1 Introduction

Nonrenewable fuel shortages and environmental problems are becoming increasingly pressing global issues, leading the world to turn towards renewable energy sources [1, 2], and clean, efficient systems [3]. As a result, motor vehicles are receiving more attention for their efficiency and ability to adopt clean energy concepts, as traditionally nonrenewable fuels power them.

Motor vehicles can be cleaner for the environment by improving engine efficiency or using alternative fuels. Mallouh, et al. [4] studied the usage of the fuel cell (FC) in rickshaws (three-wheeled vehicles) rather than conventional internal combustion engines (ICE), and Liu, et al. [5] performed an experimental study on using alcoholic fuel instead of conventional gasoline. Szíki, et al. [6] performed a dynamic analysis for a race car with different electric motors configurations.

Another key component that can improve the vehicle's efficiency is vehicle transmission. The standard geometry of the teeth for the mesh gears can be modified to improve their performance [7, 8], and on the other hand, the performance of the gear shifting elements can be improved.

The development of motor vehicles started with the introduction of internal combustion engines (ICE), and the drive requirements added the need to include multi-gear ratio transmission. A mechanism to shift smoothly between the different gear ratios is introduced when the synchronizer is invented. The synchronizer has been studied by many researchers [9–11], and lasted for many decades as the usual gearshift mechanism. Recently, the fuel shortage and environmental issues caused the need for a more efficient system, since the synchronizer employs friction for input–output speed synchronization and has a large mass. Dog teeth clutch has been replacing the synchronizer because it provides quicker shifting time, simpler structure, larger power transmitting capacity, and has lower cost [12–14]. Commercial vehicles equipped with automated manual transmission (AMT) employ the dog clutch as a gearshift element [15]. Also, electric vehicles (EV) employ clutchless AMTs used where the friction cone is removed, and the speed synchronization is achieved by electric motor control [16–19].

However, as the mechanical speed synchronization mechanism (the friction mechanism) is missing, the speed synchronization must be a part of the gearshift control. EVs and commercial vehicles utilize an external synchronization strategy for speed synchronization. In many cases, such as motorcycles, these strategies are inapplicable, and the gearshift process occurs without speed synchronization. So, the shifting process has to be investigated to achieve successful dog clutch engagement.

To the authors' knowledge, these problems were first studied by Laird [20], who studied the radial and face dog clutch (discussed later) for heavy-duty commercial vehicles. Bóka, et al. [21] studied the application of the dog clutch shiftability in AMT for commercial vehicles. They used the notion of engagement probability to find a certain successful region depending on the initial mismatch speed and formulated the necessary equations to guarantee successful engagement at low mismatch speed (below 4 rad s⁻¹). In [22], we developed a dynamic model for the dog clutch engagement process. We divided the engagement process into four discrete stages and modelled the dynamics of each stage. The discrete stages were integrated into one continuous hybrid automata (HA) model to obtain the trajectories of the continuous states. Three gearshift cases are considered to verify the HA model, and Simulink simulation showed that the HA model could capture the continuous states' dynamics inside each discrete stage. Understanding the system dynamics of a known dog clutch geometry is crucial, but it is also essential to comprehend the gearshift quality and the dynamic performance sensitivity concerning the parameters of the dog clutch system. The gear-shifting mechanism controls the dog clutch motion, and it is a part of the vehicle transmission and the overall powertrain. Therefore, the dog clutch system parameters significantly impact the overall performance at a higher level.

The dog clutch geometry affects the static torque transmitted through the dog clutch, which primarily affects the required actuator force [23]. Nowadays, many electric powertrains (EPTs) and motorcycles employ an electromechanical gearshift actuator to move the gear shifting element (GSE), where the rotary motion from the electric motor (EM) is transferred as a linear one to GSE through a mechanical coupling mechanism. Increasing the force at GSE, which is the dog clutch, will increase the required torque at the EM side. According to Sakama, et al. [24], EM mass and power have an approximated direct linear relationship with the torque. In the case of EV, inappropriate dog clutch geometry will add unnecessary mass and power consumption. In the case of motorcycles, there is limited space for the mechanical components, and higher EM mass means larger required space and larger mechanical components and fixtures to comply with the transmitted forces.

Moreover, for EVs equipped with AMT, studies like [25] conduct multi-objective optimization for the entire powertrain to identify optimal gear ratios and component sizing for the lowest cost and energy consumption. However, the dog clutch geometry parameters are typically poorly treated and not included in the design space of the optimization problem. The tooth width and height affect the maximum transmitted torque, which, in turn, affects the maximum achievable acceleration. Acceleration is often considered a constraint in the optimization problem. Other studies, such as [26, 27], conduct optimal control research, where a control strategy and powertrain parameters are combined in one optimization problem to determine the optimal control parameters of the gearshift actuator resulting in the lowest energy consumption, while keeping an eye on the dynamic behavior and comfort. Again, some parameters, such as the axial gap, backlash, and teeth number, affect the ease of dog clutch engagement, and time and quality of the gearshift.

The dog clutch geometry affects either the required actuator force, maximum transmitted torque, or the gearshift quality, but additionally, it affects the dog clutch shiftability, as will be explained in section 4. The two effects shall be considered, and a parametric study is crucial here for two reasons. Firstly, the study provides a complete understanding of the dog clutch shiftability. Secondly, considering the dog clutch parameters increases the number of design variables and the design space, leading to increased computation time to solve the optimization problem. Finally, understanding the impact of dog clutch parameters helps researchers and development engineers to reduce the design space and variables by selecting the essential parameters affecting their system optimization objectives and constraints.

Additionally, several studies, such as [16, 18, 28], propose a gearshift strategy for electric powertrain equipped with AMT, where these strategies have three main phases: disengagement, synchronization, and engagement. During the synchronization phase, the process relies on EM angular speed and position control for synchronization. The angular position and speed difference between the EM and the transmission output shaft should be zero before engaging the next gear. The whole gearshift process time was 1,050 ms in [18] and reduced to 424 ms in [28], while the synchronization phase shares the highest percentage. According to Liu, et al. [18], the synchronization phase accounts for 47.2% of the whole gearshift process time, while it is 55% according to Walker, et al. [16], and 70.8% according to Xu, et al. [28]. As shown in section 4, engaging the sliding dog clutch without needing position and speed synchronization is possible by setting the appropriate gearshift actuator speed for the relative angular position and speed at the gearshift start time. This finding leads to a significant reduction in the synchronization phase time. However, development engineers should reduce the mismatch speed to a lower level that results in acceptable torsional vibration for their design comfort requirements or apply a torsional vibration damper.

Many researchers studied the dog clutch shiftability and the effect of system parameters. In his thesis, Eriksson, et al. [29] performed a multibody dynamic parametric study for the dog clutch used in the truck transfer case. They studied the influence of the gear tooth geometry, mass, material stiffness, and engagement speed on the dog clutch. They used three different designs to study the effect of chamfer distance, chamfer angles, teeth angle, and the number of teeth on the results and developed eight sets of parameter combinations for simulation. Andersson and Goetz [30] performed dynamic FEA using Abaqus on the dog clutch to investigate the effect of the chamfer angle, chamfer distance, tooth angle, and axial force. This work aimed to find each tooth geometry's maximum possible engagement mismatch speed.

Echtler, et al. [31] studied the saving potential of the alternative form fit shifting element compared with conventional multi-friction surface clutch (MFSC) in an automatic transmission using multibody simulation for a transmission and vehicle model. The new clutch, called TorqueLINE, consists of a cone surface and a dog clutch. Mileti, et al. [32] analyzed the form-fit engagement and the performance rating of six dog clutch design variants used in TorqueLINE. The authors created a multibody simulation using SIMPACK and applied different axial and mismatch speeds to find the successful engagement area. They tested several axial and mismatch speed pairs to find the maximum possible connection speed at a given axial speed. They found a linear relationship between the two aforementioned parameters. In a later work, Mileti, et al. [23] performed a multibody simulation for different dog clutch geometries to understand their dynamics behavior and the effect of the wear on the gearshift quality. The dynamic simulations, FEA methods, and even experimental test rigs [33, 34] can be used to study the parameters' influence, but they are time consuming and do not provide a fixable way to study a wide range, and a large number of the parameters.

To overcome the drawbacks of the aforementioned study methods, we introduced in a previous work a kinematic model for dog clutch shiftability [35]. Also, in the same work, we obtained a condition for a successful gearshift, called shiftability condition, which guarantees an impact-free gearshift process, but contains many parameters. This paper intends to employ this condition to study the system sensitivity efficiently. It introduces a method for the parametric study and analyzes the effect of these parameters on the shiftability map and the engagement probability.

Firstly, let us introduce a summary of the dog clutch system. A dog clutch, Fig. 1a, is a coupling used to transmit power. It consists of two parts having complementary geometry. These complementary shapes are referred to as dog teeth. The main dog clutch system parameters are listed in Table 1.

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Dog clutch geometry, a) 3D model, b) 2D schematic in linear representation, and c) 2D schematic in angular representation (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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For easier understanding, the dog teeth geometries in Fig. 1c are rolled out and visualized as having linear motion, Fig. 1b. At the beginning of the shifting, the sliding sleeve and the shifted gear have an axial gap x₀ and an initial relative angular position ξ₀ between the marked teeth. Here the sliding dog can slide axially with a speed of v₀, while the target gear has angular rotation regarding the dog teeth's position. The relative angular rotation during time is called the initial mismatch speed Δω₀. The engagement of the complementary geometries is eased with an angular backlash Φ_b.

2 Parametric study method

A large number of free parameters makes the parameter study difficult. We propose the following process to handle the problem. For explanation, let us consider two variables called x and y. Their values can change in the [x_min; x_max], and [y_min; y_max] ranges as free parameters, or can be fixed values at x_fxd and y_fxd, respectively. Also, a third variable called z is selected to study the overall system sensitivity for z variation in the range [z_min; z_max]. The continuous spaces for x, y, and z are discretized with a fixed step for each variable, and then each variable range is converted to a vector of discrete points.

The parameter sensitivity is computed in the following process. First, the shiftability condition in Eq. (1) is checked at a given value for z, for each point (x_i; y_j). This provides the shiftability map shown in Fig. 2a, with a parameter z_i, where points satisfying the shiftability condition are painted blue, and points not satisfying the shiftability condition are painted red.

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Illustration of the study procedure: a) shiftability map; b) engagement probability curve triplet (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Then, for computing P_y(z) at z = z_i, this is the variable x fixed at x_fxd. Similarly, to get the probability value, the number of blue points (N_bP) along the vertical solid line in Fig. 2a is divided by the total number of points (N_bP + N_rP) along this line.

When both x and y are free variables, we got a field of discrete points. To obtain probability P(z), we divide the number of blue points by the total number of points in this field. This is equivalent to the surface probability for the given field. Repeating this procedure for P_x(z), P_y(z), and P(z) at each z value, the system of three curves shown in Fig. 2b can be obtained.

The overall goal is to check the parameter variation effect on the shifting probability of the system. This is why the vertical axis is the probability in Fig. 2b. The horizontal axis is always the third variable, also called a parameter.

When comparing the curves, we can see that the probability containing both variables as free parameters is close to one curve with a fixed parameter. This helps us to consider the dominancy (influence) of the free parameters. In Fig. 2b, this is the y variable which is dominant in the (x; y) parameter pair. Table 1 lists all studied parameters' fixed values and the considered ranges.

In what follows, triplets will be selected from the free variable set to study the relative effect on shiftability.

3 Results of the parameter study

In this chapter, we examine the effect of the variables on the dog clutch shiftability. For the first variables, let us choose the angular velocity difference Δω₀, and the axial shifting velocity v₀, while the third variable, the parameter, is the relative angular position between the teeth at the beginning of the shifting, noted ξ₀.

Figure 3a shows the shiftability map. The number of blue bands (possible shifting zone) is constant but the width of the band changes depending on the parameter value. With increasing ξ₀, the variation is periodic, the zones are identical at ξ₀ = 0, ξ₀ = ϕ and ξ₀ = 2ϕ. This behavior is visualized better in Fig. 3b. At this given geometry, at fixed v₀ the uncertainty of the mismatch speed Δω₀ alone has a major effect on the probability compared to the axial speed, although both have the same uncertainty range according to Table 1. This can be clearly seen from the probability curve where both the mismatch speed Δω₀ and the axial shifting velocity v₀ are free variables, as this curve follows the probability curve where the mismatch speed Δω₀ is the free variable. Here, the mismatch speed Δω₀ has a range of [0; 500 rad s⁻¹] which is equivalent to [0; 4,777 min⁻¹]. This is a wide range compared to the axial shifting speed range [0; 500 mm s⁻¹], resulting in smaller probability values. Further on, if the axial speed v₀ is a free variable alone at Δω_0,fxd = 100 rad s⁻¹ (955 min⁻¹), the probability of successful shifting reaches higher values compared to the former case due to the narrower uncertainty range in the axial speed.

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System sensitivity for the initial relative position, ξ₀: a) shiftability map; b–c) engagement probability curve triplets (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Moreover, Fig. 3c shows the probability curves when the mismatch speed Δω₀ has a range of [0; 40 rad s⁻¹], equivalent to [0; 382 min⁻¹]. This range is much narrower than in Fig. 3b, and the axial speed range is kept unchanged. This figure shows that the axial speed is now the dominant variable, and the successful shifting probability reaches a higher value.

A comparison between Figs 3b and 3c shows that the dominancy within a selected parameter pair is not absolute but depends on the selected value range.

Additionally, it is clear from Fig. 3a, that a successful gearshift can be achieved at many non-zero angular speed and position since we can see blue regions at a nonzero mismatch speed for a given initial relative position. However, these successful gearshifts require the application of specific actuator axial velocities. When in application, this can reduce the gearshift duration for EVs equipped with AMT.

Figure 4a shows the effect of the fixed parameter values. Here the axial speed $v_0$ is the free variable and the mismatch speed is fixed at three different values: 477 min⁻¹ (solid line), 1910 min⁻¹ (dashed line), and 3,820 min⁻¹ (dash-dotted line). At a lower mismatch speed fixed value (solid line or 477 min⁻¹), there is a larger intersection zone between this line and the successful shifting region compared to the dashed line and dash-dotted line (highest fixed mismatch speed value, 3,820 min⁻¹). This means that when the axial speed $v_0$ is the free variable, it will have a higher connection probability with the mismatch speed fixed at 477 min⁻¹, compared to 1,920 min⁻¹ and 3,820 min⁻¹ values.

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Effect of the variables fixed values for a) Δω₀ and b) v₀, and engagement probability sensitivity for variables fixed values for: a) Δω₀, and b) v₀ (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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This behavior is also presented in Fig. 4b. Here, the diagram shows the shifting probability variation depending on the fixed value of the mismatch speed Δω_0,fxd. It can be seen that the variation is periodic along the horizontal axis, and its value decreases with the increase of the fixed value. Moreover, the probability oscillation amplitude decreases with increasing fixed value.

Figure 4c also shows the effect of the fixed parameter values. Here the mismatch speed Δω₀ is the free variable, and the axial speed v_0,fxd is fixed at three different values: 50 mm s⁻¹ (solid line), 200 mm s⁻¹ (dashed line), and finally, 400 mm s⁻¹ (dash-dotted line). In contrast to Fig. 4a, at a higher fixed axial speed value (dash-dotted line), there are larger intersection line segments between this line and the successful engagement region compared to the dashed line (200 mm s⁻¹), and solid line (50 mm s⁻¹) cases. This means that at higher fixed axial speed values, there is a higher connection probability.

This behavior is also presented in Fig. 4d. The diagram shows the shifting probability variation depending on the fixed value of the axial speed v_0,fxd. Here again, it can be seen that the variation is periodic along the horizontal axis, but its value increases with the increase of the fixed value. Moreover, the probability oscillation amplitude increases with increasing fixed value. Note that the amplitude variation is much smaller here than in the previous figure.

Figure 5a shows the shiftability field for ξ₀, Δω₀ set at different axial speeds. Below a certain axial speed, there is no possible connection, while a higher shifting area exists at higher axial speeds. This is clearly shown in Fig. 5b; at a given mismatch speed, there is a limited time –since there is limited tangential space to prevent tooth to tooth contact after the axial gap removal – to cover the feed distance, so with higher speeds, the clutch parts can successfully cover the overlap distance and increase the engagement probability. The probability has less sensitivity for axial speed variation at higher values when ξ₀ is the free variable. The lower limit for axial speed has an important role in the practical side; in case $\xi$ is the free variable, if axial speed is large enough, there is a possible connection, and a sensor to measure the initial relative position can be eliminated.

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System sensitivity for axial speed, v₀: a) shiftability map; b) engagement probability (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Figure 6 shows the shiftability field for ξ₀, v₀ combination at different mismatch speeds. The shiftability field narrows and starts to vanish with higher mismatch speeds. In contrast to the axial speed, there is no possible connection above a certain mismatch speed, as shown in Fig. 6b, and this result agrees with practical results. Moreover, Fig. 6a shows that at low mismatch speed (less than 4 [rad s⁻¹]) there is a certain successful engagement at lower fixed axial speed when ξ₀ is a random variable and v₀ is fixed.

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System sensitivity for mismatch speed, Δω₀: a) shiftability map; b) engagement probability (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Figure 7 shows the shiftability field for ξ₀ with Δω₀ combination at different teeth number values. In Fig. 7a, the bands are close to each other, and new bands appear in the empty areas. In Fig. 7b, the probability increases with higher teeth number, but it has a higher value if $\xi$ is a free variable since it has narrower uncertainty range compared to Δω₀. In general, the shifting probability increases with higher teeth numbers; since we consider the system periodicity and a tooth on the sliding dog has a higher chance to meet a tooth on the target gear. Again, Fig. 7b shows that Δω₀ is the dominant variable.

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System sensitivity for teeth number, Z: a) shiftability map; b) engagement probability (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Figure 8a shows the shiftability field for ξ₀ with v₀ at different overlap distance values. Higher overlap distance narrows the successful engagement region. This is due to the limited time to cover longer overlap distances. This behavior is clear in Fig. 8b where the probability decays with higher overlap distance. There is no possible connection above a certain value because the required time to cover x_fed becomes larger than the available time. However, this value is higher in the case when v₀ is a free variable. For a given Δω₀ and available tangential space, v₀ determines if the overlap distance can be covered or not. When $\xi$ is a free variable, v₀ is fixed at 250 mm s⁻¹ but v₀ has a range [0–500] mm s⁻¹, so when v₀ is a free variable, there is a probability for axial velocity to be higher than 250 mm s⁻¹. According to this, in the case when ξ₀ is the free variable, the overlap distance limit increases with higher v_0,fxd. Comparing Figs 6b and 8b, the system has similar sensitivity for both Δω₀ and x_fed variation.

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System sensitivity for overlap distance x_fed a) shiftability map, b) engagement probability (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Figure 9a shows the shiftability field for ξ₀ with Δω₀, combinations at different backlash values. It shows a higher shiftability field with higher backlash values. Figure 9b shows a higher shifting probability with higher backlash. Higher backlash gives higher tangential space for teeth engagement at the overlap distance covering phase. This gives higher available time at given Δω₀. It also shows that no connection is possible below a certain backlash value in the case when ξ₀ is the free variable. This limit can be reduced with higher v_0,fxd, since it reduces the required time to cover the overlap distance and reduces the relative rotation during this phase.

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System sensitivity for backlash, Φ_b: a) shiftability map; b) engagement probability (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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4 Discussion

In Figs 6b and 8b, it has been shown that there is a maximum connection mismatch speed, and maximum possible overlap distance, respectively. Moreover, Figs 5b and 9b showed that there is a minimum required axial speed and backlash, respectively, for a successful connection. In what follows, we aim to express these quantities analytically, and compare them with the literature.

Figure 10a shows the response surface for Δω_0,max as a function of $\xi$ and v₀, the range for $\xi$ is between zero and Φ_b. Δω_0,max shows a linear relationship with both $\xi$ and v₀, when other parameters are fixed. This surface separates the successful engagement region from the unsuccessful one.

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Maximum connection mismatch speed for a) no tooth pass case, and b) tooth passing case (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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On the other hand, for the tooth pass case, Eq. (1) has no analytical solution, similar to Eq. (3), and the response surface, similar to Fig. 10, is obtained numerically. This is achieved by finding maximum Δω₀ that satisfy Eq. (1) for different (ξ₀, v₀) pairs. The obtained surface is shown in Fig. 10b, where it can be seen that the linear relationship still exists between Δω_0,amx and v₀. Compared to Fig. 10a, ξ₀ has a linear-periodic relationship with maximum connection speed. Moreover, the surface is not continuous, compared to Fig. 10a. Unlike Fig. 10a, this surface guarantees no possible connection above it but does not guarantee a successful connection below since the successful engagement zones are not continuous, as seen in Fig. 4a and c.

Figure 11 shows that relationship between the limit angle and the backlash is inverse, and ${\theta }_{\mathit{lim}}$ is more sensitive to the change in Φ_b at lower values. Smaller θ_lim increases Δω_0,max, but these θ_lim values are achieved at the higher backlash. On the other hand, increasing the backlash decreases the dog teeth' thickness and strength. So, a compromise shall be considered between the performance and tooth strength during the design phase.

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Limit angle (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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In Eq. (5), the initial relative position has been assumed to be zero, but in reality, it has a random value. Thus, validation is required to show that Eq. (5) is valid regardless the value of the initial relative position. This is achieved by using the statistical calculation based on Eq. (3). By taking the initial relative position as the free variable, the lowest mismatch speed that results in zero probability at a given axial speed gives the pair (Δω_0,max, v₀).

Figure 12a illustrates the results for the followed procedure and shows the maximum possible connection speed at each axial speed. The curve is not perfectly linear, and it shows a stairs shape because the followed methods convert the system's continuous ranges to discretized ranges. However, the best fitting is a line. The linear relationship between the axial speed and the maximum possible connection speed does not depend on the initial relative position but only on the geometry parameters shown in Eq. (6), and this agrees with the experimental results in [32], where in the study, the initial relative position is considered free.

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Validation for the linear relationship for: a) no tooth passing case, and b) tooth passing case (Own source)

Citation: International Review of Applied Sciences and Engineering 15, 1; 10.1556/1848.2023.00644

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Moreover, to check if the linear relationship still exists for the tooth pass case with unknown initial relative position, the procedure utilized in Fig. 12a is used but based on Eq. (1). Figure 12b shows that the curve is almost linear, and a linear best-fit line can be obtained. This shows that there is a limit angle for both no tooth pass and tooth passing cases, which again matches the experimental results in [32] since low and high mismatch speeds are tested. Since the relationship is linear in both cases, the limit angle only depends on the geometric parameters.

The developed condition could detect the same system's linear relationship in [32] with much less modelling effort compared to the multibody simulation.

5 Conclusion

A sensitivity analysis for the dog clutch system is introduced. The base of the analysis is the shiftability equation that ensures successful shifting. However, it contains many independent parameters that are often missing in the powertrain performance optimization and control algorithm design.

The shiftability maps based on the shiftability equation can help in the choice of dog teeth geometry during the gearbox design case. They can also help in the choice of actuation algorithm and devices when automating a given classical gearbox.

For a given combination of x and y as a free variable, the chosen ranges for both variable and the fixed value for x when y is a free variable highly affects the probability of shifting. The effect of the fixed values can be anticipated from the shiftability field for x and y by drawing a line normal to the fixed axis and deducing how the successful regions intersecting this line change.

While the successful shifting area borders have a higher angle with the x-axis, x tends to have higher dominancy in the probability if both x and y are free, provided that both x and y have comparable uncertainty ranges. A higher randomness range in the free variable reduces the shifting probability and raises its dominance for the other free variable.

The axial speed and backlash positively affect the engagement probability, while the overlap distance and the mismatch speed negatively affect the engagement probability. The initial relative position showed a periodic effect on the connection probability. For successful engagement, there is a minimum axial speed but a maximum mismatch speed. The system parameters can be categorized into two groups; one group affects the available time to cover the overlap distance, and another group affects the required time to cover this distance. The former contains the mismatch speed and the overlap distance, while the latter contains the axial shifting speed and the overlap distance. This categorization explains the similarity in the probability curve when the initial relative position is a free variable between the parameters, mismatch speed, and the backlash and between the parameters, axial speed, and the overlap distance but in an opposite sense. This result has a role in reducing the efforts for system simulation, including the dynamic effect.

Finally, the maximum possible mismatch speed is identified for a given axial speed and a linear relationship is present between them. These results were verified upon literature. This condition has an important role in either dog clutch design or gearshift controller design for automatic transmission.

Further study is required to clarify the influence of parameters, as well as that of the real inertias.