Moving in is provided by generating signals on the valves V1, V2 and right coil of V, so all three throttles – T1, T2, T3 – are connected with the exhaust chamber. When the signal from sensor S1 is reached, throttle T1 is switching off. And when the signal from sensor S2 is reached, throttle T2 is switching off. As a result, piston reaches the front cap with only one connected throttle –  T3.

Res1, Res2 – additional volumes in both pneumatic chambers. Using of such kind of additional volumes helps to minimize the dynamic forces in the process of braking and  starting.

3. Mathematical model

The mathematical model used for analyze of electro-pneumatic drives for VMMR is based on the research results dedicated to industrial pneumatics [5, 8-10], cyclic and servo pneumatic drives with computer control working in both position and tracking modes [3, 4, 11-13]. This non-linear model considers special internal processes in pneumatic, electric, mechanical and computer parts of the mechatronic electro-pneumatic   drive.

This model is based of four differential equations that describe dynamic of pressures in pneumatic chambers and specify the process of movement of the mechanical control target. Below you can see the system of differential equations in standard form:

where x – coordinate of the piston; v – speed of the piston; P2, P2 – pressures in chambers; A1, A2 – area of piston in both chambers; m – mass of control target; R – universal gas constant; T – temperature of the compressed air; V01, V02 – start volumes of the chambers; n - coefficient of polytropy; F – the force on the output of the special developed algorithm that calculate the acting force on the piston. This algorithm allows the correct description of the Coulomb friction influence on the pneumatic cylinder piston movement and stop  processes.

Compressed air mass consumptions G1 и G2 are calculated in function of the pressures on the according throttle element inlet and outlet using of non-linear  equations:

where G – Compressed air mass consumption through the orifice (valve, throttle etc.); PIN, POUT – pressures on the inlet and outlet of the orifice; fE – effective square of the orifice. The equations consider the direction of the flow through the orifice and therefore are more suitable for computer  modeling.

The mathematical model also considers the delay of the on-off valve, computer control device and the electro- pneumatic control unit, description of time sampling which is typical for computer control   devices.

4. Computer simulation program

The developed non-linear mathematical model is realized as computer simulating program. The code is written on the programming language C++ in the Microsoft Visual Studio software. This program is solving the system of non-linear equations, saving and printing results. Program user-friendly interface provides the means of interactive user-machine communication. The main program window allows changing any electro-pneumatic drive parameter. The calculation results are shown in two diagrams: «position in time» and «velocity in time». Fig. 4 shows the program main window example. It includes the input data and the position in time diagram. As an option user can zoom necessary limit of the modeling process and study the process in details. Also the user can find detailed calculations in the saved file in .XLS format. The file stores every variable of a mechatronic system conditions. This data could be used for reports or presentations. User can also change the step of integration and the step of data saving.

Fig. 4. The main window of the program with basic data and the results of simulation.

This simulation program was tested as a tool for developing the electro-pneumatic drives for different industries. It is noted that using of this program can help to reduce drive development time. Design engineer can examine the wide range of the structures (cylinder with standard cushion system, one stage braking algorithm, multistage braking algorithm) and choose the optimal structure and control algorithm that will help to reach required cycle time or quality of the braking processes. The program helps to adjust required flow rates of the throttles and identify the positions of the sensors.