How can a tiny piezoelectric move an object with micrometric precision?

Piezoelectric devices are tiny components that expand or contract by a few microns when powered. Imagine using them to slide a miniature cart along a guide: the result is a stick‑slip actuator, capable of ultra‑fast motion with almost invisible steps. To grasp how it works, engineers built a dynamic model called PRBM‑LuGre. The PRBM (Pseudo‑Rigid Body Model) turns the hinge’s flexibility into a set of springs and masses, allowing the use of Lagrangian mechanics. The LuGre model, on the other hand, captures the complex friction behavior, including the switch between “stick” (cart stays still) and “slip” (cart moves). Combined, the model predicts the actuator’s response with good accuracy, matching experimental data with a 20 % average error. Moreover, thanks to closed‑form equations, simulations run up to a hundred times faster than traditional methods. Tests revealed that the most influential parameters are preload force, flexure stiffness, friction coefficients, and tangential stiffness. Adjusting these values can extend the linear frequency‑velocity regime, making the actuator more reliable for micro‑robotics or precision optics.

The PRBM‑LuGre model starts by representing the flexible hinge as a pseudo‑rigid body: the deformation is replaced by a limited rotation around a virtual point, with an equivalent stiffness constant. The Lagrangian approach allows writing the kinetic and potential energy of the system, including the piezo mass, the slider mass, and the elastic energies of equivalent springs. The LuGre friction law, derived from a microscopic contact model, introduces a state variable \(z\) representing the adhesion density in the contact zone. The equation \(\dot{z}=v-\sigma_0|v|z/|z|\) captures the dependence of friction force on velocity \(v\) and contact history, enabling modeling of both the “stick” phase (\(v\approx0\)) and the “slip” phase (\(v\neq0\)). Numerical integration is performed in Simulink with time steps on the order of \(10^{-8}\) s, ensuring stability even with high stiffness. Comparison with the MSDM (mass‑spring‑damper) model shows that PRBM‑LuGre reduces RMS error from 51.65 % to 20.19 %, thanks to the correct description of nonlinear friction and distributed flexibility. The parametric study, varying preload, flexure stiffness, static and dynamic friction coefficients, and tangential stiffness, reveals that extending the linear regime depends mainly on balancing contact forces and vibration absorption capacity. These results suggest that PRBM‑LuGre can replace costly FEM simulations, offering an optimal trade‑off between accuracy and computation time for designing high‑performance piezoelectric actuators.