To calculate the force due to the pressure delta it is necessary to know the pressure in both chambers. To obtain these values a whole model of the pneumatic piston has been made (Fig. 6). In the chamber blocks are calculated the values of pressure and temperature depending on volume of the chamber and mass flow by first law of thermodynamics and ideal gas law. The value of mass flow is provided by the clearance block based on the equation [7]: ( ) 2 1 3 1 2 12 p p h d c m r − = → µ ρ π & (9) where ρ is the air density, rc is the radial clearance, µ is the dynamic viscosity and h the height of the moving magnet. All the temperature dependant parameters (constant pressure specific heat, constant volume specific heat, density and dynamic viscosity) are recalculated at any integration step. The model in Fig. 6 also calculates the force due to viscous friction proportional to the relative velocity between guide and magnet: ( ) ( ) inp r inp vis air vis z z c dh z z F & & & & − =− − =− π µ β (10) Since the considered fluid is air the viscous force calculated above is smaller of several orders of magnitude if compared to the force due to pressure delta and can be neglected. In order to simplify the model is possible to calculate the effect of the force due to pressure delta through a Maxwell model [8] where the spring represents the air elasticity and the dashpot the effect of the airflow in the clearance (Fig. 7). Fig. 7 The simplified Maxwell model of the pressure delta effect The dashpot is connected to the guide and the spring to the moving magnet. To the point between spring and dashpot is assigned the coordinate air z . The force equilibrium gives: ( ) ( ) − =− =− − ∆ ∆ inp air air air p air air air p z z F k z z F & & β (11) ( ) inp air air p air air air p z z F k F & & & =− − + ∆ ∆ β β (12) 345
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