Casimir Effect on Frequency Response of Superharmonic Resonance of NEMS Resonators

2017 ◽  
Author(s):  
Martin Botello ◽  
Christian Reyes ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Frequency Response ◽  
Multiple Scales ◽  
Casimir Effect ◽  
Electrostatic Force ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Casimir Forces ◽  
Superharmonic Resonance ◽  
Reference Length

This paper investigates the frequency response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the MEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance insinuates that the AC voltage is operating in a frequency near one-fourth the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir force. For the electrostatic force, the AC voltage is in the category of hard excitation in order to induce a bifurcation phenomenon. For Casimir forces to affect the system, the gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm. 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) is 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. The influences of parameters (i.e. Casimir, damping, second voltage and fringe) were also investigated.

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.

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.

Casimir Effect on Superharmonic Resonance of Bio-NEMS Circular Plate Resonator Sensors

2018 ◽  
Author(s):  
Martin Botello ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Circular Plate ◽  
Casimir Effect ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Amplitude Response ◽  
Partial Differential ◽  
Superharmonic Resonance ◽  
Mems Resonator

Casimir effect on superharmonic resonance of electrostatically actuated bio-nano-electro-mechanical system (Bio-NEMS) circular plate resonator sensor is investigated. The plate sensor resonator is clamped at the outer end and suspended over a parallel ground plate. The sensor can be used for detecting human viruses. Superharmonic resonance of the second order, frequency near one-fourth the natural frequency of the resonator, is induced using Alternating Current (AC) voltage. The magnitude of the AC voltage is also large enough to be consider hard excitation acting on the resonator. Beside Casimir effect, other external forces (i.e. electrostatic force and viscous air damping) acting on the MEMS resonator create a nonlinear behaviors such as bifurcation and pull-in instability. Hence, numerical models, such as Method of Multiple Scales (MMS) and Reduced Order Model (ROM), are used to predict the frequency-amplitude response for MEMS resonator. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. While, ROM, based on the Galerkin procedure which uses the mode shapes of vibration of the resonator as a basis of functions, transforms the nonlinear partial differential equation of motion into a system of ordinary differential equation with respect to dimensionless time. The frequency-amplitude response allows one to observe the behavior of the system for a range of frequencies near the superharmonic resonance. The effects of parameters such as Casimir effect, voltage, and damping on the frequency-amplitude 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.

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.

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

2019 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christopher Reyes
Keyword(s):  
Frequency Response ◽  
Cantilever Beam ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Beam Theory ◽  
Driving Frequency ◽  
Third Order ◽  
Reduced Order ◽  
Superharmonic Resonance ◽  
Amplitude Frequency Response

Abstract This work deals with amplitude frequency response of MEMS cantilever resonators undergoing superharmonic resonance of third order. The cantilever resonator is parallel to a ground plate and under alternating current (AC) voltage that excites the cantilever into vibrations. The driving frequency of the AC voltage is near one sixth of the first natural frequency of the cantilever beam resulting into superharmonic resonance of third order. The cantilever beam is modeled using Euler-Bernoulli beam theory. The electrostatic force is modeled using Palmer’s formula to include the fringe effect. In order to investigate the amplitude frequency behavior of the system reduced order models (ROMs) are developed. Three methods are used to solve these ROMs they are 1) the method of multiple scales (MMS) for ROM with one mode of vibration, 2) homotopy analysis method (HAM) for ROM with one mode of vibration, and 3) direct numerical integration for 2 modes of vibration Reduced Order Model (2T ROM) producing time responses of the tip of the cantilever resonator. In this work the limitations of MMS and HAM are highlighted when considering large voltage values i.e hard excitations. For large voltage values MMS and HAM cannot accurately predict the amplitude frequency response; the results from 2T ROM time responses disagree significantly with the MMS and HAM solutions. The effect of voltage on the frequency response is investigated. As the voltage values in the system increase the responses shift to lower frequencies and larger amplitudes.

Sensitivity of MEMS/NEMS Biosensors Near Twice Natural Frequency

2010 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin W. Knecht
Keyword(s):  
Natural Frequency ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Casimir Effect ◽  
Electrostatic Force ◽  
Equation Of Motion ◽  
Direct Approach ◽  
Mass Detection ◽  
Electrostatically Actuated ◽  
Phase Amplitude

Bio-MEMS/NEMS resonator sensors near twice natural frequency for mass detection are investigated. Electrostatic force along with fringe correction and Casimir effect are included in the model. They introduce parametric nonlinear terms in the system. The partial-differential equation of motion of the system is solved by using the method of multiple scales. A direct approach of the problem is then used. Two approximation problems resulting from the direct approach are solved. Phase-amplitude relationship is obtained. Numerical results for uniform electrostatically actuated micro resonator sensors are reported.

Boundary Value Problem Method to Investigate Superharmonic Resonance of MEMS Resonators

2017 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Christian Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
The Other ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Amplitude Frequency Response ◽  
Problem Method ◽  
Mems Resonators

This paper deals with two different methods to analyze the amplitude frequency response of an electrostatically actuated micro resonator. The methods used in this paper are the method of multiple scales, which is an analytical method with one mode of vibration. The other method is based on system of odes which is derived using the partial differential equation of motion, as well as the boundary conditions. This system is then solved using a built in matlab function known as BVP4C. Results are then shown comparing the two methods, under a variety of parameters, including the influence of damping, voltage, and fringe.

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).

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.

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