## Abstract

The dog clutch offers advantages in mass and efficiency, but faces challenges in mismatch speed synchronization, which affects its shiftability. Face impact between dog teeth also reduces its lifespan. Our previous work introduced the kinematical shiftability condition that ensures impact-free gearshift but had limitations due to analysis assumptions. This study eliminates those assumptions, but uses a similar approach. Based on our previous work for dog clutch engagement dynamics, we develop the dynamical shiftability condition. Validation with full dog clutch dynamics showed an agreement. Employing another previous work that introduced shiftability map and parametric study method, we study system parameters impact on shiftability but based on the dynamical shiftability condition.

## 1 Introduction

Global concerns over nonrenewable fuel shortages and environmental issues are driving a shift to renewable energy [1, 2] and cleaner systems [3]. Motor vehicles are now a focal point for efficiency improvements and adopting clean energy concepts. Enhancing powertrains, primarily the engine and transmission, is crucial for making vehicles environmentally cleaner. The powertrain, including the engine and transmission, is a critical subsystem in vehicles. Improving engine efficiency and exploring alternative fuels are avenues for enhancing environmental sustainability [4–6]. Another crucial element that can contribute to enhancing a vehicle's efficiency is the vehicle transmission. For the transmission, modifying gear tooth geometry in power transmission gears [7, 8] and the performance of the gearshift elements can further improve vehicle efficiency.

The gearshift element enables shifting between gear ratios and synchronizes speed differences (mismatch speed) between the transmission's input and output sides. The synchronizer is used in manual transmissions and automated manual transmission (AMT), while the multi-disk wet clutch (MDWC) is used in conventional automatic transmission (AT). Synchronizer and MDWC face challenges meeting new clean, efficient system requirements because both rely on a friction-based mechanism for speed synchronization, which causes energy waste. Also, AMTs are widely used in heavy-duty and commercial vehicles due to their high transmission torque, efficiency, and low manufacturing and maintenance costs [9, 10]. However, the friction-based mechanism in synchronizers limits their lifespan, posing challenges for heavy-duty use, where a long lifespan is required. The challenges come from limitations of materials strength and manufacturing technology [11, 12].

The dog teeth clutch is replacing the synchronizer and MDWC due to its quicker shifting time, simpler structure, larger power capacity, and lower cost [13, 14]. Heavy-duty vehicles with AMT use the dog clutch as a gearshift element [15]. Some conventional ATs employ the dog clutch as an interlocking element [16, 17], or gearshift element [18]. Electric vehicles (EVs) use clutchless AMTs with the dog clutch as a gearshift element [19–22]. The dog clutch is similar to the synchronizer but without the friction mechanism, which offers its advantages. However, the removal of the speed synchronization mechanism causes a speed synchronization problem, which affects dog clutch engagement capability (or shiftability). This requires further investigation into the dog system.

These investigations originated from Laird's research [23] on shifting characteristics of radial and face dog clutches installed into a test vehicle. Laird observed that the radial clutch has a longer shifting time and bounces 5–6 times for successful engagement, while the face clutch engages successfully on the first attempt. The bounce number correlates linearly with engagement time. The bounce number depends on the engagement probability, which depends on the shifting speed and backlash. Shifting probability is minimal for backlashes below 15° and increases with higher backlash.

Echtler et al. [24], explored the energy-saving potential of TorqueLINE, an alternative shifting element, for MDWC in ATs. TorqueLINE, consists of a conical friction element, a dog clutch, and a consecutive form-fit for high-torque transmission. They showed an 85% saving potential at mismatch speeds below 2000 RPM and decreases to 50% at 5000 RPM. In a later study, Mileti et al. [25], examined the engagement capability of five dog clutch design variants, varying in backlash, teeth number, and tooth flank angle. Utilizing SIMPACK for multibody simulation and applying different axial and mismatch speeds, they identified that a dog clutch with a high angular gap, low teeth number, and low flank angle has the largest successful engagement area, while the opposite configuration has the smallest one.

Eriksson et al. [26], conducted a parametric multibody dynamic study on the dog clutch in a truck transfer case. Their research examined how dog clutch geometry, mass, material stiffness, engagement mismatch speed, and actuator force influenced performance. Three tooth designs with variations in chamfer distance, angles, tooth angle, and tooth number were used. With ten combinations derived from clutch mass, material stiffness, engagement mismatch speed, and actuator force, multibody simulations in MSC ADAMS were performed for each design. The combination with a mass of 8 kg, mismatch speed of 539 RPM, material stiffness of 81,441 N/m, and actuator force of 198 N demonstrated the best performance. It has the lowest engagement time and bounce count among all three designs.

Andersson and Goetz [27] employed dynamic finite element analysis (FEA) using Abaqus to study the impact of chamfer angle, distance, tooth angle, and axial force on the dog clutch's performance. They created 11 alternative designs. The primary goal was to determine the maximum possible mismatch speed for each tooth geometry. The results showed that the coupling is easier as the chamfer angle is smaller, and a longer chamfer distance positively affects the maximum mismatch speed.

Experimental, multibody dynamic simulation, and dynamic FEA methods provide means to explore parameter influences, but they have limitations. Primarily, these approaches are time-consuming and limit the capability to study a vast parameter space. Additionally, the face impact between dog clutch teeth, causing back-and-forth bouncing, substantially diminishes the clutch's lifespan. To the author's knowledge, prior research lacks an analysis ensuring a face impact-free gearshift process of the dog clutch, particularly at high mismatch speeds.

To overcome the drawbacks of the aforementioned study methods, in previous work [28], we followed a different approach to study dog teeth clutch shiftability. Our approach involved a kinematic analysis of the dog clutch engagement process. This kinematic analysis assumes constant axial and mismatch speeds throughout the gearshift process. From this analysis, we derived a kinematical shiftability condition, which ensures a face impact-free gearshift process without the need for complex multibody dynamic or dynamic FEA simulations. Additionally, we visually presented the successful gearshift area through the developed shiftability map. In another study [29], we created a method to calculate shifting probability and developed a parametric study method to explore the sensitivity of the shiftability map and engagement probability to system parameters. Our analysis indicated that the number of teeth and tangential backlash positively affect engagement probability, while initial mismatch speed and overlap distance have a negative impact.

In [28], we studied dog clutch shiftability kinematically, assuming constant axial and mismatch speeds. This kinematical shiftability condition is applicable in transmissions with negligible system dynamics. However, in scenarios where gearbox losses impact mismatch speed or the gearshift actuator cannot maintain constant axial speed, its applicability is limited.

In our work [30], we developed a dynamic model for the dog clutch engagement process. This model described all the engagement stages dynamically, considering the time trajectories of the axial and mismatch speeds. So, based on [30], this work aims to develop a so-called dynamical shiftability condition to overcome the limitation of the kinematical shiftability condition. Here, the mismatch and axial speeds are considered not constant. Moreover, we utilize the dynamical shiftability condition and the parametric study method developed in [29] to investigate the system sensitivity for some selected parameters and their effect on dog clutch shiftability.

## 2 Dynamical shiftability model

In ref. [30] we have introduced a full dynamic model for the dog clutch engagement process. However, we briefly mention the dog clutch geometry and its engagement stages. 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.

Dog clutch shiftability parameters

Parameter | Unit | Parameter | Unit |

[rad/s] ( | [mm] | ||

[rad/s] ([min^{−1}]) | [mm] | ||

[N] | |||

z | [–] | m | [kg] |

*ϕ*and an angular backlash given by Eq. (1) and Eq. (2), respectively. Here,

*x*

_{0}, and an initial relative angular position, denoted

*ξ*

_{0}, between the red-marked teeth, between the sliding sleeve (s) and the meshing gear (g). Figure 2a shows further parameters. The sliding sleeve has the capability to move axially, while it concurrently undergoes relative angular rotation in relation to the target gear. The relative angular rotation is referred to as the mismatch speed

*,*and mismatch speed

The engagement of the complementary geometries is eased with an angular backlash

*ξ*can be much larger than

*ξ*shall be transferred to the first cycle (or between

The engagement process can be broken down into four primary stages: 1) free fly axial motion, 2) axial (face) impact stage, 3) tangential (side) impact stage, and 4) full engagement stage.

At the start of the shifting, time *t*_{0} (Fig. 2a), the constant actuator force *F*_{act} drives the sliding sleeve axially until the axial gap is eliminated at *t*_{1}. During this time, the dog clutch components undergo relative rotation, and the relative angular position changes from

As stage 2 begins, the teeth of the sliding sleeve and the gear meet at the face impact position *x*_{0} (or *x*_{s,0}). At this position, the sleeve motion is subject to three possibilities, as illustrated in Fig. 3 for

It is assumed that if a tooth on the sliding sleeve can overlap a certain distance with a tooth on the gear sleeve without face impact, the sliding sleeve will not rebound, which ensures successful engagement. This overlapping distance is referred to as *x*_{fed}. The face impact can possibly happen until the overlap distance is fully covered at time *t*_{2} (Fig. 2c). During this phase, a relative rotation between the dog clutch and the target gear occurs, and the relative angular position changes from

In stage 3, the sliding sleeve continues its axial movement until it completely covers the full tooth height, reaching a position denoted as *x*_{0}+*h*_{t} (or *x*_{0,f}) in the axial direction (Fig. 2d). Throughout this stage, the

Stage 4 marks the completion of the gear shift process. It is characterized by the synchronization of mismatch speed and the sliding sleeve while the sliding sleeve reaches its ultimate axial position

If *,* a face impact will occur during the overlap distance coverage, and the sliding sleeve will bounce back.

The notion of mismatch speed states that one part rotates quicker than the other. In our case, if the gear is the quickest, then we consider the mismatch speed has a positive sign. This is shown in Fig. 4, when the red-blue part (the gear) is moving upwards regarding to the standing green part (sleeve). Generally, there are cases when the green part (sleeve) is the one moving quicker. In this case, we consider that the mismatch speed has a negative sign.

*,*can be within is shrunk according to:

If

Equations (7) and (8) have *0* or *1* values: *1* for face impact-free gearshift and *0* for gearshift with impact, or unsuccessful gearshift. As our study focuses on face impact-free gearshift, the later tow cases are referred to as unsuccessful gearshift, while the first case is just a successful gearshift.

In Eqs (7) and (8), *.* The aim now is to find *,* respectively.

As seen from Fig. 3,

*m*, then the linear acceleration is given according to:

*,*and at

*r*

_{1}, and

*r*

_{2}can be obtained. Also, substituting Eq. (20) in the nonhomogeneous ODE (Eq. (18)) and comparing the coefficients of the polynomial terms,

*a*

_{2},

*a*

_{1}, and

*a*

_{0}can be determined. This yield:

*C*

_{1}, and

*C*

_{2}can be determined from the initial conditions of zero initial linear position and speed. This gives:

*t*as a function of

*W*

_{0}is the principal branch of the Lambert

*W*function. The Lambert W function-also referred to as the product logarithm-consists of a set of functions denoted as

*W*function is provided in [32]. The Lambert

*W*function has applications in many fields, such as control theory [33] and signal processing [34]. One of the most common forms of the Lambert

*W*function is the Lambert

*W*(y) for real numbers with the branch

*y*, ignoring the substituted parameters values

*.*Let us write the term for Lambert

*W*function in Eq. (26) as:

*u*in Eq. (27) are positive, so, it is clear that

*u*, so, it is clear that

*u > 1*. The rest of the proof is introduced below.

*u*and

*y*are the same as defined in Eq. (27). We already proved that

*y*is in the allowable range. Also

So, the condition in Eq. (29) is satisfied, ignoring the system parameters substituted. Since, the conditions

Equation (24) for *x*_{s}*(t)* and its inverse Eq. (26) for *t(x*_{s}*)* are plotted against each other in Fig. 6 for the constant values in Table 2 except with *500 N* actuator force and *c*_{r} of 4 *N.s/m*. It is well known the function inverse is a mirror of the original function along the *45°* line, and Fig. 6 shows that Eq. (26) is a mirror of Eq. (24) along the *45°* line.

Fixed values and the study ranges for the variables and parameters

Parameter | Unit | Fixed value | Range |

Initial relative position | 4 | 0– | |

Mismatch speed | [rad/s] ([min^{−1}]) | 20 (200) | 2–120 (19–1146) |

Actuator force | [N] | 1000 | 100–2000 |

Number of teeth Z | [–] | 5 | 2–10 |

Feed distance | [mm] | 0.5 | 0.5–3 |

Backlash | 15 | 2–20 | |

Axial gap | [mm] | 10 | – |

Mass M | [kg] | 3 | – |

Inertia J_{g} | [kg m^{2}] | 0.2 | – |

Columb friction T_{cl} | [N.m] | 0 | – |

Rotational viscous friction c_{r} | [N.m.s/rad] | 0.02 | – |

Columb friction force F_{cl} | [N] | 0 | – |

Linear viscous friction c_{l} | [N.s/m] | 0 | – |

*,*and

WF stands for the case with linear viscous friction. Now, for

*T*

_{Loss}). If the engine side has an inertia

*J*

_{g}, and the wheels side has infinite inertia compared to the engine side, the angular speed of the wheel side or sleeve (s) is constant but the engine side or gear (g) angular speed decreases due to the friction torque. According to this, the gear angular acceleration is given as:

*c*

_{r}and

*T*

_{cl}are rotational viscous friction coefficient and Coulomb friction torque, respectively. Eq. (4) can be solved for

*T*

_{Loss}as a function of

*T*

_{Loss}in Eq. (33), the differential equation for

*r*

_{3}, and

*r*

_{4}can be obtained. Also, substituting Eq. (38) in the nonhomogeneous ODE (Eq. (36)) and comparing the coefficients of the polynomial terms,

*b*

_{2},

*b*

_{1}, and

*b*

_{0}can be determined. This yield:

*C*

_{3}, and

*C*

_{4}can be determined from the initial conditions

*,*we substitute the values

*t*

_{1}and

*t*

_{2}from Eq. (15) and Eq. (16), respectively, into Eq. (42).

*,*by substituting the values

*t*

_{1}and

*t*

_{2}from Eq. (31) and Eq.(32), respectively, into Eq. (42).

Having *,* these can be substituted in Eq. (7) or (8), which gives the full dynamical shiftability condition. Later on, wherever no linear viscous friction case (*c*_{l} = *0*) is considered, Eqs (44) and (45) are implicitly used in Eqs (7) and (8), while for viscous friction case, Eqs (46) and (47) are implicitly used in Eqs (7) and (8).

## 3 Parametric study method

The geometric condition outlined in Eqs (7) and (8) for shiftability encompasses numerous parameters. They can be divided into three sets of parameters: the dog geometry parameters, including *m*, the system friction parameters, including *c*_{r}, *T*_{cl}, and *c*_{l}, and *F*_{cl} and the dynamic and kinematic parameters including, *.*

The presence of many parameters can pose challenges for conducting a parameter study. In [29], we proposed a parametric study method to study the parameters effect based on the kinematical shiftability condition. As summarized below, the same method is used here but based on the dynamical shiftability condition.

Let us consider two variables, *,* and *, re*spectively. Additionally, a third variable, denoted as *y*, is selected to investigate the overall system sensitivity over a range of

The parameter sensitivity analysis goes in the following manner: Initially, the shiftability condition described in Eq. (7) is evaluated for a given value of y, across each point (

This shiftability map is plotted at a fixed initial relative position *y*. A convenient method to study this is to find the ratio of the successful region area (sum of the blue regions areas), to the whole studied area. In other words, we are calculating the successful region portion. At each point

*0, 2π/z*]. So, the engagement process must be handled based on probability. The engagement probability at each point (

*y*

_{j}) shall be calculated according to:

*G*is the shiftability condition given in Eq. (7) or (8). However, as the study method discretized all the parameters ranges, Eq. (48) shall be modified to be used in the discretized space. Let us assume the

*0, 2π/z*]), is discretized to

*n*discrete points, the engagement probability can be calculated as:

*G*has

*0*or

*1*values only, so, the number of points with

*G*=

*1*is the same as the sum of

*G*values at all points in

The higher the discrate points number n is, the more accurate the probability is. We discretized this interval to 2000 points.

In what follows, various parameters and variables will be selected to study their effect one by one on the shiftability map and engagement probability.

## 4 Results of the parameter study

In this chapter, we examine the effect of the variables on the dog clutch shiftability in two parts. The first part examines the effect of some selected geometry parameters on the dog clutch shiftability while ignoring the linear viscous friction. The second part investigates the effect of some selected system friction parameters. For the first part, Table 2 lists the fixed values and study ranges for the parameter.

However, firstly, let us validate the developed shiftability condition with our work for dog clutch dynamics [30]. Similar dynamics are employed in our previous paper [30] to describe the dog clutch during each stage. Ten cases are chosen to compare the actual dynamics with the shiftability conditions, listed in Table 3. In Table 3, the actuator force, the mismatch speed coulomb friction force, and the linear viscous friction coefficient are changed, while other system parameters are fixed according to the fixed value in Table 2. For cases 1 and 2 in Fig. 8a, case 5 in Fig. 8b, and case 8 in Fig. 8c, the sleeve has been blocked by the gear at the end of the axial gap (*x*_{s} *= 10 mm*) due to the face impact. So, these gearshifts were not successful. The shiftability condition could detect the same results for the aforementioned cases, as Table 3 shows. For case 4 in Fig. 8a, cases 6 and 7 in Fig. 8b, and cases 9 and 10 in Fig. 8c, the sleeve could reach its final position at *15 mm*, without impact, and the shiftability condition detected the same results as Table 3 shows. For case 3 in Fig. 8a, the sleeve impacted with the gear and is blocked by the gear for a short time, then continues to move until it reaches the final position. Even though the sleeve reached its final position, this was considered an unsuccessful gearshift due to the presence of the face impact, and the shiftability condition detected the same results.

Shiftability condition validation cases

case | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

N] | 100 | 250 | 300 | 500 | 200 | 350 | 500 | 200 | 350 | 250 |

100 | 160 | 120 | 50 | 140 | 200 | 240 | 160 | 100 | 160 | |

c_{l} [N.s/m] | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 4 | 8 |

F_{cl} [N] | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 10 | 5 |

Full dynamic simulation Successful (1) Unsuccessful (0) | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |

Shiftability condition Successful (1) Unsuccessful (0) | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |

After we validated our work, we show the results of the parametric study. Firstly, we examine the effect of different overlap distances

The shiftability map in Fig. 9a is plotted at a fixed *4°*. To better understand how the overlap distance affects the shiftability map, we calculate the ratio of the successful region area (Sum of Blue areas) to the entire domain area at various

The negative effect of the overlap distance comes from its effect on the relative rotation *∆t* to cover, as shown in Eq. (17), resulting in an increased relative rotation

The previous explanation assumed that the initial relative position was known. However, in cases where the initial relative position is unknown and randomly distributed within the range [*0, 2π/z*], we consider the engagement probability (denoted in Fig. 9b). Figure 9b presents the probability plot at different overlap distances, revealing a negative effect on the probability. As *∆t* increases due to higher

Figure 10a provides the shiftability map at various backlash values. It is observed that the blue regions rapidly expand with increasing backlash values.

Furthermore, Fig. 10c and d demonstrate that backlash positively affects the portion of the successful region. At a constant *backlash* increases, as Fig. 10e shows. Secondly, the curve trends -within the backlash interval corresponding to the dark blue region-for the rate of change of the portion with respect to *backlash* increases, which stimulates the portion value growth.

Increasing the backlash will increase the left side of Eq. (7), which creates more relief on the shiftability condition, allowing it to be satisfied at more points. As a result, the blue regions of the shiftability map and the portion increase. From a system perspective, as the backlash increases,

Figure 10b illustrates that the probability is highly sensitive to changes in backlash and exhibits a direct linear relationship. Higher backlash provides a greater tangential space for teeth engagement during the overlap distance covering phase, thereby increasing the available time to cover this distance. This extended duration enhances the chance of a successful gearshift.

Figure 11a illustrates the shiftability map at different teeth number values, revealing a high sensitivity of the successful region to changes in the teeth number. Additionally, Fig. 11c and d demonstrate that the portion of the successful region increases with the teeth number, *z*. In Fig. 11c and d, the surface and contour plots are cropped down as the teeth number increases, since *0, 2π/z*]. Figure 11d shows that the dark blue region shrinks as the teeth number increases, similar to Fig. 10d. With the aid of Fig. 11e, this can be explained for the same two reasons explaining this behavior for the system sensitivity for backlash.

In Eq. (7), a higher teeth number decreases the maximum value that the middle side can reach, which is *2π/z.* This increases the chance for the condition to be satisfied, even though the left-hand side remains constant. So, the teeth number positively affects dog clutch shiftability.

Figure 11b shows that the probability increases with the teeth number and exhibits a direct linear relationship, similar to the relationship observed in Fig. 10b. A higher tooth number decreases the randomness of *0, 2π/z*].

In the previous part, the focus was on the system sensitivity to its geometry parameters. For this reason, the rational friction parameters are kept constant, and the linear friction is ignored. In the following part, we investigate the effect of the system friction on the dog clutch shiftability. In the following explanation, the same fixed values in Table 2 are used, except with *10* teeth to show better the results.

Figure 12 shows the linear viscous friction coefficient *c*_{l} effect on the shiftability map (or portion of the successful region). Figure 12a shows that the system is not sensitive to the change in *c*_{l} since the contour lines are almost parallel to *c*_{l} axis. In fact, the max *c*_{l} value is *40 N.s/m* while the max *F*_{act} is *2000 N,* and the friction is very low compared to the maximum available actuator force. Figure 12a is extended to 400 *N.m/s* as Fig. 12b shows. The contour lines are not parallel to *c*_{l} axis, so, the system is sensitive to *c*_{l} change, and this dependency increases at higher *c*_{l} values since the contour lines deviate from the *c*_{l} axis. On the other hand, *F*_{act} range in Fig. 12a is reduced, with *500 N* as the maximum available actuator force, as Fig. 12c shows. Compared to Fig. 12a, the counter lines are not parallel to *c*_{l} axis. However, the comparison between Figure 12b and Figure 12c shows that the system is more sensitive to *c*_{l} change in Fig. 12c. However, both Fig. 12b and c show low dependency on *c*_{l}, especially at lower values, since the contour lines are nearly parallel to the *c*_{l} axis. Figure 12c is extended to 400 *N.m/s* as Fig. 12d shows. Figure 12d shows that the system, in this case, is highly dependent on *c*_{l}, especially at higher values. However, in Fig. 12d, the friction values are comparable to the actuator force, which is an unrealistic case. Figure 12a and c show the most realistic cases for the friction in dog clutch systems and gearshift actuators. Both aforementioned cases show low dependency on the linear viscous friction.

Besides the dependency, Fig. 12b–d shows that the linear viscous friction negatively affects the dog clutch shiftability; while holding *c*_{l} values. Also, the dark blue region is wider, in the vertical direction, at higher *c*_{l} values, as clearly seen in Fig. 12b–d. At a given actuator force, the linear viscous friction slows down the system. This will increase the time required to cover the axial gap, which means a higher *t*_{1} value. As time passes, the mismatch speed decreases due to the loss torque, and with higher *t*_{1}, the mismatch speed *c*_{l} helps in decreasing this relative rotation during the overlap distance coverage. On the other hand, slowing down the system will increase the time to cover the overlap distance, and increasing this time increases the relative rotation *c*_{l} has two conflicting effects on the relative rotation *10* teeth and *500 N* actuator force-shows that the negative effect outfitted the positive effect; crossing Fig. 14a vertically,

Figure 13 shows the rotational viscous friction coefficient *c*_{r} effect on the shiftability map. Figure 13a shows that the system is not sensitive to the change in *c*_{r} since the contour lines are almost parallel to *c*_{r} axis. *F*_{act} range in Fig. 13a is reduced, with *500 N* as the maximum available actuator force, as Fig. 13b shows. The contour lines are not parallel to *c*_{r} axis, so, the system is sensitive to *c*_{r} change. Both the actuator force and rotational viscous friction positively affect the shiftability condition. Higher *F*_{act} reduces the time to cover the overlap distance, and higher *c*_{r} reduces the mismatch speed *c*_{r}. For this reason, *c*_{r} change is more sensible at a lower maximum available force.

Moreover, the effect of both rotational and linear viscous friction on the successful region portion is combined and illustrated in Fig. 14b, at fixed *c*_{r} positively affects while *c*_{l} negatively affects the portion value, as concluded above. However, the portion is more sensitive to *c*_{r} as the contour lines are closer to *c*_{l} (vertical) axis. This can be explained by the relative rotation *c*_{r} and *c*_{l}, as these two parameters are included in *c*_{r} as the contour lines are closer to *c*_{l} (vertical) axis.

## 5 Conclusion

In this study, we developed a dynamic model for dog clutch shiftability. This work aims to overcome the limitation of the previous work for dog clutch shiftability based on a kinematical model. This model considers the system dynamics regarding the mismatch speed and the axial velocity, where they are considered not constant variables. The developed condition is based on our previous work for the dog clutch engagement dynamics. Based on this condition, and our previously developed method for parametric study, we analyzed the system parameters' effect on the dog clutch shiftability, including the shiftability map, the successful region portion, and the engagement probability.

The study included two main parts: one with a known initial relative angular position and the other with a random initial position. The first part has two significant points. Firstly, it can be employed for the design of the dog clutch system to meet the requirements Secondly, it can be used to design a gearshift algorithm. The second part has significance in identifying the best actuator force that guarantees the highest engagement probability, knowing the mismatch speed.

The application of this parametric study method is demonstrated by applying it to different selected parameters. The overlap distance negatively affected both the successful region portion and the probability. The backlash and the teeth number positively affected both the successful region portion and the probability. The linear friction negatively affects, but the rotational friction positively affects the successful region portion.

Lastly, the employed models for the linear and rotational friction are linear. However, other models can be used, provided that an analytical solution exists for the differential equation of the linear and angular dynamics.

Later on, this model will be employed with our current test rig to design an algorithm for the gearshift in dog clutch system. At the beginning of the gearshift process, the mismatch speed and relative position can be measured, and based on these two measures, an algorithm can be designed to identify the optimal gearshift parameters. As seen, the shiftability condition contains three sets of parameters, the dog geometry parameters, the system friction parameters, and the dynamic and kinematic parameters. Once the dog clutch geometry and its system are identified, the geometry and system friction parameters can be considered fixed in the shiftability condition. The only left parameters are

## Acknowledgements

The authors acknowledge that this research has received no external funding.

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