Modeling Complex Nonminimum Phase Zeros in Flexure Mechanisms

Abstract

This paper presents a model to explain complex nonminimum phase (CNMP) zeros seen in the noncollocated frequency response of a large-displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFMs) as building-blocks. Geometric nonlinearities associated with large displacement along with the kinematic under-constraint in the DPFM lead to a coupling between the X and Y direction displacements. Via a lumped-parameter model that captures the most relevant geometric nonlinearity, it is shown that specific combinations of the operating point (i.e., flexure displacement) and mass asymmetry (due to manufacturing tolerances) give rise to CNMP zeros. This model demonstrates the merit of an intentionally asymmetric design over an intuitively symmetric design in avoiding CNMP zeros. Furthermore, a study of how the eigenvalues and eigenvectors of the flexure mechanism vary with the operating point and mass asymmetry indicates the presence of curve veering when the system transitions from minimum phase to CNMP. Based on this, the hypothesis of an inherent correlation between CNMP zeros and curve veering is proposed.