Further, the function h(x?) denotes the Stribeck effect curve, given byequation(3)σ0h(x?)=FC+(FS-FC)e-(x?/x?S)2where FC A-674563 the Coulomb friction level, FS is the level of the stiction force, and x?S is the Stribeck velocity.
Consequently, the hysteresis friction function can be further derived by placing Eqs. (1) and (2) in order to obtain the following formequation(4)FH=σ0z¯-σ11h(x?)z¯ x? +(σ1+σ2)x?.A block diagram model of the piezo-actuated positioning mechanism with implemented hysteresis effect is shown in Fig. 1.
Fig. 1. Block diagram model of the piezo-actuated mechanism with hysteresis.Figure optionsDownload full-size imageDownload as PowerPoint slide
The overall dynamics of the PEA positioning mechanism with 1-DOF and incorporated hysteresis effect can be represented asequation(5)mx¨+FH+FL+KEU=0where m denotes the equivalent mass of the controlled PEA positioning mechanism, x is capillary bed the displacement of the PEA, x¨ denotes the second-order derivative of the state x with respect to time, FL is the external load, KE denotes the voltage-to-force coefficient of PEA and U denotes the applied voltage to PEA.