It has been demonstrated for other empirical formula that this kind of system with a pure magnetic suspension has a nonlinear softening behavior [5]. Similar results can be evaluated with force in form of Eq. (2) with 0 1 < ≤n . The dissipative effect is due to the power supplied to the electric circuit: ( ) input em diss z z e e i F & & − − = 2 1 (3) where 1e and 2e are the electromotive-forces induced in the coils. Their coupled contributions can be described as a function of z and z& : ( ) ( ) ( ) ( ) ( ) input input input input input z z d z z d d z z d dt d dt d e e e z z z z & & & & − − + − − = − =− + = − − 2 1 2 1 1 2 , λ λ λ λ (4) Coulomb friction The friction between moving magnet and guide is calculated by an exponential model of the coefficient of friction depending on the relative velocity between the two components: ( ) ( ) ( ) ( ) inp inp z z k z z s inp f e e z z & & & & & & − − − − + − − = β α µ µ µ 1 (5) where constant sµ and kµ are respectively the adhesion and friction coefficient, while α and β are two parameters controlling the transition between static and kinematic behavior. An example of the trend of the friction model is shown in Fig. 5 (b). Fig. 5 Magneto-elastic force due to the interaction between moving magnet and preload magnet (a) and coefficient of friction vs. relative velocity (b) End stroke bumpers End stroke elastic bumpers are represented through a Kelvin-Voigt model [6] with a bi-linear spring. The resulting elastodissipative force bump F is shown in equation (6): 343
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