Frequency Response of Electrostatically Actuated MEMS Resonators Under Superharmonic Resonance Using MMS

2016 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Frequency Response ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Second Order ◽  
Voltage Response ◽  
Order Problem ◽  
Plate Electrode ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Mems Resonator

This work investigates the voltage response of superharmonic resonance of second order of electrostatically actuated Micro-Electro-Mechanical Systems (MEMS) resonator cantilevers. The results of this work can be used for mass sensors design. The MEMS device consists of MEMS resonator cantilever over a parallel ground plate (electrode) under Alternating Current (AC) voltage. The AC voltage is of frequency near one fourth of the natural frequency of the resonator which leads to the superharmonic resonance of second order. The AC voltage produces an electrostatic force in the category of hard excitations, i.e. for small voltages the resonance is not present while for large voltages resonance occurs and bifurcation points are born. This solution is then used in the first-order problem to find the voltage-amplitude response of the structure. The influences of frequency and damping on the response are investigated. This work opens the door of using smaller AC frequencies for MEMS resonator sensors. The frequency response of the superharmonic resonance of the structure is investigated using the method of multiple scales (MMS).

Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonator Cantilevers Using the Method of Multiple Scales

2016 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Equation Of Motion ◽  
Zero Order ◽  
Voltage Response ◽  
Order Problem ◽  
Superharmonic Resonance ◽  
First Order ◽  
Mems Resonator

This work investigates the voltage response of superharmonic resonance of second order of electrostatically actuated Micro-Electro-Mechanical Systems (MEMS) resonator cantilevers. The results of this work can be used for mass sensors design. The MEMS device consists of MEMS resonator cantilever over a parallel ground plate (electrode) under Alternating Current (AC) voltage. The AC voltage is of frequency near one fourth of the natural frequency of the resonator which leads to the superharmonic resonance of second order. The AC voltage produces an electrostatic force in the category of hard excitations, i.e. for small voltages the resonance is not present while for large voltages resonance occurs and bifurcation points are born. The forces acting on the resonator are electrostatic and damping. The damping force is assumed linear. The Casimir effect and van der Waals effect are negligible for a gap, i.e. the distance between the undeformed resonator and the ground plate, greater than one micrometer and 50 nanometers, respectively, which is the case in this research. The dimensional equation of motion is nondimensionalized by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The resulting dimensionless equation includes dimensionless parameters (coefficients) such as voltage parameter and damping parameter very important in characterizing the voltage-amplitude response of the structure. The Method of Multiple Scales (MMS) is used to find a solution of the differential equation of motion. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. In this work, since the structure is under hard excitations the electrostatic force must be in the zero-order problem. The assumption made in this investigation is that the dimensionless amplitudes are under 0.4 of the gap, and therefore all the terms in the Taylor expansion of the electrostatic force proportional to the deflection or its powers are small enough to be in the first-order problem. This way the zero-order problem solution includes the mode of vibration of the structure, i.e. natural frequency and mode shape, resulting from the homogeneous differential equation, as well as particular solutions due to the nonhomogeneous terms. This solution is then used in the first-order problem to find the voltage-amplitude response of the structure. The influences of frequency and damping on the response are investigated. This work opens the door of using smaller AC frequencies for MEMS resonator sensors.

ROM of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Resonators

2015 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Second Order ◽  
Voltage Response ◽  
Order Model ◽  
Reduced Order ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Method Effects ◽  
Ground Plate

This paper deals with the voltage-amplitude response (or voltage response) of superharmonic resonance of second order of MEMS resonator sensors under electrostatic actuation. The system consists of a MEMS flexible cantilever above a parallel ground plate. The AC frequency of actuation is near one fourth the natural frequency. The voltage response of the superharmonic resonance of second order of the structure is investigated using the Reduced Order Model (ROM) method. Effects of voltage and damping voltage response are reported.

Amplitude-Frequency Response for Superharmonic Resonance of Fourth Order of Electrostatically Actuated MEMS Cantilever Resonators

2019 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christopher Reyes
Keyword(s):  
Frequency Response ◽  
Fourth Order ◽  
Microelectromechanical Systems ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Damping Force ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Homotopy Analysis ◽  
Ground Plate

Abstract This paper deals with the frequency response of superharmonic resonance of order four of electrostatically actuated MicroElectroMechanical Systems (MEMS) cantilever resonators. The MEMS structure in this work consists of a microcantilever parallel to an electrode ground plate. The MEMS resonator is elelctrostatically actuated through an AC voltage between the cantilever and the ground plate. The voltage is in the category of hard excitation. The AC frequency is near one eight of the natural frequency of the resonator. Since the electrostatic force acting on the resonator is proportional to the square of the voltage, it leads to superharmonic resonance of fourth order. Besides the electrostatic force, the system experiences damping. The damping force in this work is proportional to the velocity of the resonator, i.e. it is linear damping. Three methods are employed in this investigation. First, the Method of Multiple Scales (MMS), a perturbation method, is used predictions of the resonant regions for weak nonlinearities and small to moderate amplitudes. Second, the Homotopy Analysis Method (HAM), and third, the Reduced Order Model (ROM) method using two modes of vibration are also utilized to investigate the resonance. ROM is solved through numerical integration using Matlab in order to simulate time responses of the structure. All methods are in agreement for moderate nonlinearities and small to moderate amplitudes. This work shows that adequate MMS and HAM provide good predictions of the resonance.

Frequency Response for MEMS Circular Plate Resonators Under Superharmonic Resonance of the Second Order

2018 ◽  
Author(s):  
Martin Botello ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Circular Plate ◽  
Software Package ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Micro Electro Mechanical System ◽  
Second Order ◽  
Mode Shapes ◽  
Amplitude Response ◽  
Superharmonic Resonance ◽  
Mems Resonator

A nonlinear dynamics investigation is conducted on the frequency-amplitude response of electrostatically actuated micro-electro-mechanical system (MEMS) clamped plate resonators. The Alternating Current (AC) voltage is operating in the realm of superharmonic resonance of second order. This is given by an AC frequency near one-fourth of the natural frequency of the resonator. The magnitude of the AC voltage is large enough to be considered as hard excitation. The external forces acting on the MEMS resonator are viscous air damping and electrostatic force. Two proven mathematical models are utilized to obtain a predicted frequency-amplitude response for the MEMS resonator. Method of Multiple Scales (MMS) allows the transformation of a partial differential equation of motion into zero-order and first-order problems. Hence, MMS can be directly applied to obtain the frequency-amplitude response. Reduced Order Model (ROM), based on the Galerkin procedure, uses mode shapes of vibration for undamped circular plate resonator as a basis of functions. ROM is numerically integrated using MATLAB software package to obtain time responses. Also, ROM is used to conduct a continuation and bifurcation analysis utilizing AUTO 07P software package in order to obtain the frequency-amplitude response. The time responses show the movement of the center of the MEMS circular plate as a function of time. The frequency-amplitude response allows one to observe bifurcation and pull-in instabilities within the nonlinear system over a range of frequencies. The influences of parameters (i.e. damping and voltage) are also included in this investigation.

Reduced Order Model of Frequency Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonators

2015 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Frequency Response ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Ground Plate

This paper deals with superharmonic resonance of electrostatically actuated MEMS resonator sensors. The system consists of a MEMS cantilever on top of a parallel ground plate. An AC voltage of frequency near one fourth the natural frequency of the resonator provides the electrostatic force of actuation. The frequency response of the superharmonic resonance of the structure is investigated using two term Reduced Order Model (ROM) method.

Electrostatically Actuated MEMS Circular Plate Resonators: Frequency Response of Superharmonic Resonance of Third Order

2019 ◽  
Author(s):  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Frequency Response ◽  
Circular Plate ◽  
Microelectromechanical Systems ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Damping Force ◽  
Third Order ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Ground Plate

Abstract This paper deals with the frequency response of superharmonic resonance of order three of electrostatically actuated MicroElectroMechanical Systems (MEMS) circular plate resonators. The MEMS structure in this work consists of an elastic circular microplate parallel to an electrode ground plate. The microplate is elelctrostatically actuated through an AC voltage between the microplate and the ground plate. The voltage is in the category of hard excitations. The AC frequency is near one sixth of the natural frequency of the resonator. Since the electrostatic force acting on the resonator is proportional to the square of the voltage, it leads to superharmonic resonance of third order. Besides the electrostatic force, the system experiences damping. The damping force in this work is proportional to the velocity of the resonator, i.e. it is linear damping. Three methods are employed in this investigation. First, the Method of Multiple Scales (MMS), a perturbation method, is used predictions of the resonant regions for weak nonlinearities and small to moderate amplitudes. Second, the Reduced Order Model (ROM) method using two modes of vibration are also utilized to investigate the resonance. ROM is solved numerically integrated using Matlab in order to simulate time responses of the structure, and third, the ROM is used to predict the frequency response using AUTO, a software for continuation and bifurcation analysis. All methods are in agreement for moderate nonlinearities and small to moderate amplitudes. For relatively large amplitudes, when compared to the gap between the microplate and the ground plate, ROM more accurately predicts the behavior of the system. Effects of the parameters of the system on the frequency response are reported.

NEMS Circular Plates Under Hard Electrostatic Excitations: Amplitude-Frequency Response of Superharmonic Resonance of Second Order to Include Casimir Effect

2020 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Julio Beatriz ◽  
Benjamin Huerta
Keyword(s):  
Frequency Response ◽  
Circular Plate ◽  
Quantum Dynamics ◽  
Multiple Scales ◽  
Casimir Effect ◽  
Electrostatic Force ◽  
Primary Resonance ◽  
Second Order ◽  
Amplitude Response ◽  
Superharmonic Resonance

Abstract This work deals with the frequency-amplitude response of the superharmonic resonance of second order of electrostatically actuated clamped NEMS circular plate resonators. The NEMS system consists of a circular plate parallel to a ground plate. Hard excitations (large AC voltage) due to the electrostatic force of frequency near one fourth of the natural frequency of the plate resonator leads the plate into a superharmonic resonance of second order. Hard excitations are excitations significant enough to produce resonance although far from the primary resonance zone. There is no DC component in the voltage applied. For the partial differential equation of motion two reduced order models are developed. The first one uses one mode of vibration and it is solved using the Method of Multiple Scales (MMS), and the frequency-amplitude response is predicted. Hard excitations were modeled by keeping the first term of the Taylor polynomial of the electrostatic force as a large term. The second model uses two modes of vibration, and it is solved using numerical integration. This produces time responses of the resonator. In this work, the quantum dynamics effect such as Casimir effect is considered significant. The two branches, one unstable and one stable, with a saddle node bifurcation point are predicted. Both methods are in agreement for amplitudes up to 0.7 of the gap. The effect of damping and voltage on the frequency response are reported.

Circular Plates Under Hard Excitations to Include Casimir Effect: Amplitude-Voltage Response of Superharmonic Resonance of Second Order

2020 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Julio Beatriz ◽  
Jonathan Perez
Keyword(s):  
Circular Plate ◽  
Casimir Effect ◽  
Second Order ◽  
Peak Amplitude ◽  
Voltage Response ◽  
Voltage Amplitude ◽  
Circular Plates ◽  
Amplitude Response ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated

Abstract This paper deals with voltage-amplitude response of superharmonic resonance of second order of electrostatically actuated clamped MEMS circular plates. A flexible MEMS circular plate, parallel to a ground plate, and under AC voltage, constitute the structure under consideration. Hard excitations due to voltage large enough and AC frequency near one fourth of the natural frequency of the MEMS plate resonator lead the MEMS plate into superharmonic resonance of second order. These excitations produce resonance away from the primary resonance zone. No DC component is included in the voltage applied. The equation of motion of the MEMS plate is solved using two modes of vibration reduced order model (ROM), that is then solved through a continuation and bifurcation analysis using the software package AUTO 07P. This predicts the voltage-amplitude response of the electrostatically actuated MEMS plate. Also, a numerical integration of the system of differential equations using Matlab is used to produce time responses of the system. A typical MEMS silicon circular plate resonator is used to conduct numerical simulations. For this resonator the quantum dynamics effects such as Casimir effect are considered. Also, the Method of Multiple Scales (MMS) is used in this work. All methods show agreement for dimensionless voltage values less than 6. The amplitude increases with the increase of voltage, except around the dimensionless voltage value of 4, where the resonance shows two saddle-node bifurcations and a peak amplitude significantly larger than the amplitudes before and after the dimensionless voltage of 4. A light softening effect is present. The pull-in dimensionless voltage is found to be around 16. The effects of damping and frequency on the voltage response are reported. As the damping increases, the peak amplitude decreases. while the pull-in voltage is not affected. As the frequency increases, the peak amplitude is shifted to lower values and lower voltage values. However, the pull-in voltage and the behavior for large voltage values are not affected.

scholarly journals Voltage–Amplitude Response of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Cantilever Resonators

2019 ◽  
Vol 14 (3) ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin A. Botello ◽  
Christian A. Reyes ◽  
Julio S. Beatriz
Keyword(s):  
Numerical Integration ◽  
Multiple Scales ◽  
Low Voltage ◽  
Second Order ◽  
Voltage Amplitude ◽  
Amplitude Response ◽  
Superharmonic Resonance ◽  
Bifurcation Points ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper investigates the voltage–amplitude response of superharmonic resonance of second order (order two) of alternating current (AC) electrostatically actuated microelectromechanical system (MEMS) cantilever resonators. The resonators consist of a cantilever parallel to a ground plate and under voltage that produces hard excitations. AC frequency is near one-fourth of the natural frequency of the cantilever. The electrostatic force includes fringe effect. Two kinds of models, namely reduced-order models (ROMs), and boundary value problem (BVP) model, are developed. Methods used to solve these models are (1) method of multiple scales (MMS) for ROM using one mode of vibration, (2) continuation and bifurcation analysis for ROMs with several modes of vibration, (3) numerical integration for ROM with several modes of vibration, and (4) numerical integration for BVP model. The voltage–amplitude response shows a softening effect and three saddle-node bifurcation points. The first two bifurcation points occur at low voltage and amplitudes of 0.2 and 0.56 of the gap. The third bifurcation point occurs at higher voltage, called pull-in voltage, and amplitude of 0.44 of the gap. Pull-in occurs, (1) for voltage larger than the pull-in voltage regardless of the initial amplitude and (2) for voltage values lower than the pull-in voltage and large initial amplitudes. Pull-in does not occur at relatively small voltages and small initial amplitudes. First two bifurcation points vanish as damping increases. All bifurcation points are shifted to lower voltages as fringe increases. Pull-in voltage is not affected by the damping or detuning frequency.

Casimir Effect on Voltage Response of Superharmonic Resonance of NEMS Resonators

2017 ◽  
Author(s):  
Martin Botello ◽  
Christian Reyes ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Second Order ◽  
Mode Shapes ◽  
Voltage Response ◽  
Casimir Forces ◽  
Superharmonic Resonance ◽  
Reference Length

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. The differential equation of motion is converted to dimensionless by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

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