scholarly journals Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers

Micromachines ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 575 ◽  
Author(s):  
Aymen Jallouli ◽  
Najib Kacem ◽  
Joseph Lardies
Keyword(s):  
Differential Equations ◽  
Dynamic Behavior ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Ultrasonic Transducers ◽  
Initial Deflection ◽  
Geometric Imperfections ◽  
Resonance Frequencies

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

Effects of Squeeze Film and Initial Deflection on the Resonance Frequencies and Modal Damping of Circular Microplates

2018 ◽  
Author(s):  
Aymen Jallouli ◽  
Najib Kacem ◽  
Fehmi Najar ◽  
Joseph Lardies
Keyword(s):  
Plate Theory ◽  
Differential Quadrature Method ◽  
Static Solution ◽  
Initial Deflection ◽  
Mode Shapes ◽  
Squeeze Film ◽  
Modal Damping ◽  
Resonance Frequencies ◽  
Grid Points ◽  
Air Film

We investigate the effects of squeeze air film and initial deflection on the resonance frequencies and modal damping of capacitive circular microplates. The equation of motion of a circular microplate, which are derived from the von kármán plate theory, coupled with the Reynolds equation are discretized using the Differential Quadrature Method (DQM). The eigenvalues and eigenvectors of the multiphysical problem are determined by perturbing the system of equations around a static solution. Therefore, the resonance frequencies, modal damping coefficients and mode shapes of the plate and the fluid can be determined. The advantage of using DQM is that the solution of the system can be obtained with only few grid points. The obtained numerical results are compared with the experimental data for the case of a capacitive circular microplates with an initial deflection. The increase of the static pressure leads to a shift in the resonance frequencies due to the increase in the stiffness of the plate. Also the initial deflection change the resonance frequencies due to the change in the effective gap distance. The developed model is an effective tool to predict the dynamic behavior of a microsystem under the effect of air film and with initial deflection.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

scholarly journals Dynamic Analysis of a Tapered Composite Thin-Walled Rotating Shaft Using the Generalized Differential Quadrature Method

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Weiyan Zhong ◽  
Feng Gao ◽  
Yongsheng Ren ◽  
Xiaoxiao Wu ◽  
Hongcan Ma
Keyword(s):  
Differential Equations ◽  
Dynamic Model ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Rotating Shaft ◽  
Generalized Differential Quadrature Method ◽  
Thin Walled ◽  
Generalized Differential

A dynamic model of a tapered composite thin-walled rotating shaft is presented. In this model, the transverse shear deformation, rotary inertia, and gyroscopic effects have been incorporated. The equations of motion are derived based on a refined variational asymptotic method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the generalized differential quadrature method (GDQM). The validity of the dynamic model is proved by comparing the numerical results with those obtained in the literature and by using ANSYS. The effects of taper ratio, boundary conditions, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the natural frequencies and critical rotating speeds are investigated.

Control of Vibration Amplitude, Frequency and Damping of an Electrostatically Actuated Microbeam Using Capacitive, Inductive and Resistive Elements

2010 ◽  
Author(s):  
Abdolreza Pasharavesh ◽  
Y. Alizadeh Vaghasloo ◽  
A. Fallah ◽  
M. T. Ahmadian
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vibration Amplitude ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Electrical Current ◽  
Oscillatory System ◽  
Electrostatically Actuated ◽  
In Series

In this study vibration amplitude, frequency and damping of a microbeam is controlled using a RLC block containing a capacitor, resistor and inductor in series with the microbeam. Applying this method all of the considerable characteristics of the oscillatory system can be determined and controlled with no change in the geometrical and physical characteristics of the microbeam. Euler-Bernoulli assumptions are made for the microbeam and the electrical current through the microbeam is computed by considering the microbeam deflection and its voltage. Considering the RLC block, the voltage difference between the microbeam and the substrate is calculated. Two coupled nonlinear partial differential equations are obtained for the deflection and the voltage. The one parameter Galerkin method is employed to transform the equations of motion to a set of nonlinear coupled ordinary differential equations. Differential quadrature method (DQM) is implemented to solve the governing nonlinear ordinary differential equations. The effect of the controller parameters such as capacitance, resistance and inductance on the amplitude, frequency and damping is studied. Also the internal resonance between the electrical and mechanical parts of the system is studied. Results indicate using these elements, amplitude, frequency and damping can be controlled as desired by the user.

Analysis of Electromechanical Performance of Energy Harvesting Skin Based on the Kirchhoff Plate Theory

2014 ◽  
Author(s):  
Heonjun Yoon ◽  
Byeng D. Youn ◽  
Heung S. Kim
Keyword(s):  
Energy Harvesting ◽  
Differential Equations ◽  
Analytical Model ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Electrical Circuit ◽  
Kirchhoff Plate ◽  
Mode Shapes ◽  
Kirchhoff Plate Theory

As a compact and durable design concept, energy harvesting skin (EH skin), which consists of piezoelectric patches directly attached onto the surface of a vibrating structure as one embodiment, has been recently proposed. This study aims at developing an electromechanically-coupled analytical model of the EH skin so as to understand its electromechanical behavior and get physical insights about important design considerations. Based on the Kirchhoff plate theory, the Hamilton’s principle is used to derive the differential equations of motion. The Rayleigh-Ritz method is implemented to calculate the natural frequency and the corresponding mode shapes of the EH skin. The electrical circuit equation is derived by substituting the piezoelectric constitutive relation into Gauss’s law. Finally, the steady-state output voltage is obtained by solving the differential equations of motion and electrical circuit equation simultaneously. The results of the analytical model are verified by comparing those of the finite element analysis (FEA) in a hierarchical manner.

Vibrations of Square and Hexagonal Cylinders in a Liquid

1981 ◽  
Vol 103 (3) ◽  
pp. 233-239 ◽  
Author(s):  
Y. Shinohara ◽  
T. Shimogo
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fuel Assembly ◽  
Dynamic Behavior ◽  
Nuclear Fuel ◽  
Equations Of Motion ◽  
Hydrodynamic Forces ◽  
Partial Differential ◽  
Nuclear Fuel Assembly

A mathematical model is proposed to describe the dynamic behavior of square and hexagonal cylinder bundles immersed in a liquid. First, the hydrodynamic forces associated with cylinder motions are examined, and then equations of motion of the spring-mounted cylinders including liquid coupling are derived. When the number of cylinders is very large, these equations are replaced by partial differential equations on the assumption that the cylinder bundles form a continuum. The results of this study have application in the modeling of vibration of a nuclear fuel assembly under the excitation of earthquakes.

Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1998 ◽  
Vol 120 (2) ◽  
pp. 475-483 ◽  
Author(s):  
L. Yang ◽  
S. G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Thin Plate Theory ◽  
Force Equilibrium

An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

Numerical Formulation of Nonlinear Vibrations of Elastically-Constrained Rotating Disks

1995 ◽  
Author(s):  
Longxiang Yang ◽  
Stanley G. Hutton
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Disk ◽  
Transverse Displacement ◽  
Rotating Disks ◽  
Numerical Formulation ◽  
Force Equilibrium

Abstract An analysis of nonlinear vibrations of an elastically-constrained rotating disk is developed. The equations of motion, which are two coupled nonlinear partial differential equations corresponding to the transverse force equilibrium and to the deformation compatibility, are first developed by using von Karman thin plate theory. Then the stress function is analytically solved from the compatibility equation by assuming a multi-mode transverse displacement field. Galerkin’s method is applied to transform the force equilibrium equation into a set of coupled nonlinear ordinary differential equations in terms of time functions. Finally, numerical integration is used to solve the time governing equations, and the effects of nonlinearity on the vibrations of a rotating disk are discussed.

scholarly journals Geometrical Non-Linear Periodic Vibration of Plates

2000 ◽  
Author(s):  
M. Petyt ◽  
P. Ribeiro
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Internal Resonances ◽  
Resonance Frequencies ◽  
Modal Coupling ◽  
Non Linear ◽  
The Stability

Abstract Periodic, geometrically non-linear free and steady-state forced vibrations of fully clamped plates are investigated. The hierarchical finite element method (HFEM) and the harmonic balance method are used to derive the equations of motion in the frequency domain, which are solved by a continuation method. It is demonstrated that the HFEM requires far fewer degrees of freedom than the h-version of the FEM. Internal resonances due to modal coupling between modes with resonance frequencies related by a rational number, are discovered. In free vibration, internal resonances cause a very significant variation of the mode shape during the period of vibration. A similar behaviour is observed in steady-state forced vibration. The stability of the steady-state solutions is studied by Floquet’s theory and it is shown that stable multi-modal solutions occur.

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