Modelling Internal Leakage in the Automatic Transmission Electro-Hydraulic Controller, Taking into Account Operating Conditions

Abstract

The basic malfunction of automatic transmissions (ATs) is oil flow through hydraulic precision pair clearances called an “internal leakage”, leading to difficulties in controlling the AT. There are no sufficiently accurate methods for assessing the impact of “internal leakage” on the AT technical condition in the course of operation. A proprietary hydraulic precision pair internal leak flow model has been proposed in the paper. The novelty of the model is applying electro-hydraulic controller precision pair clearance values as data that was determined through measurements involving an actual object with a specific AT operation period. The authors conduct variant tests of the model to determine the total AT hydraulic system controller leakage. Reduced oil viscosity (approx. 20%) results in internal leakage increasing by 25%. Significant wear of the controller’s precision pair and increased oil temperature (above 80 °C) lead to internal leakage increasing by more than 50% and oil pressure decreasing below the permissible value.

1. Introduction

Hydraulic systems, due to the possibility of ensuring high pressure, high rotation forces, and torques with high accuracy and low energy consumption, have numerous industrial applications, including robotics and manipulators [1,2], machine tools [3], active car suspension systems [4], hydraulic positioning systems [5,6], hydraulic load simulators [7], in aircraft, cranes and lathes [8], in construction machinery [9], and in automatic transmission of motor vehicles [10]. Electro-hydraulic servomechanisms are currently deemed state-of-the-art solutions for regulating primary flight control systems in civilian and military aircraft [11]. A hydraulic cylinder, owing to its structural simplicity, reliable operation, and high load capacity, is the primary actuator widely applied in hydraulic systems.

Despite their advantages, hydraulic systems exhibit several downsides, such as leakages between matching components, variable friction and fluid properties, as well as supply pressure changes, which lead to deteriorated operation quality and system efficiency. If the undesirable changes exceed standard boundaries, they can be recognized as defects. Not detecting these defects may ultimately lead to a complete equipment failure [12].

Out of the aforementioned defects, leakages constitute the ones that are the most common and hardest to detect in hydraulic systems. There are two types of leakages, namely, internal and external [13,14].

The phenomenon of fluid being displaced between gaps (clearances) in a different circuit section within a hydraulic system is called an internal leakage. The term internal leakage primarily refers to hydraulic fluid leaking from cylinders, couplers, and conduits connecting system components. It is easy to detect due to its visibility.

The main causes of this leakage are damage to the piston rod or seal, extreme operating temperatures, chemicals, and impurities in the hydraulic oil. These are mainly mineral grains: Quartz (SiO₂) and alumina (Al₂O₃), whose hardness on the Mohs scale is 7 and 9, respectively, and significantly exceeds the hardness of the materials used on the spool valve parts. The influence of mechanical impurities present in the hydraulic fluid is presented in the article [15]. Experimental studies of the effect of a particle size of 10 ± 2 µm and a mass concentration of impurity of 40 mg/L showed wear and jamming of the hydraulic distributor. Such a phenomenon was not observed for particle sizes of about 2 ± 1 and 25 ± 5 µm, suggesting that abrasive particles with a size commensurate with the radial clearance between the friction parts of the manifold are dangerous for the operation of the hydraulic manifold. The results of the above-mentioned studies are in line with those presented in [16].

According to the authors of the paper [17], surface topography plays an important role in the proper operation of a hydraulic system and is often the cause of its failure. The quality of the surface topography is determined by impurities in the hydraulic fluid, which can cause adverse effects in the form of surface abrasion and particle erosion impact, and impeded movement in the form of jamming. After a certain period of operation of the hydraulic system, there is a deterioration of its operating efficiency in the form of an increase in leakage and a decrease in pressure build-up. The authors of the paper [17] believe that the surface topography of components is crucial to the proper operation of a hydraulic system. Indeed, surface wear causes 10% of component failures.

The term internal leakage refers to a pressure difference between the HP and LP chambers and damage to internal cylinder leaks, which cause hydraulic fluid to flow from the high-pressure side to the low-pressure side within the cylinder [18].

Internal leakages are caused by damaged or worn cylinder gaskets that close the gap between the cylinder piston and wall, as well as excessive wear of the electro-hydraulic controller precision pair. Excessive wear is caused by operational impurities in the ATF. Impurities in the form of a mineral fluid penetrating from the surroundings into the systems are a product of the abrasive wear of metal and plastic components. In the course of AT operation, ATF is subject to ageing processes that include oxidation and thermal decomposition, which increase fluid impurities. Impurities deteriorate its lubricating properties and intensify the tribological and corrosive wear of AT hydraulic control system components. Approximately 75% of all hydraulic system failures are caused by hydraulic fluid impurities [19]. The filtration system (in the form of a porous filter on the suction conduit and an energy filter in the ATF tank) applied within a hydraulic system is aimed at maintaining the impurity concentration at the lowest possible level, and hence, extending the ATF operating time and minimizing precision pair wear.

Both internal and external leakage reduce the operating efficiency of a hydraulic cylinder. Untight pressure conduits enable easy, direct observation. An internal leakage occurs within a hydraulic cylinder and is difficult to detect at its initial stage. An internal leakage is visible only when it is severe, namely, when the cylinder does not respond, its response to a control signal is slower, or there are any other operating-state abnormalities. If an internal leakage is not detected or in the event of a failure to take preventive action at its early stage, the leak flow will consume the energy available to the cylinder and gradually damage other components of the hydraulic system. As the tightness deteriorates, the system will ultimately fail [20].

A sudden leakage can also cause severe consequences when a system is operating under load and may pose a serious threat to machine operators. A leakage is the most significant and unavoidable form of a hydraulic cylinder failure, which directly impacts the system’s performance and service life. Therefore, detecting leakages at their early stage is crucial to energy efficiency, safe operation, and limited machine downtime.

However, detecting an internal leakage is impossible until there is a total failure of the cylinder seal and the cylinder is unresponsiveness to a control signal. In addition, internal leakage detection is difficult due to its concealed location. Whereas non-destructive online diagnostics of an internal leakage is not easily implemented due to the dynamic operating conditions of hydraulic cylinders.

From the perspective of mechanical failure diagnostic technology, methods for detecting leakages in hydraulic cylinders can be divided into three categories, namely, model-based diagnostics, signal processing-based diagnostics, and AI-based diagnostics. Model-based diagnostics involves creating precise mathematical and physical models through analyzing the test subject and applying standardized mathematical and physical methods.

Mathematical and physical models were applied to describe leakage in hydraulic motors and pumps [21,22].

For example, the authors of [23] presented the impact of changes in hydraulic fluid characteristics on the hydraulic operating parameters of a gear pump mounted on a hydraulic press. These parameters are related to pressure, flow rate, and temperature, as well as hydraulic oil viscosity, which affects the pump’s volumetric capacity. Hydraulic oil filtration and replacing materials with appropriate sealing components resulted in reducing “internal leakage” from 31% to 7%, for a maximum pressure of 200 bar, which corresponds to the transmission gear efficiency standard (η = 0.93). For an operating pressure of 120 bar, the operating efficiency has been brought to η = 0.971, which can already be deemed an ideal state of the pump.

However, most studied objects include non-linear problems during actual manufacturing operations. It is very difficult to develop mathematical models with precise parameters for non-linear hydraulic systems. The authors of [24] studied issues associated with real-time detection of leaks in non-linear, multi-box flow devices. The document also demonstrates that model-based detection methods are susceptible to sensor noise, which leads to the inability of the mathematical models to determine precise parameters. Kalman filter was used in [25] to identify leaks in hydraulic cylinders; however, this method only enables determining the leakage type.

Signal processing-based diagnostic methods include such detection methods as, among others, wavelet analysis, spectral analysis, and Fourier transform. The main principle of such method types is comparing the characteristic values of a normal signal and a damaged signal through analyzing and processing the system output signal to assess whether there is a fault.

The authors of [26,27,28,29,30] employed wavelet transform and fast Fourier transform to extract signal features and precisely determine a leakage pattern.

In [26], a wavelet-based method is presented for online detection of internal leaks in a valve-controlled hydraulic cylinder. The authors conducted extensive validation tests to demonstrate the effectiveness of the proposed technique in detecting internal leakage, considering load conditions. The method requires neither models of a cylinder nor leak-related failure, nor does it require any basic information on the operation of a functional cylinder. It is suitable for detecting small internal leakages, ranging from 0.2 to 0.25 L/min.

The work [28] focused on the diagnosis of piston seal wear and the resulting internal leakage from the double-acting seal combination used in the oil-bearing cylinder of a mobile crane. The structural simplicity and easy practical implementation of the method for detecting failures caused by internal leakage and a tolerant control system for single-rod hydraulic cylinders have been reviewed in [30]. Whereas the authors of [27] focused on searching for failure features that could predict premature hydraulic cylinder untightness. Inlet and outlet pressure signals, as well as the hydraulic cylinder piston rod position signal, have been generated at a test bench that simulates various hydraulic cylinder internal leakage levels. It was demonstrated that there are five features that can be applied to distinguish internal leakage levels. The inlet and outlet pressures, as well as the pressure differences in two chambers, were used in [30] as monitoring parameters for leakage identification. The aforementioned diagnostic methods were based on studying systems under high load and pressure conditions. It is far more difficult to diagnose an internal leakage under low loads. That is why the authors of [31] proposed a method for automatic diagnosis of internal leakage under low loads based on optimizing a deep belief network (DBN).

AI-based diagnostic methods include an expert system [32], neural network [33,34,35,36], fuzzy logic [37], and support vector machine methods [38,39].

An internal leakage is the most common fault of a hydraulic cylinder. Increased leakage rates reduce the volumetric capacity, pressure, and speed of the hydraulic cylinder, which may significantly impact the normal operation of a hydraulic system. Therefore, it is extremely important to accurately identify the leakage and its magnitude, especially to measure it online.

Such an internal leakage online measurement method and principle, including a strain gauge measuring system, was proposed in [35]. The authors developed an experimental system for collecting internal leakage and strain values, followed by processing strain values using a created mathematical model for converting a flow-strain signal. A convolutional neural network (CNN) is used to predict hydraulic cylinder leakages and compare experimental results.

A similar approach to assessing internal leakage was presented in [36]. A model for simulating a small internal leakage from hydraulic cylinders was proposed to convert the hydraulic oil leak volume into strain signals using precise strain gauges, and to convert acquired strain signals using various neural networks, for the purposes of developing a computational model and obtaining model prediction results.

The authors of [40] presented a method for estimating the durability of a power hydraulics hydraulic propulsion assembly based on monitoring changes in its technical condition. Technical condition monitoring enables appropriately early detection of a condition prior to a hydraulic assembly failure. The novel feature of the method is employing the principle of determining a control parameter advance tolerance to detect a pre-failure state of the assembly. Advance tolerances are a set of control parameters between threshold and permissible levels.

In [41], it was shown that even a small angular deflection of the spool under an imbalance force is a major cause of valve seal failure. Therefore, a predictive model of valve seal leakage was developed considering the cross-sectional shape, leakage channel height, and spool deflection. The results of the model tests showed that spool deflection was a key factor influencing the leakage rate. The results of the model tests were verified on the built experimental platform. The experimental results confirmed the accuracy of the theoretical calculations.

An internal leakage in a hydraulic cylinder is currently considered one of the most critical issues within the hydraulic industry. It contributes to significant losses in company profits due to additional energy consumption, deteriorated equipment performance, and a contaminated environment. In light of the fact that internal leakages are easy to locate and diagnose, research conducted in the field of internal leakage identification and control seems justified. This paper supplements and expands on existing studies on simulating and diagnosing failures associated with the issues of untightenings and leakages in electro-hydraulic controller precision pairs. However, non-destructive online detection of internal leakages is difficult to conduct because of the dynamic operating conditions of hydraulic cylinders. Therefore, the authors of this article proposed a proprietary and unprecedented in-source literature approach towards the impact of operational wear of precision pairs and working fluid operating parameters (primarily temperature) on the flow rate of an AT electro-hydraulic controller precision pair internal leakage. The model was verified using data constituting electro-hydraulic controller precision pair clearance values as determined through measurements involving an actual object with a specific AT operation period, which is an original model component.

2. Automatic Transmission (AT) Operational Properties

A contemporary automatic transmission (AT) is a mostly maintenance-free assembly, providing users with driving comfort that is the consequence of its following advantages:

• no traditional clutch releases the driver from the need to operate it while driving,
• there is no need to engage gears, therefore, the driver keeps both hands on the steering wheel, thus increasing driving safety and comfort,
• possible travel at very low speeds (creep) in high urban traffic,
• changing gears with virtually no temporary loss of driving force,
• high durability and reliability, as well as low failure rate.

The AT is an assembly with a high degree of structural complexity. ATs are controlled through a complex hydraulic system, the primary components of which are an electro-hydraulic controller, a hydraulic pump with a filter, hydraulic cylinders for clutches and multi-disc wet brakes, a converter, and a hydraulic fluid (ATF) cooler. The ATF has two major functions. That of a hydraulic fluid and a lubricating oil.

There are several dozen distribution components in an electro-hydraulic controller. Each distribution component may include several precision pairs (spool plunger—cylindrical housing) constituting couplings, supplied with high-pressure hydraulic fluid. Figure 1 shows a diagram of a passenger car AT hydraulic control system with a marked ATF flow direction for the first gear ratio D/1.

Effective functioning of a hydraulic control system, and hence, correct AT operation, is preconditioned upon an appropriate ATF pressure value, which depends on oil quality (viscosity) and its losses due to flow through gaps resulting from clearances in the precision pairs of electro-hydraulic controller distribution components and other couplings operating under high pressure. The phenomenon of fluid flow through a gap (clearance) between two coupling components (hydraulic precision pair) is characterized by flow rate through an untight section and defined as an “internal leakage” [42].

One distribution component may have several precision pairs (couplings) supplied with pressurized hydraulic fluid; therefore, several areas with hydraulic fluid under varying pressure are separated by a hydraulic gap. The total “internal leakage” value for all precision couplings with a different pressure of hydraulic fluid within the gap between the distribution spool and the sleeve is called a total leakage. It is the total flow rate of the fluid through untight sections (clearances) of all hydraulic components (precision pairs) participating in the implementation of a specific electro-hydraulic controller function.

The phenomenon of fluid flow through a gap (clearance) between two coupling components (hydraulic precision pairs) of an AT is present in each electro-hydraulic controller design and at each of its operating stages. These clearance values are made up of the mounting clearance and clearance increment resulting from the operational abrasive wear of precision pair precise components:

$$c_r=c_{m }+\mathsf{\Delta }{c}_e [\mathrm{mm}],$$

(1)

where: c_m—radial mounting clearance [mm], Δc_e—radial clearance increment resulting from abrasive wear during operation [mm].

The actual radial clearance in the course of hydraulic precision pair operation, so-called effective radial clearance c_re is impacted by [43,44]:

• pressure acting upon a control component (mechanical strains) leading to reduced clearance values Δc_c,
• thermal expansion of a distribution valve spool—reduced clearance value Δc_Ts,
• thermal expansion of electro-hydraulic controller body (sleeve) leading to increased clearance value Δc_Tk.

A change in the clearance (Δc_c) caused by pressure can be neglected due to its minor values (≤1000 kPa). The impact of thermal expansion of hydraulic precision pair components on a change in the radial clearance is significant since the operating temperature T_e of ATF and transmission components ranges from 70 to 90 °C. The temperature difference ΔT between the radial clearance measurement temperature c_r (taken at an ambient temperature t_o = 20 °C) and the temperature of hydraulic precision pair components in operation may be up to 70 °C. As a result, the relationship describing the effective radial clearance of a hydraulic precision pair (c_re) has the following form:

$$c_{re}=c_{r }+\mathsf{\Delta }{c}_{Tk}-\mathsf{\Delta }{c}_{Ts}=c_{r }+(\frac{\left({\beta }_k·D_k·∆T_k\right)-\left({\beta }_s·d_s·∆T_s\right)}{2}) [\mathrm{mm}],$$

(2)

where: Δc_Tk, Δc_Ts—clearance change resulting from thermal expansion of, respectively, controlled body and distribution spool [mm], β_k, β_S—linear thermal expansion coefficient for, respectively, body and spool material [1/°C], D_k, d_s—diameter of, respectively, opening in body and distribution spool [mm], ΔT_S, ΔT_S—temperature difference between the actual operating temperature of, respectively, body and distribution spool, and their temperature during the measurement.

Increased ATF flow occurs in the course of vehicle operation due to increasing clearances in precision pair couplings of the AT hydraulic system. This leads to reduced pressure and may cause electro-hydraulic controller operation interference and, hence, hinder AT operation.

The working medium in ATs is the AFT, which simultaneously acts as a hydraulic fluid, working medium for hydrokinetic torque converters, transmission oil, oil for wet clutches, and cooling medium for automatic transmission components. ATF operates in an adverse, oxidizing environment and is subjected to intensive shear. The ATF operating temperature is in the 80–90 °C range and is maintained for approximately 80% of the AT operation time. High operating temperatures in the clutch and brake lining friction area lead to operation at near-to-overheating temperatures, which favors exceeding the permissible temperature of oil and, hence, the oil’s thermal decomposition and irreversible chemical changes. High oil flow rates and contact with air promote its oxidation. As a result, ATF physical and chemical properties undergo changes in the course of operation, which often entails a decrease in its viscosity.

Kinematic viscosity being lower relative to fresh ATF is the reason for the increased flow rate of precision pair internal leakages. Reduced ATF viscosity also entails a reduction in the volumetric capacity of a hydraulic gear pump, which in consequence leads to a decrease in its capacity. There may be a phenomenon at a certain stage of AT operation and under specific vehicle driving conditions wherein the demand for the working fluid stream (increased due to high losses through precision pair leakages) exceeds the hydraulic pump capacity. Such a situation may entail interference in the operating processes of the hydraulic control process and consequently lead to AT malfunction. According to the manufacturers of modern automatic transmissions, they are maintenance-free assemblies that do not require periodic replacement of ATF throughout the entire service life. If necessary, the fluid level can be topped up.

An assembly that ensures appropriate pressure within the hydraulic system of the analyzed AT is an internal-gear pump with a sickle insert that constitutes one unit with the pump body (Figure 2a). This pump does not have a clearance compensation system. The task of the stationary crescent insert is to separate the suction space from the supply space. Such a solution reduces hydraulic fluid stream pulsations. The theoretical speed characteristic Q_p = f(n_p) of a pump can be determined based on geometrical dimensions and structural relationships. In the case of positive-displacement pumps, Q_p can be calculated from the relationship:

$$Q_p=\frac{q_p}{{10}^6}\cdot n_p\cdot {\eta }_{\upsilon p} [{\mathrm{dm}}^3/\min ],$$

(3)

where: q_p—pump unit capacity (geometrical capacity) [mm³], n_p—pump rotor speed (equal to engine speed) [rpm], η_vp—pump volumetric capacity factor, η_vp = 0.95.

Pump unit capacity q_p is determined using the relationship:

$$q_p=\pi \cdot m\cdot z\cdot b\cdot h, [{\mathrm{mm}}^3],$$

(4)

where: m—tooth module, z—pinion teeth number, b—pinion and rotor gear width, h—tooth height.

The theoretical speed characteristics Q_p = f(n_p) of a passenger car transmission hydraulic pump for the following data—m = 4.5 mm, z = 12, b = 12 mm, h = 8.4 mm—is shown in Figure 2b.

The objective of the study involves model research and assessing the impact of increased clearance (wear) of distribution component precision pairs and the reduced fluid viscosity caused by its increased operating temperature on the rate of ATF flow through internal untight sections (leakages) of an AT electro-hydraulic controller and other couplings operating under pressure. Parameters from measuring the precision pairs of an actual object were used as input data for model computations and validations. This enabled calculating the internal leakage flow rates (total internal leakage) in the electro-hydraulic controller hydraulic precision pairs for various operating conditions and states of an automatic transmission.

3. Model of Fluid Flows through Internal Gaps of AT Hydraulic Precision Pairs

The impact of AT hydraulic system precision pair wear on the internal leakage ATF flow rate can be assessed based on analyzing the results of automatic transmission electro-hydraulic controller internal leakage model tests. The input data required for the model include characteristics of changes in ATF kinematic viscosity and density, along with temperature, as well as operating process parameter waveforms for the hydraulic system—pressure in different sections of the transmission hydraulic control system. The computation block (Figure 3) was created based on a mathematical model for internal leakage of a sample electro-hydraulic controller of an automatic transmission. Developing the mathematical model required prior study of the structural model and the functional relationships between its components. Furthermore, the prerequisite was a decomposition of the actual object to obtain the physical model parameters through determining the characteristic dimensions of the actual object’s precision pairs.

The internal leakage value constitutes information that enables determining, which part of the hydraulic pump instantaneous capacity is lost to the electro-hydraulic controller internal leakage flow.

3.1. Hydraulic Gap Models

Minimum fluid flow in the friction area ensures lubrication of hydraulic precision pair components. Hydraulic fluid flows from an area of higher pressure to an area of lower pressure. The nature of the flow is laminar, and its rate depends on the geometric dimensions of the gap and the hydraulic fluid properties. Leakage volume is proportional to the gap pressure drop. Assuming an ideal cylindricity of the hydraulic spool, the existing relationships for internal leakages have been determined based on empirical studies. These formulas do not include the mating surface roughness profile, since one execution tolerance class resulting from the type of applied removal machining (lapping) is assumed for all hydraulic precision pairs. Existing theoretical models enable determining the total flow rate for internal leakage in the precision pairs of an electro-hydraulic controller of an automatic transmission in various operating states. These models can be employed to determine the impact of precision pair wear and hydraulic fluid properties on the operating processes of an automatic transmission hydraulic system. Hydraulic precision pair internal leakage values can be calculated theoretically. The equation for internal leakage for hydraulic precision steam with a concentric ring gap is of the form [45]:

$$q_{ve}={10}^6·\frac{{(p}_l-p_0)·\mathsf{\pi }·d_1·c_r^3·\rho (\frac{p_l+p_0}{2})}{12·\mathsf{\mu }·l_c· \rho \left(0\right)} [{\mathrm{mm}}^3/\mathrm{s}],$$

(5)

where: p_l—supplied side pressure [kPa], p₀—leakage side pressure [kPa], (p_l − p₀)—gap pressure drop [kPa], d₁—hydraulic precision pair component (shaft) diameter [mm], c_r—hydraulic precision pair radial clearance [mm], ρ(0)—hydraulic fluid density at atmospheric pressure [g/cm³], $\rho \left(\frac{p_l+p_0}{2}\right)$—hydraulic fluid density taking into account pressure action [g/cm³], µ—hydraulic fluid dynamic viscosity, µ = ν·ρ [Pa∙s], ν—kinematic viscosity [mm²/s], l_c—length of contact between precision pair component and cylindrical surface [mm].

The following needs to be taken into account when determining a hydraulic precision pair internal leakage value:

• temperature-related change in kinematic viscosity,
• hydraulic fluid thermal expansion—density change.

Figure 4 shows a diagram of a hydraulic precision pair with a concentric annular gap and pressure distribution in the gap in the longitudinal and transverse directions.

For a hydraulic fluid pressure in the range of 100–30,000 kPa, it is assumed that the density of the hydraulic fluid in a hydraulic gap $\rho \left(\frac{p_l+p_0}{2}\right)$ under pressure is equal relative to the fluid density determined for atmospheric pressure. Hydraulic fluid density for an adopted pressure range (100–30,000 kPa) does not change significantly; therefore, the change in its density under pressure action is neglected. Assuming that the pressure in the automatic transmission hydraulic pressure ranges from 100 to 30,000 kPa, Equation (3) for hydraulic precision pair internal leakage flow rate adopts a simplified form:

$$q_{ve}={10}^6·\frac{(p_l-p_0)·\mathsf{\pi }·{ d}_1·c_r^3}{12·v·\mathsf{\rho }\left(0\right)·l_c} [{\mathrm{mm}}^3/\mathrm{s}],$$

(6)

The value of internal leakages q_ve decreases linearly along with increasing kinematic viscosity v, and increases to the third power along with growing radial clearance c_r. The value of the fluid stream flowing through a hydraulic gap is impacted by the eccentric position of the distribution spool in the sleeve (Figure 5). In consequence, the equation for flow rate q_ve through internal leakages in a hydraulic precision pair with an eccentric annular gap has the form:

$$q_{ve}={10}^6·\frac{\left(p_l-p_0\right)·\mathsf{\pi }·{ d}_1·c_r^3}{12·\mathsf{\mu }·l_c}·\left(1+1.5·\frac{e^2}{c_r^2}\right) [{\mathrm{mm}}^3/\mathrm{s}],$$

(7)

where: e—eccentricity size [mm].

In the case of an eccentric annular gap, the nature of pressure distribution is the same in the longitudinal direction (linear), as for a concentric gap, while the internal leakages are greater [46].

A different case are internal leakages in components with an O-ring rubber seal (circular cross-section) in a moving connection—Figure 6. The case in question enables determining the internal leakage flow rate only in an experimental manner. According to theoretical considerations, it is assumed that there is no internal leakage for such a coupling [46,47].

In reality, internal leakages in such couplings occur, and their value depends on a number of factors, such as seal flexibility (rubber hardness), structural pre-clamp, clearance in the annular groove, degree of wear of the outer rubber ring part, and relative component speed.

In the course of operating a hydraulic component with a rubber seal, hydraulic fluid pressure causes the seal to be pressed to the cylindrical surface with a pressure force N₁ (Figure 6). The interface surface depends on the degree of seal strain through hydraulic fluid pressure. As the component is used, the outer part of the seal is abraded (flattened) and permanently deformed. The consequence is an increased actual area of contact between the seal and the cylindrical surface, hence, higher unit pressure, which forces the ring into the gap, thus increasing its destruction rate. The greater the pressure and gap size, and the lower the sealing material hardness, the more intense the sealing ring is forced into the gap. As the operation time of such a connection elapses, the rubber sealing ring loses its pre-clamp and flexibility.

The phenomenon of obliteration occurs in the case of narrow gaps with a radial clearance range of 4–20 µm (capillary). The boundary layer thickness, which conditions gap obliteration for commonly used hydraulic fluids, is 4–5 µm. When fluid flows through narrow gaps, this layer may constitute a significant part of a nominal cross-section. Obliteration process intensity largely depends on the period the hydraulic component is impacted by fluid pressure and the gap pressure drop value (it increases as it grows). The working motion of the hydraulic component leads to gap obliteration and destruction. Total gap obliteration occurs only in narrow gaps below 5 µm. In the case of larger gaps (c_r ≥ 20 µm), we can observe active gap cross-section reduction, leading to a decreased flow rate of the hydraulic fluid flowing through the gap. The obliteration process is also dependent on the degree of hydraulic fluid contamination. The greater the concentration of hydraulic fluid impurities, the more intensive this process is. In the case of gaps larger than 20 µm, obliteration is virtually non-present, and it can be neglected in calculating the flow rate of a hydraulic fluid flowing through a precision pair gap [48].

3.2. Assumptions to the Electro-Hydraulic Controller Internal Leakage Model

Figure 7 shows an example structural model of automatic transmission electro-hydraulic controller internal leakages in second gear. Developing structural models for specific gears involves separating these components from an automatic transmission electro-hydraulic controller model, which are currently in contact with pressurized hydraulic fluid. These structures can be described mathematically in terms of internal leakages for specific precision pair couplings of a given hydraulic component.

The transmission electro-hydraulic controller model has been divided into four structural models of the internal leakage flow rate, according to the attached transmission gear criterion. The model covers four forward gears (selector level in position D). The following assumptions were adopted when modelling electro-hydraulic controller internal leakages:

• the model takes into account the stationary states of distribution valve spool positions,
• pressure relationships (p_s, p_m) in other sections, besides primary pressure p_g, were determined analytically (object measurement impossible),
• pressure relationship for controller discharge channels was determined theoretically, as structurally assumed—a maximum 10% of the primary pressure p_g,
• the phenomenon of obliteration was neglected in the calculation due to high (c_r > 20 µm) hydraulic precision pair clearances obtained during the measurements on actual objects and minor (≤1000 kPa) hydraulic gap pressure drops,
• the impact of material thermal expansion on hydraulic precision pair radial clearance was taken into account (effective clearance c_e).

Pressures of the hydraulic fluid in individual system sections are required to model controller internal leakages:

• main supply pressure p_g,
• control pressure p_s,
• modulated pressure p_m,
• converter p_k.

The scope of primary p_g and converter p_k pressures were determined experimentally during road tests, using a specially designed diagnostic tool, the main components of which were two piezoelectric ATF pressure sensors [10]: a temperature sensor, a rotational speed sensor employing the Hall phenomenon, and a signal processing module that enables recording, reading, and processing of measurement results on a PC computer.

However, it is technically impossible to measure control p_s and modulated p_m pressures in a working transmission. The values of control p_s and modulated p_m pressure in an actual system should fall within a range of 0–500 kPa, since these parameters describe MV solenoid valves. The relationship for control pressure p_s, which is directly proportional to primary pressure p_g can be determined analytically using a model of a distribution valve spool in a force equilibrium state (Figure 8). Then, the force equilibrium equation takes the form:

$$\mathsf{\Sigma }{F}_x=F_{s1}-F_{g1}+F_{g2}-F_{s2}+F_{s3}-F_{sp17}=0,$$

(8)

where: $F_{si}$—forces longitudinal to control pressure,$F_{gi}$—forces longitudinal from the primary pressure, $F_{sp17}$—longitudinal force from the distribution valve spring.

Distribution spool longitudinal force F_s1 $p_s$ resulting from the impact of control pressure on the spool is calculated from the relationship:

$$F_{s1}=p_s·A_1 [\mathrm{N}],$$

(9)

Distribution spool longitudinal forces F_s2(3) generated due to the impact of control pressure p_s on surfaces perpendicular to the spool axis are calculated from the relationship:

$$F_{s2\left(3\right)}=p_s·A_{2\left(3\right)W} [\mathrm{N}],$$

(10)

where: A_2(3)W—area of surface perpendicular to the spool axis [mm²].

Spool surface area in consideration of the spool axis is calculated from the relationship:

$${ A}_{2\left(3\right)W}=\frac{\pi ·{(d}_{2\left(3\right)}^2-d_w^2)}{4} [{\mathrm{mm}}^2],$$

(11)

where: $d_{2\left(3\right)}$—spool outer diameter [mm], $d_w$—spool axis diameter [mm].

The longitudinal force $F_{sp17}$ caused by the spring supporting the spool in an equilibrium position should be determined from the relationship:

$$F_{sp17}=k_{17}·x_{17} [\mathrm{N}],$$

(12)

where: k₁₇—spring constant [N/mm], $x_{17}$—spring deflection for a spool in the equilibrium position.

The spring constant can be determined from the relationship [48]:

$$k_{17}=\frac{G·d_{d17}^4}{8·z_{17}·D_{p17}^3} [\mathrm{N}/\mathrm{mm}],$$

(13)

where: G—Kirchoff modulus for spring steel:${ G=0.8·10}^5$ [GPa], $d_{d17}—$spring wire diameter [mm], $z_{17}—$number of active spring coils, $D_{p17}—$spring pitch diameter [mm],

The spool ring is pre-tensioned during installation. Spring deflection $x_{17}$ for a spool in the equilibrium position is calculated from the relationship:

$${ x}_{17}=l-l_n [\mathrm{mm}],$$

(14)

where:$l$—pre-tensioned spring length [mm], $l_n$—spring length in a preset distribution spool position [mm].

The relationship for control pressure p_s of the controller relative to the primary pressure p_g takes the form:

$$p_s=\frac{p_g·\left(A_{2W}-A_{1W}\right)+F_{sp17}}{A_1} [\mathrm{kPa}].$$

(15)

The relationship for modulated pressure p_m of the controller relative to primary pressure p_g was determined to be linearly dependent on primary pressure p_g. The maximum value of modulated pressure limited by the MV valve, p_m = 500 kPa, corresponds to the maximum value of primary pressure value p_g. Such an assumption enabled determining the relationship for modulated pressure p_m in the form of:

$$p_m={\alpha }_m·p_{g, }{p}_m\le 500 \left[\mathrm{k}\mathrm{P}\mathrm{a}\right],$$

(16)

where: ${\alpha }_m—$coefficient of proportionality relative to primary pressure $p_g$ determined experimentally during road tests [10].

The mathematical description was developed based on distribution and actuation components separated from a hydraulic diagram with their connection system. In addition, structural models take into account the method of hydraulic fluid supply and distribution spool position within the electro-hydraulic controller body. This enables using a mathematical relationship to describe the total internal leakage of a hydraulic precision pair.

The total leakage equations for the second gear take the form:

$$\mathsf{\Sigma }{q}_{D/2}=\mathsf{\Sigma }{q}_{p_{g_2}}+\mathsf{\Sigma }{q}_{p_{s_2}}+\mathsf{\Sigma }{q}_{p_{m_2}}+\mathsf{\Sigma }{q}_{p_{k_2}} [{\mathrm{mm}}^3/\mathrm{s}],$$

(17)

where: $\mathsf{\Sigma }{q}_{D/2}$—total hydraulic system leakage in second gear, $\mathsf{\Sigma }{q}_{p_{g2}}$, $\mathsf{\Sigma }{q}_{p_{s_2}}$, $\mathsf{\Sigma }{q}_{p_{m_2}}$, $\mathsf{\Sigma }{q}_{p_{k_2}}$—total leakage in the section supplied with primary, modulated and converter pressures, respectively, in second gear.

The total leakage equation for the section supplied with primary pressure in second “2” gear takes the form:

$$\begin{gathered} \mathsf{\Sigma }{q}_{p_{g2}}=\mathsf{\Sigma }{q}_{p_{g2}5}+\mathsf{\Sigma }{q}_{p_{g2}7}+\mathsf{\Sigma }{q}_{p_{g2}8}+\mathsf{\Sigma }{q}_{p_{g2}9}+\mathsf{\Sigma }{q}_{p_{g2}11}+\mathsf{\Sigma }{q}_{p_{g2}14}+\mathsf{\Sigma }{q}_{p_{g2}15}+\mathsf{\Sigma }{q}_{p_{g2}16}+ \\ \mathsf{\Sigma }{q}_{p_{g2}17}+\mathsf{\Sigma }{q}_{p_{g2}20}+\mathsf{\Sigma }{q}_{p_{g2}23}+\mathsf{\Sigma }{q}_{p_{g2}25}+\mathsf{\Sigma }{q}_{p_{g2}26} [{\mathrm{mm}}^3/\mathrm{s}], \end{gathered}$$

(18)

where: $\mathsf{\Sigma }{q}_{p_{g2}i}$—total leakage in the section supplied with primary pressure p_g for hydraulic dampers (i = 5, 7, 11) and distribution slide valves (i = 8, 9, 14, 15, 16, 17, 20, 23, 25, 26) in second gear.

For example, the equation for total leakage in the section supplied with primary pressure for hydraulic dampers “5” and “8” in second gear takes the form:

$$\mathsf{\Sigma }{q}_{p_{g2}5}=q_{p_{g2}5-1}={10}^6·\frac{\left(p_g-p_m\right)·\mathsf{\pi }·d_{1-5}·c_{re1-5}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c1-5}} [{\mathrm{mm}}^3/\mathrm{s}].$$

(19)

Whereas the equation for total leakage in the section supplied with primary pressure for distribution valve “8” and “25” in second gear takes the corresponding form:

$$\begin{gathered} \mathsf{\Sigma }{q}_{p_{g2}8}=q_{p_{g2}8-1}+q_{p_{g2}8-2}={(10}^6·\frac{\left(p_g-p_m\right)·\mathsf{\pi }·d_{1-8}·c_{re1-8}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c1-8}})+ \\ {(10}^6·\frac{\left(p_g-0.1·p_g\right)·\mathsf{\pi }·d_{2-8}·c_{re2-8}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c2-8}}) [{\mathrm{mm}}^3/\mathrm{s}], \end{gathered}$$

(20)

$$\begin{gathered} \mathsf{\Sigma }{q}_{p_{g2}26}=q_{p_{g2}26-1}+q_{p_{g2}26-2}+q_{p_{g2}26-3}+q_{p_{g2}26-4}= \\ {(0.5·10}^6·\frac{p_g·\mathsf{\pi }·d_1·c_{re1}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c1}})+{(0.5·10}^6·\frac{p_g·\mathsf{\pi }·d_1·c_{re2}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c2}})+ \\ {(0.5·10}^6·\frac{p_g·\mathsf{\pi }·d_3·c_{re3}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c3}})+{(0.5·10}^6·\frac{p_g·\mathsf{\pi }·d_4·c_{re4}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c4}}) [{\mathrm{mm}}^3/\mathrm{s}]. \end{gathered}$$

(21)

It is similarly possible to express the equations for the total leakage in the section supplied with primary pressure for the other hydraulic dampers and distribution slide valves.

The total leakage equation for the section supplied with control pressure in second “2” gear takes the form:

$$\begin{gathered} \mathsf{\Sigma }{q}_{p_{s2}}=\mathsf{\Sigma }{q}_{p_{s2}13 }+\mathsf{\Sigma }{q}_{p_{s2}14}+\mathsf{\Sigma }{q}_{p_{s2}15}+\mathsf{\Sigma }{q}_{p_{s2}16}+ \\ \mathsf{\Sigma }{q}_{p_{s2}17}+\mathsf{\Sigma }{q}_{p_{s2}18}+\mathsf{\Sigma }{q}_{p_{s2}19} [{\mathrm{mm}}^3/\mathrm{s}], \end{gathered}$$

(22)

where: $\mathsf{\Sigma }{q}_{p_{s2}i}$—theoretical total leakage in the section supplied with control pressure p_s for distribution slide valves (i = 13, 14, 15, 16, 17, 18, 19) in second gear.

For example, the equation for total leakage in the section supplied with control pressure for distribution valve “13” in second gear takes the form:

$$\mathsf{\Sigma }{q}_{p_{s2}13 }=q_{p_{s2}13-1}={10}^6·\frac{\left(p_s-{0.1·p}_g\right)·\mathsf{\pi }· d·c_{re1-13}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c1-13}} [{\mathrm{mm}}^3/\mathrm{s}].$$

(23)

It is similarly possible to express the equations for the total leakage in the section supplied with control pressure for the other distribution slide valves.

The total leakage equation for the section supplied with modulated pressure in second gear takes the form:

$$\mathsf{\Sigma }{q}_{p_{m2}}=\mathsf{\Sigma }{q}_{p_{m2}7 }+\mathsf{\Sigma }{q}_{p_{m2}8 }+\mathsf{\Sigma }{q}_{p_{m2}12 }+\mathsf{\Sigma }{q}_{p_{m2}{12}^{’} } [{\mathrm{mm}}^3/\mathrm{s}],$$

(24)

where: $\mathsf{\Sigma }{q}_{p_{m2}i}$—total theoretical leakage in the section supplied with modulated pressure p_m for hydraulic dampers (i = 7, 12′) and distribution slide valves (i = 8, 12) in second gear.

For example, the equation for total leakage in the section supplied with modulated pressure for hydraulic dampers “7” in second gear takes the form:

$$\mathsf{\Sigma }{q}_{p_{m2}7}=q_{p_{m2}7-1}={10}^6·\frac{\left(p_m-{0.1·p}_m\right)·\mathsf{\pi }·d_{1-7}·c_{re1-7}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c1-7}} [{\mathrm{mm}}^3/\mathrm{s}].$$

(25)

The equation for total leakage in the section supplied with modulated pressure for distribution valve “8” in second gear takes the form:

$$\mathsf{\Sigma }{q}_{p_{m2}8}=q_{p_{m2}8-1}={10}^6·\frac{\left(p_m-{0.1·p}_m\right)·\mathsf{\pi }·d_{1-8}·c_{re1-8}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c1-8}} [{\mathrm{mm}}^3/\mathrm{s}].$$

(26)

It is similarly possible to express the equations for the total leakage in the section supplied with modulated pressure for the other hydraulic dampers and distribution slide valves.

The total leakage equation for the converter section supplied in second gear takes the form:

$$\mathsf{\Sigma }{q}_{p_{k2}}=\mathsf{\Sigma }{q}_{p_{k2}20 }+\mathsf{\Sigma }{q}_{p_{k2}22}+\mathsf{\Sigma }{q}_{p_{k2}23} [{\mathrm{mm}}^3/\mathrm{s}],$$

(27)

where: $\mathsf{\Sigma }{q}_{p_{k2}i}$—theoretical total leakage in the converter section p_k of distribution slide valves (i = 20, 22, 23) in second gear.

For example, the equation for total leakage in the converter section for distribution valve “20” in second gear takes the form:

$$\begin{gathered} \mathsf{\Sigma }{q}_{p_{k2}20}=q_{p_{k2}20-1}+q_{p_{k2}20-2}= \\ {10}^6·\frac{p_k·\mathsf{\pi }·d_{2-20}·c_{re2-20}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c2-20}}+{10}^6·\frac{\left(p_k-{0.1·p}_k\right)·\mathsf{\pi }·d_{2-20}·c_{re2-20}^3}{12·\mathsf{\nu }·\mathsf{\rho }\left(0\right)·l_{c2-20}} [{\mathrm{mm}}^3/\mathrm{s}]. \end{gathered}$$

(28)

It is similarly possible to express the equations for the total leakage in the converter section for the other distribution slide valves.

3.3. Determination of Hydraulic Precision Pair Geometric Dimensions

The geometry of hydraulic precision pairs, resulting from their abrasive wear in the course of operation, is significant from the perspective of the size of the electro-hydraulic controller coupling internal leakages. Therefore, the input data for model tests of the total flow rate of electro-hydraulic controller internal leakages of ATF was the geometric parameters of the actual object’s precision pairs. To this end, the authors analyzed the functions of individual controller components and the controller fluid channel supply diagram when implementing specific AT gears. Determining significant features and properties of the test object required its decomposition and a number of automatic transmission electro-hydraulic controller precision pair geometry measurements, the typical diagrams of which are shown in Figure 9:

• distribution spool outer diameter—d_i [mm],
• distribution spool cylindrical opening (sleeve) outer diameter—D_i [mm],
• length of contact between cylindrical openings and the distribution spool working surface—l_c [mm],
• hydraulic damper piston rod outer diameter—d_i [mm],
• hydraulic damper cylindrical opening (sleeve) outer diameter—D_i [mm],
• length of contact between hydraulic damper precision pair components and a mating cylindrical opening surface—l_c [mm].

Outer diameter of distribution valves, hydraulic damper piston rods, and the sleeve was taken using a workshop microscope with an accuracy of up to 0.001 mm. They were employed to calculate the radial clearance c_r of hydraulic precision pairs using the relationship:

$$c_{r\left(i\right)}=\frac{(D_i-d_i)}{2} [\mathrm{mm}],$$

(29)

where: $D_i$—distribution valve or hydraulic damper cylindrical opening outer diameter, $d_i—$distribution spool or hydraulic damper piston rod outer diameter.

The measurements were taken in two or three planes “1”, “2” and “3”, and two mutually perpendicular directions x₁-x₂ and y₁-y₂ (Figure 10).

Each of the measurements was repeated twice. In addition, a slide caliper (equipped with a dial gauge with a gauge vernier scale interval equal to 0.02 mm) was used to measure the length of contact l_c between cylindrical openings and the working surface of distribution spools and the length of contact between hydraulic damper piston rod precision pair components and the sleeve surface. The measurements were taken in a laboratory where the ambient temperature was 20 ± 0.5 °C. Spool and damper piston rod diameter was measured in a plane perpendicular to the spool axis, approximately at the half of the cylindrical length—minor cylindrical section lengths. Examples of measurement results are listed in

Table 1. Sample results of controller precision pair measurements.

No. Acc. to Figure 8	Component	Measurement Plane	Measurement Direction:	Coordinate x1 (y1) [mm]	Coordinate x2 (y2) [mm]	Diameter: x1-x2 (y1-y2) [mm]	Diameter Mean Value di (Di) [mm]
6	spool	1	0°	89.605	79.666	9.939	9.939
				79.666	89.605	9.939
			90°	86.422	76.483	9.939
				76.483	86.422	9.939
		2	0°	84.371	74.430	9.941	9.942
				74.430	84.371	9.941
			90°	85.121	75.178	9.943
				75.178	85.121	9.943

.

Figure 11 shows electro-hydraulic controller blocks in a passenger car gearbox. The electro-hydraulic controller block body and certain distribution spools are made of aluminum alloys. The remaining distribution spools and hydraulic damper plungers are made of steel.

The diameter measurements of the cylindrical openings (sleeve) in the body were taken in the direction perpendicular to the opening plane. In reality, it is a position on the sleeve that is not subject to operational wear since it has no contact in the course of operation with a second precision pair component—the distribution valve spool or hydraulic damper piston rod. Mainly, the cylindrical sleeves of controller aluminum blocks are subject to operational abrasive wear in places most frequently in contact with a second precision pair component. For the purposes of measuring the cylindrical opening (sleeve) diameter, the controller block body was positioned on the planned closing cover faying surface.

Figure 12 shows the results of clearance measurements and calculations for selected typical precision pairs of a passenger car transmission electro-hydraulic controller. The length of contact l_c between a precision pair component and the cylindrical surface is mostly determined by the width of the fin supporting the distribution valve spool (actual contact length—Figure 9a) and not the cylindrical surface length, as in the case of the hydraulic damper (Figure 9b).

4. Controller Internal Tightness Model Test Result Analysis

Internal leakages through hydraulic precision pair couplings of the controller were calculated for several ATF temperatures and degrees of wear of controller components. Hydraulic fluid temperature T_e change entails changes to such input values for calculating internal leakage ATF flow rate ∑q_i as kinematic viscosity v and hydraulic fluid density ρ, as well as effective radial clearance c_re described by the relationship.

Table 2. Modelling conditions for hydraulic controller internal leakage flow rates.

Variant No.	W1	W2	W3	W4	W5	W6	W7	W8
ATF fluid temperature: Te [°C]	80	80	80	80	−40	20	40	100
Nominal degree of wear Zsb [%]	0	0	10	50	50	50	50	50
Kinematic viscosity: ν [mm2/s]	10.93	8.40	10.93	10.93	1923.0	145.0	42.0	7.0
Density: ρc [g/cm3]	0.783	0.783	0.783	0.783	0.870	0.826	0.812	0.768

shows 8 modelling variants for internal leakages in hydraulic precision pair couplings of the electro-hydraulic controller:

• Variant (W1)—assumed viscosity of fresh ATF, ATF operating temperature of 80 °C, effective radial clearance c_re in hydraulic precision pairs at the values measured on an actual object. The impact of ATF temperature on component thermal expansion was taken into account in relationship (2).
• Variant (W2)—assumed value of kinematic viscosity v as obtained in the measurements for the sample after an operational mileage S = 106,315 km [10].
• Variant (W3)—data as in (W1). Changed modelled wear by increasing radial clearances in all hydraulic precision pairs by 10%.
• Variant (W4)—data as in (W1). Changed modelled wear by increasing radial clearances in all hydraulic precision pairs by 50%.
• Variant (W5)—ATF operating temperature −40 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.
• Variant (W6)—ATF operating temperature 20 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.
• Variant (W7)—ATF operating temperature 40 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.
• Variant (W8)—ATF operating temperature 100 °C, modelled consumption 50%, kinematic viscosity, and density in accordance with oil temperature.

Variants (W5–W8) were aimed at demonstrating the impact of ATF temperature on changing leakage flow rate, i.e., electro-hydraulic controller internal leakages.

The properties of ATF, which according to the manual of the studied transmission should be of the DEXRON ATF IID specification, were modelled using data from the MSDS of HIPOL ATF IID [49]. Changes in kinematic viscosity v and density ρ of the IID ATF together with temperature were determined using source literature graphs and formulas [46,50].

The changes in ATF hydraulic fluid kinematic viscosity over the temperature range of T_e = −40 ÷ 100 °C were described by an approximated relationship [10]:

ν = 343.92·e^−0.043·Tp [mm²/s],

(30)

where: regression coefficient R² = 0.9853.

ATF hydraulic fluid density changes over the temperature range of T_e = −40 ÷ 100 °C were described by a linear relationship [2].

ρ = −0.0007·T_P + 0.8409 [g/mL].

(31)

In the case of all model study variants (

Table 3. Values of internal leakage q_D through hydraulic precision pair couplings—test results for model variants W1–W8.

Modelling Variant	${\mathit{q}}_{\mathit{D}/1\to \mathit{i}+1}$ [dm3/min]	Maximum Share of Internal Leakage Relative to the Stream Generated by the Hydraulic Pump Up [%]
W1	2.16	8.89
W2	2.81	11.6
W3	3.64	14.9
W4	8.56	35.2
W5	0.021	0.086
W6	0.38	1.56
W7	1.43	5.89
W8	11.23	56.2

) with modelled consumption, it was assumed that the consumption constituted the same part of nominal (measured) radial clearance in each precision pair, expressed as a percentage (%). It is assumed that the hydraulic component temperature has the same value as the ATF.

Figure 13 shows the results of model calculations involving the characteristics of flow rate for a passenger car automatic transmission electro-hydraulic controller internal leakage under conditions of a minimum acceleration test and driving in first “1” and second “2” gear.

During the test, the vehicle is accelerated from parking through all possible, successive gears, so that the engine speed at none of the gears exceeds 2500 rpm. The test involves recording pressure values at two control points within the system: the engine speed, which is identical to the hydraulic pump shaft speed, and the ATF temperature.

Over time t_p, the controller internal leakage q for individual variants (W1–W8) adopts a similar waveform but differs in terms of values. Internal leakage values grow with the increasing conventional wear of precision pairs and ATF temperature. The curves almost simultaneously shift towards higher recorded leak values. For example, in the case of W4 and the range t_p = 0–0.8 s, internal leakages occur at a level q = 6.37–6.22 dm³/min, followed by a sudden drop to q = 5.09 dm³/min, and a rapid growth to q = 8.46 dm³/min. At t_p = 2.1 s, the leakages reach their maximum value, q = 8.56 dm³/min.

Hydraulic pump capacity Q_p depends on the engine speed, which rises rapidly during vehicle acceleration, hence the sudden increase in pump capacity Q_p—Figure 13. After the engine obtains a specific speed n = 2500 rpm, pump capacity reaches its maximum value Q_p = 33.6 dm³/min, and then stabilizes.

The waveforms of total internal leakages for hydraulic controller precision pair couplings are directly proportional to the pressure in relevant channels, limited by the components of such pairs, for a given operating configuration of a control device. The proportionality results from the relationships applied for modelling (Formula (5)). The pressure control system maintains its values as per valve characteristics by relieving excess flow generated by the hydraulic pump. In the situation of a theoretical supply stream and internal leakage equilibrium, excess flow is not released, and the system becomes unstable. Please note that the stream generated by the pomp is to supply hydraulic cylinders with significant instantaneous absorption capacity appropriately quickly. In addition, the pump also supplies the torque converter. Characteristic irregular increases and decreases arise from the fact that pressure value waveforms applied in modelling have been acquired experimentally using an actual object and demonstrate the imperfection of the control device within the studied hydraulic system (Figure 1, item 25) and an adequately high sensitivity and sampling frequency of diagnostic device sensors.

Minimum flow rate q_D/1→i+1 for electro-hydraulic controller internal leakages was marked with a green vertical dashed line, and the maximum values were marked with a red dashed line (Figure 13).

Based on the analysis of the obtained model test results, it was assumed that hydraulic precision pair coupling internal leakage values are diverse—Table 3.

Significantly greater internal leakage values q_D/1→i+1 are obtained when modelling operational wear of hydraulic precision pair components—increase in radial clearance c_re by 10% in the third variant (W3) and by 50% in variants four to eight (W4–W8). The fourth variant (W4) assumes very high abrasive wear of hydraulic controller precision pair components (50% increase in clearance) and operation at the hydraulic fluid operating temperature T_e = 80 °C. Results of the model test in variant five (W5) exhibit the significant impact of hydraulic fluid temperature T_e and temperature of controller components—spool T_s and body T_k (sleeve), regardless of, even considerable, values of radial clearances (increased by 50%)—on internal leakage values q_D/1→i+1. Assuming a hydraulic fluid temperature T_e of −40 °C and radial clearance c_re increased by 50%, the obtained internal leakage values q_D/1→i+1 amount to just 0.021 dm³/min.

Therefore, it can be inferred that in the case of a hydraulic fluid temperature T_e of −40 °C during cold start-up of the transmission (when ATF has not yet reached the operating temperature value), there are virtually no internal leakages q_D/1→i+1 through the radial clearances c_re of the electro-hydraulic controller hydraulic precision pairs. However, as ATF temperature increases, entailing a reduction in its viscosity and density (assuming a constant value of conventional degree of wear Z_sb = 50%), the maximum internal leakage values increase (Figure 14). Temperature growth within the range of T_e = −40 ÷ 20 °C does not cause a significant increase in internal leakages. A sudden increase in the maximum leakages takes place upon exceeding temperature T_e = 40 °C (q_D = 1.41 dm³/min), and when the fluid temperature is T_e = 100 °C, leakages reach values severalfold higher than at the level of q_D = 11.23 dm³/min.

It was assumed for the W6 model tests that the hydraulic fluid temperature Te was 20 °C. Internal leakages q_D/1→i+1 for this model variant (W6) were virtually non-existent—maximum q_D/1→i+1 = 0.38 dm³/min. The objective of the model studies for the assumptions set out in variant seven (W7) was to demonstrate that an increase in ATF temperature and the heating of transmission components initiates the increase in internal leakage values q_D/1→i+1. Assuming a 50% wear of the hydraulic precision pair coupling (increased radial clearance c_re) and hydraulic fluid temperature T_e = 40 °C, internal leakages q_D/1→i+1 take insignificant values—a maximum of 1.41 dm³/min. The objective of the W8 model tests was to demonstrate an extreme case—wherein the hydraulic fluid overheats at T_e = 100°C and hydraulic precision pair coupling wear is 50% (increased radial clearance c_re). Internal leakages q_D/1→i+1 in this model variant take high values ranging from 6.67 to 11.23 dm³/min, which may constitute more than 50% of the instantaneous capacity Q_p of the hydraulic pump.

5. Conclusions

The objective of the model study was to determine the impact of electro-hydraulic controller component wear and hydraulic fluid temperature T_e on the electro-hydraulic controller hydraulic precision pair internal leakage flow rate q_D/1→i+1. The following assumptions were drawn based on the obtained model study results:

(1)

The value of internal leakages q_D/1→i+1 is significantly impacted by hydraulic fluid temperature T_e and the temperature of hydraulic precision pair components above 40 °C. A temperature increase in the range of 40–100 °C leads to a considerable reduction in the kinematic viscosity v and density ρ of the fluid, and hence, an eightfold increase in internal leakages.

(2)

AT operation at low ambient temperatures makes electro-hydraulic controller hydraulic precision pair internal leakages q_D/1→i+1 adopt low values, which do not exceed a 3% share of the internal leakage relative to the stream generated by the hydraulic pump.

(3)

Excessive internal leakages q_D/1→i+1 caused by high wear of electro-hydraulic controller hydraulic precision pairs in the event of AT overheating (above 80 °C) may cause malfunctioning of the automatic transmission. In such a case, internal leakages account for a significant part (above 50%) of the instantaneous hydraulic pump capacity Q_p, which leads to primary pressure p_g decreasing below the value required for the correct operation of the automatic transmission hydraulic controller system (variant W8).

(4)

ATF viscosity v reduced in the course of operation (approx. 20%) causes significantly greater electro-hydraulic controller precision pair internal leakages q_D/1→i+1 (approximately 25% of the instantaneous hydraulic pump capacity Q_p). Controller precision pair leakages, combined with internal leaks in other system components and reduced pump volumetric capacity factor η_vp, may lead to a decrease in the primary pressure p_g below the value required for the correct operation of an automatic transmission hydraulic control system.

(5)

The wear of controller hydraulic precision pair components (increased radial clearance c_re) significantly affects the internal leakage flow rate q_D/1→i+1, especially when the temperature T_e of ATF in the transmission reaches or exceeds the operating value (T_e ≥ 80 °C), which has been evidenced by a model test for an ATF with a temperature value of T_e = 100 °C.

(6)

Minor wear (radial clearance c_re up by 10%) of controller hydraulic precision pair components (between the distribution spool and sleeve) and the operation of the AT hydraulic system under operating temperature conditions (T_e = 80 °C) means that the hydraulic precision pair internal leakage value flow rate q_D/1→i+1 may adversely impact AT operation. In such a case, internal leakages may reach a value of up to 20% of the hydraulic pump instantaneous capacity Q_p—variant W3.

Assessing the technical condition (diagnostic tests) of a hydraulic system should involve measuring radial clearance c_r between the spool and sleeve of a distribution valve, and its value should be compared with the assembly clearance c_m value in the hydraulic precision pair couplings of an electro-hydraulic controller. Measurements of controller precision pair components should be taken after analysing the results of road tests and assessing the degree of hydraulic fluid (ATF) contamination.

In the case of diagnosing a transmission, where conducting road tests is impossible (due to the technical condition), the procedure requires starting with emptying the oil pan of hydraulic fluid and assessing the type and degree of contamination. This should be followed by dismantling the electro-hydraulic controller, a verification of its technical condition (clearance measurements), and a check for potential spool and sleeve surface scratches. Such a case always requires the disassembly of the electro-hydraulic controller and cleaning its components. Control precision pair components are re-mounted (if their technical condition enables so), lubricating them with fresh hydraulic fluid (ATF).