Response of Electrostatically Actuated Flexible MEMS Structures to the Onset of Low-Velocity Contact

2009 ◽  
Author(s):  
Bryan Wilcox ◽  
Harry Dankowicz ◽  
Walter Lacarbonara
Keyword(s):  
Steady State ◽  
Contact Region ◽  
Response Amplitude ◽  
Physical Contact ◽  
Model Description ◽  
System Response ◽  
Scaling Effects ◽  
Lumped Mass ◽  
Low Velocity ◽  
Beam Elements

Near-grazing, low-velocity contact in vibro-impacting systems has been shown to result in dramatic changes in steady-state system response following rapid transient growth of deviations away from the pre-grazing steady-state response. In low-dimensional example systems such transitions are often associated with large jumps in response amplitude. Coupled with the rapidity of the transient dynamics, this phenomenology supports the design of limit-switch sensors that trigger at the onset of grazing contact. A particularly exciting area of application of such sensors, and one in which their implementation might offer particular advantages, is in the context of microelectromechanical structures. Here, desirable scaling effects, such as increased system frequencies, low damping, batch fabrication, and decreased packaging size, can be leveraged. Fabricating simple beam structures at the microscale is relatively easier than fabricating proof-mass-based lumped-parameter systems with elaborate suspension structures. Consequently, it often becomes necessary to account for the flexibility of participating mechanical members, for example doubly-clamped, silicon-based beam elements. Physical contact further poses modeling challenges, as the flexibility of the beam elements and that of the contact region necessitate a compliant, but very stiff model description. The present work investigates a sequence of reduced-order models for such a doubly-clamped beam, subject to capacitive electrostatic actuation and a low-compliance physical constraint localized at a point along the span of the beam. The objective is to determine whether grazing-induced transitions, characteristic of lumped-mass models, are retained in the flexible structure. Specifically, numerical simulations are employed to quantify the variations in the response amplitude following the onset of contact and to contrast these to a spreading of system energy across mechanical modes.

Design of Limit-Switch Sensors Based on Discontinuity-Induced Nonlinearities

2009 ◽  
Author(s):  
Bryan Wilcox ◽  
Harry Dankowicz
Keyword(s):  
Steady State ◽  
Threshold Value ◽  
System Response ◽  
Dramatic Improvement ◽  
Low Velocity ◽  
Response Characteristics ◽  
Sensor Performance ◽  
Grazing Response ◽  
And Control ◽  
Limit Switch

Limit-switch sensors are input-output devices that switch operating state in reaction to the crossing of a threshold value of their input. These are used to monitor and control critical values of temperature, voltage, pressure, etc. in both consumer and industrial settings. This paper argues for exploiting nonsmooth fold bifurcations in the design of ultrafast and robust, resettable, electromechanical limit switches. Specifically, the discussion emphasizes the dramatic changes in system response associated with the onset of near-grazing, low-velocity contact in vibro-impacting systems. These include rapid transient dynamics away from a pre-grazing, periodic, steady-state trajectory following the onset of impacts and post-grazing steady-state trajectories with distinctly different amplitude and frequency content. The results reported here include an experimental and computational verification of the ultrafast transient growth rates that show a significant potential for dramatic improvement in sensor performance. Moreover, two novel candidate sensor designs are discussed that rely on the post-grazing response characteristics for device function. In the first instance, transduction of a change in the response periodicity following grazing in a mechanical device is detected in a coupled electromagnetic circuit. In the second instance, a snap-through post-grazing response forms the operating principle of a capacitively-driven circuit protection device.

Limit-Switch Sensor Functionality Based on Discontinuity-Induced Nonlinearities

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Bryan Wilcox ◽  
Harry Dankowicz
Keyword(s):  
Steady State ◽  
Threshold Value ◽  
System Response ◽  
Dramatic Improvement ◽  
Low Velocity ◽  
Response Characteristics ◽  
Sensor Performance ◽  
Grazing Response ◽  
And Control ◽  
Limit Switch

Limit-switch sensors are input-output devices that switch operating state in reaction to the crossing of a threshold value of their input. These are used to monitor and control critical values of temperature, voltage, pressure, etc., in both consumer and industrial settings. This paper argues for exploiting nonsmooth fold bifurcations in the design of ultrafast and robust, resettable, electromechanical limit switches. Specifically, the discussion emphasizes the dramatic changes in system response associated with the onset of near-grazing, low-velocity contact in vibro-impacting systems. These include rapid transient dynamics away from a pre-grazing, periodic, steady-state trajectory following the onset of impacts and post-grazing steady-state trajectories with a distinctly different amplitude and frequency content. The results reported here include a review of an experimental and computational verification of the ultrafast transient growth rates that show a significant potential for dramatic improvement in sensor performance. Moreover, two novel candidate sensor designs are discussed that rely on the post-grazing response characteristics for device function. In the first instance, transduction of a change in the response periodicity following grazing in a mechanical device is detected in a coupled electromagnetic circuit. In the second instance, a snap-through post-grazing response forms the operating principle of a capacitively driven circuit protection device.

Steady-State Unbalance Response of Continuous Rotors on Anisotropic Supports

1993 ◽  
Author(s):  
Graziano Curti ◽  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Steady State ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Analytical Investigation ◽  
Dynamic Stiffness Matrix ◽  
Rotor Systems ◽  
Beam Elements ◽  
Unbalance Response ◽  
Constant Coefficients ◽  
Linearized Model

Abstract An analytical investigation of the steady-state unbalance response of axisymmetric rotor systems with anisotropic, flexible and damped bearings is presented. According to the exact approach of the dynamic stiffness method, the rotor is modelled by means of continuous beam elements. In this work, the expression of the 8 × 8 dynamic stiffness matrix of a rotating Timoshenko beam is derived and it is shown that it is related, by means of a simple law, to the previously published 4 × 4 dynamic stiffness matrix, which holds for the isotropic bearings case. The effects of concentrated disks and bearings are included into the formulation; in particular, each bearing is described by eight constant coefficients, according to the well-known linearized model of the bearing forces. The unbalance response of a typical rotor system taken from the literature is analyzed. A comparison is presented with the finite element results reported by other authors.

New Insight Into Energy Harvesting via Axially-Loaded Beams

2009 ◽  
Author(s):  
Mohammed F. Daqaq
Keyword(s):  
Steady State ◽  
Energy Harvesting ◽  
Axial Load ◽  
Excitation Frequency ◽  
Axial Loading ◽  
Response Amplitude ◽  
Resistive Load ◽  
Steady State Response ◽  
State Response ◽  
Axially Loaded

Driven by the study of Leland and Wright [1], this manuscript delves into the qualitative understanding of energy harvesting using axially-loaded beams. Using a simple nonlinear electromechanical model and the method of multiple scales, we study the general nonlinear physics of energy harvesting from a piezoelectric beam subjected to static axial loading and traversal dynamic excitation. We obtain analytical expressions for the steady-state response amplitude, the voltage drop across a resistive load, and the output power. We utilize these expression to study the effect of the axial loading on the overall nonlinear behavior of the harvester. It is demonstrated that, in addition to the ability of tuning the harvester to the excitation frequency via axial load variations, the axial load aids in i) increasing the electric damping in the system thereby enhancing the energy transfer from the beam to the electric load, ii) amplifying the effect of the external excitation on the structure, and hence, increases the steady-state response amplitude and output voltage, and iii) increasing the bandwidth of the harvester by enhancing the effective nonlinearity of the system.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

A new simplified method for anti-symmetric mode in-plane vibrations of frame structures with column cracks

2016 ◽  
Vol 24 (24) ◽  
pp. 5794-5810 ◽  
Author(s):  
Kemal Mazanoglu ◽  
Elif C Kandemir-Mazanoglu
Keyword(s):  
Beam Theory ◽  
Equivalent Model ◽  
Frame Structures ◽  
Symmetric Mode ◽  
Lumped Mass ◽  
Beam Elements ◽  
Local Flexibility ◽  
Mode Vibration ◽  
Frame Systems ◽  
Lumped Masses

This paper is on the natural frequency and mode shape computation of frame structures with column cracks. First, a model of intact frame structures is built to perform vibration analysis. Beam elements are considered as lumped masses and rotational springs at the storey levels of frames. Equivalent model of columns and lumped mass-stiffness effects of beams have been combined to carry out continuous solution for the anti-symmetric mode in-plane vibrations of frames. In addition, frame systems with multiple column cracks are analyzed in terms of anti-symmetric mode vibration characteristics. Cracks are considered as massless rotational springs in compliance with the local flexibility model. Compatibility and continuity conditions are satisfied at crack and storey locations of the equivalent column, modeled using the Euler–Bernoulli beam theory. The proposed method is tested for single-storey single- and multi-bay, H-type and double-storey single-bay frame systems with intact and cracked columns. Results are validated by those given in the current literature and/or obtained by the finite element analyses.

Transient Vibration and Feedback Control of an Inductrack Maglev System

2020 ◽  
Author(s):  
Ruiyang Wang ◽  
Bingen Yang ◽  
Hao Gao
Keyword(s):  
Steady State ◽  
Feedback Control ◽  
Dynamic System ◽  
Permanent Magnets ◽  
Preliminary Investigation ◽  
Lookup Table ◽  
Lumped Mass ◽  
Transient Vibration ◽  
Transient Model ◽  
Halbach Arrays

Abstract As a new strategy for magnetic levitation envisioned in 1990s, the Inductrack system with permanent magnets (PMs) aligned in Halbach arrays has been intensively studied and applied in many projects. Due to the nonlinear, time-varying electro-magneto-mechanical coupling in such a system, the dynamic behaviors are complicated with transient responses, which in most cases can hardly be predicted with fidelity by a steady-state Inductrack model. Presented in this paper is a benchmark 2-DOF transient Inductrack model, which is derived from the first laws of nature, without any assumed steady-state quantities. It is shown that the dynamic response of the Inductrack dynamic system is governed by a set of nonlinear integro-differential equations. As demonstrated in numerical simulations with the transient model, unstable vibrations in the levitation direction occur when the traveling speed of the vehicle exceeds a threshold. To resolve this instability issue, feedback control is implemented in the Inductrack system. In the development, an assembly of Halbach arrays and active coils that are wound on the PMs is proposed to achieve a controllable source magnetic field. In this preliminary investigation, the proposed control system design process takes two main steps. First, a PID controller is set and tuned based on a simple lumped-mass dynamic system. Second, the nonlinear force-current correlation is obtained from a lookup table that is pre-calculated by steady-state truncation of the full transient Inductrack model. With the implemented feedback control algorithms, numerical examples display that the motion of the vehicle in levitation direction can be effectively stabilized at different traveling speeds. Although only a 2-DOF transient model is used here, the modeling technique and the controller design approach developed in this work are potentially applicable to more complicated models of Inductrack Maglev systems.

Air-Conduction Auditory Steady-State Response: Comparison of Interchannel Recording Using Two Modulation Frequencies

2008 ◽  
Vol 19 (09) ◽  
pp. 696-707 ◽  
Author(s):  
Wafaa A. Kaf ◽  
Ali A. Danesh
Keyword(s):  
Steady State ◽  
Carrier Frequency ◽  
Young Female ◽  
Response Amplitude ◽  
Normal Female ◽  
Steady State Response ◽  
State Response ◽  
Auditory Steady State Response ◽  
The Mean ◽  
Contralateral Response

Background: Two-channel auditory steady-state response (ASSR) recording at high and low MF (modulation frequency) most likely provides an insight about the response amplitude and latency from different directions at the brainstem level and at the thalamus or cortical level. Little is known about the combined relationship between MF (39 and 79 Hz) and electrode montages (ipsilateral and contralateral) to single AM (amplitude modulation) tones on the ASSR amplitude and latency. Purpose: To determine if ipsilateral versus contralateral response asymmetries are present at the brainstem level (79 Hz ASSR) and at the thalamus or cortical levels (39 Hz ASSR). Research Design: Descriptive and inferential statistics for interchannel ipsilateral and contralateral ASSR amplitude and latency to 79 and 39 Hz. Study Sample: Twenty-five normal-hearing, right-handed young female adults participated in the study. All participants were right-handed, and their age ranged between 18 to 28 years (mean 24.5 ± 1.6 years). Data Collection and Analysis: Ipsilateral and contralateral ASSR to 39 and 79 Hz MF and 100% AM stimuli were recorded at 500, 2000, and 4000 Hz carrier frequencies at 65 dB SPL. The ASSR amplitudes and phases were determined for each MF across Fc (carrier frequency) for the two channels to the test (right) ear. ASSR amplitude and latency between recording montages for each MF and across carrier frequency were compared by computing two-way repeated measures ANOVA. Results: The mean ipsilateral ASSR amplitudes to 39 Hz across frequency were slightly larger (228.6 ± 61.6 µV) than the contralateral response amplitude (223.2 ± 78 µV) while the mean ipsilateral 79 Hz amplitudes were smaller (127.3 ± 114.8) compared to contralateral 79 Hz amplitude (154.6 ± 112.7 µV). For latency response, the mean ipsilateral/contralateral latency difference, on average, was 1 msec or less for both MFs. Results, in normal female adults, indicated no significant interchannel ASSR asymmetries for amplitude and latency (p > 0.05) at the brainstem (79 Hz ASSR) and at the thalamus or cortical levels (39 Hz ASSR). Conclusions: Interchannel ipsilateral and contralateral ASSR amplitude and latency to 79 and 39 Hz are not significantly different in normal, young female adults. Two-channel recording of ASSR to different MFs may be of clinical value in otoneurologic assessment.

scholarly journals A Tri-Stable Piezoelectric Vibration Energy Harvester for Composite Shape Beam: Nonlinear Modeling and Analysis

Sensors ◽  
2020 ◽  
Vol 20 (5) ◽  
pp. 1370 ◽  
Author(s):  
Xuhui Zhang ◽  
Meng Zuo ◽  
Wenjuan Yang ◽  
Xiang Wan
Keyword(s):  
Energy Harvester ◽  
Response Amplitude ◽  
Vibration Energy ◽  
System Response ◽  
Variation Principle ◽  
Potential Well ◽  
Restoring Force ◽  
Vibration Energy Harvester ◽  
Piezoelectric Vibration Energy Harvester ◽  
Piezoelectric Vibration

To reveal the nonlinear mechanism of the tri-stable piezoelectric vibration energy harvester based on composite shape beam (TPEH-C) and its influence on the system response, the nonlinear restoring force and the nonlinear magnetic force are discussed and analyzed in this paper. The nonlinear magnetic model is acquired by using equivalent magnetizing current theory, and the nonlinear resilience model is obtained by fitting experimental data. The corresponding distributed parameter model based on generalized Hamiltonian variation principle has been established. Frequency response functions for the TPEH-C are derived according to harmonic balance expansion, and the influence of different magnet distances and different excitation accelerations on the response amplitude and bandwidth of the TPEH-C are investigated. More importantly, the correctness of the theoretical analysis is verified by experiments. The results reveal that the spectrum of composite beam shows hard characteristic and the depth of potential well is changed, which provides a new way to ameliorate the potential well of the TPEH-C. A suitable magnet distance enables the TPEH-C to improve the response amplitude and the effective frequency range. The results in this paper have a theoretical guiding significance for the optimal design and engineering application of the TPEH-C.

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