Andrew J. Dick

Utilizing Off-Resonance and Dual-Frequency Excitation to Distinguish Attractive and Repulsive Surface Forces in Atomic Force Microscopy

A beam model is developed and discretized to study the dynamic behavior of the cantilever probe of an atomic force microscope. Atomic interaction force models are used with a multimode approximation in order to simulate the probes response. The system is excited at two-and-a-half times the fundamental frequency and with a dual-frequency signal consisting of the AFM probes fundamental frequency and two-and-a-half times the fundamental frequency. A qualitative change in the response in the form of period doubling is observed for the harmonic off resonance excitation when significantly influenced by repulsive surface forces. Through the use of dual-frequency excitation, standard response characteristics are maintained, while the inclusion of the off-resonance frequency component result in an identifiable qualitative change in the response. By monitoring specific frequency components, the influence of attractive and repulsive surface forces may be distinguished. This information could then be used to distinguish between imaging regimes when bistability occurs or to operate at the separation distance between surface force regimes to minimize force levels.

Tip trajectories of a smart micro-cantilever beam: analysis and design

The tip trajectories of a smart micro-cantilever beam consisting of an atomic force microscope probe with an additional segment of piezoelectric material on top of the probe are studied. A precise model with inhomogeneous partial differential equations and compatible inhomogeneous boundary conditions is developed to describe the dynamics of the smart micro-cantilever beam. The forced vibration solution of this model with respect to two independent inputs from the piezoelectric actuator and the base excitation is derived. By using this solution and the geometric relationship, the trajectory of the end of the tip is obtained from the motion of the free end of the cantilever beam. On the basis of the resonant response to harmonic inputs at the second dynamic mode, nano-scale elliptical and linear tip trajectories are predicted. Within this paper, a smart micro-cantilever beam is shown to produce nano-scale curved tip trajectories for the first time. Analytical and simulation findings indicate that the characteristics of the resulting trajectories are influenced by the magnitudes of two inputs. Potential applications of the elliptical and linear trajectories for nanomanipulation are proposed.

Localization in Microresonator Arrays: Influence of Natural Frequency Tuning

Intrinsic localized modes are localization events caused by intrinsic nonlinearities within an array of perfectly periodic coupled oscillators. Recent developments in microscale fabrication techniques have allowed for the studies of this phenomenon in micro-electromechanical systems. Studies have also identified a relationship between the spatial profiles of intrinsic localized modes and forced nonlinear vibration modes, as well as a potential sensitivity to fundamental frequency relationships of one-to-one and three-to-one between adjacent oscillators. For the system considered, the one-to-one frequency relationship is determined to provide nonideal conditions for studying intrinsic localized modes. The influence of the three-to-one frequency relationship on the behavior of the intrinsic localized modes is studied with analytical methods and numerical simulations by tuning the fundamental frequencies of the oscillators. While the perfect tuning condition is not determined to produce a unique phenomenon, the number and energy concentration of the localization events are found to increase with the increased frequency ratio, which results in a decrease in the effective coupling stiffness within the array.

Real-Time State Detection in Highly Dynamic Systems

This article investigates the feasibility of real-time state detection on a microsecond timescale for use in highly dynamic systems and presents an experimental setup of a high-rate dynamic system coupled with real-time measurement architecture with the goal of detecting changes in the interfacial state of the system. The feasibility of microsecond state detection is assessed through a preliminary timing study. The experimental setup consists of two colliding aluminum bars and includes the option of changing the bar’s boundary conditions and the interface material between the bars. A piezoelectric transducer will be used for detecting changes in dynamic interfacial state by employing electromechanical impedance monitoring and the measurement data from this will be acquired and processed at high speeds using deterministic real-time tools and methodology. Damage detection algorithms from the structural health monitoring community will be used for rapid detection of changes in state. The eventual goal of this work is to adapt currently used methods or to develop entirely new high speed state detection algorithms to be implemented on the real-time system for state detection. This technology has the potential to be used in many applications, including the aerospace, civil, and energy industries among others.

High Fidelity Methods for Modeling Nonlinear Wave Propagation in One-Dimensional Waveguides

In this paper, two new methods are proposed to study wave propagation in materials with constitutive law that have nonlinear terms. In the first method, the gauge transformation is used to derive the dynamic shape function. A perturbation method is then applied in order to derive an equation for the wavenumber. The influence of the nonlinearity takes the form of a dependence of the wavenumber on the magnitude of the corresponding frequency component. Under the small amplitude and weak nonlinearity assumptions of the perturbation method, the wavenumber is incorporated into the spectral finite element method (SFEM). The second approach is a numerical method based on alternating frequency-time (AFT) iterations. The nonlinear term represented as a residual nonlinear force term is reduced through the alternating iterations between the time-domain and the frequencydomain. Finally, response behaviors under impact loading predicted with these methods are studied and compared to equivalent linear response behavior. Copyright

Asymmetric Solutions of SDOF System with Wire Rope Vibration Isolator Subjected to Harmonic Excitation

A new modification of homotopy analysis method (HAM) is proposed for capturing asymmetric solutions of wire rope isolation systems. Analytical expressions of asymmetric solutions to wire rope isolation systems are obtained. A dynamic system with quadratic polynomial restoring force is investigated specifically. Then the analytical results are applied to a single-degree-of-freedom (SDOF) system with wire rope vibration isolator to investigate the response curve and other dynamic characteristics. The analytical approximations match satisfactorily with the numerical results. The presented analytical approximation is a useful method to derive the response curves and examine limit cycles without resorting to numerical simulations.

Spectral Domain Force Identification of Impulsive Loading in Beam Structures

In this paper, an identification method is presented for calculating impulsive loads from the propagating mechanical wave which are produced in beam-like structures. This method uses a spectral finite element method (SFEM) model of a segment of the structure to calculate force information from the measured response. The SFEM model is prepared from the Euler-Bernoulli beam equation in the frequency domain. The method is studied using simulated response data and then applied to data collected from an experimental system. Excellent performance is observed for nominal conditions and a parametric study is performed to determine how different factors affect accuracy. Factors studied include structure size, loading location, and loading duration. When limited to only acceleration data, the use of finite differencing methods to obtain the required slope response information is determined to provide the most significant source of error in the identified force information.

Near-Grazing Based Intermittent Contact Mode Atomic Force Microscopy

The dynamic behavior of an atomic force microscope cantilever probe is studied for dual frequency excitation. By using the Euler-Bernoulli beam equation with a multi-mode approximation, the system is modeled with base excitation and tip-sample interaction forces obtain from molecular dynamics simulations. The dynamic response of the cantilever probe is simulated for a range of separation distance values and analyzed using Poincare sections, bifurcation diagrams, and spectral analysis. The response of the cantilever probe is found to display a qualitative change when influenced by surface forces. The frequency component at half of the fundamental frequency provides an effective way to monitor the amount of force that the probe is applying to the surface of the sample. With this frequency component, an amplitude modulation operation mode is proposed in order to maintain near-grazing behavior during imaging.Copyright

Alternating Frequency–Time Finite Element Method: High-Fidelity Modeling of Nonlinear Wave Propagation in One-Dimensional Waveguides

In this paper, a spectral finite element method (SFEM) based on the alternating frequency–time (AFT) framework is extended to study impact wave propagation in a rod structure with a general material nonlinearity. The novelty of combining AFT and SFEM successfully solves the computational issue of existing nonlinear versions of SFEM and creates a high-fidelity method to study impact response behavior. The validity and efficiency of the method are studied through comparison with the prediction of a qualitative analytical study and a time-domain finite element method (FEM). A new analytical approach is also proposed to derive an analytical formula for the wavenumber. By using the wavenumber equation and with the help of time–frequency analysis techniques, the physical meaning of the nonlinear behavior is studied. Through this combined effort with both analytical and numerical components, distortion of the wave shape and dispersive behavior have been identified in the nonlinear response. The advantages of AFT-FEM are (1) high-fidelity results can be obtained with fewer elements for high-frequency impact shock response conditions; (2) dispersion or dissipation is not erroneously introduced into the response as can occur with time-domain FEM; (3) the high-fidelity properties of SFEM enable it to provide a better interpretation of nonlinear behavior in the response; and (4) the AFT framework makes it more computationally efficient when compared to existing nonlinear versions of SFEM which often involve convolution operations.

On the Role of Boundary Conditions in the Nonlinear Dynamic Response of Simple Structures

Nonlinear responses of structures under extreme impact loading with various boundary conditions are studied. A variety of structures including rods and beams are modeled with material and geometric nonlinearities. High fidelity responses are obtained by using the alternating wavelet-time finite element method (AWT-FEM). Nonlinear distortion and dispersion are identified in the response and the influence of the boundary conditions on the nonlinearity is explored.