Sensor Egregium – An Atomic Force Microscope Sensor for Continuously Variable Resonance Amplification

Numerous nanometrology techniques concerned with probing a wide range of frequency dependent properties would benefit from a cantilevered sensor with tunable natural frequencies. In this work, we propose a method to arbitrarily tune the stiffness and natural frequencies of a microplate sensor for atomic force microscope applications, thereby allowing resonance amplification at a broad range of frequencies. This method is predicated on the principle of curvature-based stiffening. A macroscale experiment is conducted to verify the feasibility of the method. Next, a microscale finite element analysis is conducted on a proof-of-concept device. We show that both the stiffness and various natural frequencies of the device can be highly controlled through applied transverse curvature. Dynamic phenomena encountered in the method, such as eigenvalue curve veering, are discussed and methods are presented to accommodate these phenomena. We believe that this study will facilitate the development of future curvature-based microscale sensors for atomic force microscopy applications.

Modeling Complex Nonminimum Phase Zeros in Flexure Mechanisms

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.

Complex Non-Minimum Phase Zeros in the Dynamics of Double Parallelogram Flexure Module Based Flexure Mechanisms

This paper presents a model to explain complex non-minimum phase (CNMP) zeros seen in the non-collocated frequency response of a large displacement XY flexure mechanism, which employs multiple double parallelogram flexure modules (DPFM) as building-blocks. Geometric non-linearities 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 non-linearity, 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.

Mathematical Insights Into Linear Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors that contain slight mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be linearly combined to form localized modes when the mistune is present. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh–Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we develop an effective visual method—through use of the deviatoric component and the rotor mistune—to precisely identify those modes needed to form localized modes. Finally, we show that curve veering is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. Numerical examples on a disk–blade system with mistune confirm all the findings above.

Mathematical Insights of Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors with mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be easily reoriented when the mistune is present to form localized modes. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh-Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we can develop an effective visual method — through use of the deviatoric component and the rotor mistune — to precisely identify those modes needed to form localized modes. Finally, we show that curve veering with respect to engine orders is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. A numerical example confirms the findings above.

On the Quantification of Eigenvalue Curve Veering: A Veering Index

Eigenvalue curve veering is a phenomenon that has found relevance and application in a variety of structural dynamic problems ranging from localization and stability studies to material property determination. Contemporary metrics for quantifying veering can be ambiguous and difficult to interpret. This manuscript derives three normalized indices in an effort to reconcile the deficit; two of these quantify the physical conditions which produce the behavior while the third provides a definitive measure of the overall intensity of the effect. Numerical examples are provided to illustrate the application of the methods, which are expected to form a basis for the development of advanced analytical tools.