Nonlinear Modal Interactions in Clamped-Clamped Mechanical Resonators
TL;DR: 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.
Abstract: 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.
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TL;DR: In this paper, the authors discuss different techniques for sensitive position detection and give an overview of the cooling techniques that are being employed, including sideband cooling and active feedback cooling, and conclude with an outlook of how state-of-the-art mechanical resonators can be improved to study quantum mechanics.
399 citations
TL;DR: In this paper, the authors discuss different techniques for sensitive position detection and give an overview of the cooling techniques that are being employed, including sideband cooling and active feedback cooling, and conclude with an outlook of how state-of-the-art mechanical resonators can be improved to study quantum physics.
Abstract: Mechanical systems are ideal candidates for studying quantumbehavior of macroscopic objects. To this end, a mechanical resonator has to be cooled to its ground state and its position has to be measured with great accuracy. Currently, various routes to reach these goals are being explored. In this review, we discuss different techniques for sensitive position detection and we give an overview of the cooling techniques that are being employed. The latter include sideband cooling and active feedback cooling. The basic concepts that are important when measuring on mechanical systems with high accuracy and/or at very low temperatures, such as thermal and quantum noise, linear response theory, and backaction, are explained. From this, the quantum limit on linear position detection is obtained and the sensitivities that have been achieved in recent opto and nanoelectromechanical experiments are compared to this limit. The mechanical resonators that are used in the experiments range from meter-sized gravitational wave detectors to nanomechanical systems that can only be read out using mesoscopic devices such as single-electron transistors or superconducting quantum interference devices. A special class of nanomechanical systems are bottom-up fabricated carbon-based devices, which have very high frequencies and yet a large zero-point motion, making them ideal for reaching the quantum regime. The mechanics of some of the different mechanical systems at the nanoscale is studied. We conclude this review with an outlook of how state-of-the-art mechanical resonators can be improved to study quantum {\it mechanics}.
316 citations
TL;DR: This work shows that all studies of frequency stability report values several orders of magnitude larger than the limit imposed by thermomechanical noise, and proposes a new method to show this was due to the presence of frequency fluctuations, of unexpected level.
Abstract: Frequency stability is key to the performance of nanoresonators. This stability is thought to reach a limit with the resonator's ability to resolve thermally induced vibrations. Although measurements and predictions of resonator stability usually disregard fluctuations in the mechanical frequency response, these fluctuations have recently attracted considerable theoretical interest. However, their existence is very difficult to demonstrate experimentally. Here, through a literature review, we show that all studies of frequency stability report values several orders of magnitude larger than the limit imposed by thermomechanical noise. We studied a monocrystalline silicon nanoresonator at room temperature and found a similar discrepancy. We propose a new method to show that this was due to the presence of frequency fluctuations, of unexpected level. The fluctuations were not due to the instrumentation system, or to any other of the known sources investigated. These results challenge our current understanding of frequency fluctuations and call for a change in practices.
220 citations
Cites background from "Nonlinear Modal Interactions in Cla..."
...The tension depends on transverse displacement and this is the origin of non-linear mode coupling[6], [7]:...
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TL;DR: Clear signatures of nonlinear resonance in these atomically thin resonators are demonstrated and these resonators behave as membranes with resonance frequencies in between 10 and 30 MHz and quality factors in between 16 and 109.
Abstract: Mechanical resonators are fabricated from freely suspended single-layer MoS2 . Their dynamics have been studied by optical interferometry. These resonators behave as membranes with resonance frequencies in between 10 and 30 MHz and quality factors in between 16 and 109. We also demonstrate clear signatures of nonlinear resonance in these atomically thin resonators.
220 citations
TL;DR: In this article, the Dirac Delta Function has been used to model the relationship between stress and strain in two-dimensional lattices and elasticity relations in a three-dimensional manifold.
Abstract: 1 Introduction: Linear Atomic Chains- 2 Two- and Three-Dimensional Lattices- 3 Properties of the Phonon Gas- 4 Stress and Strain- 5 Elasticity Relations- 6 Static Deformations of Solids- 7 Dynamical Behavior of Solids- 8 Dissipation and Noise in Mechanical Systems- 9 Experimental Nanostructures- 10 Nanostructure Fabrication I- 11 Nanostructure Fabrication II- A Mathematical Tools- A1 Scalars, Vectors, Tensors- A11 Vectors- A12 Tensors- A2 Eigenvectors and Eigenvalues- A3 The Dirac Delta Function- B Compatibility Relations for Stress and Strain- C Notation
182 citations
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TL;DR: In this article, the Dirac Delta Function has been used to model the relationship between stress and strain in two-dimensional lattices and elasticity relations in a three-dimensional manifold.
Abstract: 1 Introduction: Linear Atomic Chains- 2 Two- and Three-Dimensional Lattices- 3 Properties of the Phonon Gas- 4 Stress and Strain- 5 Elasticity Relations- 6 Static Deformations of Solids- 7 Dynamical Behavior of Solids- 8 Dissipation and Noise in Mechanical Systems- 9 Experimental Nanostructures- 10 Nanostructure Fabrication I- 11 Nanostructure Fabrication II- A Mathematical Tools- A1 Scalars, Vectors, Tensors- A11 Vectors- A12 Tensors- A2 Eigenvectors and Eigenvalues- A3 The Dirac Delta Function- B Compatibility Relations for Stress and Strain- C Notation
182 citations
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01 Jan 2005
174 citations