Vibration Mode Localization in Randomly Disordered Weakly Coupled Two-Dimensional Systems

1997 ◽  
Author(s):  
Wei-Chau Xie
Keyword(s):  
Eigenvalue Problems ◽  
Vibration Mode ◽  
Numerical Approach ◽  
Logarithmic Scale ◽  
Two Dimensional ◽  
Regular Perturbation ◽  
Mode Localization ◽  
Weakly Coupled ◽  
Exponential Rates ◽  
Rates Of Decay

Abstract In this paper, a general method of regular perturbation for linear eigenvalue problems is presented, in which the orders of perturbation terms are extended to infinity. The method of regular perturbation is applied to study vibration mode localization in randomly disordered weakly coupled two-dimensional cantilever-spring arrays. Localization factors, which characterize the average exponential rates of decay or growth of the amplitudes of vibration, are defined in terms of the angles of orientation. First-order approximate results of the localization factors are obtained using a combined analytical-numerical approach. For the systems under consideration, the direction in which vibration is originated corresponds to the smallest localization factor; whereas the “diagonal” directions correspond to the largest rate of decay or growth of the amplitudes of vibration. When plotted in the logarithmic scale, the vibration modes are of a hill shape with the amplitudes of vibration decaying linearly away from the cantilever at which vibration is originated.

Localization of Vibration Propagation in Two-Dimensional Systems With Multiple Substructural Modes

2003 ◽  
Vol 70 (1) ◽  
pp. 119-128
Author(s):  
W.-C. Xie
Keyword(s):  
Two Dimensional ◽  
Regular Perturbation ◽  
Matrix Formulation ◽  
One Dimensional ◽  
Weakly Coupled ◽  
Exponential Rates ◽  
Rates Of Growth ◽  
The Given ◽  
Localization Behavior ◽  
Localization Of Vibration

Localization of vibration propagation in randomly disordered weakly coupled two-dimensional cantilever-mesh-spring arrays, in which multiple substructural modes are considered for each cantilever, is studied in this paper. A method of regular perturbation for a linear algebraic system is applied to determine the localization factors, which are defined in terms of the angles of orientation and characterize the average exponential rates of growth or decay of the amplitudes of vibration in the given directions. Iterative formulations are derived to determine the amplitudes of vibration of the cantilevers. In the diagonal directions, a transfer matrix formulation is obtained. For a given direction of orientation, the localization behavior is similar to that of a one-dimensional cantilever-spring-mesh chain. The effect of the stiffnesses and the disorder in the stiffnesses of the cantilevers on the localization behavior of the system is investigated.

Vibration Mode Localization in Two-Dimensional Systems

AIAA Journal ◽  
10.2514/2.5 ◽  
1997 ◽  
Vol 35 (10) ◽  
pp. 1653-1659 ◽  
Author(s):  
Wei-Chau Xie ◽  
Xing Wang
Keyword(s):  
Vibration Mode ◽  
Two Dimensional ◽  
Mode Localization

Vibration mode localization in two-dimensional systems

AIAA Journal ◽  
1997 ◽  
Vol 35 ◽  
pp. 1653-1659 ◽  
Author(s):  
Wei-Chau Xie ◽  
Xing Wang
Keyword(s):  
Vibration Mode ◽  
Two Dimensional ◽  
Mode Localization

scholarly journals A Low-g MEMS Accelerometer with High Sensitivity, Low Nonlinearity and Large Dynamic Range Based on Mode-Localization of 3-DoF Weakly Coupled Resonators

Micromachines ◽  
2021 ◽  
Vol 12 (3) ◽  
pp. 310
Author(s):  
Muhammad Mubasher Saleem ◽  
Shayaan Saghir ◽  
Syed Ali Raza Bukhari ◽  
Amir Hamza ◽  
Rana Iqtidar Shakoor ◽  
...  
Keyword(s):  
Microelectromechanical Systems ◽  
Dynamic Range ◽  
Proof Mass ◽  
Amplitude Ratio ◽  
Silicon On Insulator ◽  
Coupled Resonators ◽  
Mode Localization ◽  
Mems Accelerometer ◽  
Weakly Coupled ◽  
Input Acceleration

This paper presents a new design of microelectromechanical systems (MEMS) based low-g accelerometer utilizing mode-localization effect in the three degree-of-freedom (3-DoF) weakly coupled MEMS resonators. Two sets of the 3-DoF mechanically coupled resonators are used on either side of the single proof mass and difference in the amplitude ratio of two resonator sets is considered as an output metric for the input acceleration measurement. The proof mass is electrostatically coupled to the perturbation resonators and for the sensitivity and input dynamic range tuning of MEMS accelerometer, electrostatic electrodes are used with each resonator in two sets of 3-DoF coupled resonators. The MEMS accelerometer is designed considering the foundry process constraints of silicon-on-insulator multi-user MEMS processes (SOIMUMPs). The performance of the MEMS accelerometer is analyzed through finite-element-method (FEM) based simulations. The sensitivity of the MEMS accelerometer in terms of amplitude ratio difference is obtained as 10.61/g for an input acceleration range of ±2 g with thermomechanical noise based resolution of 0.22 and nonlinearity less than 0.5%.

Moment Lyapunov Exponents of a Two-Dimensional Viscoelastic System Under Bounded Noise Excitation

2002 ◽  
Vol 69 (3) ◽  
pp. 346-357 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Stochastic Fluctuation ◽  
Viscoelastic System ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered.

scholarly journals Perturbation of eigenvalues due to gaps in two-dimensional boundaries

2006 ◽  
Vol 463 (2079) ◽  
pp. 759-786 ◽  
Author(s):  
Anthony M.J Davis ◽  
Stefan G Llewellyn Smith
Keyword(s):  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Neumann Boundary ◽  
Diffusion Problem ◽  
Length Scale ◽  
Two Dimensional ◽  
Constructive Approach ◽  
Term Outcome ◽  
Long Term Outcome

Motivated by problems involving diffusion through small gaps, we revisit two-dimensional eigenvalue problems with localized perturbations to Neumann boundary conditions. We recover the known result that the gravest eigenvalue is O (|ln  ϵ | −1 ), where ϵ is the ratio of the size of the hole to the length-scale of the domain, and provide a simple and constructive approach for summing the inverse logarithm terms and obtaining further corrections. Comparisons with numerical solutions obtained for special geometries, both for the Dirichlet ‘patch problem’ where the perturbation to the boundary consists of a different boundary condition and for the gap problem, confirm that this approach is a simple way of obtaining an accurate value for the gravest eigenvalue and hence the long-term outcome of the underlying diffusion problem.

Sign in / Sign up

Export Citation Format

Share Document