Nonlinear vibration analysis of a ┴-shaped mass attached to a clamped–clamped microbeam under electrostatic actuation

2016 ◽  
Vol 231 (11) ◽  
pp. 2147-2158 ◽  
Author(s):  
M Zamanian ◽  
A Karimiyan ◽  
SAA Hosseini ◽  
H Tourajizadeh
Keyword(s):  
Nonlinear Vibration ◽  
Electrostatic Force ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Electrostatic Actuation ◽  
Concentrated Force ◽  
Assumed Mode Method ◽  
Lagrange’S Equation ◽  
Comparison Functions ◽  
Horizontal Part

This article studies the nonlinear vibration of a ┴-shaped mass attached to a clamped–clamped microbeam under electrostatic actuation considering the effect of stretching. The DC and AC electrostatic force is applied to the horizontal part of ┴-shaped mass. The dynamic solution is studied using two methods of modeling. In the first model, the ┴-shaped mass is considered as a rigid body between two flexible microbeams. Then, the discretized equation of motion is derived using Lagrange’s equation combined with assumed mode method. The vibration mode shape of linear system is used as the comparison functions. In the second model, the dynamical effect of ┴-shaped mass is modeled as a concentrated force and moment, and it is introduced in the equation of motion by the Dirac function. Afterwards, the equation of motion is discretized using Galerkin method. In both methods of modeling, the equations of motion are solved using two methods. The first method is approximate analytical perturbation and the other one is Runge–Kutta numerical method. The effect of geometrical dimension of ┴-shaped mass on the nonlinear shift of resonance frequency and dynamic pull-in voltage is studied. The efficiency and accuracy of the presented formulations is verified by comparing the obtained results by two methods of modeling and two methods of solution.

Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

1963 ◽  
Vol 30 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Free Vibrations ◽  
Layered Plates ◽  
Variational Approximations ◽  
Elastic Systems ◽  
Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

On the Lagrange’s Method for Floating Bodies

2006 ◽  
Author(s):  
Keyvan Sadeghi
Keyword(s):  
Kinetic Energy ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
The Body ◽  
Floating Body ◽  
Floating Bodies ◽  
Lagrange’S Equation ◽  
Lagrange’S Method ◽  
Linear Radiation

It is shown that, in the context of a linear theory, all fluid radiation actions on a floating body can be solely represented by a part of the fluid mechanical energy corresponding to the wetted surface of the body. In this regard, it is indicated that the linear radiation damping can be expressed by a fluid kinetic energy which has a bilinear form. Then from the Lagrange’s equations of motion, an equation of motion is derived that is called the conjugate Larange’s equation of motion. A variant of Hamilton’s principle is also introduced as the variational generator of the conjugate Lagrange’s equation of motion.

Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part II—System Equations

1990 ◽  
Vol 112 (2) ◽  
pp. 215-224 ◽  
Author(s):  
S. Nagarajan ◽  
D. A. Turcic
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Elastic Response ◽  
Mutual Dependence ◽  
Generalized Coordinates ◽  
Lagrange’S Equation ◽  
Elastic Mechanism

The first step in the derivation of the equations of motion for general elastic mechanism systems was described in Part I of this work. The equations were derived at the elemental level using Lagrange’s equation and the generalized coordinates were both the rigid body degrees of freedom, and the elastic degrees of freedom of element ‘e’. Each rigid body degree of freedom gave rise to a scalar equation of motion, and the elastic degrees of freedom of element e gave rise to a vector equation of motion. Since both the rigid body degrees of freedom and elastic degrees of freedom are considered as generalized coordinates, the equations derived take into account the mutual dependence between the rigid body and elastic motions. This is important for mechanisms that are built using lightweight and flexible members and which operate at high speeds. A schematic diagram of how the equations of motion are obtained in this work is shown in Fig. 1 in Part I. The transformation step in the figure refers to the rotational transformation of the nodal elastic displacements (which were measured in the element coordinate system), so that they are measured in terms of the reference coordinate system. This transformation is necessary in order to ensure compatibility of the displacement, velocity and acceleration of the degrees of freedom that are common to two or more links during the assembly of the equations of motion. This final set of equations after assembly are obtained in closed form, and, given external torques and forces, can be solved for the rigid body and elastic response simultaneously taking into account the mutual dependence between the two responses.

scholarly journals Pole-skipping and hydrodynamic analysis in Lifshitz, AdS2 and Rindler geometries

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Haiming Yuan ◽  
Xian-Hui Ge
Keyword(s):  
Boundary Condition ◽  
Green’S Function ◽  
Green's Function ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Boundary Theory ◽  
Hydrodynamic Analysis ◽  
Dimensionless Variables ◽  
Pass Through ◽  
Special Points

Abstract The “pole-skipping” phenomenon reflects that the retarded Green’s function is not unique at a pole-skipping point in momentum space (ω, k). We explore the universality of pole-skipping in different geometries. In holography, near horizon analysis of the bulk equation of motion is a more straightforward way to derive a pole-skipping point. We use this method in Lifshitz, AdS2 and Rindler geometries. We also study the complex hydrodynamic analyses and find that the dispersion relations in terms of dimensionless variables $$ \frac{\omega }{2\pi T} $$ ω 2 πT and $$ \frac{\left|k\right|}{2\pi T} $$ k 2 πT pass through pole-skipping points $$ \left(\frac{\omega_n}{2\pi T},\frac{\left|{k}_n\right|}{2\pi T}\right) $$ ω n 2 πT k n 2 πT at small ω and k in the Lifshitz background. We verify that the position of the pole-skipping points does not depend on the standard quantization or alternative quantization of the boundary theory in AdS2× ℝd−1 geometry. In the Rindler geometry, we cannot find the corresponding Green’s function to calculate pole-skipping points because it is difficult to impose the boundary condition. However, we can still obtain “special points” near the horizon where bulk equations of motion have two incoming solutions. These “special points” correspond to the nonuniqueness of the Green’s function in physical meaning from the perspective of holography.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

ASYMPTOTIC NUMERICAL METHOD FOR THE DYNAMIC STUDY OF NONLINEAR VIBRATION ABSORBERS

2014 ◽  
Vol 06 (05) ◽  
pp. 1450053 ◽  
Author(s):  
FATHI DJEMAL ◽  
FAKHER CHAARI ◽  
JEAN LUC DION ◽  
FRANCK RENAUD ◽  
IMAD TAWFIQ ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Vibration Control ◽  
Numerical Method ◽  
Nonlinear Vibration ◽  
Degrees Of Freedom ◽  
Equation Of Motion ◽  
Vibration Absorbers ◽  
Asymptotic Numerical Method ◽  
Dynamic Absorber ◽  
Newton Raphson

Vibrations are usually undesired phenomena as they may cause discomfort, disturbance, damage, and sometimes destruction of machines and structures. It must be reduced or controlled or eliminated. One of the most common methods of vibration control is the use of the dynamic absorber. The paper is interested in the study of a nonlinear two degrees of freedom (DOF) model. To solve nonlinear equation of motion a high order implicit algorithm is proposed. It is based on the introduction of a homotopy, an implicit scheme of Newmark and the use of techniques of Asymptotic Numerical method (ANM). We propose also a regularization of the contact force to overcome the difficulty of the singularity in this model. A comparison will be presented between the results obtained by the proposed algorithm and those using the classical Newton–Raphson and Newmark time scheme.

The Influence of Tank Orientation Angle on a 2D Structure-Tuned Liquid Damper System

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
J. S. Love ◽  
M. J. Tait
Keyword(s):  
Mechanical Model ◽  
Vibrational Energy ◽  
Equations Of Motion ◽  
Orientation Angle ◽  
Structural Response ◽  
Tuned Liquid Damper ◽  
Lagrange’S Equation ◽  
Equivalent Mechanical Model ◽  
Principal Axes ◽  
2D Structure

Tuned liquid dampers (TLDs) utilize sloshing fluid to absorb and dissipate structural vibrational energy, thereby reducing wind induced dynamic motion. By selecting the appropriate tank length, width, and fluid depth, a rectangular TLD can control two structural sway modes simultaneously if the TLD tank is aligned with the principal axes of the structure. This study considers the influence of the TLD tank orientation on the behavior of a 2D structure-TLD system. The sloshing fluid is represented using a linearized equivalent mechanical model. The mechanical model is coupled to a 2D structure at an angle with respect to the principal axes of the structure. Equations of motion for the system are developed using Lagrange’s equation. If the TLD and structure are not aligned, the system responds as a coupled four degree of freedom system. The proposed model is validated by conducting structure-TLD system tests. The predicted and experimental structural displacements and fluid response are in agreement. An approximate method is developed to provide an initial estimate of the structural response based on an effective mass ratio. The results of this study show that for small TLD orientation angles, the performance of the TLD is insensitive to TLD orientation.

Nonlinear Dynamic Study on Effects of Rub-Impact Caused by Oil-Rupture in a Multi-Shafts Turbine With a Squeeze Film Damper

2014 ◽  
Author(s):  
T. N. Shiau ◽  
C. R. Wang ◽  
D. S. Liu ◽  
W. C. Hsu ◽  
T. H. Young
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Rotor System ◽  
Squeeze Film ◽  
Speed Ratio ◽  
Frequency Spectra ◽  
Squeeze Film Damper ◽  
Assumed Mode Method ◽  
Nonlinear Dynamic Behavior

An investigation is carried out the analysis of nonlinear dynamic behavior on effects of rub-impact caused by oil-rupture in a multi-shafts turbine system with a squeeze film damper. Main components of a multi-shafts turbine system includes an outer shaft, an inner shaft, an impeller shaft, ball bearings and a squeeze film damper. In the squeeze film damper, oil forces can be derived from the short bearing approximation and cavitated film assumption. The system equations of motion are formulated by the global assumed mode method (GAMM) and Lagrange’s approach. The nonlinear behavior of a multi-shafts turbine system which includes the trajectories in time domain, frequency spectra, Poincaré maps, and bifurcation diagrams are investigated. Numerical results show that large vibration amplitude is observed in steady state at rotating speed ratio adjacent to the first natural frequency when there is no squeeze film damper. The nonlinear dynamic behavior of a multi-shafts turbine system goes in its way into aperiodic motion due to oil-rupture and it is unlike the usual way (1T = >2T = >4T = >8T etc) as compared to one shaft rotor system. The typical routes of bifurcation to aperiodic motion are observed in a multi-shafts turbine rotor system and they suddenly turn into aperiodic motion from the periodic motion without any transition. Consequently, the increasing of geometric or oil parameters such as clearance or lubricant viscosity will improve the performance of SFD bearing.

Design of Electrostatic Micro Mirrors for Improved Stability Though Surface Contact

1998 ◽  
Author(s):  
Timothy Moulton ◽  
G. K. Ananthasuresh
Keyword(s):  
Design Method ◽  
Electrostatic Force ◽  
Electrostatic Actuation ◽  
Mems Devices ◽  
Complex Dynamic ◽  
Micro Electro Mechanical Systems ◽  
Electrostatically Actuated ◽  
Surface Contact ◽  
Improved Stability ◽  
Mems Process

Abstract There exists a need to stabilize the electrostatic actuation commonly used in Micro-Electro-Mechanical Systems (MEMS). Most electrostatically actuated MEMS devices act as variable capacitors with varying gap between the charged conductors. Electrostatic force in these devices is a nonlinear attractive force between the conductors resulting in a complex dynamic system. These systems are stable for only a small portion of the initial gap. In this paper a design method is presented for electrostatic micro-mirrors with improved stability. Controllable, stable electrostatic actuation can be achieved through surface contact between the two conductors. Once in contact with the surface, the compliance of the structure is used to stabilize the electrostatic actuation over a long range of motion. Beam based variable angle mirrors were designed and fabricated using the Multi-User MEMS Process at MCNC technology center. The design methods for stable electrostatic actuation were tested on these mirrors. Some characteristics are noted and their implementation into future designs is discussed.

Sign in / Sign up

Export Citation Format

Share Document