Nonlinear beam mechanics

In this Thesis, nonlinear dynamics and nonlinear interactions are studied from a micromechanical point of view. Single and doubly clamped beams are used as model systems where nonlinearity plays an important role. The nonlinearity also gives rise to rich dynamic behavior with phenomena like bifurcations, stochastic switching and amplitude-dependent resonance frequencies. The theoretical background of micromechanical systems involving the relevant nonlinearities for beams clamped on one (cantilever) or two sides (clamped-clamped beam) are discussed in chapter 2. First, the linear response of a mechanical resonator is discussed. Then, the linear equations are extended with nonlinear terms accounting for geometric and inertial effects. Specifically, the origin of the Duffing nonlinearity in the equation of motion of a clamped-clamped beam is shown. It is shown that the nonlinearity couples the flexural vibration modes of a beam. Microcantilevers are widely used in mass sensing and force microscopy. At small resonance amplitudes, cantilever motion is described by a harmonic oscillator model, while at high amplitudes, the motions is limited by nonlinearities. In chapter 3, the intrinsic mechanical nonlinearity in microcantilevers is studied. It is shown that although the origin is different, the nonlinearity resembles a Duffing nonlinearity resulting in hysteresis and bistable amplitudes. This bistability is then used to implement a mechanical memory. The bistability of microcantilevers can also be used to study the switching characteristics when noise is applied. Chapter 4 shows the experimental implementation of this stochastic switching of microcantilever motion. It is shown that upon increasing the noise intensity, the switching rate rises exponentially as expected from Kramer's law. However, at higher noise intensities, the switching rate saturates and eventually even decreases, which suggests that the noise influences the dynamical parameters of the system. In chapter 5, we investigate in detail the coupling between the flexural vibration modes of a clamped-clamped beam. The coupling arises from the displacement-induced tension. A theoretical model based on the nonlinearity is developed, which is experimentally verified by driving two modes of the beam at high amplitudes and reading out their motion at the two frequencies. The experiments show that the resonance frequency of one flexural mode depends on the amplitude of another flexural mode and the theory is in excellent agreement with the experiments. The nonlinearity not only couples the flexural modes in a clamped-clamped beam, but we show in chapter 6 that also the cantilever modes are coupled. Here, the mechanism causing the nonlinearity is different, as there is no displacement-induced tension. The microcantilever is driven using a piezo actuator and its motion is read out using an optical setup. At high vibration amplitudes, the resonance frequency of one mode depends on the amplitude of the other modes. The torsional modes also show nonlinear behavior as evidenced by a frequency stiffening of the response. The modal interactions in a microcantilever can also be used in a all-mechanical analogue of a cavity-optomechanics, where one mode is used as a cavity mode, which influences the probe mode. In chapter 7, we show that by exciting at the sum and difference frequencies of the two modes, the $Q$ factor of the probe mode could be suppressed over a wide range. In chapter 8, the interaction between a directly- and parameterically-driven resonance mode is studied. Parametric driving means that the spring constant of the beams is modulated at twice the resonance frequency. Clamped-clamped beams with an integrated piezo-actuator on top, designed for applications as gas sensors, are used in the experiments. First, the parametric amplification and oscillation of the beam is studied, then the motion of a parametrically-driven mode is detected by a change in resonance frequency of the directly-driven mode. There is a linear dependence of the oscillation frequency of the parametrically-driven mode and the resonance frequency of the directly-driven mode. A potential application as a linear frequency converter is suggested.Kavli Institute of Nanoscience DelftApplied Science

Interactions between directly- and parametrically-driven vibration modes in a micromechanical resonator

The interactions between parametrically- and directly-driven vibration modes of a clamped-clamped beam resonator are studied. An integrated piezoelectric transducer is used for direct and parametric excitation. First, the parametric amplification and oscillation of a single mode are analyzed by the power and phase dependence below and above the threshold for parametric oscillation. Then, the motion of a parametrically-driven mode is detected by the induced change in resonance frequency in another mode of the same resonator. The resonance frequency shift is the result of the nonlinear coupling between the modes by the displacement-induced tension in the beam. These nonlinear modal interactions result in the quadratic relation between the resonance frequency of one mode and the amplitude of another mode. The amplitude of a parametrically-oscillating mode depends on the square root of the pump frequency. Combining these dependencies yields a linear relation between the resonance frequency of the directly-driven mode and the frequency of the parametrically-oscillating mode.QN/Quantum NanoscienceApplied Science

Size-dependent effective Young’s modulus of silicon nitride cantilevers

The effective Young’s modulus of silicon nitride cantilevers is determined for thicknesses in the range of 20–684 nm by measuring resonance frequencies from thermal noise spectra. A significant deviation from the bulk value is observed for cantilevers thinner than 150 nm. To explain the observations we have compared the thickness dependence of the effective Young’s modulus for the first and second flexural resonance mode and measured the static curvature profiles of the cantilevers. We conclude that surface stress cannot explain the observed behavior. A surface elasticity model fits the experimental data consistently.Kavli Institute of NanoscienceApplied Science