The model of the transmission system with a PCFRS is shown in Figure 11, where s represents the sleeve, r represents the synchro ring, and g represents the target gear. *I*_{1} is the moment of inertia of the motor. *I*_{g} is the moment of inertia of the target gear. *I*_{x}*, I*_{e}*, I*_{f} are the moment of inertia of the meshing gear, *I*_{v} is the moment of inertia of the wheel. *T*_{1}*, T*_{v} are drive torque of the motor and wheel torque. *c*_{i} is the damping coefficient of the corresponding component, where *i *= 0, 1, 2, 3, 4. *k*_{i} is the stiffness coefficient of the corresponding component, where *i *= 0, 2. The PCFRS is mounted on the input shaft. This section will combine the transmission system to analyze the dynamics characteristics of the PCFRS.

In order to illustrate the analysis of the shifting process of the PCFRS, the sleeve, the wave spring, the synchro ring and the friction cone ring are used to articulate the working mechanism. The whole process of the shifting of the sleeve is divided into multiple stages. In order to express the dynamics relationship of the gear meshing process more clearly, the gear contact model is adopted [32], as shown in Figure 12. Where *R*_{a} and *R*_{b} are the basic radius of the gear, \({\theta }_{a}\) and \({\theta }_{b}\) are the torsional freedom of the driving gear and the driven gear, *c*_{y1}, *c*_{y2}, *k*_{y1} and *k*_{y2} are the supporting damping coefficient and supporting stiffness of the driving gear and the driven gear, respectively. *c*_{m} and *k*_{m} are the damping coefficient and stiffness of the driving gear and driven gear meshing. *I*_{a}, *I*_{b} are the moment of inertia of the meshing gear. *e* is the static transmission error of the gear pair.

The relative displacement of the driving gear and the driven gear due to torsion in the normal coordinate system is

$${y}_{0}={R}_{a}{\theta }_{a}-{R}_{b}{\theta }_{b} .$$

(12)

In fact, the process of synchronously eliminating the speed difference and the timing of shifting are both random. This section selects the most representative stages that can cover all situations. Combined with the dynamics model establishment method proposed [33], it is expressed as follows.

**Stage 1:** The sleeve moves to make the synchro ring contact the friction cone ring.

The state of PCFRS in the first stage is shown in Figure 13. At this stage, the sleeve is in the middle position of the synchronizer, and the wave spring is in the free length state. The sleeve moves axially toward the target gear under the guiding action of the hub, pushing the wave spring to contact the synchro ring. The spring force pushes the synchro ring to move axially toward the target gear under the guidance of the hub and the synchro ring quickly contact the friction cone surface on the outside of the friction cone ring. When the sleeve continues to move and reaches the set position, the wave spring gets a certain degree of compression and the shifting process is in stage 2 or stage 3.

$${I}_{1}{\ddot{\theta }}_{1}={T}_{1}-{c}_{1}{\dot{\theta }}_{1}-{c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)-{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right),$$

(13)

$${(I}_{s}+{I}_{r}){\ddot{\theta }}_{s}={c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)+{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right),$$

(14)

$${I}_{g}{\ddot{\theta }}_{g}=-{c}_{3}{\dot{\theta }}_{g}-{R}_{g}{[c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)+ {k}_{m}\left({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1}\right)],$$

(15)

$${(I}_{x}+{I}_{e}){\ddot{\theta }}_{x}=-{c}_{4}{\dot{\theta }}_{x}+{R}_{x}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)\\ {+k}_{m}\left({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1}\right)\end{array}\right]-{ R}_{e}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right],$$

(16)

$${I}_{f}{\ddot{\theta }}_{f}={R}_{f}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right]-{c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)-{ k}_{0}\left({\theta }_{f}-{\theta }_{v}\right),$$

(17)

$${I}_{v}{\ddot{\theta }}_{v}={c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)+{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right)-{T}_{v,}$$

(18)

where *I*_{s} is the moment of inertia of the hub and the sleeve. *I*_{r} is the moment of inertia of the synchro ring. *θ*_{i} is the rotation angle of the corresponding component, \({\dot{\theta }}_{i}\) is the angular velocity of the corresponding component, \({\ddot{\theta }}_{i}\) is the angular acceleration of the corresponding component, where *i* = s, g, x, f, v, 1. *R*_{j} is the basic radius of the corresponding gear, *y*_{j} are the translational displacement of the corresponding gear. \(\dot{y}\)_{j} are the translational velocity of the corresponding gear, where *j* = g, x, e, f. *y*_{01} and *y*_{02} are the relative displacement of meshing gear. \(\dot{y}\)_{01} and \(\dot{\mathrm{y}}\)_{02} are the relative velocity of meshing gear. *e*_{y1,}* e*_{y2} are the static transmission error of the gear pair. \(\dot{e}\)_{y1,} \(\dot{e}\)_{y2} are the static transmission velocity error of the gear pair. *F*_{shift} is the force of shifting.

**Stage 2**: Synchro ring and friction cone ring contact for synchronization.

The state of PCFRS at this stage is shown in Figure 14. The sleeve stops moving in the axial direction, and the spring force generated by the spring compression makes the friction cone surface of the synchro ring and the friction cone ring keep in touch. The friction surface will be produced friction torque, because there is a speed difference between the synchro ring and the friction cone ring. The synchro ring is constrained by the hub, so that the rotational speed of the PCFRS and the input shaft gradually become consistent. At this stage, the shifting motor stops driving, the stop of the sleeve is realized by the self-locking characteristic of the worm gear. When the speed difference is reached the allowable teeth-entering range by the friction torque, this stage ends and the shifting process will proceed to stage 3.

$${I}_{1}{\ddot{\theta }}_{1}={T}_{1}-{c}_{1}{\dot{\theta }}_{1}-{c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)-{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right),$$

(19)

$${(I}_{s}+{I}_{r}){\ddot{\theta }}_{s}={c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)+{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right)-{F}_{f}{R}_{m,}$$

(20)

$${I}_{g}{\ddot{\theta }}_{g}={F}_{f}{R}_{m}-{c}_{3}{\dot{\theta }}_{g}-{R}_{g}{[c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)+{k}_{m}({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1})],$$

(21)

$${(I}_{x}+{I}_{e}){\ddot{\theta }}_{x}=-{c}_{4}{\dot{\theta }}_{x}+{R}_{x}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)\\ {+k}_{m}\left({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1}\right)\end{array}\right] - {R}_{e}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right],$$

(22)

$${I}_{f}{\ddot{\theta }}_{f}={R}_{f}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right]-{c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)-{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right),$$

(23)

$${I}_{v}{\ddot{\theta }}_{v}={c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)+{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right)-{T}_{v},$$

(24)

where *F*_{f} is the friction force between the synchro ring and the friction cone ring. *R*_{m} is the friction equivalent radius.

**Stage 3**: Before the speed difference is eliminated, the sleeve enters the teeth and the friction cone ring teeth contact.

The state of PCFRS at this stage is shown in Figure 15. According to the shifting characteristics of the PCFRS, the sleeve can move again without completely eliminating the speed difference. When the speed difference reaches the allowable range, the sleeve moves to contact the internal teeth of the friction cone ring to turn teeth, and the contact between the sleeve and the friction cone ring generates surface contact force. At this time, there is a friction torque between the friction cone ring and the synchro ring. There is also a turn-teeth torque between the friction cone ring and the sleeve. Both friction torque and turn-teeth torque promote the elimination of the speed difference. Under the action of the helical teeth, the sleeve moves to the internal teeth hole of the friction cone ring. At the end of this phase, the shifting process proceeds to stage 4 or stage 5.

$${I}_{1}{\ddot{\theta }}_{1}={T}_{1}-{c}_{1}{\dot{\theta }}_{1}-{c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)-{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right),$$

(25)

$${(I}_{s}+{I}_{r}){\ddot{\theta }}_{s}={c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)+{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right)-{F}_{f}{R}_{m}-{F}_{t}{R}_{t} ,$$

(26)

$${I}_{g}{\ddot{\theta }}_{g}={F}_{t}{R}_{t}+{F}_{f}{R}_{m}-{c}_{3}{\dot{\theta }}_{g}-{R}_{g}{[c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)+{k}_{m}({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1})],$$

(27)

$${(I}_{x}+{I}_{e}){\ddot{\theta }}_{x}=-{c}_{4}{\dot{\theta }}_{x}+{R}_{x}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)\\ {+k}_{m}\left({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1}\right)\end{array}\right]-{R}_{e}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right],$$

(28)

$${I}_{f}{\ddot{\theta }}_{f}={R}_{f}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right]-{c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)-{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right),$$

(29)

$${I}_{v}{\ddot{\theta }}_{v}={c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)+{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right)-{T}_{v},$$

(30)

where *F*_{t} is the tangential component of the turn-teeth force. *F*_{N} is the positive pressure on the bevel. *R*_{t} is the turn-teeth radius. *γ* is the teeth surface angle.

**Stage 4**: After the speed is eliminated, the sleeve teeth moves in contact with the friction cone ring teeth.

The state of PCFRS at this stage is shown in Figure 16. This situation exists when the speed difference between the sleeve and the friction cone ring is eliminated, but the sleeve has not been fully meshed. Therefore, stage 4 is required. The sleeve continues to turn the friction cone ring teeth and advance the teeth. The gear shifting method at this stage is that the sleeve teeth are slowly advanced so that the teeth surface is in continuous contact, so that no excessive great speed difference is generated, and finally a successful meshing without a speed difference is achieved.

$${I}_{1}{\ddot{\theta }}_{1}={T}_{1}-{c}_{1}{\dot{\theta }}_{1}-{c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)-{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right),$$

(31)

$${(I}_{s}+{I}_{r}){\ddot{\theta }}_{s}={c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)+{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right)+{F}_{f}{R}_{m}-{F}_{t}{R}_{t},$$

(32)

$${I}_{g}{\ddot{\theta }}_{g}={F}_{t}{R}_{t}-{F}_{f}{R}_{m}-{c}_{3}{\dot{\theta }}_{g}-{R}_{g}{[c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)+{k}_{m}({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1})],$$

(33)

$${(I}_{x}+{I}_{e}){\ddot{\theta }}_{x}=-{c}_{4}{\dot{\theta }}_{x}+{R}_{x}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)\\ {+k}_{m}\left({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1}\right)\end{array}\right]-{R}_{e}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right],$$

(34)

$${I}_{f}{\ddot{\theta }}_{f}={R}_{f}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right]-{c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)-{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right),$$

(35)

$${I}_{v}{\ddot{\theta }}_{v}={c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)+{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right)-{T}_{v},$$

(36)

**Stage 5**: After the speed is eliminated, the contact between the sleeve and the teeth surface of the friction cone produces a speed difference. The generated speed difference is used to complete the teeth advancement.

The state of PCFRS at this stage is shown in Figure 17. It exists in the situation of stage 4, but it is not the slow gear advancement. The teeth advancement according to the teeth advancement state of the stage 3, resulting in the speed difference, and then the slow teeth advancement is used. The difference in speed reduces the relative angle between the sleeve and the teeth hole of the friction cone ring to realize shifting. At this stage, there may be impact during meshing, but the difference in speed is small, the impact produced is within an acceptable range. In addition, the speed difference can also be eliminated when the teeth is advanced to realize the speed difference-free shifting.

$${I}_{1}{\ddot{\theta }}_{1}={T}_{1}-{c}_{1}{\dot{\theta }}_{1}-{c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)-{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right),$$

(37)

$${(I}_{s}+{I}_{r}){\ddot{\theta }}_{s}={c}_{2}\left({\dot{\theta }}_{1}-{\dot{\theta }}_{s}\right)+{k}_{2}\left({\theta }_{1}-{\theta }_{s}\right)+{F}_{f}{R}_{m},$$

(38)

$${I}_{g}{\ddot{\theta }}_{g}=-{F}_{f}{R}_{m}-{c}_{3}{\dot{\theta }}_{g}-{R}_{g}{[c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)+{k}_{m}({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1})],$$

(39)

$${(I}_{x}+{I}_{e}){\ddot{\theta }}_{x}=-{c}_{4}{\dot{\theta }}_{x}+{R}_{x}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{01}+{\dot{y}}_{g}-{\dot{y}}_{x}-{\dot{e}}_{y1}\right)\\ {+k}_{m}\left({y}_{01}+{y}_{g}-{y}_{x}-{e}_{y1}\right)\end{array}\right]-{R}_{e}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right],$$

(40)

$${I}_{f}{\ddot{\theta }}_{f}={R}_{f}\left[\begin{array}{c}{c}_{m}\left({\dot{y}}_{02}+{\dot{y}}_{e}-{\dot{y}}_{f}-{\dot{e}}_{y2}\right)\\ {+k}_{m}\left({y}_{02}+{y}_{e}-{y}_{f}-{e}_{y2}\right)\end{array}\right]-{c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)-{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right) ,$$

(41)

$${I}_{v}{\ddot{\theta }}_{v}={c}_{0}\left({\dot{\theta }}_{f}-{\dot{\theta }}_{v}\right)+{k}_{0}\left({\theta }_{f}-{\theta }_{v}\right)-{T}_{v} ,$$

(42)

The relative angle of the friction cone ring teeth hole is also random, and there may be a small or zero relative angle to the friction cone ring teeth hole during the teeth engagement process, so the shifting is completed directly after the gear is dialed or the gear is directly engaged.