Modal Density Research Articles

The characterization of dynamic behavior of Li-ion battery packs for enhanced design and states identification

The dynamic responses of a Li-ion battery pack deployed on hybrid electric vehicles are studied with a high fidelity finite element model and a parametric reduced-order model. The effects of microstructure transformation in the electrode materials caused by lithium-ion intercalation/deintercalation on the evolution of dynamic responses are investigated including the effects of the state of charge, aging, and cell-to-cell variations. The dynamic responses obtained from a finite element analysis show two interesting phenomena. First, the high modal density is controllable with the design modification of a pack component. Second, dynamic responses, especially the evolution of the natural frequencies of the fixed-boundary modes, of a Li-ion battery pack provide useful information to estimate the dynamic states or health states of the battery. A probabilistic analysis is also carried out considering stochastic operational conditions of hybrid electric vehicles with a parametric reduced-order model. The probabilistic analysis not only suggests appropriate modes and locations for monitoring its dynamic responses, but also determines the maximum response level of every cell in the battery pack. The proposed modeling approach can improve the safety and reliability of the structural design of battery cells and packs. Furthermore, it can be useful for the identification of the battery states during the operation.

Modal density of rectangular structures in a wide frequency range

A novel approach to investigate the modal density of a rectangular structure in a wide frequency range is presented. First, the modal density is derived, in the whole frequency range of interest, on the basis of sound transmission through the infinite counterpart of the structure; then, it is corrected by means of the low-frequency modal behavior of the structure, taking into account actual size and boundary conditions. A statistical analysis reveals the connection between the modal density of the structure and the transmission of sound through its thickness. A transfer matrix approach is used to compute the required acoustic parameters, making it possible to deal with structures having arbitrary stratifications of different layers. A finite element method is applied on coarse grids to derive the first few eigenfrequencies required to correct the modal density. Both the transfer matrix approach and the coarse grids involved in the finite element analysis grant high efficiency. Comparison with alternative formulations demonstrates the effectiveness of the proposed methodology.

Enriched reproducing kernel particle method for fractional advection–diffusion equation

The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection–diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

Effect of Changes in Fiber Orientations on Modal Density of Fiberglass Composite Plates

As the manufacturing technologies for composites materials are advanced, the use of composites for engineering applications has increased significantly. The purpose of this work is to study the response of the idealized subsystem made up of composite material. To study the vibrational response theoretically, Statistical Energy Analysis (SEA) method is used. The S.E.A. techniques are used to predict the vibrational parameters of coupled structural elements and acoustic volumes and it depends largely on an accurate estimate of three parameters: the modal densities, damping loss factors, and coupling loss factors of the subsystems. In this paper the modal density of rectangular composite plate made up of fiberglass is obtained by experimentally and theoretically. Also experimentally analyzed the effect of changes in fiber orientations on modal density the same sized plates, with different fiber orientations.

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.

Mistuned Higher-Order Mode Forced Response of an Embedded Compressor Rotor—Part II: Mistuned Forced Response Prediction

This paper is the second part of a two-part paper that presents a comprehensive study of the higher-order mode (HOM) mistuned forced response of an embedded rotor blisk in a multistage axial research compressor. The resonant response of the second-stage rotor (R2) in its first chordwise bending (1CWB) mode due to the second harmonic of the periodic passing of its neighboring stators (S1 and S2) is investigated computationally and experimentally at three steady loading conditions in the Purdue three-stage compressor research facility. A nonintrusive stress measurement system (NSMS, or blade tip-timing) is used to measure the blade vibration. Two reduced-order mistuning models of different levels of fidelity are used, namely, the fundamental mistuning model (FMM) and the component mode mistuning (CMM), to predict the response. Although several modes in the 1CWB modal family appear in frequency veering and high modal density regions, they do not heavily participate in the response such that very similar results are produced by the FMM and the CMM models of different sizes. A significant response amplification factor of 1.5–2.0 is both measured and predicted, which is on the same order of magnitude of what was commonly reported for low-frequency modes. In this study, a good agreement between predictions and measurements is achieved for the deterministic analysis. This is complemented by a sensitivity analysis which shows that the mistuned system is highly sensitive to the discrepancies in the experimentally determined blade frequency mistuning.

Nonlinear Parametric Reduced-Order Model for the Structural Dynamics of Hybrid Electric Vehicle Batteries

Battery packs used in electrified vehicles exhibit high modal density due to their repeated cell substructures. If the excitation contains frequencies in the region of high modal density, small commonly occurring structural variations can lead to drastic changes in the vibration response. The battery pack fatigue life depends strongly on their vibration response; thus, a statistical analysis of the vibration response with structural variations is important from a design point of view. In this work, parametric reduced-order models (PROMs) are created to efficiently and accurately predict the vibration response in Monte Carlo calculations, which account for stochastic structural variations. Additionally, an efficient iterative approach to handle material nonlinearities used in battery packs is proposed to augment the PROMs. The nonlinear structural behavior is explored, and numerical results are provided to validate the proposed models against full-order finite element approaches.

Agency and Co-production: A Multimodal Perspective

AbstractBuilding on multimodal (inter)action analysis as a theoretical and methodological framework, this article introduces and develops the theoretical/methodological tool called primary agency. Taking the mediated action as a unit of analysis, agency can be analysed as a feature of action. However, there is a lack of empirical approaches for the study of agency, and an overemphasis on language as the most important site for identifying agentive action. I develop primary agency through an analysis of three co-produced higher-level actions from a research project into high school tutoring. These are the higher-level actions of conducting research, tutoring and reading a text. Applying co-production and the modal density foreground/background continuum I explore how the researcher, the tutor and the student co-produce these higher-level actions. Through this analysis, I identify the most significant mediational means for each higher-level action, and the social actor with ownership or agency over these mediational means. I define this social actor as the one with primary agency over the co-produced higher-level action. Finally, my analysis outlines the implications of primary agency for co-produced higher-level actions, including the role of the researcher, the attention/awareness participants pay to overarching research projects, and links between primary agency and successful learning.

Vibratory behavior prediction of mistuned stator vane clusters: An industrial application

Geometric and material dispersions of bladed disks imply some mistuning effects, which result in both the loss of cyclic symmetry properties and a vibratory response amplification in areas of high modal density. These conditions complicate the prediction of vibratory behavior, causing high modal density and extreme sensitivity to mistuning. A non-intrusive spectral stochastic method has been developed to predict the vibratory behavior of a mistuned stator vane. To apply this method to an industrial model, a Karhunen-Loève expansion and a double modal synthesis method have been combined with the stochastic method. The complete method was first tested on a simple academic model and then on an industrial stator vane sector.

SEA-FEM hybrid analysis for predicting Inter-floor impact noise

A hybrid method for predicting inter-floor impact noise in an apartment house is developed. Statistical Energy Analysis (SEA) is useful for predicting mid- to high-frequency noise by utilizing simple equivalent models of structures that have complex geometry and material properties. For low frequencies with concomitant low modal density, however, the error can be significant. Finite Element Method (FEM) is useful for predicting low frequency noise but entails complex modelling process and can be computation intensive. The inter-floor impact noise contains a significant low frequency component arising from floor vibration engendered by a person walking on the floor. The force of impact at 30–300Hz frequency range excites the first and second bending modes of the floor and causes sound to radiate, and for the component of the sound less than 100Hz, the SEA prediction error can be quite significant. In the present study, a modular hybrid method in which the floor structure is modelled using FEM and the vibration energy transmission through walls is modelled using SEA is proposed. The low frequency vibration of the floor due to impact is analyzed by modelling the floor asa clamped uniform plate while the propagation of vibratory energy to adjacent walls is analyzed by determining CLF and DLF values. The proposed SEA-FEM hybrid analysis method is capable of providing accurate estimation of the impact noise for low-, medium- and high-frequencies. Compared with a FEM−based method, the proposed method is found to yield results that are similar while reducing computation time by 70%.

Infinite-dimensional Kalman filter design for flexible curved beams based on PDE models

State observer design is a crucial step in active vibration control of large flexible space structures (LFSSs). Since the LFSSs often have lower natural frequencies, high modal density and low damping, the traditional observer design methods which use the modal truncation model are easily leading to the problem of observation spillover. Instead, the approaches directly using the partial differential equation (PDE) models of flexible structures can avoid the spillover problem, and are becoming the hotspot of research in this field. The presented research aimed at the flexible curved beams which are often used as support components in space antennas, and designed the infinite dimensional state observer for the in-plane vibration of cantilevered curved beams with weakly internal damping based on the theory of Kalman filtering. Firstly, a second order PDEs model for the in-plane vibration of the curved beam is established which considers the shear deformation and rotary inertia of the curved beam simultaneously, then by using the variable substitution method, the second order PDEs model is reduced to first order and transformed to an abstract state space model (SSM) in Hilbert space. The infinite-dimensional Kalman filter for the curved beam is designed using the SSM, and the explicit formula for the feedback gain operators of the Kalman filter are obtained based on the spectral factorization method in frequency domain. Finally, a numerical example is carried out to validate the effect of the presented infinite dimensional Kalman filter design method, and to reveal the effectiveness of the method on preventing the observation spillover.

Improved Metrological Traceability of Particle Size Values Measured with Line-Start Incremental Centrifugal Liquid Sedimentation.

Line-start incremental centrifugal liquid sedimentation (disc-CLS) is a powerful method to determine particle size based on the principles of Stokes' law. Because several of the input quantities of the Stokes equation cannot be easily determined for this case of a rotating disc, the disc-CLS approach relies on calibrating the sedimentation time scale with reference particles. To use these calibrant particles for establishing metrological traceability, they must fulfill the same requirements as those imposed on a certified reference material, i.e., their certified Stokes diameter and density value must come with a realistic measurement uncertainty and with a traceability statement. As is the case for several other techniques, the calibrants do not always come with uncertainties for the assigned modal diameter and effective particle density. The lack of such information and the absence of a traceability statement make it difficult for the end-user to estimate the uncertainty of the measurement results and to compare them with results obtained by others. We present the results of a collaborative study that aimed at demonstrating the traceability of particle size results obtained with disc-CLS. For this purpose, the particle size and effective particle density of polyvinyl chloride calibrants were measured using different validated methods, and measurement uncertainties were estimated according to the Guide to the Expression of Uncertainty in Measurement. The results indicate that the modal Stokes diameter and effective particle density that are assigned to the calibrants are accurate within 5% and 3.5%, respectively, and that they can be used to establish traceability of particle size results obtained with disc-CLS. This conclusion has a great impact on the traceability statement of certified particle size reference materials, for which the traceability is limited to the size and density values of the calibrant particles.

Broad frequency band full field measurements for advanced applications: Point-wise comparisons between optical technologies

Abstract The progress of optical systems gives nowadays at disposal on lightweight structures complex dynamic measurements and modal tests, each with its own advantages, drawbacks and preferred usage domains. It is thus more easy than before to obtain highly spatially defined vibration patterns for many applications in vibration engineering, testing and general product development. The potential of three completely different technologies is here benchmarked on a common test rig and advanced applications. SLDV, dynamic ESPI and hi-speed DIC are here first deployed in a complex and unique test on the estimation of FRFs with high spatial accuracy from a thin vibrating plate. The latter exhibits a broad band dynamics and high modal density in the common frequency domain where the techniques can find an operative intersection. A peculiar point-wise comparison is here addressed by means of discrete geometry transforms to put all the three technologies on trial at each physical point of the surface. Full field measurement technologies cannot estimate only displacement fields on a refined grid, but can exploit the spatial consistency of the results through neighbouring locations by means of numerical differentiation operators in the spatial domain to obtain rotational degrees of freedom and superficial dynamic strain distributions, with enhanced quality, compared to other technologies in literature. Approaching the task with the aid of superior quality receptance maps from the three different full field gears, this work calculates and compares rotational and dynamic strain FRFs. Dynamic stress FRFs can be modelled directly from the latter, by means of a constitutive model, avoiding the costly and time-consuming steps of building and tuning a numerical dynamic model of a flexible component or a structure in real life conditions. Once dynamic stress FRFs are obtained, spectral fatigue approaches can try to predict the life of a component in many excitation conditions. Different spectral shaping of the excitation can easily be used to enhance the comparison in the framework of any of the spectral approaches for fatigue life calculations, highlighting benefits and drawbacks of a direct experimental approach to failure and risk assessment in structural dynamics when dealing with complex patterns in real-life testing. Are optical measurements and spatially dense datasets really effective in advanced model updating of lightweight structures with complex structural dynamics? The noise shown in the raw signal of some experiments may pose issues in proficiently exploiting the added data in a fruitful model updating procedure. Model updating results are here compared between scanning and native full field technologies, with comments and details on the test rig, on the advantages and drawbacks of the approaches. The identification of EMA models highlights the increasing quality of shapes that can be obtained from native full field high resolution gears, against that (some time unexpectedly poor) of SLDV tested.

High Frequency Analysis of a Point-Coupled Parallel Plate System

The root-mean-square (RMS) response of various points in a system comprised of two parallel plates coupled at a point undergoing high frequency, broadband transverse point excitation of one component is considered. Through this prototypical example, asymptotic modal analysis (AMA) is extended to two coupled continuous dynamical systems. It is shown that different points on the plates respond with different RMS magnitudes depending on their spatial relationship to the excitation or coupling points in the system. The ability of AMA to accurately compute the RMS response of these points (namely, the excitation point, the coupling points, and the hot lines through the excitation or coupling points) in the system is shown. The behavior of three representative prototypical configurations of the parallel plate system considered is: two similar plates (in both geometry and modal density), two plates with similar modal density but different geometry, and two plates with similar geometry but different modal density. After examining the error between reduced modal methods (such as AMA) to classical modal analysis (CMA), it is determined that these several methods are valid for each of these scenarios. The data from the various methods will also be useful in evaluating the accuracy of other methods including statistical energy analysis (SEA).

High frequency analysis of a point-coupled parallel plate system

The RMS response of various points in a system comprised of two parallel plates coupled at a point undergoing high frequency, broadband transverse point excitation of one component is considered. Through this prototypical example, Asymptotic Modal Analysis (AMA) is extended to two coupled continuous dynamical systems. It is shown that different points on the plates respond with different RMS magnitudes depending on their spatial relationship to the excitation or coupling points in the system. The ability of AMA to accurately compute the RMS response of these points (namely the excitation point, the coupling points, and the hot lines through the excitation or coupling points) in the system is shown. The behavior of three representative prototypical configurations of the parallel plate system: two similar plates (in both geometry and modal density), two plates with similar modal density but different geometry, and two plates with similar geometry but different modal density. After examining the error between reduced modal methods (such as AMA) to Classical Modal Analysis (CMA), it is determined that these several methods are valid for each of these scenarios. The data from the various methods will also be useful in evaluating the accuracy of other methods including SEA.

Investigation of the effect of neglecting reflections in Power-based sound in ducts

In high frequency sound propagation inside ducts, the modal density is very high that the sound starts to propagate in rays. The acoustic performance of a duct network can be simulated using power-based models. The application of such situations is for HVAC systems and large silencers for power generation. Several standards are available for the analysis of HVAC systems using the same technique, such as ASHRAE and VDI. For each element, the flow-generated sound power inside the element is added to the input sound power and the output sound power is calculated by subtracting the Insertion Loss of the element. The attenuated sound energy can be either dissipated inside the element or reflected back to the system. Standards always assume that no energy is reflected and all the attenuation happens inside the element. This assumption is investigated in this paper. Several standard HVAC elements are considered, calculating the amount of energy dissipated inside the element and that reflected back. It was found that if all the reflected energy is considered, this will affect the output power from the system, especially in the highly reflective elements. The investigation was done using the Finite Element Method with the ray tracing technique.

Bayesian modal identification with particle filtering for sediment property inversion

Sequential Bayesian filtering methods have been previously used in dispersion curve tracking for long range sound propagation in the ocean. Modal frequency probability density functions were extracted for sound speed inversion. Here, we calculate modal arrival time densities, instead, and employ them for inversion for sediment sound speed and thickness and water column depth. Bayesian mode identification is performed to this end. We investigate two methods for describing the statistical errors in power spectra, which we use in the arrival time density calculation using normal modes. We then link these densities to the parameters of interest. The approaches are tested with synthetic data as well as data collected in the Gulf of Mexico. [Work supported by ONR.]

An active control method for the spectral homogenization of airborne sound insulation laboratories at low frequencies

Low frequency sound insulation measurements are affected by large uncertainties and inaccuracies due to the low modal density of small laboratory rooms. For this reason, an automated measuring system for the active spectral homogenization of enclosed spaces at low frequencies has been developed. The aim is to achieve a quasi-diffuse field, reducing the amplitude of room modes, in order to apply standard procedures for sound insulation measurements at low frequencies. The homogenization method is based on the active control of the interference spatial patterns of a system of two loudspeakers using a phase steering technique. The room response spectrum, as function of frequency-dependent phase difference between source signals, is measured in different positions in order to achieve the optimal spectral homogenization. Such technique allows to decrease the modal sound pressure level fluctuations in the source and receiving volumes in the frequency range between 30 Hz and 120 Hz. Based on this, sound insulation measurements at low frequencies on a high performance triple glazing and steel structure are performed and compared with standard ones.

Energy based correlation criteria in the mid-frequency range

Aircraft structures are characterized by their lightweight design. As such, they are prone to vibrations. Numerical models based on the Finite Element Method often show significant deviations when the mid-frequency range is considered, where strong interaction between vibrations and acoustics is present. Model validation based on experimental modal data is often not possible due to the high modal density that aircraft fuselage structures exhibit in this frequency range. Classical correlation criteria like the Modal Assurance Criterion require mode shapes and can therefore not be applied. Other correlation criteria using frequency response data, such as the Frequency Domain Assurance Criterion, are highly sensitive to even small structural modifications and fail to indicated the correlation between test and analysis data in the mid-frequency range. Nevertheless, validated numerical models for the mid- to high-frequency ranges are a prerequisite for acoustic comfort predictions of aircraft cabin. This paper presents a new method for the correlation of response data from test and analysis in the mid-frequency range to support model validation in the mid-frequency range and to enable the usage of finite element models in this frequency range. The method is validated on a stiffened cylindrical shell structure, which represents a scale-model of an aircraft fuselage. The correlation criterion presented here is inspired by Statistical Energy Analysis and is based on kinetic energies integrated over frequency bands and spatially integrated over surface areas of the structure. The objective is to indicate frequency bands where the finite element model needs to be adjusted to better match with experimental observations and to locate the areas where these adjustments should be applied.

Evolution of pore size distribution in deforming granular materials

This letter presents the results of a numerical exercise in discrete element method focusing on the evolution characteristics of pore size distribution (psd) in granular materials under uniform deformation. The evolution has resulted from the microstructural rearrangement, in which contraction leads to disintegration of big pores into small pores and vice versa for dilation besides the swelling and shrinkage of individual pore spaces during shearing. The numerical analysis indicates the evolution of psd's towards a unique distribution when the granular assemblies are subjected to an identical boundary constraint. Primarily the variation in porosity during deformation governs the pore evolution characteristics, but the stress path also dictates the modal pore density. The nature of any psd, however, is found to be independent of the initial packing of the assembly and the current stress state.