( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) > − <− − − + − − + − − > − <− − − + − − < − > − − − − − < − > − − − − − − − − = 0 , if 0 , if otherwise 0 0 , if 0 , if lim2 lim2 2 lim1 1 lim1 lim1 1 lim1 lim1 1 lim2 lim2 2 lim1 1 bump inp inp inp inp bump inp inp inp bump inp inp inp bump inp inp inp inp bump z z z F c z z z k z z k z z z z z z F c z z k z z z z z z F c z z k z z z z z z F c z z k z z z k z z z F & & & & & & & & (6) where lim1 z± are the values of the relative coordinate inp z z − in which the moving magnet hits the bumpers and starts acting the first linear spring and lim2 z± are the values beyond which acts the second linear spring. The force is equal to zero within lim1 z± and when the magnet is moving away from the bumpers. Pneumatic effect When sliding into the guide the moving magnet acts as a piston into a pneumatic cylinder dividing it in two chambers. A displacement of the magnet creates a pressure delta that reacts to the movement. The two chambers are connected by the clearance between the magnet and the guide and the airflow in the clearance reduces the pressure delta. Moreover another reaction to the movement is due by the viscous friction: air vis air p air F F F = + ∆ (7) The main pneumatic contribute is the force due to pressure delta: ( )1 2 2 4 p p d Fair p − = ∆ π (8) where 1p and 2p are the pressure in the chambers and d is the diameter of the moving magnet. Fig. 6 Pneumatic sub-model 344
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