Development of a hybrid SmEdA/SEA model for predicting the power exchanged between low and high modal density subsystems

Statistical modal Energy distribution Analysis (SmEdA) was developed from classical Statistical Energy Analysis (SEA). It allows computing power flow between coupled subsystems from the deterministic modes of uncoupled subsystems without assuming the SEA modal energy equipartition. SmEdA is well adapted in mid-frequency when the subsystems have not a very high modal density. However, for some systems e.g. the plate-cavity system, one subsystem can exhibit a low modal density while the other one a high one. The goal of the paper is then to propose an extension of SmEdA formulation that allows describing one subsystem by its deterministic modes, and the other one as a diffuse field statistically supposing modal energy equipartition. The uncertain subsystem is then characterized by sets of natural frequencies and mode shapes constructed based on Gaussian Orthogonal Ensemble matrix and the cross-spectrum density of a diffuse field, respectively. This formulation permits not only the computation of mean noise response but also the variance generated by the uncertainties and furthermore without bringing in much computation. It is demonstrated that the obtained analytical results from the proposed hybrid SmEdA/SEA are consistent with numerical results computed by FEM with an appropriate degree of uncertainty.

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.

High-Frequency Vibration Analysis of Thin Plate Based on B-Spline Wavelet on Interval Finite Element Method

For the dynamic analysis of thin plate bending problems, the Finite Element Methods (FEMs) are the most commonly used numerical techniques in engineering. However, due to the deficiency of low computing efficiency and accuracy, the FEMs can’t be directly used to effectively evaluate dynamic analysis of thin plate with high modal density within low-high frequency domain. In order to solve this problem, the Wavelet Finite Element Methods (WFEMs) has been introduced to solve the problem by improving the computing efficiency and accuracy in this paper. Due to the properties of multi-resolution, the WFEMs own excellently high computing efficiency and accuracy for structure analysis. Furthermore, for the destination of predicting dynamic response of thin plate within high frequency domain, this paper introduces the Multi-wavelet element method based on c1 type wavelet thin plate element and a new assembly procedure to significantly promote the calculating efficiency and accuracy which aim at breaking up the limitation of frequency domain when using the existing WFEMs and traditional FEMs. Besides, the numerical studies are applied to certify the validity of the method by predicting state response of thin plate within 0∼1000Hz based on a special numerical example with high modal density. According to the literature, the frequency domain between 0 to 1000Hz contains the low-high frequency domain aiming at the numerical example. The numerical results show excellent agreement with the reference solutions captured by FEM and analytical expressions respectively. Among these, it is noteworthy that the relative errors between the analytical solutions and numerical solution are less than 0.4% when the dynamic response involved with 1000 modes.

Numerical Application of Double Modal Synthesis to an Industrial Mistuned Clustered Stator Vane and Experimental Validation

Bladed disks are subjected to dynamically fluctuating high-pressure loads. To ensure their robust design, current models must be continuously improved to take phenomena such as mistuning into account. Geometric and material dispersions imply some mistuning effects, which result in both the loss of cyclic symmetry properties and a vibratory response amplification in frequency ranges of high modal density. These conditions complicate the prediction of vibratory behavior, causing high modal density and extreme sensitivity to mistuning. It becomes thus even more essential to maintain structural strength and prevent fatigue. In a first section, a substructuring method has been adopted to reduce finite elements models. It has been applied on an industrial stator vane model. This reduction method is a double modal synthesis based on Craig-Bampton method coupled to a static modes basis reduction through a second modal synthesis. This method yields excellent results and allows dividing computation costs roughly by a five hundred factor. Thanks to an appropriate choice of substructures it is then possible to generate thousands of mistuning configurations and compute them. Thus, 50,000 mistuning draws have been performed, covering a range of 5% maximum variation of blades and platforms Young modulus with a significant statistical meaning. A virtual modal characterization of the industrial stator vane has finally been made in a context of statistical treatment. In a second section, a modal identification has been performed on an industrial stator vane. The part has been subjected to white noise excitation and frequency response functions vibrations have been obtained through scanning vibrometer measurements. The numerical model has then been adjusted, matching whole system modes as they are less or not subjected to misutning. Experimental results have finally been compared to the readjusted statistical modal characterization previously computed. What stands out from this comparison is that most modal frequencies of the industrial stator vane used for experimental analyses are in the envelope predicted by statistical calculation.

A Simple Estimation of Coupling Loss Factors for Two Flexible Subsystems Connected via Discrete Interfaces

A simple formula is proposed to estimate the Statistical Energy Analysis (SEA) coupling loss factors (CLFs) for two flexible subsystems connected via discrete interfaces. First, the dynamic interactions between two discretely connected subsystems are described as a set of intermodal coupling stiffness terms. It is then found that if both subsystems are of high modal density and meanwhile the interface points all act independently, the intermodal dynamic couplings become dominated by only those between different subsystem mode sets. If ensemble- and frequency-averaged, the intermodal coupling stiffness terms can simply reduce to a function of the characteristic dynamic properties of each subsystem and the subsystem mass, as well as the number of interface points. The results can thus be accommodated within the theoretical frame of conventional SEA theory to yield a simple CLF formula. Meanwhile, the approach allows the weak coupling region between the two SEA subsystems to be distinguished simply and explicitly. The consistency and difference of the present technique with and from the traditional wave-based SEA solutions are discussed. Finally, numerical examples are given to illustrate the good performance of the present technique.

High-Precision Positioning of Laser Beams for Vibration Measurements

Predicting the vibratory response of structures with complex geometry can be challenging especially when their properties (geometry and material properties) are not known accurately. These structures can suffer also from high modal density, which can result in small changes in structural properties creating large changes in the resonant response. To address this issue, structural properties could be accurately identified, or the structural response could be experimentally measured. Both these approaches require collecting measurements of higher order vibration modes, which have complicated shape. Consequently, high-accuracy positioning of laser beams is necessary for vibrometers based on laser Doppler velocimetry. This paper presents a methodology to address this challenge. The architecture involves a single-point vibrometer, a motion controller, translating/rotating stages, and special application software for alignment and edge detection. A key novelty of this technology is that the beam of the vibrometer is used for both detecting the edges and for measuring the vibration. Using a motion controller, the system can automatically place/scan and measure the surface of the structure with a positioning resolution of 1 μm. Experimental results are provided to demonstrate the new technique.

Integer Frequency Veering of Mistuned Blade Integrated Disks

As a result of more balanced blade aspect ratios of modern blade-integrated disks (blisks), interactions between disk-dominated and blade-dominated modes are becoming more and more important, especially if blade mistuning is considered. The specific vibration behavior in these transition regions is characterized by a mix of both fundamental mode types into “coupled” modes. In this paper, numerical and experimental investigations based on a front high-pressure compressor (HPC) blisk stage were carried out in order to determine the effect of blade mistuning on those regions in detail. At this, effects like mode localization and amplitude magnification are found to be weakened in an integer frequency-veering zone. Contrary to this, blisks are very sensitive to mistuning in regions of pure blade-dominated mode families with high modal density.

High-Precision Positioning of Laser Beams for Vibration Measurements

Predicting the vibratory response of structures with complex geometry can be challenging especially when their properties (geometry and material properties) are not known accurately. These structures can suffer also from high modal density, which can result in small changes in structural properties creating large changes in the resonant response. To address this issue, structural properties could be accurately identified, or the structural response could be experimentally measured. Both these approaches require collecting measurements of higher order vibration modes which have complicated shape. Consequently, high-accuracy positioning of laser beams is necessary for vibrometers based on laser Doppler velocimetry. This paper presents a methodology to address this challenge. The architecture involves a single-point vibrometer, a motion controller, translating/rotating stages, and special application software for alignment and edge detection. A key novelty of this technology is that the beam of the vibrometer is used for both detecting the edges and for measuring the vibration. Using a motion controller, the system can automatically place/scan and measure the surface of the structure with a positioning resolution of 1 μm. Experimental results are provided to demonstrate the new technique.