On the dynamics of tapping mode atomic force microscope probes

A?mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. A?multimode Galerkin approximation is utilized to discretize the nonlinear partial differential equation of motion governing the cantilever response and associated boundary conditions and obtain a set of nonlinearly coupled ordinary differential equations governing the time evolution of the system dynamics. A?comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A?higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors.     

Vibration confinement and energy harvesting in flexible structures using collocated absorbers and piezoelectric devices

We propose an optimal design for supplementing flexible structures with a set of absorbers and piezoelectric devices for vibration confinement and energy harvesting. We assume that the original structure is sensitive to vibrations and that the absorbers are the elements where the vibration energy is confined and then harvested by means of piezoelectric devices. The design of the additional mechanical and electrical components is formulated as a dynamic optimization problem in which the objective function is the total energy of the uncontrolled structure. The locations, masses, stiffnesses, and damping coefficients of these absorbers and capacitances, load resistances, and electromechanical coupling coefficients are optimized to minimize the total energy of the structure. We use the Galerkin procedure to discretize the equations of motion that describe the coupled dynamics of the flexible structure and the added absorbers and harvesting devices. We develop a numerical code that determines the unknown parameters of a pre-specified set of absorbers and harvesting components. We input a set of initial values for these parameters, and the code updates them while minimizing the total energy in the uncontrolled structure. To illustrate the proposed design, we consider a simply supported beam with harmonic external excitations. Here, we consider two possible configurations for each of the additional piezoelectric devices, either embedded between the structure and the absorbers or between the ground and absorbers. We present simulations of the harvested power and associated voltage for each pair of collocated absorber and piezoelectric device. The simulated responses of the beam show that its energy is confined and harvested simultaneously.     

Characterization of the flow over a cylinder moving harmonically in the cross-flow direction

The flow over a stationary cylinder is self-excited with a specific natural frequency f_N. When the cylinder is moved harmonically in the cross-flow direction, the response of the flow (in terms of the lift force) will contain two frequencies, namely, the natural frequency f_N and the excitation frequency f_E. When f_E is close to f_N, the natural flow response will be entrained by the excitation, and the response will be periodic with frequency f_E, and dynamicists refer to this phenomenon as lock-in or synchronization. When f_E is away from f_N, the flow will be either periodic with a period that is multiple of the excitation period (i.e., period-n) and dynamicists refer to this phenomenon as secondary synchronization or quasiperiodic consisting of two incommensurate frequencies, or chaotic. We use modern methods of non-linear dynamics to characterize these responses and show that the route to chaos is torus breakdown.     

Nonlinear analysis of time-delay position feedback control of container cranes

Time-delay feedback control of container cranes is robustly stable and insensitive to initial conditions for most of the linearly stable region. To better understand this robustness and any limitations of the technique, we undertake a nonlinear analysis of the system. To this end, we develop a nonlinear model of the crane system by modeling the crane-hoist-payload assembly as a double pendulum. Then, we derive a linear approximation specific to this model. Finally, we derive a cubic model of the dynamics for nonlinear analysis. Using linear analysis, we determine the gain and time delay factors for stabilizing controllers. Also, we show that the controller undergoes a Hopf bifurcation at the linear stability boundary. Using the method of multiple scales on the cubic model, we determine the normal form of the Hopf bifurcation. We then show that for practical operating ranges, the controller undergoes a supercritical bifurcation that helps explain the robustness of the controller.     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

Measurement of the oscillator strength of the E2 transition Cs(6S-5D)

We have measured the transition probability of the electric quadrupole transition Cs(6S-5D) as 21 ± 1.5 s^?1 using two-photon ionization of the ground state 6S, and the 5D as an intermediate state. It was previously measured by a number of methods with conflicting results. Our measurement is in agreement with a laser absorption-fluorescence measurement, and in disagreement with the results of anomalous dispersion, emission, and electron impact techniques. Our result is in agreement with a very recent relativistic calculation.     

Nonlinear interactions in a parametrically excited system with widely spaced frequencies

An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.     

Acoustic propagation in partially choked converging-diverging ducts

A computer model based on the wave-envelope technique is used to study acoustic propagation in converging-diverging hard walled and lined circular ducts carrying near sonic mean flows. The influences of the liner admittance, boundary layer thickness, spinning mode number, and mean Mach number are considered. The numerical results indicate that the diverging portion of the duct can have a strong reflective effect for partially choked flows.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation

The method of multiple scales is used to analyze the response of two-degree-of-freedom systems with quadratic non-linearities to a parametric harmonic excitation having the frequency Ω. Four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when ω₂ ≈ 2ω₁ and either $\text{Ω ≈ 2ω}{}_1$ or $\text{Ω ≈ 2ω}{}_2$, where ω₁ and ω₂ are the linear undamped natural frequencies of the system. Two critical values ζ₁ and ζ₂ of the amplitude F of the excitation are identified in the analysis. When F >ζ₂, the amplitude of the directly excited mode grows exponentially with time according to the linear analysis, whereas the amplitudes of both modes achieve steady state constant values, irrespective of the initial amplitudes, according to the non-linear analysis. When F < ζ₁, the motion decays to zero according to both the linear and non-linear analyses. When ζ₁ ? F ? ζ₂, the motion decays to zero according to the linear analysis, whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non-linear analysis. This is an example of subcritical instability. When $\text{Ω ≈ 2ω}{}_2$, the steady state value of the higher mode, which is directly excited, is a constant that is independent of the excitation of amplitude F, whereas the amplitude of the lower mode, which is indirectly excited through internal resonance, grows with the excitation amplitude F. This is another example of saturation.