Analytical stiffness estimation for short flexures

Accurate stiffness determination is critical for flexures used in high accuracy positioning—such as inchworm® piezomotors—because stiffness significantly affects the positioner’s response time, resonant frequency and range of motion. The flexures are often short beams while the standard formulae for stiffness are for long, slender beams based only on bending stress and can over predict the stiffness of the short flexures. An improved model for a single, short beam is presented which incorporates shear stress and a comparison to FEA results shows almost exact agreement. A study of a more complex flexure showed that simple stiffness models were inaccurate because of end effects in the supporting structure. Development of a multi-section analytical model to include the end effects is presented in this paper and again a comparison to FEA results shows sufficient accuracy to be used in design to quickly optimize the positioner’s performance.