skip to main content

Title: Strong internal resonance in a nonlinear, asymmetric microbeam resonator
Abstract Exploiting nonlinear characteristics in micro/nanosystems has been a subject of increasing interest in the last decade. Among others, vigorous intermodal coupling through internal resonance (IR) has drawn much attention because it can suggest new strategies to steer energy within a micro/nanomechanical resonator. However, a challenge in utilizing IR in practical applications is imposing the required frequency commensurability between vibrational modes of a nonlinear micro/nanoresonator. Here, we experimentally and analytically investigate the 1:2 and 2:1 IR in a clamped–clamped beam resonator to provide insights into the detailed mechanism of IR. It is demonstrated that the intermodal coupling between the second and third flexural modes in an asymmetric structure (e.g., nonprismatic beam) provides an optimal condition to easily implement a strong IR with high energy transfer to the internally resonated mode. In this case, the quadratic coupling between these flexural modes, originating from the stretching effect, is the dominant nonlinear mechanism over other types of geometric nonlinearity. The design strategies proposed in this paper can be integrated into a typical micro/nanoelectromechanical system (M/NEMS) via a simple modification of the geometric parameters of resonators, and thus, we expect this study to stimulate further research and boost paradigm-shifting applications exploring the various benefits more » of IR in micro/nanosystems. « less
; ;
Award ID(s):
Publication Date:
Journal Name:
Microsystems & Nanoengineering
Sponsoring Org:
National Science Foundation
More Like this
  1. Metal-mediated cross-coupling reactions offer organic chemists a wide array of stereo- and chemically-selective reactions with broad applications in fine chemical and pharmaceutical synthesis.1 Current batch-based synthesis methods are beginning to be replaced with flow chemistry strategies to take advantage of the improved consistency and process control methods offered by continuous flow systems.2,3 Most cross-coupling chemistries still encounter several issues in flow using homogeneous catalysis, including expensive catalyst recovery and air sensitivity due to the chemical nature of the catalyst ligands.1 To mitigate some of these issues, a ligand-free heterogeneous catalysis reaction was developed using palladium (Pd) loaded into a polymeric network of a silicone elastomer, poly(hydromethylsiloxane) (PHMS), that is not air sensitive and can be used with mild reaction solvents (ethanol and water).4 In this work we present a novel method of producing soft catalytic microparticles using a multiphase flow-focusing microreactor and demonstrate their application for continuous Suzuki-Miyaura cross-coupling reactions. The catalytic microparticles are produced in a coaxial glass capillary-based 3D flow-focusing microreactor. The microreactor consists of two precursors, a cross-linking catalyst in toluene and a mixture of the PHMS polymer and a divinyl cross-linker. The dispersed phase containing the polymer, cross-linker, and cross-linking catalyst is continuously mixed and thenmore »formed into microdroplets by the continuous phase of water and surfactant (sodium dodecyl sulfate) introduced in a counter-flow configuration. Elastomeric microdroplets with a diameter ranging between 50 to 300 micron are produced at 25 to 250 Hz with a size polydispersity less than 3% in single stream production. The physicochemical properties of the elastomeric microparticles such as particle swelling/softness can be tuned using the ratio of cross-linker to polymer as well as the ratio of polymer mixture to solvent during the particle formation. Swelling in toluene can be tuned up to 400% of the initial particle volume by reducing the concentration of cross-linker in the mixture and increasing the ratio of polymer to solvent during production.5 After the particles are produced and collected, they are transferred into toluene containing palladium acetate, allowing the particles to incorporate the palladium into the polymer network and then reduce the palladium to Pd0 with the Si-H functionality present on the PHMS backbones. After the reduction, the Pd-loaded particles can be washed and dried for storage or switched into an ethanol/water solution for loading into a micro-packed bed reactor (µ-PBR) for continuous organic synthesis. The in-situ reduction of Pd within the PHMS microparticles was confirmed using energy dispersive X-ray spectroscopy (EDS), X-ray photoelectron spectroscopy (XPS) and focused ion beam-SEM, and TEM techniques. In the next step, we used the developed µ-PBR to conduct continuous organic synthesis of 4-phenyltoluene by Suzuki-Miyaura cross-coupling of 4-iodotoluene and phenylboronic acid using potassium carbonate as the base. Catalyst leaching was determined to only occur at sub ppm concentrations even at high solvent flow rates after 24 h of continuous run using inductively coupled plasma mass spectrometry (ICP-MS). The developed µ-PBR using the elastomeric microparticles is an important initial step towards the development of highly-efficient and green continuous manufacturing technologies in the pharma industry. In addition, the developed elastomeric microparticle synthesis technique can be utilized for the development of a library of other chemically cross-linkable polymer/cross-linker pairs for applications in organic synthesis, targeted drug delivery, cell encapsulation, or biomedical imaging. References 1. Ruiz-Castillo P, Buchwald SL. Applications of Palladium-Catalyzed C-N Cross-Coupling Reactions. Chem Rev. 2016;116(19):12564-12649. 2. Adamo A, Beingessner RL, Behnam M, et al. On-demand continuous-flow production of pharmaceuticals in a compact, reconfigurable system. Science. 2016;352(6281):61 LP-67. 3. Jensen KF. Flow Chemistry — Microreaction Technology Comes of Age. 2017;63(3). 4. Stibingerova I, Voltrova S, Kocova S, Lindale M, Srogl J. Modular Approach to Heterogenous Catalysis. Manipulation of Cross-Coupling Catalyst Activity. Org Lett. 2016;18(2):312-315. 5. Bennett JA, Kristof AJ, Vasudevan V, Genzer J, Srogl J, Abolhasani M. Microfluidic synthesis of elastomeric microparticles: A case study in catalysis of palladium-mediated cross-coupling. AIChE J. 2018;0(0):1-10.« less
  2. A single micro-electromechanical (MEMS) resonator can be shown to exhibit behaviors unexpected in a simple resonant structure. For small driving forces, the resonator displays typical simple harmonic oscillator re- sponse. As the driving force is increased, the resonator shows the slightly more complex, but well understood, Duffing response. Rather unexpected response behavior can appear when the resonator frequency is detuned by nonlinear- ity to where two oscillatory modes of the resonator begin to interact through nonlinear coupling due to an internal resonance. The paper focuses on how the resonator response changes as the internal resonance is approached in the operating parameter space and how that behavior is conveniently represented in a bifurcation diagram. This behavior is accurately captured by a generic mathematical model. We describe an analysis of the model which shows how this coupled response varies with the system and drive parameters, especially focusing on the nonlinear coupling strength between the two modes.
  3. A locally resonant meta-surface for preferential excitation of a guided mode in a hollow pipe can improve ultrasonic guided wave inspection of pipelines. The proposed meta-surface comprises a periodic arrangement of bonded prismatic rod-like resonators in the circumferential and axial directions of the pipe. We demonstrate the presence of bandgaps for the low-frequency axisymmetric longitudinal modes L(0,1) and L(0,2) and the torsional mode T(0,1). The generated bandgaps can be used to filter the higher harmonics associated with the system nonlinearity to improve nonlinear ultrasonic measurements on pipes. These bandgaps exist even for the non-axisymmetric flexural modes but with their hybridized dispersion curves exhibiting mode-coupling for higher circumferential orders. Moreover, a “partial” bandgap is obtained where preferential transmission of the L(0,2) mode over L(0,1) is possible. We discuss the potential advantages of this partial bandgap to improve pipeline inspections using the L(0,2) mode. Time-domain finite element analyses are used to validate the presence of these bandgaps under radial, circumferential, and axial excitation that mimics the excitation using a ring of piezoelectric transducers. Finally, we discuss the influence of resonator spacing, filling fraction, and the number of resonator rings on the bandgaps for an informed meta-surface design.
  4. Modulation-based control and locking of lasers, filters and other photonic components is a ubiquitous function across many applications that span the visible to infrared (IR), including atomic, molecular and optical (AMO), quantum sciences, fiber communications, metrology, and microwave photonics. Today, modulators used to realize these control functions consist of high-power bulk-optic components for tuning, sideband modulation, and phase and frequency shifting, while providing low optical insertion loss and operation from DC to 10s of MHz. In order to reduce the size, weight and cost of these applications and improve their scalability and reliability, modulation control functions need to be implemented in a low loss, wafer-scale CMOS-compatible photonic integration platform. The silicon nitride integration platform has been successful at realizing extremely low waveguide losses across the visible to infrared and components including high performance lasers, filters, resonators, stabilization cavities, and optical frequency combs. Yet, progress towards implementing low loss, low power modulators in the silicon nitride platform, while maintaining wafer-scale process compatibility has been limited. Here we report a significant advance in integration of a piezo-electric (PZT, lead zirconate titanate) actuated micro-ring modulation in a fully-planar, wafer-scale silicon nitride platform, that maintains low optical loss (0.03 dB/cm in a 625 µmmore »resonator) at 1550 nm, with an order of magnitude increase in bandwidth (DC - 15 MHz 3-dB and DC - 25 MHz 6-dB) and order of magnitude lower power consumption of 20 nW improvement over prior PZT modulators. The modulator provides a >14 dB extinction ratio (ER) and 7.1 million quality-factor (Q) over the entire 4 GHz tuning range, a tuning efficiency of 162 MHz/V, and delivers the linearity required for control applications with 65.1 dB·Hz2/3and 73.8 dB·Hz2/3third-order intermodulation distortion (IMD3) spurious free dynamic range (SFDR) at 1 MHz and 10 MHz respectively. We demonstrate two control applications, laser stabilization in a Pound-Drever Hall (PDH) lock loop, reducing laser frequency noise by 40 dB, and as a laser carrier tracking filter. This PZT modulator design can be extended to the visible in the ultra-low loss silicon nitride platform with minor waveguide design changes. This integration of PZT modulation in the ultra-low loss silicon nitride waveguide platform enables modulator control functions in a wide range of visible to IR applications such as atomic and molecular transition locking for cooling, trapping and probing, controllable optical frequency combs, low-power external cavity tunable lasers, quantum computers, sensors and communications, atomic clocks, and tunable ultra-low linewidth lasers and ultra-low phase noise microwave synthesizers.

    « less
  5. Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($\alpha ,\beta ,k_1,k_2$) linear boundary feedback law at the right end. The $2 \times 2$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$. The role of the control parameters is examined and the following results have been proven: (i) when $\beta \neq 0$, the set of vibrational modes is asymptotically close to the vertical line on the complex $\nu$-plane given by the equation $\Re \nu = \alpha + (1-k_1k_2)/\beta$; (ii) when $\beta = 0$ and the parameter $K = (1-k_1 k_2)/(k_1+k_2)$ is such that $\left |K\right |\neq 1$ then the following relations are valid: $\Re (\numore »_n/n) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iii) when $\beta =0$, $|K| = 1$, and $\alpha = 0$, then the following relations are valid: $\Re (\nu _n/n^2) = O\left (1\right )$ and $\Im (\nu _n/n) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iv) when $\beta =0$, $|K| = 1$, and $\alpha>0$, then the following relations are valid: $\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$.« less