 9–10) and the knowledge resulting from experimental research (Chapters 7– 8). Reading through the consecutive chapters of the book one finds out how to design various kinds and types of hydraulic gear machines. The validity and efficiency of such an approach is proved in the last, namely eleventh chapter of the book, in which various machines designed according to the principles specified in the book are presented. Some of the machines function as industrial products, and others, in a form of prototypes or models, are ready to be introduced in the industrial practice. The knowledge presented in this book can be used also in other areas of machine design. For instance, the contents of Chapter 3, concerning the cycloidal gearing, could be applied in the designing of the mechanical cycloidal transmissions. Similarly, the visualisation research methods utilized to study the phenomena and process occurring in the channels and clearances of the hydraulic gear machines presented in Chapter 7 could be applied for the research into other types of fluid power machines and appliances such as valves, directional valves, or even entire hydraulic systems. The author will be satisfied if his book grasps the readers’ interest and becomes useful in solving their professional problems.

The illustrations given in a form of design drawings, diagrams or curves play a significant role in the publication. For the preparation of those figures, the author would like to express his gratitude to Aleksander Czarny, a mechanical engineering technician, who has been an ardent member of the Fluid Power Research Group for many years.

My special thanks go to Piotr Storta, the translator of this book from Polish into English. A thorough analysis of the source text, searching for the right English terminology and the appropriate language form reflecting the author’s thoughts are his great merit.

The publication of the book is an opportunity to remember my father, Prof. Stefan Stryczek, the author of a recognised academic handbook ‘Napęd hydrostatyczny’ (‘Hydrostatic drive’) and the Dean of the Mechanical Engineering Faculty of Wroclaw University of Science and Technology, a scientist and engineer, who for a few decades had been a leader of the Polish fluid power engineers. I dedicate this book to his memory.

Author

1. Definition, systematics and methodology of designing hydraulic gear machines

1.1. Definition and systematics

The concept of ‘hydraulic (fluid power) gear machines’ (HGM) refers to such a type of machines utilized in the fluid power drive and control systems, where gears constitute the basic system both in terms of the principles of design and operation.

Figure 1.1 presents the simplified diagram of HGM. The machine consists of a gear system and a housing. Between the gear teeth, the intertooth displacement chambers are created, and in the housing the channels and the clearances are formed.

Fig. 1.1. Simplified diagram of the hydraulic gear machine (HGM)

The intertooth displacement chambers co-operate with the channels and clearances system, transferring the working medium from the inlet side towards the outlet side of the machine. The HGM can perform two basic functions:

– the function of a pump, in which mechanical energy from the motor is transformed into hydraulic energy accumulated in the discharged working fluid – see the solid line in Figure 1.1.

– the function of a motor, in which hydraulic energy accumulated in the working fluid supplied to it, is transformed into mechanical energy of the rotational motion of the shaft driving the working unit – see the dashed line in Figure 1.1.

The design solution of HGM results from the selection of two basic systems, that is a gear system and a system of channels and clearances created in the housing of the pump.

In the HGM, the following systems are mainly employed:

– the involute and cycloidal systems,

– the internal and external gearing systems,

– the fixed or the moveable rotation axes systems.

Gear systems can co-operate with:

– a fixed system of internal chambers and channels,

– a moveable system of internal chambers and channels.

Taking the basic function performed by a machine as well as the design solutions of a gear system and an internal channels and clearances system as the criteria of division, Figure 1.2 presents the systematics of hydraulic gear machines. Four groups of machines are distinguished.

Fig. 1.2. Systematics of hydraulic gear machines (HGM)

Machines of the first group include pumps or motors featuring the external involute gearing, fixed rotation axes and the fixed channels and clearances system. Of all the HGMs, machines of the first group are most frequently applied in practice [68].

Machines of the second group include pumps or motors featuring the internal involute gearing, fixed rotation axes and the fixed channels and clearances system. Machines of that group are of more compact design, smaller size and mass than the machines of the first group [66, 67].

Machines of the third group include pumps or motors featuring the internal cycloidal gearing, fixed rotation axes and the fixed channels and clearances system. Machines of that group are of an even more compact design, smaller size and mass than the machines of the second group. They are often referred to as ‘gerotor machines’ [80, 82, 83, 84, 85].

Machines of the fourth group are predominantly low speed, high-torque machines with the external or internal involute or cycloidal gearing, moveable axes and the system of moveable channels and clearances. Machines of that group are of small size and mass relatively to the power generated. They are often referred to as orbital or planetary machines [80, 82, 83, 84, 85].

That group also includes the orbitrol control units and the multifunctional hydraulic gear machines.

The systematics of the HGM can also be carried out depending on other criteria of division. Regarding their displacement (capacity), the HGM can be divided into the constant and variable displacement (capacity) machines.

From the viewpoint of independent streams flowing through the machines, the single- or multi-stream units can be distinguished.

Regarding the way the machines are connected, the single-stage or multi-stage machines can be distinguished.

1.2. Designing methodology

Hydraulic gear machines are designed in a systematized way. Basing, among others, on the source [60, 62], in Figure 1.3 a diagram of methodology for designing and studying hydraulic gear machines is presented. The methodology includes eleven stages of design and study. Each of the stages is symbolically illustrated by a diagram of a machine, which shows what design problem is solved within its framework. The problem is marked in the diagram with a solid line.

At the first stage of design, a concept of a particular design solution for an HGM is developed. Within the framework of that concept it is necessary to select a machine type out of four machine groups shown in Figure 1.2, and consequently, the following:

– the basic function performed by the machine,

– the design solution for the gear system,

– the design solution for the system of the channels and clearances.

It is also necessary to determine the values of the basic technical parameters which the machine is supposed to feature.

At the second stage, an analysis of energy transformation in the HGM is carried out, which Figure 1.3 depicts with the arrow-head line on the inlet and the outlet, which goes through the inside of the machine. This stage consists in developing an energy model of the machine, analysing its energy balance, and in developing anticipated characteristics, which are related to the technical parameters of the machine assumed at stage 1.

At the third stage, a gear system of the HGM is designed, which Figure 1.3 presents in a form of a circle with the blackened intertooth displacement chamber.

The designing includes the geometry and kinematics of the gear system, in order to minimize the dimensions of the gear system and to maximize the volume of the intertooth displacement chambers.

At the fourth stage, the system of channels and clearances of the HGM is designed, which Figure 1.3 presents as lines edging the gear system. The task of the system is to supply and discharge the working fluid to and from the intertooth displacement chambers while the machine is operating. As it has already been mentioned above, it can be either a fixed or a moveable system.

The system consists of:

– a channel, a chamber and an inlet bridge,

– a channel, a chamber and an outlet bridge,

– an axial and radial clearance.

The questions of the primary concern at the designing of the internal chambers and clearances system, are to secure continuity of the flow, to minimize the flow resistance, to eliminate the cavitation phenomena and to minimize internal leakage.

At the fifth stage, the displacement (capacity) Q and of the displacement pulsation (capacity) ΔQ of the HGM are calculated, which in Figure 1.3 are presented in a form of a reference mark directed towards the outlet port of the machine. The calculations are carried out basing on the gear system with its geometry and kinematics defined at stage 3. According to the designed gear system, specific formulae are applied. The formulae allow to determine the effect of the gear system on the displacement (capacity) and on the pulsation of the displacement (capacity).

At the sixth stage, a theoretical analysis of pressure p and pulsation of pressure Δp in the intertooth displacement chamber of the HGM in the full working cycle is carried out, which Figure 1.3 presents in a form of a reference mark directed towards the outlet port of the machine. The analysis is carried out to test the accuracy of co-operation of the gear system and the system of the internal chambers and clearances.

Fig. 1.3. Methodology of design and research of hydraulic gear machines (HGM)

At the seventh stage, a visual study of the flow processes and phenomena in the channels and clearances of the HGM is conducted, which in Figure 1.3 is presented in a form of a reference mark directed towards the inside of the machine. At the beginning of that stage, an experimental machine is constructed and a test stand equipped with a fast camera for photo recording is prepared.

Basing on the results of the designing work obtained in stages 1– 6, an experimental machine with a technical glass housing is constructed. The operating machine then is monitored and, by means of the fast camera, the flow processes and phenomena occurring in the machine are recorded. By changing the design solutions and the operational parameters of the machine, it is possible to monitor their influence on the processes and phenomena inside the machine. Based on the analysis of the processes, it is possible to correct the design solution of the entire HGM.

At the eighth stage, an experimental research into the pressure in the channels and clearances of the HGM is conducted, which Figure 1.3 presents as a reference mark directed towards the inside of the machine. Similarly to how it is carried out at stage 7, an experimental machine and a test stand are constructed, equipped with a system for the measuring of the dynamic pressure in the intertooth displacement chambers during the operation of the machine. The pressure curves are then drawn. By changing the design solutions and the operational parameters of the machine, it is possible to monitor their influence on the processes occurring in the mesh of the gear system and the system of the internal chambers and clearances. Based on the analysis of the processes, it is possible to correct the design solution of the entire HGM.

At the ninth stage, the HGM housing is designed, which in Figure 1.3 is marked with a bold line edging the gear system and the system of channels and clearances.

To design the housing the following steps need to be taken: determining the basic shape of the housing and conducting the strength analysis by means of FEM, modifying (correcting) the basic shape of the housing and carrying out the strength analysis utilizing FEM, and finally, accepting the final shape of the housing.

At the tenth stage, the axial clearance compensation system of the HGM is designed, which Figure 1.3 presents in a form of an oval compensation element working with the gear system, edged with a bold line and hatched with oblique lines. The designing process starts with the selection of the shape and size of the compensation element. Next, the pressure research results obtained at stage 8 are implemented, on the basis of which the resultant repulsive force working on the compensation element and on its point of contact are determined. Finally, on the outer surface of the compensation element, a surface is formed, which is influenced by the working pressure. As a result, the resultant pressing force is generated, which should be greater than the repulsive force. The point of contact of the pressing force is also determined. It should be placed as close to the point of contact of the repulsive force as possible.

At the eleventh stage, the final design solution of the HGM is developed based on the results obtained at the ten preceding stages, their synthesis is carried out, and the design documentation of the HGM is prepared.

2. The process of energy transformation in hydraulic gear machines

As section 1.1 and Figure 1.2 indicate, hydraulic gear machines can be divided into four groups. At the same time, among those machines, gear pumps and gear motors can be distinguished.

Based on the source [94], the general models of a gear pump and a gear motor have been developed along with their ideal and real characteristics, as it is presented below.

2.1. General models of a pump and a motor

2.1.1. General model of a pump

Figure 2.1 presents the general model of a gear. It consists of a shaft (1), which drives a gear system (2) located in a housing (3). In the drawing of the model, neither bearings supporting the shaft and the gears in the rotational movement nor minor sealing elements are marked, but they are there in the real model. In the gear system (2), there are the intertooth displacement chambers T which transport the working medium from the inlet zone I of the pump to the outlet zone O. In the housing of the pump, a system of channels CL and clearances G is created in order to direct the flow of the working medium going through the pump.

The transformation of mechanical energy Emech supplied by the shaft (1), into hydraulic energy Ehydr stored in the working fluid is performed in the following way: the pump sucks the working fluid from the tank through the inlet I into the intertooth displacement chambers T. Next, the chamber T shifts by the rotational movement (φ angle) in the system of channels CL and clearances G, transporting the working medium to the outlet O. On reaching the outlet, the chamber pumps the working fluid into the hydraulic system. Finally, the chamber T returns to the inlet O in order to start another work cycle. In the pump, there can be even several dozen chambers T which, while working one after another respectively, secure high displacement Ogt of the pump.

The motion of the chambers is generated by torque Mgt working on the drive shaft (1), and causes compression of the medium in the displacement chambers T. Consequently, in the process of displacing the working medium from the inlet zone I into the outlet zone O, the pressure of the medium increases from low value pI to high value po.

Fig. 2.1. General model of the hydraulic pump of groups 1–4.1 – shaft, 2 – gear system, 3 – pump housing, CL – internal channels, G – clearances, T – intertooth displacement chamber, I – inlet, O – outlet

The model is acceptable for all four groups of the hydraulic gear machines presented in Figure 1.2. which work as pumps. It is necessary, however, to explain certain issues concerning machines of the fourth group. Gears of those machines rotate with a planetary motion. It results in a number of suction-charging cycles performed by the chamber T during one revolution of the drive shaft (1). Therefore, a pump working in such a way could be referred to as a multiple-action pump. The model presented in Figure 2.1, however, refers to a single-action pump. Nevertheless, it can be applied to the multiple-action pump, yet it then will refer to a part of the revolution of the shaft, and to one suction-charging cycle. In the situation when the charging cycle is repeated a number of times, it is necessary to apply a complex system of channels CL for supplying and receiving working fluid from the displacement chamber T. During the flow through the channels, high resistance of the movement and problems with ‘self-sucking’ occur.

Thus, it is critical to provide overpressure on the inlet I, which is performed by the so-called charging pump.

In practice, such a solution is applied only in Orbitrol control block featured by the fourth group of the machines (presented in Figure 11.10).

2.1.2. General model of a motor

Figure 2.2 presents the general model of a motor. The model includes a shaft (1), a gear system (2), and a housing (3). The drawing of the model does not show bearings and minor sealing elements, which, as it is assumed, are there. The gear system (2) features the intertooth displacement chambers T and in the housing (3) there is a system of channels CL and of internal clearances G. The transformation of hydraulic energy Ehydr accumulated in the stream of the working fluid into mechanical energy Emech is performed by the shaft (1) in the following way: the pump of the hydraulic system supplies the motor with the working medium through the inlet port I, and the medium flows into the intertooth displacement chambers T. Next, the chamber T moves with rotational motion (φ angle) in the system of channels CL and clearances G towards the outlet port O. On reaching the outlet port, the working fluid flows down into the tank. In a motor, there can be even several dozen displacement chambers T which, acting in sequence one after another, provide high torque on the shaft of the motor Mst.

Fig. 2.2. General model of the hydraulic motor of groups 1–3.1 – shaft, 2 – gear system, 3 – housing, CL – internal channels, G – clearances, T – intertooth displacement chamber, I – inlet, O – outlet

While the working fluid is flowing from the inlet zone I to the outlet zone O, its pressure falls from a high value pI down to a low value po. The expansion of the pressure results in the rotation of chamber T, the rotation of gears (2) and the generating of torque Mst on the drive shaft (1) of the motor.

The model is acceptable for all the four groups of the hydraulic gear machines presented in figure 1.2., working as motors. However, when considering machines of the fourth group working as motors, it is observed that the gears move with a planetary motion. It causes the intertooth displacement chamber T being filled in and discharged a number of times in order to generate one revolution of the drive shaft (1). A motor working in such a way is a multiple-action motor. The model presented in Figure 2.2 applies to a single-action motor, though. Nevertheless, that model can apply to a multiple-action motor, yet it refers to a single supplying-discharging cycle and a part of the shaft (1) revolution.

The problem is additionally analysed in Figure 2.3 where a separate model of the fourth group motor featuring multiple-action moveable axes is presented. The model, similarly to the fixed axes model shown in Figure 2.2, consists of a shaft (1), of a gear system (2) and of the housing (3). The gear system features displacement chambers T. The displacement chamber T shifts by φ angle corresponding to one working cycle. First, it connects with the supply pump through the inlet port I and the inlet channel CLI, and the supply with the working fluid at the high pressure pI is performed, as it Figure 2.3 depicts with a solid line. Next, the working fluid expands in the chamber T, which results in the revolution of the gear system (2) and the shaft (1).

Fig. 2.3. General model of the hydraulic motor of the fourth group.1 – shaft, 2 – gear system, 3 – housing, CL – internal channels, G – internal chambers, T – intertooth displacement chamber, I – inlet, O – outlet

Finally, the chamber T connects to the outlet port O through the channel CLO, and the working fluid flows down into the tank at the low pressure po, which is presented in Figure 2.3 with a dashed line.

The comparison of the fixed axes motor model presented in Figure 2.2 with the model of the moveable axes motor in Figure 2.3 proves similar processes taking place in both. The difference lies only in the fact that in the fixed axes motor, the energy transformation process occurs within a complete revolution of the shaft, whereas in the moveable axes motor, the same process occurs within a part of the revolution of the shaft, and then it is repeated a number of times. In order to make the repetition process happen, a more complex system of supply channels and chambers CLI, G and relief channels and chambers CLo, G are necessary.

2.2. Ideal and real characteristics of a pump and a motor

2.2.1. Characteristics of a pump

As Figure 2.1 has already presented, in the pump, mechanical energy Emech delivered by the shaft from the motor is transformed into hydraulic energy of pressure Ehydr accumulated in the working fluid. Hence, following the energy conservation law:

Emech = Ehydr

In the ideal pump, there is no energy loss, therefore:

Mgt · φg = Vgt · Δpg

(2.1)

where:

Mgt – theoretical torque on the shaft of the pump (the torque of the generator),

φg – angle of the revolution of the shaft of the pump,

Vgt – theoretical volume of the working fluid displaced from the pump,

Δpg = po - pI – difference of pressure at the outlet po and the inlet pI (suction and charging pressure difference).

After differentiating relative to time, Equation (2.1) takes the following form:

(2.2)

As a result, the following dependency is obtained:

Mgt · ωg = Qgt · Δpg

(2.3)

where:

ωg – angular velocity of the shaft of the pump,

Qgt – flow generated by the pump, i.e. the theoretical displacement of the pump.

Theoretical displacement of the pump Qgt regardless of the volumetric loss, is defined by the dependency:

Qgt = qg · ng

(2.4)

where:

qg – specific delivery understood as the maximum obtainable delivery of the working fluid, expressed in cm³, which is generated by the real pump after one revolution at the outlet pressure equal to the inlet pressure, namely Δp = po - pI ≈ 0, i.e. with no volumetric loss,

ng – rotational velocity of the shaft of the pump.

By implementing dependency (2.4) in the Formula (2.3), taking ωg = 2πng into account, and modifying the formula, the following theoretical torque Mgt on the shaft of the pump formula is created:

(2.5)

The process of energy transformation in the pump is described by two characteristic quantities:

– theoretical delivery Qgt of the pump (see Formula 2.4),

– theoretical torque Mgt delivered onto the drive shaft of the pump (see Formula 2.5). In that case it is assumed that the shaft of the pump revolves at a constant speed ng, and in the working fluid delivered from the inlet to the outlet of the pump, an increase in the Δpg = po - pI is generated.

The process of energy transformation should be analysed for two kinds of the pump:

– for the ideal pump featuring no energy loss,

– for the real pump featuring energy loss.

Figure 2.4a presents the energy balance for the ideal pump. The figure shows that the streams representing theoretical delivery Qgt and theoretical torque Mgt flow through the pump with no loss.

Fig. 2.4. Balance of the characteristic quantities of the pump.a) ideal pump, b) real pump.

What stems from Formula (2.4) is that theoretical delivery Qgt of the ideal pump, assuming constant rotational velocity of the shaft ng = const, is constant and does not depend on loading the pump with pressure Δpg. Therefore, Figure 2.5a depicts characteristics of theoretical delivery Qgt of the ideal pump depending on its load Δpg, namely Qgt = f (Δpg). The characteristics is illustrated with the straight horizontal line. Formula (2.5) shows that theoretical torque Mgt, assuming constant specific delivery qg changes in direct proportion to loading of the pump Δpg. Hence, Figure 2.5b shows the characteristics of theoretical torque Mgt of the ideal pump depending on load Δpg, namely Mgt = f (∆pg). The characteristic is a straight line coming from the beginning of the coordinate system.

Fig. 2.5. Characteristics of the pump.a) theoretical delivery Qgt and real delivery Qg, b) theoretical torque Mgt and real torque Mg,c) volumetric efficiency ηvg, hydraulic – mechanical efficiency ηh–mg, total efficiency ηg

Figure 2.4b presents the energy balance for the real pump. The figure shows that the stream indicating theoretical delivery Qgt is reduced by the value of volumetric loss ∆Q and, consequently, on the outlet of the pump, real delivery Qg is generated. The volumetric loss depends on the leakage of the working fluid through the clearances G in the pump, formed between the rotating displacement chamber T and the fixed elements of the housing. Hence, Figure 2.5a depicts the characteristics of real delivery Qg depending on the load of the pump ∆pg, namely Qg = f (∆pg). The characteristics resembles the shape of a parabola. Between theoretical characteristics Qgt and real characteristics Qg, volumetric loss ∆Qg is marked. Figure 2.4b shows that the stream indicating theoretical torque Mgt is increased by hydraulic-mechanical loss ∆Mg and, as a result, the shaft of the pump has to be loaded with real torque Mg. Hydraulic-mechanical loss ∆Mg torque results both from the resistance of the working fluid flow through the channels and the clearances in the pump, and the friction between those parts of the pump which remain in the relative motion during the operation of the pump. Taking that into consideration, Figure 2.5b presents the characteristic of real torque Mg depending on pressure ∆pg working on the pump, namely Mg = f (∆pg).

Characteristic Mg is still a straight line but it is shifted relative to the characteristic of theoretical torque Mgt by the value of torque loss ∆Mg.

The analysis of Figure 2.5a allows to observe the following dependency:

Qg = Qgt - ∆Qg

(2.6)

The figure shows that volumetric loss ∆Qg grows proportionally to the growth of pressure ∆pg working on the pump. It results from the theory of flow through the clearances, which states that the flow through the clearances of the displacement chamber and volumetric loss ∆Qg grow proportionally to the growth of the pressure difference ∆p inside and outside the displacement chamber.

By juxtaposing real delivery Qg with theoretical delivery Qgt, the volumetric efficiency of the pump can be defined as:

(2.7)

The analysis of Figure 2.5b allows to the determine the following dependency:

Mg = Mgt + ∆Mg

(2.8)

The figure shows that loss torque ∆Mg grows along with the growth of pressure ∆pg in the pump. This occurs mainly due to an increase in the mechanical friction and an increase in the resistance of the working fluid flow in the channels of the pump.

The comparison of theoretical torque Mgt with real torque Mg enables the determination of the hydraulic-mechanical efficiency of the pump:

(2.9)

Considering the real pump as a machine which collects power from the motor and transfers it to the hydraulic system, it is necessary to determine the power balance.

Driving power Ng, which should be supplied from the motor, can be defined knowing real torque Mg, which ought to be applied on the shaft of the pump, as well as the angular velocity ωg of the shaft:

Ng = Mg · ωg

(2.10)

Effective power Ne, which can be utilized in a hydraulic system, is determined based on the knowledge of real delivery Qg, and load ∆pg of the pump, namely:

Ne = Qg · ∆pg

(2.11)

By comparing effective power Ne to driving power Ng, the total efficiency ηg of the pump can be defined as:

(2.12)

By implementing dependencies (2.10), (2.11) and then (2.7), (2.9) and (2.5) in the formula, the following equation is obtained:

ηg = ηvg · ηhmg

(2.13)

It means that the total efficiency of the pump is a product of the volumetric efficiency ηvg and the hydraulic-mechanical efficiency ηhmg.

The curves ηhmg of all the three, namely of efficiency ηvg, ηhmg, ηg relating to load ∆pg of the pump, are presented in Figure 2.5c.

2.2.2. Characteristics of a motor

As Figure 2.2 presents, in the motor hydraulic energy Ehydr accumulated in the working fluid is transformed into mechanical energy Emech transferred onto the shaft of the working unit. It is a reverse transformation than the one observed in the pump. In an ideal motor, there is no energy loss, and based on the energy conservation law the following equation is created:

Mst · ωs = Qst · ∆ps

(2.14)

where:

Mst – theoretical torque on the shaft of the motor (torque of the motor),

ωs – angular velocity of the shaft of the motor,

Qst – flow rate of the working fluid through the motor, namely the theoretical capacity of the motor,

Δps = pI - po – pressure difference on the inlet and the outlet (supply and discharge pressure difference).

Theoretical capacity Qst of the motor regardless of the volumetric loss is determined by the dependency:

Qst = qs ns

(2.15)

where:

qs – specific capacity of the motor, understood as the minimum volume of the working fluid expressed in cm³ which should be delivered to the motor in order the shaft to perform one revolution, at the supply pressure equal to the discharge pressure Δps = pI - po = 0,

ns – rotational velocity of the shaft of the motor.

After implementing dependency (2.15) in Formula (2.14), and taking into account that ωs = 2πns, and after all the necessary transformations, the formula for calculating torque Mst on the shaft of the motor is as follows:

(2.16)

When analysing the process of energy transformation in the motor, it is necessary to consider two characteristic quantities:

– theoretical capacity Qst of the motor (see Formula 2.15),

– theoretical torque Mst on the shaft of the motor (see Formula 2.16).

At the same time, it is assumed that the shaft revolves at the constant speed ns, and in the working fluid flowing through the motor, a decrease in pressure Δps = pI - po is observed.

The process of energy transformation should be analysed regarding two kinds of motors:

– an ideal motor without the energy loss,

– a real motor with energy loss.

Figure 2.6a presents the energy balance for the ideal motor. The figure depicts streams, standing for theoretical capacity Qst and theoretical torque Mst, which flow through the motor without any loss.

Fig. 2.6. Balance of the characteristic quantities of the motor.a) ideal motor, b) real motor.

What results from Formula (2.15) is that theoretical capacity Qst of the ideal motor, assuming the constant rotational velocity of the shaft (ns = const), is constant and does not depend on the decrease in pressure δps. Figure 2.7a presents the characteristic of the theoretical capacity of the motor relating to the pressure decrease in the motor, in a form of a straight horizontal line. What results from Formula (2.16), however, is that theoretical torque Mst on the shaft of the motor, assuming constant capacity qs, changes in direct proportion to the decrease in pressure Δps. The characteristic of theoretical torque relating to the decrease in the pressure is shown in Figure 2.7b in a form of a straight line coming from the beginning of coordinate system.

Figure 2.6b presents the energy balance for the real. The figure shows that the stream which indicates the real capacity Qs of the motor is larger than the stream of theoretical capacity Qst by the value of volumetric loss ΔQs. The volumetric loss are, similarly to the loss of in pumps, caused by the leakage through the clearances. Hence, Figure 2.7a shows the characteristic of real capacity Qs depending on the load of the motor, namely Qs = f (Δps). The characteristic is of a parabolic shape. Between real and theoretical characteristics the volumetric loss ΔQs is marked. Figure 2.6b shows also that the stream representing real torque Ms of the motor is smaller than theoretical torque Mst by the value of hydraulic–mechanical torque ΔMs.

The hydraulic-mechanical loss torque, as in the case of pumps, is a consequence of resistance of the motor. Respectively, Figure 2.7b shows the characteristics of real torque Ms depending on the pressure which loads the motor (Δps), namely Ms = f (Δps). The characteristics is still similar to a straight line, however shifted in relation to the characteristics of the theoretical torque Mst by the value of loss ΔMs.

Based on Figure 2.7a, the following dependency is derived:

Qs = Qst + ΔQs

(2.17)

The figure shows that in the motor, just as in the pump, the volumetric loss ΔQs increases along with an increase in the loading pressure Δps.

By comparing theoretical capacity Qst to real capacity Qs, volumetric efficiency ηvs of the motor can be defined as:

(2.18)

From the analysis of Figure 2.7b, the following dependency is derived:

Ms = Mst - ΔMs

(2.19)

The figure shows that the characteristics is shifted in relation to the coordinate system origin by value Δps min. Hence, the motor will be able to start only if the pressure Δps min necessary to deal with the resistance of the working fluid flow through the channels, and the resistance of the motion of the moveable elements of the motor

Fig. 2.7. Characteristics of the motor.a) theoretical capacity Qst and real capacity Qs, b) theoretical torque Mst and real torque Ms,c) volumetric efficiency ηvs, hydraulic – mechanical efficiency ηh–ms, total efficiency ηs

is provided. The figure also shows that loss torque ΔMs increases along with an increase in the pressure difference Δps.

By comparing real torque Ms to theoretical torque Mst of the motor, the hydraulic – mechanical efficiency of the motor can be expressed as:

(2.20)

Power is calculated for both the pump and for the motor.

The inlet power Ns delivered by the working fluid stream flowing into the motor is defined as a product of real capacity Qs and pressure difference Δps, namely:

Ns = Qs · Δps

(2.21)

The effective power Ne transferred into the system is defined as a product of torque Ms on the shaft of the motor and angular velocity ωs, namely:

Ne = Ms · ωs

(2.22)

By comparing effective power Ne to inlet power Ns, the total efficiency ηs of the motor is:

(2.23)

After transformations similar to the ones utilized for the pump, the following dependency is obtained:

ηs = ηvs · ηhms

(2.24)

It means that the total efficiency ηs of the motor is a product of volumetric efficiency ηvs and hydraulic – mechanical efficiency ηhms. Figure 2.7c presents the curves of the efficiency of the motor which are similar to the curves of the efficiency of the pump shown in Figure 2.5c.

2.3. Conclusions

To sum up, a few design conclusions can be drawn. First of all, it is necessary to formulate a concept of the design solution for a hydraulic gear machine. It is necessary to decide whether the machine is supposed to work as a pump or as a hydraulic motor. It is related to the proper energy transformation in machines. In the case of a pump, the transformation is of mechanical energy into hydraulic energy, and in the case of a motor – the transformation of hydraulic energy into mechanical energy. Next, it is necessary to create a general model of the machine, which allow to carry out the analysis of energy transformation in the machine. Finally, it is crucial to decide what types of characteristics describing the transformations and what formulae for the determining of those characteristics should be employed.

3. Gears in the hydraulic gear machines

Figures 2.1, 2.2, 2.3 show that displacement chambers T formed in the intertooth space of gears play the basic role in the process of energy transformation in HGM.

In order to determine the volume of the intertooth displacement chambers, and consequently, delivery Qgt, or alternatively, capacity Qst of the machine, it is necessary to tackle the problem of designing gears for an HGM.

According to Chapter 1 and Figure 1.2, the following gear systems are normally utilized for the design of HGMs:

– external involute gear systems – machines of the first group,

– internal involute gear systems – machines of the second group,

– internal cycloidal gear systems – machines of the third group.

Moreover, gears in a system can co-operate at fixed axes (machines of the first ÷third group) and at moveable axes (machines of the fourth group).

General design requirements concerning the designing of gear systems can be formulated depending on the geometry, kinematics and hydraulic parameters of a given gear system.

As far as its geometry is concerned, the smallest and the most compact gear system possible is intended. Respectively, in the case of external gear systems, two gears with the same number of teeth and usually smaller than the boundary number, namely z₁ = z₂ < zg are assumed. However, in the case of internal gears, a requirement of the minimum tooth difference z₁ - z₂ → min. is additionally assumed so as to ensure a small size and compactness of the gear system.

In terms of kinematics, it is absolutely necessary to ensure fluency and continuity of the mesh but, at the same time, it is important to try to assure that the meshing zone is not too large, and to prevent the situation of creating too large closed spaces which generate numerous disadvantageous hydraulic phenomena occurring in the machine. In the case of the involute gear systems, this is achieved by providing contact number 1 < ε < 1.2. In the case of the internal cycloidal gear systems, the fluency and continuity of the mesh is provided through the designing of proper tooth shapes.

Due to hydraulic parameters, it is desired to obtain displacement chambers of the highest possible volume, which ensure the smallest possible change of that volume during operation of the machine, namely the flow pulsation or, alternatively, the capacity pulsation for the machine. It is performed mainly through the selection of proper values of the gearing and mesh parameters.

For the description of the geometry and kinematics of all the considered gear systems, a uniform system of parameters is assumed, which includes:

– number of teeth z₁, z₂,

– modulus m,

– angle of the outline αo of involute gears

– involute gear tooth depth ratio y and cycloidal gear tooth depth ratio λ,

– involute gear outline shift ratio x, or cycloidal gear outline shift ratio v,

– tooth width b.

The assumption of such a system of parameters enables a uniform approach to designing of gear systems featuring various mesh and tooth shapes, as well as it creates a possibility to compare the gear systems.