Experimental Investigation of 2:1 and 3:1 Internal Resonances in Nonlinear MEMS Arch Resonators

2016 ◽  
Author(s):  
Abdallah Ramini ◽  
Amal Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Experimental Investigation ◽  
Frequency Response ◽  
Electrical Current ◽  
Vibrational Modes ◽  
Response Curves ◽  
Internal Resonances ◽  
Mems Resonators ◽  
Resonance Frequencies ◽  
Doped Silicon

We demonstrate experimentally internal resonances in MEMS resonators. The investigation is conducted on in-plane MEMS arch resonators fabricated with a highly doped silicon. The resonators are actuated electrostatically and their stiffness are tuned by electrothermal loading by passing an electrical current though the microstructures. We show that through this tuning, the ratio of the various resonance frequencies can be varied and set at certain ratios. Particularly, we adjust the resonance frequencies of two different vibrational modes to 2:1 and 3:1. Finally, we validate the internal resonances at these ratios through frequency-response curves and FFTs.

Modal Analysis to Interpret Localization Phenomena of Harmonic Oscillations in Nonlinear Oscillator Arrays

2019 ◽  
Author(s):  
Yuji Harata ◽  
Takashi Ikeda
Keyword(s):  
Coordinate System ◽  
Frequency Response ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Vibrational Modes ◽  
Response Curves ◽  
Internal Resonances ◽  
Harmonic Oscillations ◽  
Oscillator Arrays ◽  
Phase Angles

Abstract This paper investigates localization phenomena in a nonlinear array with N Duffing oscillators connected by weak, linear springs when the array is subjected to harmonic excitation. In the theoretical analysis, the equations of motion are derived for: (1) the physical coordinate system, and (2) modal coordinate system. The modal equations of motion form an autoparametric system, i.e., the excitation acts directly on the first mode of vibration, and the other modes are indirectly excited because they are nonlinearly coupled with the first mode. Van der Pol’s method is employed to obtain the solutions of the harmonic oscillations, and then the expressions of the frequency response curves are given. In the numerical calculations, the frequency response curves of the amplitudes and phase angles in the cases of N = 2 and 3 are presented. The frequency response curves, obtained in the modal coordinate system, demonstrate that localization phenomena occur in the physical coordinate system when multiple vibrational modes simultaneously appear. When imperfections exist in the N Duffing oscillators, the modal equations of motion do not form an autoparametric system because the external excitation directly acts on all modes. Instead, internal resonances may occur in such systems.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

scholarly journals Vibration Parameters Estimation by Blade Tip-Timing in Mistuned Bladed Disks in Presence of Close Resonances

2020 ◽  
Vol 10 (17) ◽  
pp. 5930
Author(s):  
Saeed Bornassi ◽  
Christian Maria Firrone ◽  
Teresa Maria Berruti
Keyword(s):  
Frequency Response ◽  
Optimization Technique ◽  
Numerical Test ◽  
Parameters Estimation ◽  
Least Square ◽  
Response Curves ◽  
Bladed Disks ◽  
Bladed Disk ◽  
Resonance Frequencies ◽  
Blade Tip

The present paper is focused on the post processing of the data coming from the Blade Tip-Timing (BTT) sensors in the case where two very close peaks are present in the frequency response of the vibrating system. This type of dynamic response with two very close peaks can occur quite often in bladed disks. It is related to the fact that the bladed disk is not perfectly cyclic symmetric and the so called “mistuning” is present. A method based on the fitting of the BTT sensors data by means of a 2 degrees of freedom (2DOF) dynamic model is proposed. Nonlinear least square optimization technique is employed for identification of the vibration characteristics. A numerical test case based on a lump parameter model of a bladed disk assembly is used to simulate different response curves and the corresponding sensors signals. The Frequency Response Function (FRF) constructed at the resonance region is compared with the traditional Sine fitting results, the resonance frequencies and damping values estimated by the fitting procedure are also reported. Accurate predictions are achieved and the results demonstrate the considerable capacity of the 2DOF method to be used as a standalone or as a complement to the standard Sine fitting method.

scholarly journals Experimental and Theoretical Study of Two-to-One Internal Resonance of MEMS Resonators

2018 ◽  
Author(s):  
Amal Z. Hajjaj ◽  
Feras K. Alfosail ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Perturbation Technique ◽  
Vibrational Modes ◽  
Reduced Order ◽  
Simultaneous Excitation ◽  
Direct Excitation ◽  
Mems Resonators ◽  
Resonance Frequencies ◽  
Good Agreement

In this paper, we investigate experimentally and theoretically the two-to-one (2:1) internal resonance between the first two symmetric vibrational modes of microelectromechanical (MEMS) arch resonator electrothermally tuned and electrostatically driven. Applying electrothermal voltage across the beam anchors controls its stiffness and then its resonance frequencies. Hence the ratio between the two frequencies can be tuned to a ratio of two. Then, we study the dynamic response of the arch beam during internal resonance. In the studied case, the presence of high AC bias excitation leads to the direct simultaneous excitation of the 1st and 3rd frequencies in addition to the activation of the internal resonance. A reduced order model and perturbation techniques are presented to analyze the nonlinear response of the structure. In the perturbation technique, the direct excitation of the 3rd resonance frequency is taken into consideration. Results show the presence of Hopf bifurcations, which can lead to chaotic motion at higher excitation. A good agreement among the theoretical and experimental results is shown.

Frequency Response Curves and Dynamic Characteristics of a Ginkgo Tree in Different Growth Periods

2020 ◽  
Vol 63 (6) ◽  
pp. 1673-1684
Author(s):  
Jie Zhou ◽  
Linyun Xu ◽  
Guanhua Liu ◽  
Yan Xuan ◽  
Hongping Zhou ◽  
...  
Keyword(s):  
Frequency Response ◽  
Frequency Domain ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Fruit Trees ◽  
Modal Parameters ◽  
Response Curves ◽  
Resonance Frequencies ◽  
Modal Parameters Identification

HighlightsThe frequency domain modal parameters identification method was applied to a ginkgo tree.Dynamic characteristics of the ginkgo tree were tested during five phenological periods.Almost all resonance frequencies were near the peaks of the frequency response curves.Leaves caused the number of natural frequencies of the ginkgo tree to be greatly reduced.Abstract. Understanding the dynamic characteristics of fruit trees is the premise of effective mechanized harvesting. This study performed a tracking test on a ginkgo tree in five phenological periods from the dormant period to the leaf-unfolding period. The frequency domain modal parameters identification method was applied to the ginkgo tree, and the relationship between the natural frequencies and resonance frequencies of the ginkgo tree was obtained. The main factors affecting the fundamental frequency and damping ratio of the ginkgo tree were not the elastic modulus and moisture content but rather the growth of the leaves. The growth of leaves caused the number of natural frequencies in the low-frequency band to be greatly reduced, and the value of the natural frequencies exhibited a slightly decreasing tendency. The damping caused by leaves had a significant weakening effect on the transmission of vibrational energy on the lateral branches. The resonance frequencies that caused strong response of the ginkgo tree were mostly near the peak frequencies of the frequency response curves (natural frequencies), but eccentric motor excitation could not effectively stimulate all the natural frequencies of the ginkgo tree to reach resonance. In the frequency response curves of the ginkgo tree, the main natural frequency with the maximum energy might not cause the maximum vibration response of the ginkgo tree, even if this excitation frequency could induce resonance. Resonance could be used to maximize the transfer of excitation energy, but each position of the tree had its own independent frequency spectrum characteristics. A single excitation frequency could not cause all positions of the ginkgo tree to resonate simultaneously. Changing the excitation frequency of harvesting equipment within a small frequency range could achieve the maximum resonance response of most positions on fruit trees. Keywords: Dynamic characteristics, Growth periods, Leaves, Natural frequencies, Resonance.

Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

2020 ◽  
pp. 107754632093818
Author(s):  
Ashraf T EL-Sayed
Keyword(s):  
Frequency Response ◽  
Time Delays ◽  
Degrees Of Freedom ◽  
Primary Resonance ◽  
Delayed Feedback ◽  
Harmonic Oscillators ◽  
Feedback Signal ◽  
Response Curves ◽  
Internal Resonances ◽  
Van Der Pol

It has been revealed in the proposed work that a pair of delay positive position feedback control can lessen the vibration response of double Van der Pol oscillators with external forces. We also studied the effects of both the control and the delayed feedback signal gains to illustrate the low vibration amplitudes. The averaging perturbation process has been used to consider the frequency-response equations of amplitudes and modulation phases at the primary resonance and one-to-one internal resonances. According to the perturbation solutions for the four-degrees-of-freedom system, we presented the frequency response curves that were periodic in the time delays. The stability analysis presented in this study has shown optimum stable ranges. If the time delays increase, the steady-state amplitudes of the oscillator’s system will periodically result in few stable regions and more unstable ones. The numerical simulation has been introduced to check the analytical approximation. It was also found to be almost identical after presenting the comparison of the results.

Study of plate vibrating in fluids having part-through crack at random angles and locations

2021 ◽  
pp. 095440622110127
Author(s):  
Ruqia Ikram ◽  
Asif Israr
Keyword(s):  
Frequency Response ◽  
Multiple Scales ◽  
Peak Amplitude ◽  
Nonlinear Frequency ◽  
Response Curves ◽  
Crack Location ◽  
Cracked Plate ◽  
Nonlinear Frequency Response ◽  
Angle Crack ◽  
Crack Angle

This study presents the vibration characteristics of plate with part-through crack at random angles and locations in fluid. An experimental setup was designed and a series of tests were performed for plates submerged in fluid having cracks at selected angles and locations. However, it was not possible to study these characteristics for all possible crack angles and crack locations throughout the plate dimensions at any fluid level. Therefore, an analytical study is also carried out for plate having horizontal cracks submerged in fluid by adding the influence of crack angle and crack location. The effect of crack angle is incorporated into plate equation by adding bending and twisting moments, and in-plane forces that are applied due to antisymmetric loading, while the influence of crack location is also added in terms of compliance coefficients. Galerkin’s method is applied to get time dependent modal coordinate system. The method of multiple scales is used to find the frequency response and peak amplitude of submerged cracked plate. The analytical model is validated from literature for the horizontally cracked plate submerged in fluid as according to the best of the authors’ knowledge, literature lacks in results for plate with crack at random angle and location in the presence of fluid following validation with experimental results. The combined effect of crack angle, crack location and fluid on the natural frequencies and peak amplitude are investigated in detail. Phenomenon of bending hardening or softening is also observed for different boundary conditions using nonlinear frequency response curves.

Magnification curves of electromagnetic seismographs

1964 ◽  
Vol 54 (5A) ◽  
pp. 1459-1471
Author(s):  
S. K. Chakrabarty ◽  
G. C. Choudhury ◽  
S. N. Roy Choudhury
Keyword(s):  
General Solution ◽  
Frequency Response ◽  
Experimental Method ◽  
Phase Shifts ◽  
Response Curves ◽  
Operating Condition ◽  
Coil Inductance ◽  
The World ◽  
Electromagnetic Seismograph ◽  
Proper Design

Abstract The general solution of the equations connecting the motion of the two coupled components in an electromagnetic seismograph has been obtained in another paper and it shows that the magnification of a seismograph depend on seven instrumental constants. Using these results, equations and curves have been derived in the present paper from which the Magnification as well as Phase shifts in the response of a seismograph and their variations with damping and coil inductance can be easily obtained. Based on these curves a number of magnification curves for different combinations, which are in operation at the different seismological stations of the world, have been derived. Suitable equations and curves have also been obtained which can be used for estimating the absolute Magnification of a Seismograph. An experimental method of obtaining the frequency response curves of seismographs in their operating condition has been described and the results obtained by this method has been given. It has been indicated how the results incorporated in the present paper can be used in the proper design of seismographs required for the different purposes.

Vibrational modes in MEMS resonators

2019 ◽  
Vol 29 (12) ◽  
pp. 123001 ◽  
Author(s):  
Daniel Platz ◽  
Ulrich Schmid
Keyword(s):  
Vibrational Modes ◽  
Mems Resonators
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