Hidde J. R. Westra

Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators

A theoretical and experimental investigation is presented on the intermodal coupling between the flexural vibration modes of a single clamped-clamped beam. Nonlinear coupling allows an arbitrary flexural mode to be used as a self-detector for the amplitude of another mode, presenting a method to measure the energy stored in a specific resonance mode. The observed complex nonlinear dynamics are quantitatively captured by a model based on coupling of the modes via the beam extension; the same mechanism is responsible for the well-known Duffing nonlinearity in clamped-clamped beams.

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.

Mechanical stiffening, bistability, and bit operations in a microcantilever

We investigate the nonlinear dynamics of microcantilevers. We demonstrate mechanical stiffening of the frequency response at large amplitudes, originating from the geometric nonlinearity. At strong driving the cantilever amplitude is bistable. We map the bistable regime as a function of drive frequency and amplitude, and suggest several applications for the bistable microcantilever, of which a mechanical memory is demonstrated.

Q-factor control of a microcantilever by mechanical sideband excitation

We demonstrate the coupling between the fundamental and second flexural modes of a microcantilever. A mechanical analogue of cavity-optomechanics is then employed, where the mechanical cavity is formed by the second vibrational mode of the same cantilever, coupled to the fundamental mode via the geometric nonlinearity. By exciting the cantilever at the sum and difference frequencies between fundamental and second flexural modes, the motion of the fundamental mode of the cantilever is damped and amplified. This concept makes it possible to enhance or suppress the Q-factor over a wide range.

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.

Stochastic switching of cantilever motion

The cantilever is a prototype of a highly compliant mechanical system and has an instrumental role in nanotechnology, enabling surface microscopy, and ultrasensitive force and mass measurements. Here we report fluctuation-induced transitions between two stable states of a strongly driven microcantilever. Geometric nonlinearity gives rise to an amplitude-dependent resonance frequency and bifurcation occurs beyond a critical point. The cantilever response to a weak parametric modulation is amplified by white noise, resulting in an optimum signal-to-noise ratio at finite noise intensity. This stochastic switching suggests new detection schemes for cantilever-based instrumentation, where the detection of weak signals is mediated by the fluctuating environment. For ultrafloppy, cantilevers with nanometer-scale dimensions operating at room temperature--a new transduction paradigm emerges that is based on probability distributions and mimics nature.

Modal interactions of flexural and torsional vibrations in a microcantilever

The nonlinear interactions between flexural and torsional modes of a microcantilever are experimentally studied. The coupling is demonstrated by measuring the frequency response of one mode, which is sensitive to the motion of another resonance mode. The flexural-flexural, torsional-torsional and flexural-torsional modes are coupled due to nonlinearities, which affect the dynamics at high vibration amplitudes and cause the resonance frequency of one mode to depend on the amplitude of the other modes. We also investigate the nonlinear dynamics of torsional modes, which cause a frequency stiffening of the response. By simultaneously driving another torsional mode in the nonlinear regime, the nonlinear response is tuned from stiffening to weakening. By balancing the positive and negative cubic nonlinearities a linear response is obtained for the strongly driven system. The nonlinear modal interactions play an important role in the dynamics of multi-mode scanning probe microscopes.

Magnetomotive drive and detection of clamped-clamped mechanical resonators in water

We demonstrate magnetomotive drive and detection of doubly clamped string resonators in water. A compact 1.9 T permanent magnet is used to detect the fundamental and higher flexural modes of 200 μm long resonators. Good agreement is found between the magnetomotive measurements and optical measurements performed on the same resonator. The magnetomotive detection scheme can be used to simultaneously drive and detect multiple sensors or scanning probes in viscous fluids without alignment of detector beams.

Detecting the motion of a microresonator via mode-mode interaction

We demonstrate that a flexural resonance mode of a clamped-clamped beam resonator can be measured by using another resonance mode of the same resonator as a detector. The detector mode and the mode to be detected are coupled through the beam displacement. Displacement of the resonance mode to be measured introduces tension in the beam, which gives rise to an upwards shift in the resonance frequency of the detector mode. We experimentally show that this mechanism can be used to detect the first and second resonance mode of a silicon beam resonator. The modal interaction can also be used to increase the dynamic range of the resonator.

Dynamics of immersed clamped-clamped microresonators

Micro- and nanomechanical resonators are used in a variety of sensing applications. We investigate the dynamics of clamped-clamped micromechanical string resonators immersed in viscous fluids. The resonators are driven using a magneto-motive technique and an optical lever is used to detect their motion. The resonance frequencies and Q-factors of the odd flexural resonance modes are determined up to the 9th mode. Using liquids with varying density and viscosity, the inertial and viscous effects are discriminated by measuring the resonance frequencies and Q-factors.