p1 A1  p2 A2 m1  ρ1  xA1 ∩ m2  −ρ2  xA2 (2)

where A 1, A2 are the cylinder surfaces and ρ1, ρ2 are the air densities at intended air pressures values p1, p2.

d m γ p s − p , p (t )  p , p  A  p A ,

dt 1 1 1 0 10 10  1 20   2

m10   ρ10 A1 x10 , (3)

d m 2  −γ  p 2 − p atm , p 2 (t 0 )  p 20 ,

dt

The value of air inlet/outlet flow rates can be approximated using the nominal flow rate γ delivered by manufacturer. The initial cylinder position is marked by x10 and the full stroke is equal to L. The values of initial pressure values p10, p20, the supply pressure ps and the outlet pressure patm are parameters of the solution The compact solution of above equations is easy [2] in case of constant chamber volumes

ˆ −  ˆ − m10 − α 2 , , t  0, t1 *)

m1 m1 t 

mˆ1 , t ≥ t1*, , mˆ1  ps  V1

m1 RT

(t) 

γ RT mˆ1 − m10

, t1 * 

α  V ( p