The linearized model proposed above is effective only for small displacement. To extend the validity of the model is possible to introduce a nonlinear spring in place of the linear one. As an example in equation (13) the model is proposed with a cubic polynomial expansion. ( ) ( ) ( ) ( ) − =− − − − =− − − ∆ ∆ inp air air air p air air air air air air air p z z F k z z k z z k z z F & & β 3 ,3 2 ,2 ,1 (13) To estimate the parameters air i k , is necessary to hypothesize that there is no clearance between magnet and guide. Thus =∞ air β and inp air z z = and so the force due to the pressure delta can be calculated by the hypothesis of a generic polytropic transformation in the chambers: ( ) ( ) − − − + − =− ∆ γ γ π air C C air C C air p z z l l p z z l l p d F 2 2 20 1 1 10 2 4 (14) where C l1 and C l2 are the initial length of the chambers modified to take into account the dead volume of each chamber, 10 p and 20 p are the initial pressures and γ is an appropriate polytropic index. From equation (14) is possible to get the power series: ( ) ( ) ( ) − ⋅ + − + − = ∑ ∏ ∞ = + = ∆ 0 1 2 2 10 1 1 20 1 2 1 1 ! 1 4 i i air i C i C i C i i C i j air p z z l l l p l p j i d F γ π (15) and obtain the generic air i k , coefficient: ( ) ( ) i C i C i C i i C i j air i l l l p l p j i d k 1 2 2 10 1 1 20 1 2 , 1 1 ! 1 4 ⋅ + − + − =− + = ∏ γ π (16) In Fig. 8 (a) is shown the comparison of the ideal gas law and its linear and cubic forms in the hypothesis of no clearance between the two chambers. The characteristic depends on the initial position of the moving magnet. If the magnet is not centered, as in Fig. 8 (a), the characteristic is asymmetric and the cubic gas law has a second order term. Fig. 8 Pressure delta vs. displacement in the hypothesis of no airflow in the clearance between magnet and guide (a) and pneumatic damping vs. radial clearance (b) 346
RkJQdWJsaXNoZXIy MTMzNzEzMQ==