Frequency Response of Cylindrical Resonators in a Viscous Fluid

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Michael J. Martin ◽  
Brian H. Houston
Keyword(s):  
Resonant Frequency ◽  
Frequency Response ◽  
Quality Factor ◽  
Linear Models ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Offshore Structures ◽  
Crossover Frequency ◽  
Fluid Density ◽  
Cylindrical Resonators

The frequency response of a cylinder in a viscously damped fluid is a problem of fundamental engineering interest, with applications ranging from microsystems to offshore structures. The analytic solution for the drag in a vibrating cylinder in the laminar flow regime is combined with the equations of motion for forced vibration of a cylinder attached to a spring. The resulting model gives an analytic solution for the dynamic response of the system, including the gain, frequency lag, resonant frequency, quality factor, and stability of the system. The results show that the response of the system is nonlinear, with the phase of the system differing from the phase predicted by linear models. The gain, quality factor, resonant frequency, and crossover frequency all increase with the nondimensional natural frequency β and decrease with the ratio of the fluid density to the resonator density.

Frequency Response of a Viscously Damped Flat Plate

2011 ◽  
Vol 78 (4) ◽  
Author(s):  
Michael James Martin
Keyword(s):  
Frequency Response ◽  
Flat Plate ◽  
Driving Force ◽  
Analytic Solution ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Phase Lag ◽  
Biological Sensing ◽  
Fluid Density ◽  
Stokes Second Problem

The frequency response of a flat plate in a viscous fluid is a problem with applications in microsystems, including gyroscopes, accelerometers, viscometers, and biological sensing. To find the frequency response away from the resonant frequency, the equations of motion are combined with the solution to Stokes’ second problem to produce an analytic solution for the motion of the plate in response to a sinusoidal driving force. These results are used to determine the gain, phase lag, and dynamic stability of the system. The behavior of the system depends on an effective damping ratio ζeff, which depends on the resonator dimensions, and is proportional to the square root of the viscosity times the fluid density.

On the Frequency Response of a Spinning Disk With Large Deformations

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Nonlinear Frequency ◽  
Response Characteristics ◽  
Geometrical Nonlinear ◽  
Oscillation Frequencies ◽  
Multi Mode ◽  
Different Levels

In this paper, the effect of geometrical nonlinear terms, caused by a space fixed point force, on the frequencies of oscillations of a rotating disk with clamped-free boundary conditions is investigated. The nonlinear geometrical equations of motion are based on Von Karman plate theory. Using the eigenfunctions of a stationary disk as approximating functions in Galerkin’s method, the equations of motion are transformed into a set of coupled nonlinear Ordinary Differential Equations (ODEs). These equations are then used to find the equilibrium positions of the disk at different discrete blade speeds. At any given speed, the governing equations are linearized about the equilibrium solution of the disk under the application of a space fixed external force. These linearized equations are then used to find the oscillation frequencies of the disk considering the effect of large deformation. Using multi mode approximation and different levels of nonlinearity, the frequency response of the disk considering the effect of geometrical nonlinear terms are studied. It is found that at the linear critical speed, the nonlinear frequency of the corresponding mode is not zero. Results are presented that illustrate the effect of the magnitude of disk displacement upon the frequency response characteristics. It is also found that for each mode, including the effect of the geometrical nonlinear terms due to the applied load causes a separation in the frequency responses of its backward and forward traveling waves when the disk is stationary. This effect is similar to the effect of a space fixed constraint in the linear problem. In order to verify the numerical results, experiments are conducted and the results are presented.

Resonant frequency response of plasma wave detectors

2006 ◽  
Vol 89 (21) ◽  
pp. 213512 ◽  
Author(s):  
Sungmu Kang ◽  
Peter J. Burke ◽  
L. N. Pfeiffer ◽  
K. W. West
Keyword(s):  
Resonant Frequency ◽  
Frequency Response ◽  
Plasma Wave

Medium damping influences on the resonant frequency and quality factor of piezoelectric circular microdiaphragm sensors

2011 ◽  
Vol 21 (4) ◽  
pp. 045002 ◽  
Author(s):  
M Olfatnia ◽  
Z Shen ◽  
J M Miao ◽  
L S Ong ◽  
T Xu ◽  
...  
Keyword(s):  
Resonant Frequency ◽  
Quality Factor

A Microcontroller Approach to Measuring Transmissibility of MEMS Devices

2015 ◽  
Vol 2015 (DPC) ◽  
pp. 001564-001593
Author(s):  
Chong Li ◽  
Yixuan Wu ◽  
Haoyue Yang ◽  
Luke L. Jenkins ◽  
Robert N. Dean ◽  
...  
Keyword(s):  
Resonant Frequency ◽  
Quality Factor ◽  
Real Time ◽  
High Speed ◽  
Mechanical Quality Factor ◽  
Quantization Noise ◽  
Mems Devices ◽  
Input And Output ◽  
Mechanical Quality ◽  
Mems Device

The transmissibility reveals two very useful characteristics of a micro-electro-mechanical systems (MEMS) device, the resonant frequency and the mechanical quality factor. Real time knowledge on these two important factors can enhance application performance or avoid potential problems from environmental disturbances due to fabrication tolerances and the resulting operational differences in otherwise identical devices. Expensive laboratory equipment is typically used to measure the transmissibility. However, these test systems are not readily adaptable to field use. Therefore, it is important to be able to measure the transmissibility using a real time technique with a simplified test setup. This study proposes a technique that can compute the transmissibility in real time using a low cost microcontroller. This technique utilizes two laser vibrometers to detect the input and output motions of the proof mass in a MEMS device, which are fed to high speed 500 KHz analog to digital converters (ADC) in the microcontroller. A filtering step is performed to decrease noise. After the sampling and pre-filtering, a Fast Fourier Transform (FFT) is performed to convert the time-domain signals to frequency domain signals. The amplitude of the output signal at each frequency is divided by the amplitude of the corresponding input signal at each frequency to obtain the transmissibility. To overcome the difficulties resulting from measurement and quantization noise, a recursive calculating algorithm and a de-quantization filter are introduced. The recursive calculating process guarantees that the system updates the results continually, which results in a transmissibility plot covering the entire bandwidth. The de-quantization filter considers the validity of the data and performs the transmissibility division step accordingly. A cantilevered structure was chosen as the device-under-test to verify and evaluate this technique. The cantilevered device was attached to an electromechanical shaker system for vibratory stimulation. Two laser vibrometers were used to detect the input and output motion and this data was fed into a microcontroller. The microcontroller was STM32F407, which is 32-bit and 168 MHz controller. The tests demonstrated that this technique can measure the transmissibility and therefore the resonant frequency and mechanical quality factor accurately compared to a professional signal analyzer.

A MEMS PPA Based Active Vibration Isolator - STUDENT AWARD 3RD PLACE $500

2016 ◽  
Vol 2016 (DPC) ◽  
pp. 000853-000880
Author(s):  
Chong Li ◽  
C. Lavinia Elana ◽  
Robert N. Dean ◽  
George T. Flowers
Keyword(s):  
Resonant Frequency ◽  
Quality Factor ◽  
Proof Mass ◽  
Control Voltage ◽  
Stable Range ◽  
Vibration Isolator ◽  
Operating Range ◽  
Stable Operating ◽  
Mass Damper ◽  
Active Vibration

Several types of micro-devices are adversely affected by high frequency mechanical vibrations present in the operating environment. Examples include MEMS vibratory gyroscopes and resonators, and micro-optics. Various types of MEMS vibration isolators have been developed for use in the packaging of these vibration sensitive devices. Passive isolators consist of a spring-mass-damper MEMS device and usually have a very high mechanical quality factor, which makes them susceptible to ringing at the isolator's resonant frequency. Active isolators have been realized by using state sensing of the proof mass motion and feeding one or more of these states back through an actuator to adjust the frequency response of the isolator. For example, the technique known as skyhook damping uses velocity feedback to adjust, and typically increase, the damping of the isolator. Although these technique are doable, they require state sensing or state estimation, with feedback electronics to drive the actuator. A simpler MEMS active vibration isolator architecture employs only a parallel plate actuator (PPA) with the MEMS spring-mass-damper structure. The PPA driven with a DC voltage, in its stable operating range, displaces the proof mass, which results in a change in the effective system spring constant due to the electrostatic spring softening effect. This results in a change in the resonant frequency and the quality factor of the isolator. However, due to the nonlinearities inherent in this type of device, the stable operating range is reduced as the PPA voltage is increased. Furthermore, even when the isolator is stable in steady-state, a sufficiently large transient response can also drive it into the unstable regime, resulting in the electrodes snapping into contact. In this study, the PPA based active vibrator isolator is developed and its performance is evaluated. The characteristics of the transient instability are investigated and its stable range of operation is specified, for booth external disturbances and rapid application of the control voltage. This MEMS PPA based active vibration isolator can improve performance compared to passive isolators, while being much simpler than state feedback active isolators.

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

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