Modal Overlap Factor Research Articles

Analysis of the perceived intensity of vibratory stimuli driven by the Modal Overlap Factor

The vibro-tactile perception experiment reported in this paper aims to investigate correlations between the perceived vibratory intensity and the Modal Overlap Factor (MOF) of vibratory stimuli. The MOF is often used to physically characterise the vibratory behavior of a structure and can be estimated from the product of frequency, modal density and modal loss factor. In this experiment, stimuli are applied to the participant's hands by means of a steering wheel set in vibration by a shaker. The vibratory stimuli are synthesized from filtered white noise convolved with an impulse response defined as a sum of damped sine waves. The MOF can be controlled by selecting the number of sine waves, their frequency and damping ratio. A set of 100 stimuli, with MOF values ranging from 2 to 200%, in the 5-305Hz range, and equalized in acceleration, is used. Each of the 80 participants is asked to rate twice the perceived intensity of 10 stimuli, pseudo-randomly chosen among the 100 stimuli. Data are analyzed to establish trends between perceived vibratory intensity and the physical parameters of the stimuli.

Second-harmonic generation based on double higher-order topological corner states

Topological corner state provides a promising platform for enhancing nonlinear optical effects, since it enables the light highly localized and is topologically protected against defects. Here, we realize second-harmonic generation (SHG) based on the nonlinear interactions between two different type corner modes. These corner modes have ultrahigh Q factors and large modal overlap factor, and match the doubly resonant condition simultaneously, ensuring SHG with high efficiency. These results may expand our understanding of the nonlinear interaction between higher-order corner modes in a more general framework, and can find applications in the fields of topologically-protected imaging and sensing.

Understanding the material loss of anti-resonant hollow-core fibers.

In this paper, the material loss of anti-resonant hollow-core fiber (AR-HCF) and its properties are studied. We revisit the formula of power attenuation coefficient for the index-guiding optical fiber described by Snyder and Love in the 1980s and derive the modal overlap factor that governs the material loss of hollow-core fibers (HCF). The modal overlap factor formula predicts the material loss of AR-HCF, which agrees with numerical simulations by the finite element method. The optimization of silica-based AR-HCF design for the lowest loss at 4 µm wavelength is numerically discussed where the silica absorption reaches over 800 dB/m. Our work would provide practical guidance to develop low-loss AR-HCF at highly absorptive wavelengths, e.g. in the vacuum UV and mid/far-infrared spectral regions.

On the diffuseness of the vibrational field of a cross-laminated timber plate: Comparison between theoretical and experimental methods

The modelling of vibro-acoustic properties of plates usually requires the hypothesis of diffuse conditions of the vibrational field. For heavy-weight homogeneous isotropic plates, this condition is met starting from low frequencies. Nevertheless, the same hypotheses do not hold when applied to stiff, inhomogeneous and non-isotropic materials, like wood, thus its modal behavior needs to be addressed with more detail. The aim of the work is to benchmark theoretical methods and experimental measurements that allow to estimate when the transition from modal to diffuse field occurs in Cross Laminated Timber (CLT) elements. For this purpose, the diffuseness of the vibrational field has been investigated through the experimental evaluation of the mode count and the modal overlap factor on a pilot CLT panel. Measurements of the spatial distribution of modes, impulse responses, dispersion relations and driving point admittance were conducted in laboratory on an 8 m2 CLT plate. The analysis of the results allowed, on the one hand, to directly estimate the two parameters, and on the other, to provide input data necessary to evaluate and compare different theoretical models. Results of direct measurements, performed up to 550 Hz, show that the number of modes occurring in one-third octave bands exceeds the standard threshold of 5 only at 500 Hz, whereas the modal overlap factor is lower than the standard threshold of 1 for most of the modes. This indicates that the vibrational field is mainly non-diffuse in the mid-low frequency range, due to the small number of modes combined with a low damping of the CLT plate. Nonetheless, measurements of modal reverberation time and structural reverberation time in one-third octave band provided consistent results, implying that modal decays are the major contributors to the one-third octave band decay. Furthermore, comparisons with theoretical models show that the driving point admittance proved to be the best method of analysis, requiring simple measurement setup and signal processing, whereas the ISO 10848-4 is affected by the thin plate hypothesis and underestimates the mode count at mid-high frequencies.

Mid-to-high frequency piecewise modelling of an acoustic system with varying coupling strength

Mid-to-high frequency vibro-acoustic modelling has always been a challenging topic. The previous study shows the promise of a piecewise calculation scheme based on the Condensed Transfer Function (CTF) approach in a frequency range where the modal overlap factor roughly exceeds one. The piecewise scheme has shown its capability in terms of balancing the accuracy, efficiency and the wealth of information for the modeling of lightly coupled vibroacoustic systems. This paper extends the method beyond the weak-coupling assumption. A coupling strength factor is first proposed to quantify and adjust the coupling strength between two sub-systems. Two mutually connected sub-cavities are then used as an example to validate the piecewise scheme in relation to the changes in the coupling strength level, as well as the variations in the coupling strength factor itself. The effect of the coupling strength on the computational error of the piecewise scheme is systematically studied. Finally, the applicability of the piecewise calculation scheme is experimentally validated.

Dependence of Waveguide Properties of Anti-Resonant Hollow-Core Fiber on Refractive Index of Cladding Material

Anti-resonant hollow-core fiber is featured by the broadband and low-loss transmission of light in the hollow core thanks to the optical property of anti-resonance cladding design. In this paper, we numerically explore the antiresonance of cladding from a new angle and confirm unique and important roles of refractive index of cladding material played in the confinement, bend and material losses of modes in anti-resonant hollow-core fibers. The numerical simulations demonstrate the dependence of confinement loss in anti-resonant hollow-core fiber on the cladding material refractive index which differs from the capillary waveguide and 1.5 refractive index is found favorable to achieve the lowest confinement loss ever in anti-resonant hollow-core fiber. Meanwhile, higher material refractive index can more effectively mitigate the material absorption and consequent contribution to the overall fiber loss in the anti-resonant hollow-core fiber than in the tube waveguide, by reducing the modal overlap factor. Besides, to the best of our knowledge, it is for the first time to discover that the modal dispersion of anti-resonant hollow-core fiber can be effectively altered by the cladding material refractive index, which was usually regarded to be determined by the core dimension and filled-in gas only. The refractive index around 1.8 is preferred to achieve a flattened profile of group dispersion over the transmission spectral window in anti-resonant hollow-core fibers. Our results bring in a new perspective to investigate the guidance mechanism of anti-resonant hollow-core fibers and provides guidance for material selection in the fiber design for optimization of waveguide properties.

On-Chip Photonic Crystal Surface-Emitting Membrane Lasers

Photonic crystal lasers can be realized either based on photonic bandgap defect mode or defect-free bandedge mode, while the bandgap is not essential for the latter. We review here defect-free bandedge mode based photonic crystal surface-emitting lasers (PCSELs) for on-chip integration. We first discuss ultra-thin membrane reflector vertical-cavity surface-emitting lasers (MR-VCSELs), where single layer photonic crystal slabs can be designed as a broadband membrane reflector. Later, we discuss another type of defect-free PCSELs where the lasing cavity is formed based on evanescent coupling of gain medium with the photonic crystal bandedge mode near bandedge. Cavity designs were carried out for the optimal modal overlap and high confinement factors. Lateral cavity size scaling was also investigated both theoretically and experimentally in PCSELs. Buried tunnel junction based InGaAsP quantum well heterostructures were also designed and incorporated into electrically injected PCSELs. Finally, discussions are given toward energy efficient lasers.

On modal cross-coupling in the asymptotic modal limit

The conditions under which significant modal cross-coupling occurs in dynamical systems responding to high-frequency, broadband forcing that excites many modes is studied. The modal overlap factor plays a key role in the analysis of these systems as the modal density (the ratio of number of modes to the frequency bandwidth) becomes large. The modal overlap factor is effectively the ratio of the width of a resonant peak (the damping ratio times the resonant frequency) to the average frequency interval between resonant peaks (or rather, the inverse of the modal density). It is shown that this parameter largely determines whether substantial modal cross-coupling occurs in a given system's response. Here, two prototypical systems are considered. The first is a simple rectangular plate whose significant modal cross-coupling is the exception rather than the norm. The second is a pair of rectangular plates attached at a point where significant modal cross-coupling is more likely to occur. We show that, for certain cases of modal density and damping, non-negligible cross coupling occurs in both systems. Under similar circumstances, the constraint force between the two plates in the latter system becomes broadband. The implications of this for using Asymptotic Modal Analysis (AMA) in multi-component systems are discussed.

Shunted piezoelectric patch vibration absorber on two-dimensional thin structures: Tuning considerations

This paper investigates the flexural vibration control effects produced on a distributed two-dimensional thin structure by an electrically shunted piezoelectric patch tuned vibration absorber. The study highlights how the high modal overlap factors that characterise the response of thin two-dimensional distributed structures and the electro-mechanical, capacitive, stiffening and inertial, inherent properties of the piezoelectric patch transducer substantially influence the tuning of the electrical shunt and consequently the vibration control effects produced at frequencies near a given resonance frequency of the structure. In particular, the study shows that the classical approach considering a simplified two degrees of freedom equivalent model of the structure and shunted piezoelectric patch is inadequate to identify the optimal tuning parameters of the shunt. The study, also provides general guidelines for the dimensioning of the piezoelectric patch transducer and electrical shunt in order to maximise the electro-mechanical vibration absorption.

Single-mode instability in standing-wave lasers: The quantum cascade laser as a self-pumped parametric oscillator

We report the observation of a clear single-mode instability threshold in continuous-wave Fabry-Perot quantum cascade lasers (QCLs). The instability is characterized by the appearance of sidebands separated by tens of free spectral ranges (FSR) from the first lasing mode, at a pump current not much higher than the lasing threshold. As the current is increased, higher-order sidebands appear that preserve the initial spacing, and the spectra are suggestive of harmonically phase-locked waveforms. We present a theory of the instability that applies to all homogeneously broadened standing-wave lasers. The low instability threshold and the large sideband spacing can be explained by the combination of an unclamped, incoherent Lorentzian gain due to the population grating, and a coherent parametric gain caused by temporal population pulsations that changes the spectral gain line shape. The parametric term suppresses the gain of sidebands whose separation is much smaller than the reciprocal gain recovery time, while enhancing the gain of more distant sidebands. The large gain recovery frequency of the QCL compared to the FSR is essential to observe this parametric effect, which is responsible for the multiple-FSR sideband separation. We predict that by tuning the strength of the incoherent gain contribution, for example by engineering the modal overlap factors and the carrier diffusion, both amplitude-modulated (AM) or frequency-modulated emission can be achieved from QCLs. We provide initial evidence of an AM waveform emitted by a QCL with highly asymmetric facet reflectivities, thereby opening a promising route to ultrashort pulse generation in the mid-infrared. Together, the experiments and theory clarify a deep connection between parametric oscillation in optically pumped microresonators and the single-mode instability of lasers, tying together literature from the last 60 years.

Bending, longitudinal and torsional wave transmission on Euler-Bernoulli and Timoshenko beams with high propagation losses.

Advanced Statistical Energy Analysis (ASEA) is used to predict vibration transmission across coupled beams which support multiple wave types up to high frequencies where Timoshenko theory is valid. Bending-longitudinal and bending-torsional models are considered for an L-junction and rectangular beam frame. Comparisons are made with measurements, Finite Element Methods (FEM) and Statistical Energy Analysis (SEA). When beams support at least two local modes for each wave type in a frequency band and the modal overlap factor is at least 0.1, measurements and FEM have relatively smooth curves. Agreement between measurements, FEM, and ASEA demonstrates that ASEA is able to predict high propagation losses which are not accounted for with SEA. These propagation losses tend to become more important at high frequencies with relatively high internal loss factors and can occur when there is more than one wave type. At such high frequencies, Timoshenko theory, rather than Euler-Bernoulli theory, is often required. Timoshenko theory is incorporated in ASEA and SEA using wave theory transmission coefficients derived assuming Euler-Bernoulli theory, but using Timoshenko group velocity when calculating coupling loss factors. The changeover between theories is appropriate above the frequency where there is a 26% difference between Euler-Bernoulli and Timoshenko group velocities.

Modal Overlap Factor of a beam with an acoustic black hole termination

An Acoustic Black Hole (ABH) effect is a passive vibration reduction technique which takes advantage of properties of wave propagation in thin structures of varying thickness. A practical implementation of ABH on a uniform beam consists in an extremity whose thickness follows a power-law profile covered by a very thin layer of additional damping material. A modal analysis based on a High Resolution technique shows that the ABH significantly increases the Modal Overlap Factor (MOF) of the beam, thus reducing the resonant behaviour of the structure. Further investigations, including a two-dimensional numerical model of the structure based on the finite difference method, show that this MOF can be explained by an increase of the modal density and a high damping of a number of modes of the structure due to the ABH.

Broadband external cavity tuning in the 3-4 μm window

In this work, a continuous external cavity spectral tuning of 556 cm−1 in the 3–4 μm region is shown for a heterogeneous bound-to-continuum active region design based on the strain-compensated Sb-free material system. The two active regions have center wavelengths at 3.3 μm and 3.7 μm with a total modal overlap factor of 86%. We also show an effective multilayer broadband anti-reflectivity coating in the 3–4 μm region reducing the overall reflectivity over the whole tuning range below 1.4%.

Estimation of mechanical properties of panels based on modal density and mean mobility measurements

The mechanical characteristics of wood panels used by instrument makers are related to numerous factors, including the nature of the wood or characteristic of the wood sample (direction of fibers, micro-structure nature). This leads to variations in Young's modulus, the mass density, and the damping coefficients. Existing methods for estimating these parameters are not suitable for instrument makers, mainly because of the need of expensive experimental setups, or complicated protocols, which are not adapted to a daily practice in a workshop. In this paper, a method for estimating Young's modulus, the mass density, and the modal loss factors of flat panels, requiring a few measurement points and an affordable experimental setup, is presented. It is based on the estimation of two characteristic quantities: the modal density and the mean mobility. The modal density is computed from the values of the modal frequencies estimated by the subspace method ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques), associated with the signal enumeration technique ESTER (ESTimation of ERror). This modal identification technique is proved to be robust in the low- and the mid-frequency domains, i.e. when the modal overlap factor does not exceed 1. The estimation of the modal parameters also enables the computation of the modal loss factor in the low- and the mid-frequency domains. An experimental fit with the theoretical expressions for the modal density and the mean mobility enables an accurate estimation of Young's modulus and the mass density of flat panels. A numerical and an experimental study show that the method is robust, and that it requires solely a few measurement points.

On the applicability of the lognormal distribution in random dynamical systems

This paper is concerned with the probability density function of the energy of a random dynamical system subjected to harmonic excitation. It is shown that if the natural frequencies and mode shapes of the system conform to the Gaussian Orthogonal Ensemble, then under common types of loading the distribution of the energy of the response is approximately lognormal, providing the modal overlap factor is high (typically greater than two). In contrast, it is shown that the response of a system with Poisson natural frequencies is not approximately lognormal. Numerical simulations are conducted on a plate system to validate the theoretical findings and good agreement is obtained. Simulations are also conducted on a system made from two plates connected with rotational springs to demonstrate that the theoretical findings can be extended to a built-up system. The work provides a theoretical justification of the commonly used empirical practice of assuming that the energy response of a random system is lognormal.

Analysis of subsystem randomness effects on the mid-frequency vibrations of built-up structures

The paper concerns the analysis of subsystem randomness effects on the mid-frequency vibration responses of built-up systems. The system model considered, in the first instance, is a long-wavelength finite element (FE) subsystem connected with a short-wavelength statistical energy analysis (SEA) subsystem via discrete couplings. The randomness effects of the SEA subsystem on both the displacement response of the FE subsystem and the energy response of the SEA subsystem are then investigated under the frame of the hybrid FE/SEA theory [P. Shorter, R. Langley, Vibro-acoustic analysis of complex systems, Journal of Sound and Vibration, 288 (2005) 669–700]. It is found that the subsystem randomness effects may be well indicated by a dimensionless parameter α, which is a function of the number of coupling points, the dynamic mismatch between the FE and SEA subsystems and the modal overlap factor of the SEA subsystem. The smaller the value of α is, the more insignificant the randomness effects are. As a result, a so-called “α-criterion” is derived which states that, if a built-up structure satisfies the condition of α⪡1, the randomness effects of the SEA subsystem can be neglected. In this case, the SEA subsystem can be simply treated as an infinite (or semi-infinite as appropriate) structure regardless of its mode count being sufficiently high or not. Numerical examples are presented to illustrate the validity of the present theory.

Effects of nearfield waves and phase information on the vibration analysis of curved beams

At high frequencies, energy methods such as the statistical energy analysis and the power flow analysis have been popularly used to predict the averaged responses of vibro-acoustic subsystems. Usually, these energy methods ignore flexural nearfield components and phase information, mainly for simplicity. Such assumptions sometimes lead to an erroneous conclusion, in particular for complex structures and at medium frequencies around the Schroeder cutoff frequency. This paper deals with the effects of nearfield waves and phase information at medium to high frequencies by using the ray tracing method (RTM). A curved beam and a coupled beam system were chosen as test examples, which exhibit the typical mode conversion between various types of travelling waves. Propagation of longitudinal, flexural, and torsional waves was studied based on the Euler-Bernoulli beam theory. Analyses of the spatial distribution of vibrational energy quantities revealed that the conventional RTM could mimic the overall trend of the traveling wave solution. However, the results varied smoothly in space due to the neglect of wave interference. By considering the phase information, local fluctuations of vibration energy could be correctly described. It was confirmed that the flexural nearfield plays a significant role near boundaries and junctions. It was also shown that the accuracy of the analysis depends mainly on the modal overlap factor. Similar to other high frequency methods, the results become close to the traveling wave solutions as the modal overlap factor increases.

Room requirements for laboratory measurement of transmission loss and impact sound insulation.

Laboratory measurement of transmission loss (ASTM E90) and impact sound insulation (ASTM E492) have one thing in common: measurement of the sound power radiated into the receive room by the specimen under test. In E90 the reported quantity, transmission loss (TL), is the difference between the incident and transmitted sound power, whereas for E492 the reported quantity, normalized impact sound pressure level (NISPL), is a direct measure of the radiated sound power of the floor. A derivation of the TL and NISPL equations using a statistical energy analysis (SEA) framework shows that the ASTM equations are based on the assumption of power balance and statistical sampling of the sound field. This SEA framework is used to define the acoustical conditions of the receive room (number of modes in each frequency band, the average loss factor in each band, and hence the modal overlap factor), which are then related to the physical properties such as room shape, volume, absorption, etc. The conditions necessary for...

A scaling procedure for the response of an isolated system with high modal overlap factor

The paper deals with a numerical approach that reduces some physical sizes of the solution domain to compute the dynamic response of an isolated system: it has been named Asymptotical Scaled Modal Analysis (ASMA). The proposed numerical procedure alters the input data needed to obtain the classic modal responses to increase the frequency band of validity of the discrete or continuous coordinates model through the definition of a proper scaling coefficient. It is demonstrated that the computational cost remains acceptable while the frequency range of analysis increases. Moreover, with reference to the flexural vibrations of a rectangular plate, the paper discusses the ASMA vs. the statistical energy analysis and the energy distribution approach. Some insights are also given about the limits of the scaling coefficient. Finally it is shown that the linear dynamic response, predicted with the scaling procedure, has the same quality and characteristics of the statistical energy analysis, but it can be useful when the system cannot be solved appropriately by the standard Statistical Energy Analysis (SEA).

The Vibration Reduction Index Kij: Laboratory Measurements for Rigid Junctions and for Junctions with Flexible Interlayers

The vibration reduction index Kij expresses the attenuation of the vibrational power flow through a junction. This quantity is important because it determines the contribution of the flanking transmission to the global sound transmission between rooms. It is used in acoustic building prediction models like that proposed in the standard EN 12354. Currently a draft exists to measure the Kij in laboratory. This draft, the prEN 10848, was prepared by the Technical Committee CEN/TC 126 and has been submitted for parallel enquiry. There are also prediction formulae available for the Kij, which are included in Annex E of the standard EN 12354-1 (2000). This article presents the Kij measured in a laboratory according to the draft prEN 10848 and documents an investigation in order to validate them for rigid junctions and junctions with a flexible interlayer. The observations focus on the comprehension and the quantification of the influence of some parameters like the modal overlap factor, the number of modes and the workmanship on the accuracy of the results. Furthermore, an extensive discussion tackles the problem of the determination of the parameter f1 used in the prediction formulae for junctions with flexible interlayers and the effect of the applied load.