Mode Localization Phenomenon Research Articles

Modal dynamics of boundary-interior coupled structures. Part 1: A general approach using components Green’s function

In Part 1 of this series papers (Part 1 & Part 2 (Guo and Rega, 2020) [1]), a general operator-based boundary analysis approach is proposed for free vibrations of coupled structural systems, with the new coupled modal frequencies and modal shapes being directly extracted in a systematic manner from the uncoupled components Green’s functions. Therefore, ad-hoc re-modelling of the whole augmented structural system is avoided. It turns out that a hybrid decomposition of the original coupled system into two sub-problems at the coupling interface, i.e., one excited kinematically at the boundary and the other excited by an interaction force in its interior domain, is of vital importance. Using general operator arguments and reciprocity-based operator equality, the dynamic coupling is imposed by combining the boundary kinematic effects and interaction force effects. The orthogonality of the coupled modal shapes is further established (and in Part 2 (Guo and Rega, 2020) [1] a general full interpretation of mode localization phenomenon will be presented). Finally, two coupled structural systems, i.e., string-oscillator and string-beam, are attacked by this operator formulation as examples, with coupled frequencies/modal shapes being obtained in a systematic way.

Theoretical and experimental investigation of mode localization in electrostatically and mechanically coupled microbeam resonators

The phenomenon of mode localization is explored theoretically and experimentally on two mechanically or electrostatically coupled beam resonators. Lumped parameter models are used to simulate the response of the systems. The eigenvalue problems are solved for both case studies under different stiffness perturbations and coupling strengths. The influence of the side electrode bias on the veering points is also explored. The dynamics of the systems are studied and compared using their frequency response curves under different perturbation and damping scenarios. The effect of damping for different elements of the coupled system is studied and proposed to improve sensitivity in high damping environments. It is observed that the exploitation of mode localization depends primarily on the choice of the resonator of the coupled system to be under direct excitation, its stiffness to be perturbed, and its response to be monitored. The revealed dynamic behaviors show great potential for applications in sensing and mechanical computing. The theoretical findings are validated using experimental case studies of two silicon doubly-clamped mechanically coupled microbeams and two electrostatically coupled microcantilevers. Electrothermal voltage is applied to the mechanically coupled resonators to introduce the stiffness perturbation in order to investigate mode localization. The theoretical and experimental results show good qualitative agreement.

Effect of Noise and Nonlinearities on Thermoacoustics of Can-Annular Combustors

Abstract Can-annular combustors consist of N distinct cans setup symmetrically around the axis of the gas turbine. Each can is connected to the turbine inlet by means of a transition duct. At the turbine inlet, a small gap between the neighboring transition ducts allows acoustic communication between the cans. Thermoacoustic pulsations in the cans are driven by the respective flames, but also the communication between neighboring cans through the gap plays a significant role. In this study, we focus on the effect of the background noise intensity and of the nonlinear flame saturation. We predict how usually clusters of thermoacoustic modes are unstable in the linear regime and compete with each other in the nonlinear regime, each cluster consisting of axial, azimuthal and push-pull modes. Since linear theory cannot predict the nonlinear solution, stochastic simulations are run to study the nonlinear solution in a probabilistic sense. One outcome of these simulations is the various pulsation patterns, which are in principle different from one can to the next. We recover how not only a stronger flame response in one can gives rise to the phenomenon of mode localization, but also how the nonlinearity of the flame saturation and the competition between modes have an effect on the nonlinear mode shape. We finally predict the coherence and phase between cans on the linearized system subject to noise, and compare the predictions with engine measurements, in terms of spectra of amplitude in each can and coherence and phase, observing a good match.

A Sensitivity Tunable Accelerometer Based on Series-Parallel Electromechanically Coupled Resonators Using Mode Localization

This paper reports a novel mode-localized resonant accelerometer, which can keep high sensitivity even over a wide range. To improve the adjustability of sensitivity, a new four degree of freedom (4-DoF) series-parallel resonator array is proposed. Three sensing resonators are mechanically coupled together in series by folding beams and the fourth sensitivity-tuning resonator is electrically coupled in parallel with the sensing system. When the stiffness perturbation caused by acceleration occurs, a change of modal amplitude ratio due to the mode localization phenomenon can be detected. By tuning the voltage on the sensitivity-tuning resonator, the sensitivity of the accelerometer can be altered to keep the amplitude ratio within a reasonable range. The theoretical model of the accelerometer is established and analyzed by numerical method. To confirm the feasibility of the design, a device fabricated by SOI-MEMS technology is tested under open-loop circuit. The measured amplitude ratio sensitivity varies from 1.14/g to 23.37/g with the change of the tuning voltage, and the sensitivity adjustment range reaches 2050%. [2019-0199]

Calibration of mass sensors for surface roughness of their micro-resonators

In this study, the interplay between the sensitivity of mass sensors and the mode localization phenomenon of their resonators due to surface roughness is interpreted. A novel model of a micro-resonator with rough surfaces is developed. This model is used to calibrate mass sensors for the surface roughness of their micro-resonators. It is revealed that the resonance frequency of the micro-resonator either increases or decreases due to surface roughness. On one hand, the increase in the frequency is due to a mode localization where, because of the surface roughness, the vibration energy is confined over a small portion of the resonator’s length. On the other hand, the decrease in the frequency is due to a globalization of the vibration energy where it distributes over an effective length longer than the resonator’s length. The localization of the vibration energy enhances the sensitivity of mass sensors while the globalization of the vibration energy degrades the sensitivity of mass sensors.

Axially functionally graded beams and panels in supersonic airflow and their excellent capability for passive flutter suppression

Mode localization in the airflow direction can be observed in aeroelastic structures. On the contrary, can the aeroelastic stability of the structure be enhanced if the geometric sizes and material properties vary along the airflow direction? The present study mainly solves this problem. The mode localization phenomenon in panel flutter in supersonic airflow is displayed. Three arbitrary thickness functions according to the flutter mode are taken into account. Their panel flutter behaviors show that although the consistency between the variation of thickness and flutter mode can increase the flutter bound slightly, the increments are limited. Consequently, a novel strategy for passive control of the panel flutter is proposed by the optimal axially functionally graded (AFG) design of the panel. By investigating the sensitivity of each element in the aerodynamic stiffness matrix to the aeroelastic stability of the structure, the optimal thickness and Young's modulus functions are given out. Simulation results show that the optimal AFG design in this study can suppress the flutter essentially. It can increase the flutter bound of the structure by changing the flutter modes rather than only makes a slight extension on the original basis. Moreover, the designed thickness and Young's modulus are reasonable and applicable.

Mass sensor using mode localization in two weakly coupled MEMS cantilevers with different lengths: Design and experimental model validation

This paper presents a sensor using the mode localization phenomenon to detect a mass perturbation. It is composed of two cantilevers with different lengths and connected by a coupling beam. The short cantilever is electrostatically actuated and by changing the applied DC voltage, we can reduce its stiffness and reach the veering point, which corresponds to a balanced system. This principle allows us to overcome the manufacturing defect which perturbs the initial system. An analytical model using the Euler-Bernoulli beam theory is developed for the design. The equation of the continuous system is discretized with the Galerkin method and simulations are performed. The designed device composed of polysilicon coupled microbeams is then fabricated with the Multi-User MEMS Processes and an experimental investigation is carried out. Three devices with different coupling are considered with a length ratio of 0.98. This ratio is suitable to reach the veering point by using a DC balancing voltage around the half of the pull-in voltage. The comparison between theoretical and experimental results shows a good agreement for each device.

Mode localization phenomenon of functionally graded nanobeams due to surface integrity

This is the first study on the mode localization and surface stress-based deflection phenomena of functionally graded (FG) nanobeams due to surface integrity. A new model for FG nanobeams with engineering surfaces is developed. The engineering surface is considered a different material phase with a surface texture (i.e. waviness and roughness). The initial curvatures of cantilever, simple supported, and clamped–clamped FG nanobeams due to surface residual stresses are determined. It is revealed that the initial curvature increases with an increase in the slope of the surface texture and/or a decrease in the surface roughness. Moreover, the natural frequencies and mode shapes of FG nanobeams are derived depending on the surface’s texture and mechanical properties. It is observed that natural frequencies of FG beams may decrease or increase due surface roughness. Thus, as a first prospect, the surface roughness allows the vibration energy to propagate over the beam length and hence its natural frequency decreases resulting in a zero-frequency mode. As for the other prospect, surface roughness inhibits the propagation of the vibration energy through the beam length leading to a mode localization and an increase in the natural frequency.

Suppression on Nonlinearity of Mode-Localized Sensors Using Algebraic Summation of Amplitude Ratios as the Output Metric

Compared with the classic frequency-modulated resonant sensors, the recently emerging sensing scheme based on the mode localization phenomenon of the weakly coupled resonators provides the improved sensitivity by more than 2 orders of magnitude using amplitude ratio as the output metric. However, the inherent loci veeringphenomenon of the amplitude ratio over stiffness perturbation curvegenerates strong or weak nonlinearities in different zones, severelylimiting the linear sensing ranges. In this paper, we propose to use the algebraic summation of the amplitude ratios of two vibrational modes for eliminating the nonlinear errors caused by loci veering, extending the linear sensing range to the fullscale. This new output metric was verified as being very effective to suppress the nonlinear errors using two mode-localized stiffness sensors with 2 and 3 degrees of freedom coupled resonators. This is the first theoretically and experimentally demonstrated full-scale linear output metric for mode-localized sensors, which will push forward more applications of the mode-localized sensing scheme.

Vibroacoustic study of a point-constrained plate mounted in a duct

The vibroacoustic study of the interaction of sound with a point-constrained, simply-supported square plate is considered in this paper. The plate is mounted flush on one of the walls of an infinite duct of rectangular cross section and is backed by a cavity. The plate response and the acoustic field is predicted by solving the coupled governing equations using modal expansion with the relevant eigenmodes of the plate dynamics and acoustic fields in the duct and cavity. By varying the location of the point constraint, the frequency characteristics of the transmission loss in the duct can be tuned. The point constraint can also alter the amplitude and spectral characteristics of the plate's response. Interestingly, some new peaks are observed in the response because of the excitation of unsymmetric modes which are otherwise dormant. Mode-localization phenomenon, which is the localization of vibration in specific regions of the plate, is observed for selected constrained points.

Characterization of forced localization of disordered weakly coupled micromechanical resonators

The mode localization phenomenon of disordered weakly coupled resonators (WCRs) is being used as a novel transduction scheme to further enhance the sensitivity of micromechanical resonant sensors. In this paper, two novel characteristics of mode localization are described. First, we found that the anti-resonance loci behave as a linear function of the stiffness perturbation. The anti-resonance behavior can be regarded as a new manifestation of mode localization in the frequency domain, and mode localization occurs at a deeper level as the anti-resonance approaches closer to the resonance. The anti-resonance loci can be used to identify the symmetry of the WCRs and the locations of the perturbation. Second, by comparing the forced localization responses of the WCRs under both the single-resonator-driven (SRD) scheme and the double-resonator-driven (DRD) scheme, we demonstrated that the DRD scheme extends the linear measurement scale while sacrificing a certain amount of sensitivity. We also demonstrated experimentally that the amplitude ratio-based sensitivity under the DRD scheme is approximately an order of magnitude lower than that under the SRD scheme, that is, the amplitude ratio-based sensitivity is −70.44% (N m−1)−1 under the DRD scheme, while it is −785.6% (N m−1)−1 under the SRD scheme. These characteristics of mode localization are valuable for the design and control of WCR-based sensors.

A High-Sensitivity Micromechanical Electrometer Based on Mode Localization of Two Degree-of-Freedom Weakly Coupled Resonators

This paper reports a high-sensitivity resonant electrometer based on the mode localization of two degree-of-freedom weakly coupled resonators (WCRs). When charges are applied to the input electrodes, the effective stiffness of a specific resonator of the WCRs will be perturbed, leading to a drastic change in mode shape owing to the mode localization phenomenon. By measuring the shift of the amplitude ratio, the small charge fluctuation can be accurately sensed. The theoretical mode of the electrometer is established based on the transfer functions of the WCRs. In particular, we establish the design rules of the coupling factor according to the −3-dB bandwidth, amplitude ratio measurement errors, and frequency misalignment between the resonators. The experimental results show that the amplitude ratio-based sensitivity is $\sim 2151$ times higher than the frequency-based sensitivity. The amplitude ratio-based resolution of the electrometer is approximately 1.269 fC . [2016-0119]

An Acceleration Sensing Method Based on the Mode Localization of Weakly Coupled Resonators

This paper reports an acceleration sensing method based on two weakly coupled resonators (WCRs) using the phenomenon of mode localization. When acceleration acts on the proof masses, differential electrostatic stiffness perturbations will be applied to the WCRs, leading to mode localization, and thus, mode shape changes. Therefore, acceleration can be sensed by measuring the amplitude ratio shift. The proposed mode localization with the differential perturbation method leads to a sensitivity enhancement of a factor of 2 than the common single perturbation method. The theoretical model of the sensitivity, bandwidth, and linearity of the accelerometer is established and verified. The measured relative shift in amplitude ratio ( $\sim 312162$ ppm/g) is 302 times higher than the shift in resonance frequency ( $\sim 1035$ ppm/g) within the measurement range of ±1 g. The measured resolution based on the amplitude ratio is 0.619 mg and the nonlinearity is $\sim 3.5$ % in the open-loop measurement operation. [2015-0247]

Quantification of Vibration Localization in Periodic Structures

The phenomenon of vibration mode localization in periodic and near periodic structures has been well documented over the past four decades. In spite of its long history, and presence in a wide range of engineering structures, the approach to detect mode localization remains rather rudimentary in nature. The primary way is via a visual inspection of the mode shapes. For systems with complex geometry, the judgment of mode localization can become subjective as it would depend on visual ability and interpretation of the analyst. This paper suggests a numerical approach using the modal data to quantify mode localization by utilizing the modal assurance criterion (MAC) across all the modes due to changes in some system parameters. The proposed MAC localization factor (MACLF) gives a value between 0 and 1 and therefore gives an explicit value for the degree of mode localization. First-order sensitivity based approaches are proposed to reduce the computational effort. A two-degree-of-freedom system is first used to demonstrate the applicability of the proposed approach. The finite element method (FEM) was used to study two progressively complex systems, namely, a coupled two-cantilever beam system and an idealized turbine blade. Modal data is corrupted by random noise to simulate robustness when applying the MACLF to experimental data to quantify the degree of localization. Extensive numerical results have been given to illustrate the applicability of the proposed approach.

Advances of Research on Mode Localization in Mistuned Cyclically Periodic Structures

The mode localization phenomenon in mistuned cyclically periodic structures has been a hot spot in the past few decades. According to numbers of documents, the fundamental concepts, mechanism, main research methods, and the main results of mode localization are introduced; the current research status of the mode localization is summarized. The possiblity of mode localization in a telescope reflector model has been analyzed by using Finite Element Method. We also illustrate the problems for further studies, including: the unified modeling and analysis theory with more extensive applicability and better analytical accuracy; determining how to properly introduce mistuning parameters that can reflect the true state of the structure; constructing mode localization factor which can sensitively and effectively reflect the effect of mode localization; developing new damage identification method considering the influence of mode localization; how to make use of the localization characteristic to control the vibration level of the important sub-structure and improve the sensitivity of the periodic structure to the random detuning.

Physical consequences of a nonparametric uncertainty model in structural dynamics

One of the main claims of the nonparametric model of random uncertainty introduced by Soize (2000) [3] is its ability to account for model uncertainty. The present paper investigates this claim by examining the statistics of natural frequencies, total energy and underlying dispersion equation yielded by the nonparametric approach for two simple systems: a thin plate in bending and a one-dimensional finite periodic mass–spring chain. Results for the plate show that the average modal density and the underlying dispersion equation of the structure are gradually and systematically altered with increasing uncertainty. The findings for the mass–spring chain corroborate the findings for the plate and show that the remote coupling of nonadjacent degrees of freedom induced by the approach suppresses the phenomenon of mode localization. This remote coupling also leads to an instantaneous response of all points in the chain when one mass is excited. In the light of these results, it is argued that the nonparametric approach can deal with a certain type of model uncertainty, in this case the presence of unknown terms of higher or lower order in the governing differential equation, but that certain expectations about the system such as the average modal density may conflict with these results.

Mode localization in composite laminates

We study free vibrations of monolithic and composite thin rectangular plates; the former are made of linear elastic, homogeneous, isotropic materials and the latter of fiber reinforced laminas. The plates are clamped on all four edges and interior points on a transverse normal to the plate midsurface are rigidly tied together and have either null displacements and null rotations (Type-I constraint) or only null transverse displacements (Type-II constraint). Depending upon the location of the point on the midsurface through which the transverse normal passes, modes localize in different regions of the plate. Plates of various aspect ratios (length/width) and stacking sequences of 0°, 45° and 90° leading to symmetric and anti-symmetric configurations about their midsurfaces are considered. The problem is studied using the first order shear deformable (or the Mindlin) plate theory. It is found that both the Type-I and Type-II constraints divide the plate into two vibrating regions with amplitudes of transverse vibration localized in a particular region on either side of the clamped interior points. It is found that the mode localization in laminates is governed by the mode localization characteristics of constituent laminas. For symmetric cross ply laminates the localization of modes is found to decrease with the increase in the number of 0° plies. For anti-symmetric cross ply laminates and those made of all 45° plies the mode localization is found to be independent of the number of plies. For isotropic plates made of a monolithic material the mode localization phenomenon is stronger for Type-I constraint compared to that for Type-II constraints. Also, for these plates the mode localization occurs when lumped masses are placed at these interior points. The significance of the work lies in providing an alternative and an economical way of annulling plate vibrations in selected parts of the plate, and confining the energy of vibration in desired regions of the plate.

Mode localization phenomena and stability of disordered trains and train-like articulated systems travelling in confined fluid

The possible existence of mode localization in a chain of flexibly interconnected cylinders, representing a high-speed train travelling in a tunnel is investigated, in the presence of some disorder in the chain. Specifically, imperfections in the mass and springs in the model are considered as possible sources of mode localization, and it is shown that in general they can do so. This is part of a broader study into the dynamics of high-speed trains running in a tunnel, or more generally of a train-like system travelling in a coaxial cylindrical tube and subjected to aerodynamic forces associated with lateral motions of the cylinders. Each cylinder in the model is coupled to the other cylinders and is supported by springs, such that it has both translational and rotational degrees of freedom. Two models are used: a lumped-parameter Euler–Bernoulli beam model (LEB) and a lumped-parameter Timoshenko-beam model (LTB). The results of this study show that Timoshenko beam effects of rotational inertia and shear deformation have a considerable influence on the mode localization phenomena and that imperfections in the supporting springs have a considerable influence on the stability of the LTB model.

Identification of local damages in coupled beam systems from measured dynamic responses

This paper deals with local damages identification for strongly and weakly coupled beam systems with close and repeated natural frequencies based on the measured dynamic responses of the systems under external moving forces. The dynamic responses of the coupled beam systems are calculated from Newmark integration method and they are used for structural damage detection. The mode localization phenomenon due to local damage(s) in the weakly coupled beam system is studied. In the inverse analysis, a dynamic response sensitivity-based finite element model updating method is employed for the detection of local damage(s) in the coupled beam systems. Numerical simulation shows that the proposed method is effective in identifying the damages in the coupled beam systems with good accuracy from several measurements. It is found the proposed method is insensitive to artificial measurement noise and has the potential for practical applications.

Transition from normal to local modes in an elastic beam supported by nonlinear springs

The transient mode localization phenomenon is considered in a mechanical model combining a simply supported beam and transverse nonlinear springs with hardening characteristics. Two different approaches to the model reduction, such as normal and local mode representations for the beam's center line, are discussed. It is concluded that the local mode discretization brings advantages for the transient localization analysis. Based on the specific coordinate transformations and the idea of averaging, explicit equations describing the energy exchange between the local modes and the corresponding localization conditions are obtained. It was shown that when the energy is slowly pumped into the system then, at some point, the energy equipartition around the system suddenly breaks and one of the local modes becomes the dominant energy receiver. The phenomenon is interpreted in terms of the related phase-plane diagram which shows qualitative changes near the image of the out-of-phase mode as the total energy of the system has reached its critical level. A simple closed form expression is obtained for the corresponding critical time estimate.