A Novel Double-Sided Electromagnetic Dog Clutch with an Integrated Synchronizer Function

Abstract

Dog clutches are superior to synchromesh units due to much less wear caused by friction but require an external torque source to synchronize the rotation speeds. The current trend in e-mobility to use the driving motor of an electric vehicle as this source just creates another problem, which is known as torque holes. In this work, we propose a novel double-sided dog clutch that synchronizes the speeds electromagnetically by itself avoiding mechanical contact between the parts. A shift sleeve, two coils placed coaxially in their stators, and two complementary rings form an electromagnetic reluctance actuator, which is integrated inside the gearbox between two gearwheels and represents the double-sided clutch. Thus, intermediate parts between the shift sleeve and the actuator are not required. Both actuator sides can produce axial force and electromagnetic torque. However, torques and forces are generated simultaneously on both sides. Therefore, a special control algorithm is developed to keep the resulting axial force approximately equal to zero while the torque is generated in the neutral gear position. After the synchronization, the axial force is applied on the corresponding side to shift the required gear engaging the shift sleeve teeth directly with the slots of the complementary ring mounted on the gearwheel. So, an axial contact of the teeth at an unaligned state, which can lead to unsuccessful shifting, is avoided. A testrig, which includes a clutch prototype and a testing two-speed gearbox, has been designed and built. The developed theoretical ideas have been verified during the experiments under different conditions. The experiments confirm that the actuator can reduce positive and negative speed differences on both sides and subsequently shift the gear without a shift sleeve collision at misaligned angular positions.

1. Introduction

The growing interest to equip electric cars with multi-speed gearboxes to increase their efficiency [1,2,3] also leads to novel proposals and solutions in gear shifting itself. The speed of an electric motor can be controlled much easier than the crankshaft speed of a combustion engine, so the driving motor can directly synchronize the rotation speed of the gearwheel, which has to be engaged, with the speed of the output shaft. Thus, an additional friction clutch to decouple the motor from the gearbox is no longer required, as well as synchronizers inside the gearbox, whose operation principle (synchronization of speeds by friction) inevitably causes their wear and creates related problems [4,5,6,7]. Instead, a dual-sided dog clutch integrated between two gearwheels can be used [8].

Among various types of clutches, dog clutches stand out for their high efficiency, low complexity, and the ability to transmit high torques without slippage between the rotating parts, making friction losses and heat generation negligible. However, other types of clutches currently dominate in passenger cars. For example, hydraulic multi-plate clutches can be found in planetary gearboxes of automatic transmissions, multi-plate dry or wet clutches are applied to connect/disconnect input shafts of a dual-clutch transmission (DCT) to/from the motor, and the mentioned above combination of a single plate friction clutch and synchronizers is used to shift gears in manual transmissions and automated manual transmissions; moreover, synchromesh units are also applied to engage and disengage gearwheels in DCTs. In non-electric vehicles, dog clutches are typically found only in sequential gearboxes of motorbikes and racing cars [9]. This is caused by their disadvantages. The gear can be shifted only in a limited range of speed differences between the gearwheel and the output shaft; moreover, gear switching is abrupt and noisy. In electric vehicles, the preliminary speed synchronization by the motor eliminates these problems. The required gear can be shifted smoothly if the shift sleeve, which is placed axially movable on the output shaft of a gearbox, engages with the slots of the gearwheel at a low relative speed. Thus, gearboxes equipped with dog clutches may also become common in passenger cars [10].

This fact has made dog clutches an object of many scientific studies in the last years. Researchers from the Budapest University of Technology and Economics have intensively investigated the probability of successful shifting at different conditions in the early 2010s [11,12,13] and in recent years [14,15,16]. If the face surfaces of the shift sleeve teeth collide with the face surfaces of the teeth on the counterpart during the shifting process, the friction force between them can prevent further engagement of the teeth with the slots [11]. To avoid this problem, algorithms for a direct teeth–slot engagement based on the trajectory planning have been proposed by our institute [17] and also by other researchers [8,18,19]. A related topic of study is reducing the impact force between the side surfaces of the teeth and the slots [20].

Comparing to the considered problems, the simplification of the clutch actuation has gotten less attention. Usually, the shift sleeve is actuated with a shift fork, which is connected to a rotary motor outside the gearbox via several intermediate parts that transform the rotary motion of the motor to the linear motion of the fork [3,7,8,9,16,18,19,20,21]. Linear actuators have been proposed to avoid the use of intermediate mechanisms, but they are still designed to be placed outside the gearbox and move the sleeve indirectly via the shift fork [22,23]. To eliminate this drawback, we have proposed a direct actuation of the shift sleeve by electromagnetic actuators integrated into the double-sided dog clutch [17,24].

However, it has long been known that the synchronization of the gearwheel and sleeve speeds by the motor of an electric or hybrid vehicle just creates another problem, which is called torque holes [25]. While the speeds are synchronized, the wheels are disconnected from the motor, significantly reducing the drivability of the vehicle. The solutions proposed earlier [26] and recently [27,28] always include the use of two electric motors, one of which is connected with the wheels and propels the vehicle, while the other synchronizes the rotation speeds in the gearbox. The torque holes can be minimized with the use of only one motor if a DCT is applied [29]. Basically, the DCT is an automated manual gearbox with two countershafts, where each of them can be connected or disconnected from the motor by the corresponding clutch. One of the gearwheels on the disconnected countershaft is pre-selected and engaged while the vehicle is being propelled. Switching between the gears is realized by releasing one clutch and connecting the other to the motor. However, synchronizers are required again to equalize the rotation speed of the pre-selected gearwheel with the output shaft speed. If the shifting actuator could apply torque on the gearwheel to increase or reduce its rotation speed before the shift sleeve is axially moved under the influence of the force generated by the actuator, the synchronizers would no longer be necessary. Such an actuator can find application not only in dual-clutch transmissions but also replace synchronizers in manual transmissions and automated manual transmissions [30] to avoid a possible failure caused by wear [31].

To generate not only axial shifting force but also synchronizing torque, a linear-rotary actuator is required. Linear-rotary electromagnetic actuators (LREAs) presented in the literature are either complex structures with numerous permanent magnets (PMs) [32,33,34,35,36,37,38] or common electrical machines with multiple stator packets [39,40]. They are not suitable for use as a shifting actuator integrated into a dog clutch due to their size and complexity. Therefore, a novel LREA for application in dog clutches has first been proposed by B. Miroschnitschenko in [41]. The method of design and optimization is developed for the LREA in [42], and modified designs with additional features are described in [43]. It has a very simple construction, where no PMs are required, and only two coils are placed coaxially. The principle of torque generation is similar to that of a homopolar single-phase switched reluctance motor [44]. The shift sleeve of a one-sided dog clutch and its counterpart represent active parts of the actuator. Thus, it can be simply integrated and applied to synchronize the speeds of two rotating elements and de-/couple them mechanically.

The one-sided actuator [41,42] can be used for the smooth mechanical coupling of two rotating shafts and find its application, for example, to turn on and turn off a 4-wheel drive in an automotive area. However, automotive gearboxes regularly require double-sided dog clutches, which can switch between three shifting states, i.e., disengaged (neutral gear), gear 1 engaged, and gear 2 engaged. To allow the use of the actuator as a double-sided dog clutch, its topology can be simply repeated on the side of the second gearwheel. However, this will double its size and number of parts. To avoid this, we propose a modified design of the actuator that represents a double-sided electromagnetically actuated dog clutch in this work.

The design and operation principle of the one-sided clutch [41,42] are briefly introduced in Section 2. They underlie the novel double-sided dog clutch, which is described in Section 3. However, the control algorithm for the synchronization of rotation speeds becomes more complex in the double-sided topology. It is developed in Section 4. Moreover, a control algorithm for a direct teeth–slot engagement to eliminate the possibility of unsuccessful shifting is given in Section 4 together with the mechanics of the shifting process and simulation results. The basic idea and theoretical results of this and previous works are experimentally verified in Section 5, which also describes the developed testrig with a prototype of the double-sided clutch placed into a designed testing gearbox. The conclusions are drawn in Section 6. We note that this work focuses on the gear shifting using the developed device. Other aspects, such as operation under load and capabilities of sensorless control, are considered in separate papers [45,46].

2. Design and Operation Principle of a Novel One-Sided Dog Clutch

The one-sided clutch (Figure 1a) is represented by the LREA proposed in [41]. The actuator has two coils mounted on the stators, the shift sleeve, and the complementary ring, which is located opposite to the shift sleeve. The sleeve, the ring, and the stators are made of soft magnetic steel and placed coaxially relative to each other. The complementary ring is rigidly connected to shaft 1, and the shift sleeve is mounted axially moveable on shaft 2. It is worth mentioning that other rotating elements can be used instead of one of the shafts; for example, it can be replaced by a gearwheel. The shift sleeve and the complementary ring have the teeth and slots placed on their inner side surfaces. The number of teeth z is the same on each element, as well as the number of slots and teeth on one element. Moreover, all teeth have the same angular length ${\tau }_t$, all slots have the same angular length ${\tau }_s$, and ${\tau }_s$ is slightly larger than ${\tau }_t$. Finally, the teeth and the slots are placed angularly equidistant, forming the pitch between the teeth (and the slots) $\tau$, which is equal to 360/z degrees.

If the left coil is excited, its current creates a magnetic flux that closes via the left stator, the shift sleeve, and the ring, so that reluctance forces arise. The axial component of the reluctance force $F_{elmagl}$ works on the shift sleeve and can move it axially towards the complementary ring. If the right coil is excited simultaneously, it generates a magnetic flux, which propagates in the steel of the right stator and the shift sleeve and creates an axial reluctance force $F_{elmagr}$ with the opposite direction. As long as $F_{elmagr}$ is greater than $F_{elmagl}$, the sleeve remains at its initial axial position where the radial air gap between the teeth is much smaller than the axial distance between the sleeve and the ring. In this case, the permeance of the magnetic circuit is maximal at the position where the teeth are completely radially aligned. This position corresponds to the relative angle between the ring and the sleeve ${\theta }_{rel}=0$ (Figure 1b). Thus, if the relative angles between a tooth of the sleeve and the nearest two teeth of the ring are not equal, reluctance torques, which work on the shafts, arise. These torques tend to rotate the shafts towards each other and align the nearest teeth radially, i.e., to make the relative angle equal to zero. The axial movement of the shift sleeve can engage the teeth of one element with the slots of the opposite element in the range of the relative angular positions between ${\theta }_{rel}=-j_{\theta }/2$ and ${\theta }_{rel}=j_{\theta }/2$, where $j_{\theta }$ is the angular backlash:

$$\begin{array}{c}\hfill j_{\theta }={\tau }_s-{\tau }_t.\end{array}$$

(1)

The sleeve can be moved and engaged by de-exciting the right coil, making $F_{elmagl}$ uncompensated. If engaged, the teeth and the slots interlock and couple the shafts mechanically. To decouple the shafts, the left coil can be de-excited while the right coil is excited again creating $F_{elmagr}$ that moves the sleeve back towards the right stator disengaging the teeth.

If the shafts are rotating with different speeds, the speed difference $n_{rel}$ leads to the change of ${\theta }_{rel}$ in the range between ${\theta }_{rel}=-\tau /2$ and ${\theta }_{rel}=\tau /2$. The values between $-\tau /2$ and 0 correspond to the positions where the nearest teeth of the complementary ring are located relative to the sleeve teeth in counterclockwise direction and the values between 0 and $\tau /2$ to the positions where the nearest teeth of the complementary ring are located relative to the sleeve teeth in clockwise direction. If the left coil is excited during the relative rotation, reluctance torques with the same absolute value $|T_{elmag}|$ but opposite signs work on the sleeve and on the ring. Zero torques are generated at the relative angles ${\theta }_{rel}=\pm \tau /2$ and ${\theta }_{rel}=0$; moreover, the torques change their signs at these positions. Considering the clockwise direction as a positive direction for torques, positive torque works on the complementary ring in the range of negative ${\theta }_{rel}$, and negative torque works on the ring in the range of positive ${\theta }_{rel}$. By exciting and de-exciting the left coil in the appropriate range of ${\theta }_{rel}$, reluctance torques with the desired signs can be generated to synchronize the rotation speeds. The arising $F_{elmagl}$ can be compensated by exciting the right coil, as is described above. As soon as the rotation speeds are almost synchronized, the residual speed difference can be used to bring ${\theta }_{rel}$ into the range where the teeth can be engaged with the slots directly avoiding the contact between the teeth and the side surface of the opposite element. This state is also called the face contact phase and can lead to unsuccessful shifting [11,12,13]. After that, the right coil is de-excited, and the shift sleeve moves axially towards the complementary ring. Thus, smooth engagement is achieved, and impacts caused by significant speed difference as well as the face contact phase are avoided.

On the basis of the introduced one-sided clutch, a modified LREA with an extended functionality suitable for operation as a double-sided dog clutch has been developed. Its design and operation principle are described in Section 3.

3. Design and Operation Principle of a Novel Double-Sided Dog Clutch

The design of the double-sided dog clutch (Figure 2) is similar to that of the one-sided clutch. The stators, the coils, and the complementary rings are placed on each side. The dimensions of the parts on the left side are equal to the dimensions of the corresponding parts on the right side. The complementary rings are rigidly connected to the gearwheels of a gearbox, and the shift sleeve has teeth on both side surfaces and is placed axially movable on the splines of the hub, which is fixed on the output shaft. Since the shift sleeve is made without slots to reduce its radial size and mass, the ring teeth, which are placed above the sleeve teeth in radial direction, always remain unengaged. There are nine tooth pairs on each side, so ${\theta }_{rel}$ ranges between −20° and 20°. The sleeve teeth can be engaged with the ring slots by the axial movement of the shift sleeve, same as in the one-sided clutch. The angular lengths of the teeth ${\tau }_t$ and of the slots ${\tau }_s$ are 18° and 24°, respectively, making the angular backlash $j_{\theta }$ equal to ± 3°. The needle bearings are placed between the gearwheels and the output shaft so that the gearwheels can rotate on the shaft. Three spring-loaded balls are placed in the notches of the hub.

The gearwheel teeth are interlocked with the teeth of the countershaft (the countershaft is not shown in Figure 2) so that two gears are formed on the sides. The gear ratio of the left gear is $g_l$, and the gear ratio of the right gear is $g_r$; moreover, $g_l$ and $g_r$ are different. Thus, if the countershaft speed $n_c$ is non-zero, the gearwheels rotate with the different speeds $n_l$ and $n_r$, which are related with $n_c$ as

$$\begin{array}{c}\hfill n_l={\displaystyle \frac{n_c}{g_l}},\end{array}$$

(2)

$$\begin{array}{c}\hfill n_r={\displaystyle \frac{n_c}{g_r}}.\end{array}$$

(3)

Consequently, $n_l$ and $n_r$ relate to each other as

$$\begin{array}{c}\hfill {\displaystyle \frac{n_l}{n_r}}={\displaystyle \frac{g_r}{g_l}},\end{array}$$

(4)

and the relative rotation speeds $n_{rell}$ and $n_{relr}$ with different values are presented on the sides:

$$\begin{array}{c}\hfill n_{rell}=n_l-n_{sleeve},\end{array}$$

(5)

$$\begin{array}{c}\hfill n_{relr}=n_r-n_{sleeve},\end{array}$$

(6)

where $n_{sleeve}$ is the rotation speed of the shift sleeve, which is equal to the rotation speed of the output shaft.

The axial positions of the shift sleeve x = $-8$ mm, 0 mm, and 8 mm correspond the left gear, neutral gear, and right gear positions, respectively. Around these positions, the spring-loaded balls generate the mechanical force $F_{spring}$, which tends to keep the gear shifted. The axial reluctance force and the reluctance torque can be generated on each side with the same principle as in the one-sided clutch. If the sleeve remains at the central axial position (x = 0), the magnetic circuits on the sides are symmetrical, and the sides are magnetically decoupled so that the electromagnetic torque $T_{elmag}$ and the axial electromagnetic force $F_{elmag}$, which act on the gearwheel and on the shift sleeve, respectively, can be described for both sides by the same torque and force characteristics shown in Figure 3. There, ${\theta }_{rel}$ is the relative angular position between the shift sleeve and the complementary ring of the considered side, and I is the current of the corresponding coil. A required gear can be shifted by exciting the coil on the corresponding side creating $F_{elmag}$, which overcomes $F_{spring}$ and moves the shift sleeve towards the gearwheel engaging the teeth with the slots of the complementary ring. When engaged, the sleeve and the gearwheel rotate together so that the output shaft and the countershaft are coupled mechanically. Assuming that the countershaft is connected to the driving motor, the motor torque can be transmitted to the output shaft so that the output shaft can be rotated with a higher speed or a higher torque depending on the gear ratio of the shifted gear.

At lower torques, the self-disengagement of the sleeve is prevented by the force of the spring-loaded balls, but the coil on the engaged side has to be excited to generate an additional holding force at higher loads. Since the inductance value is increased significantly by a mechanical contact between the parts at the engaged state, significant forces can be generated at very low currents. So, the power losses in the coil are few watts, and the heat generation is negligible. The torque value that can be transmitted by the clutch is limited only by the mechanical strength of the parts. In the design shown in Figure 2, the radial width of the shift sleeve teeth is 8 mm, and their average radius is 57.5 mm, which results in the average angular width of the teeth of 18 mm for the given above ${\tau }_t$ = 18°. This is enough to withstand loads of several hundreds newton-meters, which can be presented in a vehicle.

The gear can be disengaged by exciting the coil on the opposite side. If the created axial reluctance force is lower than $F_{spring}$ and the motion speed is relatively low when the shift sleeve reaches the central position, the sleeve can be stopped at x = 0 so that the clutch is set in neutral gear again. Otherwise, the sleeve overcomes the central position and engages the opposite gearwheel so that the other gear is shifted. Consequently, an active force control is required to return the sleeve in the neutral gear position. The easiest way to do this is to place a sensor that will measure the axial position of the shift sleeve so that the required motion speed is obtained from the measured position, and the coil currents on both sides are controlled based on the difference between the measured and required speed to stop the motion at x = 0. However, the position sensor will increase the complexity and the manufacturing costs of the clutch. Moreover, the sensor must be placed inside the gearbox in the aggressive environment of transmission oil mixed with metal particles, which inevitably arise due to the wear of the parts. This might significantly reduce the clutch reliability. A preferred alternative is the development of a sensorless control method that allows us to control the motion of the shift sleeve without a direct measurement of its axial position. This method has been developed and will be published soon [46].

The main difference of the operation principle of the double-sided clutch comparing to that of the one-sided clutch is the method of speed synchronization. The axial force that arises on the side that generates synchronizing torque cannot be compensated by simply exciting the coil on the opposite side, as can be done in the one-sided clutch [42]. Both forces vary with the change of ${\theta }_{rel}$ on the corresponding side, and the force difference rises with the increase in the coil current (Figure 3). Since the relative rotation speeds $n_{rell}$ and $n_{relr}$ are different, the relative angular positions on the left ${\theta }_{rell}$ and the right ${\theta }_{relr}$ sides change asynchronously. Thus, the coil currents must be controlled differently to keep the force balance. If the difference between the forces exceeds the holding force created by the spring-loaded balls $F_{spring}$, the shift sleeve starts to move axially. If the force disbalance is not eliminated quickly, the shift sleeve leaves the neutral gear position and moves to the side with the higher force, which will lead to the collision between the sleeve and the corresponding complementary ring. This collision is especially dangerous at high relative speeds and may cause damage to the clutch. Moreover, not only an axial force but also torque will be generated by exciting the opposite coil. The direction of this torque can be positive and negative depending on the relative angle on the side. This can result in a situation where the side that is used to balance the axial force increases the rotation speed difference stronger than the side used to generate the synchronizing torque decreases it. Therefore, a special control algorithm for speed synchronization is developed in Section 4.

4. Algorithm for Gear Shifting with a Preliminary Speed Synchronization in a Gearbox Equipped with the Double-Sided Clutch

To synchronize the rotation speed of the gearwheel, which has to be engaged, with the rotation speed of the shift sleeve, positive or negative torque must be generated by the actuator depending on the sign of the relative speed. For this purpose, one actuator side is used for torque generation, while the other compensates the arisen axial force. For further considerations, we name these sides as synchronizing and compensating side, respectively. The synchronizing side produces the torque with the desired direction $T_{syn}$ and in addition to that an undesired axial force. The compensating side is excited and de-excited simultaneously to balance the axial force and also generates the collateral torque $T_{compens}$. We can assume that the absolute value of the relative speed on the synchronizing side $n_{relsyn}$ is higher than that of the relative speed on the compensating side $n_{relcompens}$. In this case, the relative angular position on the synchronizing side ${\theta }_{relsyn}$ changes faster than the relative angular position on the compensating side. Since $|T_{elmag}|$ changes with $|{\theta }_{rel}|$ (Figure 3), this will inevitably lead to the situations when the absolute value of the average torque generated on the compensating side during one excitation period is higher than the absolute value of the average torque generated on the synchronizing side during that period. If the torque signs are different, the speed difference will be increased instead of being reduced. Therefore, the side with the lower $|n_{rel}|$ must be chosen for torque generation, while the opposite side with the higher $|n_{rel}|$ is used to compensate the axial force. In this case, $T_{compens}$ has different signs during one excitation period so that its average absolute value declines with the increasing difference between $|n_{relsyn}|$ and $|n_{relcompens}|$. Over a longer time period, the integral of $T_{compens}$ becomes approximately equal to zero. If the synchronization starts when $|n_{relcompens}|$ is much higher than $|n_{relsyn}|$, the influence of the compensating side on the synchronization process is negligible. To avoid frequent switching between the sides when $|n_{rell}|$ and $|n_{relr}|$ are almost equal, a hysteresis band can be added to the comparator of the relative speeds.

The coil on the synchronizing side must be excited in the negative range of ${\theta }_{relsyn}$ to generate a positive torque, while a negative torque is generated in the range of positive ${\theta }_{relsyn}$ (Figure 3). At positive $n_{relsyn}$, ${\theta }_{relsyn}$ changes from −20° to 20° during one period, and at negative $n_{relsyn}$, it changes from 20° to −20°. Thus, to create positive $T_{syn}$ while $n_{relsyn}<$ 0, the turn-on relative angle for the coil on the synchronizing side ${\theta }_{relon}$ is 0°, and the turn-off relative angle ${\theta }_{reloff}$ is −20°. For $n_{relsyn}>$ 0 and the required $T_{syn}>$ 0, ${\theta }_{relon}$ and ${\theta }_{reloff}$ become equal to −20° and 0°, respectively. To generate negative torque, the turn-on and turn-off relative positions are simply inverted, i.e., ${\theta }_{relon}$ becomes ${\theta }_{reloff}$, and vice versa. However, the given values of the turn-on and turn-off angles assume the ideal case when the coil can be excited and de-excited instantly. In fact, the coil inductance makes an instant current change impossible. Thus, ${\theta }_{relon}$ and ${\theta }_{reloff}$ must be adjusted depending on the relative speed on the synchronized side, the inductance, and the resistance of the coil and supply voltage. Considering all these influences, the adjustment angle ${\theta }_{adj}$ is introduced.

When the coil is being excited, its current I rises according to the following equation:

$$\begin{array}{c}\hfill I={\displaystyle \frac{U}{R}}\left(1-e^{-tR/L}\right),\end{array}$$

(7)

where U is the applied voltage, R is the coil resistance, L is the coil inductance, e is Euler’s number, and t is the time elapsed after the start of excitation. Consequently, the time $t_{ex}$ required to excite the coil on the synchronizing side changing its current from I = 0 to the required value $I=I_{syn}$ can be calculated as

$$\begin{array}{c}\hfill t_{ex}=-{\displaystyle \frac{ln(1-\frac{I_{syn}{R}_{syn}}{U})L_{syn}}{R_{syn}}}{\epsilon }_t,\end{array}$$

(8)

where $R_{syn}$ and $L_{syn}$ are the resistance and inductance of the coil on the synchronizing side, respectively, and ${\epsilon }_t$ is the factor that defines the decrease in $L_{syn}$ caused by the influence of eddy currents. ${\epsilon }_t$ is always less than 1 and can be estimated experimentally using the method described in [45]. The time required to demagnetize the coil changing its current from $I=I_{syn}$ to I = 0 is approximately equal to $t_{ex}$. Finally, the adjustment angle ${\theta }_{adj}$ is calculated as

$$\begin{array}{c}\hfill {\theta }_{adj}=6\left|n_{relsyn}\right|t_{ex},\end{array}$$

(9)

where 6 is the scaling factor from rpm to degrees per second. The coil on the synchronized side is turned on and turned off ${\theta }_{adj}$ degrees earlier comparing to the idealized ${\theta }_{relon}$ and ${\theta }_{reloff}$. For example, if the calculated ${\theta }_{adj}$ is 5°, $n_{relsyn}<$ 0 and positive torque is required, ${\theta }_{relon}$ is 5°, and ${\theta }_{reloff}$ is −15°. For $n_{relsyn}>$ 0, and positive required torque, ${\theta }_{relon}$ and ${\theta }_{reloff}$ become −5° and 15°, respectively. Since ${\theta }_{adj}$ is directly proportional to $|n_{relsyn}|$, it reaches the value ${\theta }_{adj}=\tau /2$ at some $n_{relsyn}$ so that the actuator will generate only an undesired torque. To avoid this situation, ${\theta }_{adj}$ is limited with the value ${\theta }_{adj}<\tau /4$. However, the unlimited ${\theta }_{adj}$ equal to $\tau /2$ means that the de-excitation of the coil starts directly after the required current value is reached. As such, the coil cannot be excited with $I_{syn}$ if ${\theta }_{adj}$ calculated with (9) is higher than $\tau /2$. In this case, the maximum current $I_{maxex}$ with which the coil can be excited relates to $I_{syn}$ as

$$\begin{array}{c}\hfill I_{maxex}\approx {\displaystyle \frac{\tau }{2{\theta }_{adj}}}{I}_{syn}.\end{array}$$

(10)

If ${\theta }_{adj}$ obtained with (9) is less than $\tau /2$, $I_{maxex}$=$I_{syn}$.

We note that the coil inductance $L_{syn}$ changes with its current and relative position on the side, and $R_{syn}$ changes with the coil temperature. So, $t_{ex}$ becomes lower if the coil temperature rises, or if the coil is turned on or turned off at the unaligned position. To simplify the calculation, $L_{syn}$ can be assumed equal to the coil inductance at the unaligned position and average actuator current, and $R_{syn}$ can be assumed equal to the coil resistance at 20 °C. In the considered actuator, these values are 20 mH and 0.75 Ohm, respectively. This simplification causes some delayed de-/excitation on the synchronizing side when ${\theta }_{relsyn}$ is close to the aligned position ${\theta }_{rel}=0$. However, the amplitude of torque generated around the aligned position is low (Figure 3) so that this delay is insignificant. The eddy currents in the steel also delay the de-/magnetization and reduce the generated torque. However, the most important role plays the supply voltage, which value U is directly proportional to the speed of current rise/fall (Equations (7) and (8)). So, if U is not high enough, the coil current cannot reach the required value $I_{syn}$ at higher $|n_{relsyn}|$ (Equations (9) and (10)), and the synchronizing torque $T_{syn}$ decreases significantly. Thus, an efficient synchronization is possible in a limited range of $|n_{relsyn}|$.

To achieve the balance of the axial forces, we propose an algorithm based on the inverted force characteristic (Figure 4). There, the initial force characteristic $F(I,{\theta }_{rel})$ for the central axial position $x=0$ (Figure 3) is inverted to the form $I(F_{elmag},{\theta }_{rel})$. Since the axial force is the same for ${\theta }_{rel}$ and $-{\theta }_{rel}$, the absolute value $|{\theta }_{rel}|$ is enough. Interpolating the inverted characteristic using a radial basis function [47], the current values at which the force remains equal to $F_{elmag}(I_{syn},{\theta }_{rel}=0)$ can be found for different values of $I_{syn}$ and $|{\theta }_{rel}|$ and saved as a look-up table (LUT) (Figure 4). Now, adjusting the coil currents depending on the required value of the synchronizing current $I_{syn}$ and the measured value of ${\theta }_{rel}$ on the corresponding side, approximately equal axial forces can be created on both sides. However, it is not feasible to create a full force balance due to the influence of different uncertainties (eddy currents, deviations caused by manufacturing tolerances, difference between the simulated model and the real actuator, etc.) The residual force difference can be defined as $\mathsf{\Delta }{F}_{elmag}$. The axial position of the shift sleeve during the synchronization of the rotation speed is described by the equation

$$\begin{array}{c}\hfill x=\int \int {\displaystyle \frac{\mathsf{\Delta }{F}_{elmag}-F_{spring}}{m}}dt,\end{array}$$

(11)

where m is the mass of the shift sleeve. In the area of the central position, $F_{spring}$ is directed to keep $x=0$, i.e., is opposite to $\mathsf{\Delta }{F}_{elmag}$. Thus, a low residual electromagnetic force can be compensated by the spring-loaded balls.

As is shown in Figure 3, the change in $F_{elmag}$ is small in the area of lower currents. There, an active adjustment of the side currents is not required since the residual force difference can be balanced solely by the spring-loaded balls. In the proposed actuator, the current adjustment on both sides is not required if $I_{syn}$ is less than $I_{low}$ = 14 A. On the contrary, the uncertainties increase $\mathsf{\Delta }{F}_{elmag}$ at higher currents, and it acts longer on the shift sleeve at lower relative speeds since the time period of coil excitation rises. As it follows from (11), the shift sleeve will move a longer distance from the central position causing the further rise of $\mathsf{\Delta }{F}_{elmag}$ since the axial air gaps between the sleeve and the complementary rings of the sides become different. This will lead to the case when the axial forces can not be balanced any more. To avoid this problem, the required synchronizing current $I_{syn}$ must be limited at $|n_{relsyn}|$ less than the value $n_{low}$. The limitation is relatively small and can be chosen equal to 27 A in the proposed actuator, which is designed for the maximum operating current 30 A, while $n_{low}$ is chosen equal to 100 rpm. Finally, the coil voltages are controlled based on the difference between the required and measured currents using the hard chopping method common for rotary reluctance machines [48].

The entire control algorithm for the synchronization of the rotation speeds is shown in Figure 5. Summarizing, the synchronizing side is chosen based on the relative speeds $n_{rell}$ and $n_{relr}$; the adjustment angle ${\theta }_{adj}$ is calculated based on the relative speed on the synchronizing side $n_{relsyn}$, and its value is limited with ${\theta }_{adjlim}$; the coils are excited and de-excited simultaneously in the corresponding ranges of the relative angle on the synchronizing side ${\theta }_{relsyn}$, which are estimated based on the required torque direction, ${\theta }_{adj}$, and the sign of $n_{relsyn}$; the desired synchronizing current $I_{syn}$ can be chosen manually within the range between 0 and 30 A, its value is limited with $I_{synlim}$ at $|n_{relsyn}|$ lower than $n_{low}$; if the excitation command is 0, the required currents $I_{reql}$ and $I_{reqr}$ on the sides are zero, if it is 1, they are estimated based on $I_{syn}$, the relative angles ${\theta }_{rell}$ and ${\theta }_{relr}$ on the sides and the current control LUT shown in Figure 4 (at $I_{syn}$ less than $I_{low}$ = 14 A, the angle-based adjustment is not required); the voltages $U_l$ and $U_r$ are applied to the coils based on the difference between the required currents and the measured currents $I_l$, $I_r$.

Further, we shortly describe the mechanics of the shifting process. With the simultaneous excitation of both coils (Figure 5), the actuator generates electromagnetic torques $T_{elmagl}$ and $T_{elmagr}$ on the corresponding sides. The resulting torques, which work on the left $T_l$ and right $T_r$ gearwheels and on the shift sleeve $T_{sleeve}$, can be calculated as

$$\begin{array}{c}\hfill T_l=T_{elmagl}-T_{frl}-T_{frc}{g}_l+\left(T_{elmagr}-T_{frr}\right){\displaystyle \frac{g_l}{g_r}},\end{array}$$

(12)

$$\begin{array}{c}\hfill T_r=T_{elmagr}-T_{frr}-T_{frc}{g}_r+\left(T_{elmagl}-T_{frl}\right){\displaystyle \frac{g_r}{g_l}},\end{array}$$

(13)

$$\begin{array}{c}\hfill T_{sleeve}=-T_{elmagl}-T_{elmagr}-T_{load},\end{array}$$

(14)

where $T_{frl}$ and $T_{frr}$ are the friction torques of the corresponding gearwheel, $T_{frc}$ is the friction torque of the countershaft, and $T_{load}$ is the load torque on the output shaft. The friction torques are caused by friction forces in the bearings, friction between the torque transmitting elements (which are gearwheel teeth in the case of the gear train or belts, belt pulleys, and belt tensioners in the case of belt transmission), and aerodynamic resistance. Moreover, all these parameters are dependent on the rotation speed. The total values of the moment of inertia on the gearwheels $J_{suml}$ and $J_{sumr}$ are

$$\begin{array}{c}\hfill J_{suml}=J_l+\left(J_c+{\displaystyle \frac{J_r}{g_r^2}}\right)g_l^2,\end{array}$$

(15)

$$\begin{array}{c}\hfill J_{sumr}=J_r+\left(J_c+{\displaystyle \frac{J_l}{g_l^2}}\right)g_r^2,\end{array}$$

(16)

where $J_l$, $J_r$, and $J_c$ are the moments of inertia of the left gearwheel, right gearwheel, and the countershaft, respectively. The moment of inertia on the shift sleeve $J_{sleeve}$ is equal to the moment of inertia on the output shaft. The change in the angular velocities during the synchronization is then calculated from the equation of angular motion

$$\begin{array}{c}\hfill \omega ={\omega }_0+\int {\displaystyle \frac{T}{J}}dt,\end{array}$$

(17)

where $\omega$, ${\omega }_0$, T, and J are the angular velocity, the initial angular velocity at the start of the synchronization, the resulting torque, and the total moment of inertia of the corresponding part, respectively. If the rotation speed of the gearwheel that has to be engaged is higher than $n_{sleeve}$, the speeds can always be synchronized since the electromagnetic and friction torques act in the same direction. However, if $n_{rel}<$0, the actuator must be able to generate enough torque to overcome the resulting friction torque of the gearwheel. As it follows from (12) and (13), it is always harder to accelerate the bigger gearwheel.

The relative speed on the side that has to be shifted is synchronized to some small residual value $n_{relres}>$ 0, which is used to reach the relative position where the teeth can be directly engaged with the slots. This position can be named as ${\theta }_{relshift}$ and calculated based on the obtained $n_{relres}$ and the time required to move the shift sleeve from the central position to the engaged position $t_{shift}$ [17]:

$$\begin{array}{c}\hfill {\theta }_{relshift}=-6n_{relres}{t}_{shift}.\end{array}$$

(18)

The angular backlash (Equation (1)) allows an error in the calculation of ${\theta }_{relshift}$ if the absolute value of the error is less than $j_{\theta }$.

To show the entire shifting process, two different situations have been simulated in Matlab Simulink for a gearbox with the reciprocal gear ratios $g_l$ = 44/64 (overdrive) and $g_r$ = 64/44 (underdrive) assuming that the moments of inertia $J_c$, $J_l$, and $J_r$ are 0.02 kg m², 0.004 kg m², and 0.007 kg m², respectively. The first situation represents the following conditions: the left gear has to be shifted; the countershaft is at a standstill, while the rotating speed of the output shaft is controlled to keep the value 300 rpm, which results in the initial $n_{rel}$ equal to −300 rpm on both sides; the friction torques $T_{frc}$, $T_{frl}$, and $T_{frr}$ are 1 Nm, 0.25 Nm, and 0.25 Nm, respectively. In the second situation, the right gear has to be shifted, while the initial value of $n_c$ is 2180 rpm, and the output shaft is controlled to rotate at a speed of 1000 rpm, resulting in the initial relative speeds equal to 2171 rpm on the left side and 500 rpm on the right side; the friction is assumed to have the following values: $T_{frc}$ = 1.5 Nm, $T_{frl}$ = 0.7 Nm, and $T_{frr}$ = 0.4 Nm. The simulation results are shown in Figure 6 and Figure 7. It can be seen that, to generate synchronizing torque, both coils are excited simultaneously at the corresponding angular positions on the synchronizing side, which is the left side in the first case and the right side in the second case. The coil currents are adjusted depending on ${\theta }_{rel}$ on the coil side. In Figure 6, the resulting electromagnetic torque varies since $n_{rel}$ is relatively low on the compensating (right) side. In Figure 7, the collateral electromagnetic torque on the left side $T_{elmagl}$ changes very quickly since $n_{rell}$ is very high so that the resulting synchronizing torque on the right side $\Sigma T_{rightelmag}$ looks like the sum of $T_{elmagr}$ and an additional component, which has a high amplitude and frequency, but its average value is zero. In both cases, the relative speeds could be synchronized without a significant displacement of the sleeve from the central position. Further, the required gears were shifted avoiding the face contact phase at the axial positions x = −5 mm (left gear shifting) and x = 5 mm (right gear shifting). The process of the experimental verification of the theoretical results is described in Section 5.

5. Experimental Validation

5.1. Experimental Setup

To verify the theoretical results, a dual-sided clutch prototype and a two-speed gearbox for test purposes (further named as testing gearbox) were designed and built (Figure 8). The gearbox with the embedded clutch was installed on the testrig (Figure 9).

5.1.1. Actuator Prototype

The dual-sided prototype of the proposed dog clutch consists of the sleeve 1, left stator 2, right stator 3, left complementary ring 4, and right complementary ring 5. The left and right coils are mounted in the corresponding stators, and the coil terminals are soldered to the wires, which are visible in Figure 8.

5.1.2. Testing Gearbox

To imitate the operation of dual-sided dog clutches in common multi-speed gearboxes, the manufactured prototype was installed into the testing gearbox. The shift sleeve is placed on the splines of the hub, which is fixed on the output shaft 6, and the left and right rings are fixed on the left intermediate shaft 7 and right intermediate shaft 8. The left shaft is hollow and rotates on two angular contact ball bearings mounted in the left plate of the housing, and the output shaft rotates inside the left shaft on two ball bearings in which the outer rings are fixed on the inner surface of the left shaft. The right shaft rotates on two angular contact ball bearings mounted in the right housing plate, which is made removable to allow mounting and demounting of the inner parts. Additionally, the right shaft is supported by a needle bearing on the output shaft. Two pairs of belt pulleys and one belt pair are used to create a two-speed transmission with reciprocal gear ratios. The smaller pulleys have 44 teeth while the bigger have 64 teeth. One smaller pulley 9 is fixed on the left shaft and one bigger pulley 10 on the right shaft. The companion pulleys 11 and 12 are mounted on the countershaft 13 opposite to the pulleys on the intermediate shafts. The pulleys 9 and 11 form the left belt transmission together with the belt 14, and the pulleys 10 and 12 and the same belt 15 form the right belt transmission. The belt tensioners are placed under the belts between the pulleys to avoid belt slipping. The countershaft rotates on the bearing unit 16, which is fixed on the rear plate of the housing. So, if the shift sleeve is engaged with the left complementary ring, the output shaft rotates 64/44 times faster than the countershaft and 64/44 times slower if the right complementary ring is engaged. Thus, the left belt transmission represents the left gear with the gear ratio 44/64 (overdrive), while the right belt transmission represents the right gear with the gear ratio 64/44 (underdrive).

The stators of the clutch are mounted on the inner rings of the stator holders 17 and 18, whose outer rings are fixed on the housing. The inner rings are screwed to the outer rings using four spacers between them on each side. Adjustment washers can be installed between the spacers and any ring to create a desired axial air gap between the corresponding stator and the complementary ring.

5.1.3. Sensors

To measure the angular positions and the rotation speeds of the shafts, two incremental angular encoders ams AS5304 are placed on the opposite sides of the printed circuit board 19. Two rotary encoder magnetic rings Bogen RMS172-100-A-F are installed into the hubs 20 and 21, which are fixed on the output shaft and on the left shaft of the gearbox, respectively. So, the angular positions of these shafts are measured. The angular position of the right shaft can be measured indirectly multiplying the angular position of the left shaft by (44/64)². The rotation speeds are derived from the angular positions. Three linear encoders POSIC ID1102L are glued on the hub with an angular pitch of 120° opposite to the corresponding pieces of the encoder scale POSIC TPLS05-205, which are glued on the inner surface of the shift sleeve. So, not only the axial position of the shift sleeve but also its tilting around the vertical and the horizontal axis can be measured from the difference between the encoder signals. However, the tilting angles are extremely low in the designed system, and their influence on the clutch operation is not in the scope of this work. The linear encoders are connected to the rotor of the slip ring MOFLON GHS2586 marked as 22 via the wires laid inside a bore in the output shaft. The stator of the slip ring is fixed by the support 23, which also holds the board with the angular encoders. Each coil is equipped with NTC thermistor TDK B57541G1. With the described setup, the angular positions of the shafts and the axial position of the shift sleeve can be measured with the accuracy of 0.045° and 1.25 µm, respectively. The coil temperatures can be measured with a sufficient accuracy in the temperature range between −50 and 140 °C.

5.1.4. Testrig

The testrig includes two synchronous servomotors Parker NV860VAC installed on the bedplate (Figure 9) and the electronic control unit (ECU) LCM-ECU-4HB-100A-75V, which is programmed on a computer using software Matlab Simulink with an additional X2C-library and connected to the power supply Delta Elektronika SM70-22A, whose maximal long-term values of the output current and of the output voltage are 22 A and 70 V, respectively. The coils are plugged in the sealed connectors, which are placed on the inner surface of the front housing plate and connected to the power outputs of the ECU via the shielded cable. The coil currents are measured with the integrated into the ECU current sensors with the accuracy of 0.1 A. The thermistor outputs are plugged in the black connectors inside the gearbox and connected to the analogue inputs of the ECU via the black cable laid through the sealed cable gland on the outer surface of the front housing plate. Since a seal ring is laid under the removable housing plate, the gearbox is completely sealed and can be filled with transmission oil. The signals from the linear and angular encoders are transmitted to the digital inputs of the ECU via three twisted-pair cables. The gearbox is mounted on the bedplate of the testrig (Figure 9) so that the output shaft and the countershaft can be rotated at different speeds by motor 1 and motor 2, respectively. The torque sensor Kistler Typ 4503B with the measurement range between −20 and 20 Nm and the accuracy class 0.05 is coupled with the countershaft and motor 2 by the safety couplings. Also, the safety coupling connects motor 1 with the output shaft. All couplings release if the torque on the corresponding shaft becomes higher than 20 Nm. The coils and the sensors are connected to the power outputs and the signal inputs of the ECU, respectively.

5.2. Measurement of the Friction Torque

It is hard to predict the friction torques of particular rotating elements (Equations (12) and (13)) analytically. The friction forces that arise in the bearings alone are dependent on many factors and change non-linearly with the rotation speeds [49]. On the other hand, friction is a crucial parameter that determines whether the rotation speeds can be synchronized by the actuator or not. Therefore, the resulting friction torques have been estimated experimentally by rotating the countershaft in neutral gear with motor 2 at different speeds and measuring the average torque on the countershaft with the torque sensor. The measurement results represent the resulting friction torque on the countershaft $\Sigma T_{cfr}$. However, this method is not suitable to measure $\Sigma T_{cfr}$ if $n_c$ = 0. Moreover, the coefficients of static and dynamic friction are usually different. Thus, the resulting static friction on the countershaft $\Sigma T_{cfr(n_c=0)}$ has been estimated equal to ∼0.7 Nm by measuring the torque of motor 2 required to start the rotation of the countershaft in neutral gear. The measurement results are shown in Figure 10. It can be seen that $\Sigma T_{cfr}$ increases almost linearly with $n_c$ at lower rotation speeds and can be described as $\Sigma T_{cfr}$ = $\Sigma T_{cfr(n_c=0)}+0.0025n_c$ at $n_c<$ 300 rpm. At higher countershaft speeds (300 $<n_c<$ 2000 rpm), $\Sigma T_{cfr}$ can be estimated as $\Sigma T_{cfr}\approx 1.47+0.0007(n_c-300)$ omitting the non-linearity of the measurement results. The resulting friction torques on the gearwheels $\Sigma T_{lfr}$, $\Sigma T_{rfr}$ are approximately equal to the $\Sigma T_{cfr}$ multiplied by the corresponding gear factor.

5.3. Measurement of the Synchronizing Torque

To measure the average torque that can be generated by the actuator during the speed synchronization at different conditions, the countershaft was rotated until the angular position of the left complementary ring became equal to those of the right ring so that their teeth stay opposite to each other. After that, motor 1 was controlled to rotate the output shaft with the required speed $n_{sleevereq}$ of 100, 300, and 500 rpm, while motor 2 was controlled to keep the initial position of the countershaft, i.e., the required $n_c$ was zero so that the rings kept their initial positions. Thus, when the output shaft with the shift sleeve were rotated, the relative positions on both sides ${\theta }_{rell}$ and ${\theta }_{relr}$ were equal. At the same time, the actuator was controlled to generate positive torque by exciting the coils at the appropriate range of ${\theta }_{rell}$ and ${\theta }_{relr}$ with different values of the required current $I_{req}$, starting from $I_{req}$ = 3 A and finishing with $I_{req}$ = 30 A with the step of 3 A. Since the relative positions on the sides were always the same, the force balance could be achieved without current adjustment. So, the limit $I_{low}$ (Figure 5) could be set equal to 30 A, and the coils were excited with the same currents. The torque $T_{cmeas}$ measured by the torque sensor represents the motor torque required to compensate the torques generated on the actuator sides and can be described by the equation of torque balance:

$$\begin{array}{c}\hfill T_{cmeas}+\Sigma T_{cfr(n_c=0)}=-{\displaystyle \frac{T_{elmagl}}{g_l}}-{\displaystyle \frac{T_{elmagr}}{g_r}},\end{array}$$

(19)

where $\Sigma T_{cfr(n_c=0)}\sim$ 0.7 Nm (Figure 10). The negative value of $T_{cmeas}$ is plotted in Figure 11 for different $T_{sleevereq}$ and $I_{req}$ equal to 12, 21, and 30 A. We note that the value of $I_{req}$ is limited to 27 A for $n_{sleevereq}$ = 100 rpm since the capacity of the power supply is not enough to excite the coils with the currents of 30 A at this relative speed. Moreover, the distance between the peaks of $-T_{cmeas}$ increases in the plot for $n_{sleevereq}$ = 100 rpm, which means that motor 1 is no longer able to rotate the output shaft at the required speed since the actuator brakes the shift sleeve strongly.

In the experiments described above, torques of the same amplitude and direction were generated on both sides. In real conditions, the synchronizing torque is created only on one side, while the average torque generated on the opposite side is approximately zero. So, when the required current on the synchronizing side $I_{reqsyn}$ is $I_{req}$, the mean electromagnetic torque $T_{actmean}$ that the actuator can generate during the synchronization of rotation speeds can be estimated from $T_{cmeas}$ as

$$\begin{array}{c}\hfill T_{actmean}=mean\left(-{\displaystyle \frac{T_{cmeas}-\Sigma T_{cfr(n_c=0)}}{g_l+g_r}}\right).\end{array}$$

(20)

The absolute value of $T_{actmean}$ is shown in Figure 12. It can be seen that, especially at higher currents, the actuator torque decreases with the increase in the speed difference. This is related to the fact that the supply voltage U = 70 V is too low to de-/magnetize the synchronizing side fast enough. As can be seen from the characteristic for $n_{relsyn}$ = 500 rpm, $|T_{actmean}|$ does not rise after $I_{reqsyn}$ reaches the value 21 A, which means that the de-excitation begins when this current value is reached, and the required current values higher than 21 A cannot be obtained at this relative speed, meaning that $I_{maxex}$ (Equation (10)) is ∼21 A for $n_{relsyn}$ = 500 rpm.

5.4. Gear Shifting

The validation of the simulation results shown in Figure 6 was conducted, and the experimental results are presented in Figure 13. Motor 1 was controlled to rotate the output shaft with the speed of 300 rpm, and the actuator generated torque on the left side to accelerate the left shaft from the initial $n_l$ = 0 to the speed of the shift sleeve. After that, the torque was generated further to reduce the residual angle difference from ∼−15° to ∼−4°, and the shift sleeve was engaged directly (no teeth–wall collision at the axial position x = −5 mm) with the slots of the left complementary ring shifting the left gear. The simulation results have a close correlation with the experimental results: in both cases, the speed synchronization takes ∼0.6 s; the time elapsed between the start of the synchronization and the engagement of the shift sleeve is ∼0.68 s in the simulation and 0.73 s in the experiment. This essentially confirms the developed shifting algorithm in the operation range under consideration. Additionally, this experiment and the results of a common unsynchronized shifting started at the same initial speeds are demonstrated in the Supplementary Videos S2 and S1, respectively.

Figure 14 shows a different situation, when the left gear has to be shifted, while the initial $n_{rell}$ is greater than zero. Motor 1 was controlled to keep $n_{sleeve}\approx$ 500 rpm, motor 2 was turned off when $n_{rell}$ became ∼500 rpm, and the synchronization was started. The initial speed difference on the right side was much smaller in this case (−27 rpm), so the braking torque was firstly generated on the right side, and the sides were switched at the time of ∼0.43 s so that the left side became the synchronizing side. $n_{rell}$ was reduced to ∼50 rpm. After that, the actuator was turned off, and the residual speed difference was used to change ${\theta }_{rell}$ from ∼16° to −10°, where the left coil was excited again, and the left gear was shifted without the face contact phase.

Figure 15 shows the situation when the right gear has to be shifted while the initial speeds of the sleeve and of the countershaft are 1000 rpm, generating the initial $n_{rell}$ and $n_{relr}$ of 455 rpm and −312 rpm, respectively. As was mentioned in Section 4, the resulting friction torque is much higher on the side of the bigger gearwheel ($\Sigma T_{rfr}\approx$ 3 Nm in this case), and the actuator torque is not high enough to accelerate the right shaft to 1000 rpm. However, the friction torque would be much lower in a conventional gearbox, where the gearwheels are directly interlocked with the countershaft without intermediate shafts and their bearings, belt transmissions, and belt tensioners. The resulting friction torque on the right gearwheel was assumed equal to 1 Nm in the case of a conventional gearbox. So, a torque of 1 Nm was applied by motor 1 to the output shaft, and the actuator synchronized its speed with the speed of the right complementary ring reducing $n_{sleeve}$ until it became lower than $n_r$. After that, the actuator continued to generate the torque on the right side to reach the position ${\theta }_{rell}$ = −10°, where the left side was demagnetized and the right gear was shifted. The collision between the teeth and the side surface of the ring at x = 5 mm was successfully avoided.

Finally, Figure 16 shows the situation when the output shaft rotates at a speed of 1500 rpm, the initial $n_{relr}$ is 300 rpm, and the right gear has to be shifted. The relative speeds on the sides are very different in this case since the initial $n_{rell}$ is 2308 rpm. Motor 2 was turned off, and the synchronizing torque was applied on the right side to decelerate the right shaft. The friction torque and the actuator torque have the same direction in this case. After $n_{relr}$ became approximately equal to 80 rpm, the synchronizing current was strongly reduced, otherwise $n_{relr}$ would became negative at the end of the next current pulse, which must be avoided. At the lower $I_{syn}$, the adjustment of the coil currents depending on ${\theta }_{rel}$ was no longer required, and the residual force difference $\mathsf{\Delta }{F}_{elmag}$ was compensated by the spring-loaded balls. The right gear was shifted when $n_{relr}$ became lower than 30 rpm at ${\theta }_{relr}$ = −6° so that the teeth were engaged with the slots without the face contact phase (no intermediate stop of the axial motion at x = 5 mm) at unaligned angular positions.

6. Conclusions

A novel type of double-sided electromagnetic dog clutches for application in automotive gearboxes has been developed based on the LREA presented in our previous works. The clutch is represented by the double-sided actuator integrated between two gearwheels. In contrast to common non-synchronizer transmissions, a gearbox equipped with the proposed device demonstrates the possibility of the rotation speed synchronization without an external torque source, while the clutch design remains simple and reliable. Moreover, the speed is synchronized without mechanical contact between the parts, which is a big advantage comparing to common synchromesh units.

In rotary operation, the electromagnetic torque generated by the LREA reduces the initial speed difference to a small positive value, which is then used to reach the relative angle between the teeth of the sleeve and the slots of its counterpart where direct teeth engagement is possible. In linear operation, the selected gear can be engaged or disengaged using the axial reluctance force generated by the LREA. The shifting of non-neutral gears starts at the relative angle reached in the rotary operation so that the teeth are directly engaged. Thus, an axial contact of the teeth at unaligned positions, which can lead to unsuccessful shifting, is avoided.

A special control algorithm developed for rotary operation allows the compensation of the axial force, which arises together with the synchronizing torque, by a simultaneous excitation of the coil on the opposite side and adjustment of the coil currents depending on the corresponding relative angle. Thus, the shift sleeve remains in neutral gear until the appropriate shifting conditions are reached. The synchronizing torque is always generated on the side with a lower speed difference, which is named as synchronizing side, while the opposite is named as compensating side and creates the balance of the axial forces. The average value of the collateral torque on the compensating side is approximately zero. The sides can be switched during the synchronization depending on their relative speeds. The time of the current rise/fall is taken into account to adjust the excitation angles. The residual force difference can be compensated by the spring-loaded balls placed under the shift sleeve. At lower currents, the current adjustment is not required since the force changes slightly with the change in the relative angle.

Since the gearwheels have different sizes and are interlocked with the same countershaft, the resulting friction torque is always higher on the bigger gearwheel. It makes its acceleration more difficult comparing to the smaller gearwheel. The actuator must be able to generate an average torque that is higher than the resulting friction torque to accelerate the gearwheel, while its braking is always possible since the actuator torque and the friction torque work in the same direction. The simulations show that, in principle, positive and negative speed differences can be reduced by the electromagnetic torque of the actuator while the shift sleeve remains in neutral gear. Moreover, a direct engagement of the teeth with the slots during gear shifting is possible on both sides.

To verify the theoretical results, a prototype of the double-sided dog clutch and a testing gearbox were designed and manufactured. The assembled gearbox with the integrated clutch was installed on a testrig with two motors, which can rotate the output shaft and the countershaft at different speeds. The experimental results show that the average electromagnetic torque generated by the actuator strongly reduces with the increase in the relative speed on the synchronizing side. This problem can be solved if the coils will be supplied with a higher voltage (the maximum voltage was limited to 70 V by the available electronic control unit). Further, pre-synchronized gear shifting was tested experimentally for both gears. Negative and positive differences between the rotation speed of the gearwheel and the shift sleeve could be reduced by the actuator without a significant displacement of the shift sleeve from the central position. In all four cases, the required gear was shifted without an axial contact between the shift sleeve and the corresponding complementary ring at unengaged state.

This work is focused only on the possibilities of contactless speed synchronization using the proposed clutch; however, its lays a foundation for further research. The methods that allow an operation of the clutch without linear and angular encoders will be published soon [45,46]. In the future, a multi-speed gearbox equipped with double-sided clutches of the proposed type, which will be suitable for application on a passenger vehicle, can be manufactured, and effects of its use can be studied.

7. Patents

Miroschnitschenko, B.; Polschak, F.; Rafetseder, D. Electromechanical clutch and method for closing and opening an electromechanical clutch and for sensorless determination of the relative angular velocity of a first and second shaft. DE102022121714A1, February 2024.