An automatic clutch system is given to illustrate the clutch control principle. Based on the dynamic equations, the nonlinear motion characteristics and lag performances of the clutch actuator are analyzed.
2.1. Automatic Clutch System
A pull-type clutch, which is named for the moving direction of the release bearing such that the clutch disengaging motion can be achieved if the release bearing is pulled away from the clutch, is used in pneumatic AMT for heavy-duty vehicles.
Figure 1 shows an automatic clutch system for heavy-duty AMT consisting of a dry clutch assembly, a release fork, and a pneumatic clutch actuator (PCA). The dry clutch assembly is comprised of a pressure plate, a clutch disk, a diaphragm spring, and a release bearing. The PCA contains a clutch power cylinder, two engaging solenoid valves (EV1 is a quick engaging solenoid valve [QESV], and EV2 is a slow engaging solenoid valve [SESV]), and two disengaging solenoid valves (DV1 is a quick disengaging solenoid valve [QDSV], and DV2 is a slow disengaging solenoid valve [SDSV]).
Compressed air with a certain pressure is transmitted into the clutch power cylinder chamber without the piston rod if DV1 and DV2 are powered on while EV1 and EV2 are powered off. Then, the piston rod extends out. Based on the action of the release fork, the release bearing pulls the small end of the diaphragm, and the pressure plate moves away from the clutch disk, which is the clutch disengaging motion. The compressed air in the clutch power cylinder chamber without a piston rod is exhausted into the atmosphere if EV1 and EV2 are powered on while DV1 and DV2 are powered off. Then, the piston moves back if the pressure of the clutch power cylinder chamber is decreased. Based on the action of the release fork, the release bearing pushes the small end of the diaphragm, and the pressure plate moves toward the clutch disk, which is the clutch engaging motion. An external opening is designed to ensure that the cylinder chamber with a piston rod is always connected with the atmosphere. The gas is expelled during the clutch disengaging process, whereas it is filled during the clutch engaging process.
2.2. Solenoid Valve Modeling
The clutch solenoid valves are high-speed on–off valves, which are normally closed. The solenoid valve operation of the disengaging solenoid valve is shown in
Figure 2. The schematic diagrams of the other solenoid valves are similar. The disengaging solenoid valve has two positions and two ways. When the solenoid valve is powered on in
Figure 2b, the electromagnetic force generated from the solenoid overcomes the spring force that causes the spool move away from the seat until the valve is opened. When the solenoid valve is powered off in
Figure 2a, the electromagnetic force is rapidly decreased to zero, which causes the spool to move back to the seat under the action of the spring force until the valve is closed.
The voltage equation of the solenoid is expressed in Equation (1).
where
is the driving voltage;
is the inner resistance of source;
is the resistance of the solenoid;
is the current of the solenoid;
is the magnetic flux of every coil turn, and
is the numbers of coil turns.
Ignoring the magnetic flux leakage, the main gap length is excited by the spool displacement. Then, the electromagnetic force of the spool can be expressed in Equation (2).
where
is the electromagnetic force of the spool;
is the magnetic permeability;
is the sectional area of the magnetic circuits, and
and
are the spool initial position (closed position) and the spool displacement, respectively.
The forces acting on the spool mainly include the force of the compressed air, the electromagnetic force, the spring force, the friction force, and the air resistance from the cylinder chamber without a piston rod. Therefore, the spool movement can be expressed in Equation (3).
where
is the force of the compressed air expressed as
;
is the pressure of the pneumatic system;
is the spool area;
is the spring force expressed as
;
is the spring coefficient of the spool spring;
is the pre-deformation of the spool spring;
is the friction force of the spool moving;
is the air resistance expressed as
;
is the air pressure in the cylinder chamber without a piston rod;
is the spool orifice area;
is the damping coefficient of the spool moving;
is the moving mass of the spool, and
is the moving velocity of the spool.
It is shown from Equation (2) that the value of is related to the current of the solenoid and the spool displacement. The main gap length has its minimum value (holding gap) when the valve is fully open, and it has its maximum value (total gap) when the valve is fully closed. As a result, gets its maximum value if the valve is fully opened and gets its minimum value if the valve is fully closed.
The spool motion of the solenoid valve is mainly influenced by , which is decided by and , based on Equations (2) and (3). Therefore, the spool motion has stranger nonlinear characteristics, owing to the nonlinear relationships between and or .
The spool cannot move if the electromagnetic force is not big enough after the solenoid is powered on, which can be shown from Equation (3). It needs a little time for the increasing of the electromagnetic force because the process of the electromagnetic field needs a little time after the solenoid is powered on. Similarly, it still needs a little time for the decreasing of the electromagnetic force after the solenoid is powered off. Therefore, the spool motion has some lag performances. Generally, the dynamic response time of opening or closing the solenoid valve is about 2 ms [
18,
19,
20]. The dynamic response time decreases with the increasing of the voltage, the electric current, or the pressure [
21,
22,
23,
24].
Duty cycle is used to describe the percentage of the electric time for the solenoid valve. It can be expressed as Equation (4).
where
is the duty cycle;
is the cycle time of the solenoid valve, and
is the time of high level that is less than or equal to
.
Controlling PWM signals can change the average flow of the solenoid valve because the opening degree of the solenoid valve is decided by the maximum of electromagnetic force limited by the average current. Based on high frequencies of PWM voltage signals, the solenoid valve can be considered to be in only two states: one state is in a condition of full opening, and the other is in a condition of full closing. Generally, the relationship between the flow of the solenoid valve and the duty cycle is approximately linear [
25,
26,
27,
28,
29], which is shown in
Figure 3.
In this way, the relationship between the average flow and the duty cycle can be indicated as Equation (5).
where
is the average flow of the solenoid valve, and
is the maximum flow of the solenoid valve under the duty cycle of 100%.
2.4. Clutch Cylinder Modeling
The diagram of the clutch power cylinder is shown in
Figure 5. The forces during the engaging and disengaging situations are shown. A plus signal (+) denotes the engaging direction, and a minus signal (−) denotes the disengaging direction. Chamber 1 and chamber 2 are the cylinder chamber without a piston rod and one with a piston rod, respectively. The forces acting on the piston mainly include the force of the compressed air in the chamber 1, the spring force, the friction force, the force of the atmosphere in the chamber 2, and the force of the release fork acting on the piston rod.
The piston motion equation can be expressed in Equation (10) based on the force condition of the cylinder above.
where
is the mass of the piston and the piston rod;
is the piston velocity;
is the gas acting force on the piston in the cylinder chamber 1 that can be expressed as
;
is the pressure of the pneumatic system;
is the piston area;
is the gas acting force on the piston in the cylinder chamber 2 that can be expressed as
;
is the atmospheric pressure in the cylinder chamber 2;
is the gas acting area on the piston in the cylinder chamber 2 that equals the piston area minus the piston rod area;
is the cylinder spring force that can be expressed as
;
is the elastic coefficient of the cylinder spring;
is the pre-deformation of the cylinder spring;
is the friction force of the piston moving, and
is the damping coefficient of the piston moving.
The force of the piston rod is related to the dynamic characteristics of the diaphragm and the lever ratio from the diaphragm’s large end to the piston rod, which can be seen from Equation (7). The values of , , and basically stay constant if the pressure of the pneumatic system is stable. The piston motion of PCA changes with , which can be shown from Equation (10), and is influenced by the nonlinear characteristics of the diaphragm. Thus, the piston motion has stronger nonlinear characteristics decided by the nonlinear characteristics of the diaphragm.
Furthermore, the motion of the cylinder piston has some lag performances that can be seen from Equation (10). It takes little time to create the driving force enough to change the original condition. The lag time varies owing to the duty cycle, the atmospheric pressure, and the pressure of the pneumatic system.
The delay time from providing electric current to moving the piston comprises two aspects: one comes from the lag of the electric current of the solenoid, and the other one comes from the lag of the spool movement. It is difficult to control the clutch precisely for these lag performances of the actuator.