Vibration of L-shaped plates under a deterministic force or moment excitation: a case of statistical energy analysis application

Analytical and closed form solutions are presented in this paper for the vibration response of an L-shaped plate under a point force or a moment excitation. Inter-relationships between wave components of the source and the receiving plates are clearly defined. Explicit expressions are given for the quadratic quantities such as input power, energy flow and kinetic energy distributions of the L-shaped plate. Applications of statistical energy analysis (SEA) formulation in the prediction of the vibration response of finite coupled plate structures under a single deterministic forcing are examined and quantified. It is found that the SEA method can be employed to predict the frequency averaged vibration response and energy flow of coupled plate structures under a deterministic force or moment excitation when the structural system satisfies the following conditions: (1) the coupling loss factors of the coupled subsystems are known; (2) the source location is more than a quarter of the plate bending wavelength away from the source plate edges in the point force excitation case, or is more than a quarter wavelength away from the pair of source plate edges perpendicular to the moment axis in the moment excitation case due to the directional characteristic of moment excitations. SEA overestimates the response of the L-shaped plate when the source location is less than a quarter bending wavelength away from the respective plate edges owing to wave coherence effect at the plate boundary.     

Efficient parametric uncertainty analysis within the hybrid Finite Element/Statistical Energy Analysis method

This paper is concerned with the development of efficient algorithms for propagating parametric uncertainty within the context of the hybrid Finite Element/Statistical Energy Analysis (FE/SEA) approach to the analysis of complex vibro-acoustic systems. This approach models the system as a combination of SEA subsystems and FE components; it is assumed that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. The method has been recently generalised by allowing the FE components to possess parametric uncertainty, leading to two ensembles of uncertainty: a non-parametric one (SEA subsystems) and a parametric one (FE components). The SEA subsystems ensemble is dealt with analytically, while the effect of the additional FE components ensemble can be dealt with by Monte Carlo Simulations. However, this approach can be computationally intensive when applied to complex engineering systems having many uncertain parameters. Two different strategies are proposed: (i) the combination of the hybrid FE/SEA method with the First Order Reliability Method which allows the probability of the non-parametric ensemble average of a response variable exceeding a barrier to be calculated and (ii) the combination of the hybrid FE/SEA method with Laplace's method which allows the evaluation of the probability of a response variable exceeding a limit value. The proposed approaches are illustrated using two built-up plate systems with uncertain properties and the results are validated against direct integration, Monte Carlo simulations of the FE and of the hybrid FE/SEA models.     

Numerical and experimental validation of a hybrid finite element-statistical energy analysis method

The finite element (FE) and statistical energy analysis (SEA) methods have, respectively, high and low frequency limitations and there is therefore a broad class of "mid-frequency" vibro-acoustic problems that are not suited to either FE or SEA. A hybrid method combining FE and SEA was recently presented for predicting the steady-state response of vibro-acoustic systems with uncertain properties. The subsystems with long wavelength behavior are modeled deterministically with FE, while the subsystems with short wavelength behavior are modeled statistically with SEA. The method yields the ensemble average response of the system where the uncertainty is confined in the SEA subsystems. This paper briefly summarizes the theory behind the method and presents a number of detailed numerical and experimental validation examples for structure-borne noise transmission.     

Vibro-acoustic analysis of complex systems

A general method is presented for predicting the ensemble average steady-state response of complex vibro-acoustic systems that contain subsystems with uncertain, or random, properties. The method combines deterministic and statistical techniques to produce a non-iterative hybrid method that incorporates equations of dynamic equilibrium and power balance. The method is derived explicitly without reference to statistical energy analysis (SEA); however, it is seen that the wave approach to SEA can be viewed as a special case of the proposed method. The proposed method provides a flexible way to account for necessary deterministic details in a vibro-acoustic analysis without requiring that an entire system be modeled deterministically. The method therefore provides a potential solution to the mid-frequency problem (in which a system is neither entirely deterministic nor entirely statistical). The application of the method is illustrated with a numerical validation example.     

A novel hybrid ES-FE-SEA for mid-frequency prediction of Transmission losses in complex acoustic systems

The widely-used numerical modeling approaches such as the finite element method (FEM) and statistical energy analysis (SEA) often have limited applicability to the transmission loss prediction in mid-frequency range. In this paper, a novel hybrid edge-based smoothed FEM coupled with statistical energy analysis (ES-FE-SEA) method is proposed to further improve the accuracy of “mid-frequency” transmission loss predictions. The application of ES-FEM will “soften” the well-known ‘‘overly-stiff’’ behavior in the standard FEM solution and reduce the inherent numerical dispersion error. While the SEA approach deals with the physical uncertainty in the relatively higher frequency range. The plate of interest is appropriately described by an ES-FEM model, due to its relative robustness to perturbations. Its adjacent reverberation cavities are modeled by employing the SEA approach, because of their high model density. The coupling and interaction between SEA subsystems and the FE subsystem is governed by the “reciprocity relationship” theorem. A standard numerical example for benchmarking is examined and excellent agreement was achieved between the prediction and reference results. The proposed ES-FE-SEA is also verified by various numerical examples. The method is finally applied to the modeling a complicated engineering problem–acoustic fields on both sides of the front windshield in a passenger car.     

Frequency average loss factors of plates and shells

This paper describes a thorough investigation of the measurement of frequency band average loss factors of structural components for use in the statistical energy analysis (SEA) method of computation of vibration levels. The “traditional” method of measurement is to excite the structure by a random force having a flat spectral density in the frequency band of interest. The force is then cut off and the decay of the modes excited in the band is noted. The average loss factor is deduced from the decay curve. The alternative energy method is the subject of this study. In this test the power input from the band limited random force is measured and the spatial average vibration level of the structure is estimated from several surface accelerometers. It is shown that when the modes in the band have similar loss factors (as is usually the case) the energy method gives a result which is very close to that obtained from the decay method. These in turn are close to the arithmetic average of the loss factors of the individual modes in the band. It is shown that only when the band contains one or two very lightly damped modes amongst several more heavily damped modes is there a difference between the two methods. In this case it is better to use the energy result in the SEA calculations.     

STATISTICAL ENERGY ANALYSIS OF COUPLED PLATE SYSTEMS WITH LOW MODAL DENSITY AND LOW MODAL OVERLAP

Finite element methods, experimental statistical energy analysis (ESEA) and Monte Carlo methods have been used to determine coupling loss factors for use in statistical energy analysis (SEA). The aim was to use the concept of an ESEA ensemble to facilitate the use of SEA with plate subsystems that have low modal density and low modal overlap. An advantage of the ESEA ensemble approach was that when the matrix inversion failed for a single deterministic analysis, the majority of ensemble members did not encounter problems. Failure of the matrix inversion for a single deterministic analysis may incorrectly lead to the conclusion that SEA is not appropriate. However, when the majority of the ESEA ensemble members have positive coupling loss factors, this provides sufficient motivation to attempt an SEA model. The ensembles were created using the normal distribution to introduce variation into the plate dimensions. For plate systems with low modal density and low modal overlap, it was found that the resulting probability distribution function for the linear coupling loss factor could be considered as lognormal. This allowed statistical confidence limits to be determined for the coupling loss factor. The SEA permutation method was then used to calculate the expected range of the response using these confidence limits in the SEA matrix solution. For plate systems with low modal density and low modal overlap, relatively small variation/uncertainty in the physical properties caused large differences in the coupling parameters. For this reason, a single deterministic analysis is of minimal use. Therefore, the ability to determine both the ensemble average and the expected range with SEA is crucial in allowing a robust assessment of vibration transmission between plate systems with low modal density and low modal overlap.     

Response variance prediction in the statistical energy analysis of built-up systems

In the statistical energy analysis (SEA) of high frequency noise and vibration, a complex engineering structure is represented as an assembly of subsystems. The response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem, and these energies are found by employing the principle of power balance. Strictly the computed energy is an average taken over an ensemble of random structures, and for many years there has been interest in extending the SEA prediction to the variance of the energy. A variance prediction method for a general built-up structure is presented here. Closed form expressions for the variance are obtained in terms of the standard SEA parameters and an additional set of parameters alpha(k) that describe the nature of the power input to each subsystem k, and alpha(ks) that describe the nature of the coupling between subsystems k and s. The theory is validated by comparison with Monte Carlo simulations of plate networks and structural-acoustic systems.     

A quantitative criterion validating coupling power proportionality in statistical energy analysis

The response of two general spring-coupled elements is investigated to develop a unifying approach to the weak coupling criterion in Statistical Energy Analysis (SEA). First, the coupled deterministic equations of motion are expressed in the bases given by the uncoupled elements’ eigenmodes. Then, an iterative solution is expressed as a succession of exchanges between elements, where uncoupled motion provides the start approximation, converging if the ‘coupling eigenvalue’ is less than unity, in which case coupling is said to be weak. This definition is related to whether response is ‘local’ or ‘global’, encompassing a number of previously defined coupling strength definitions, applying for deterministically described structures. A stochastic ensemble is defined by that its members are equal to the investigated structure but the elements have random frequencies. It is required that the coupling eigenvalue be less than unity for all members of the ensemble. This requirement generates the title subject of the article: ‘the modal interaction strength’. It is similar to the previously defined coupling strength criterion characterising the ensemble average energy flow in uni-dimensional waveguides. Finally, SEA models are formulated in terms of the uncoupled elements’ modal data.     

微热控百叶窗驱动器的分析与模拟

提出一种适合微小卫星的微热控百叶窗驱动器,利用热电制冷片作为温控装置,利用高膨胀系数的金属作为活动部件,实现热控百叶窗的开合.建立了该驱动器的一维和二维物理数学模型,并利用有限元软件ANSYS对其工作过程进行了数值模拟.获得了其基本的性能特性.研究结果表明,与直接电加热双金属驱动元件相比,不同输入功率下,该新型驱动器的变形量分别为双金属的2.5～10倍,同时动作频率约为双金属的20倍左右.     

A miniaturization of the multi-degree-of-freedom ultrasonic actuator using a small cylinder fixed on a substrate

Multi-degree-of-freedom ultrasonic actuator has been studied for robot arms and multidimensional precision table and so on because of its simple structure, silent operation, and holding force. In this study, we aim to miniaturize multi-degree-of-freedom ultrasonic actuator for fabrication on a substrate. This actuator consists of a stainless steel cylinder and a PZT ring. The cylinder is fixed on a substrate and the PZT ring is glued to the substrate near the cylinder. The 1st longitudinal vibration and the 2nd bending vibration are simultaneously excited in the cylinder to make elliptical motion at the top of the cylinder and a ball rotor placed on the cylinder rotates because of the friction force. Length of the cylinder was decided so as to tune the resonance frequency of the 1st longitudinal vibration to the 2nd bending one. Actuator performances are evaluated experimentally using a 14 mm height and 7 mm diameter stainless steel cylinder with a 0.5 mm thickness PZT ring. The rotation about the cylinder axis is tested using the two orthogonal bending vibrations with 90 degrees phase difference. Also, the rotation about horizontal axes were investigated using the combination of the longitudinal vibration and one of two bending vibrations. We measured the rotation speed of a steel ball and obtained 15.8 rps using a 6 mm diameter ball rotor.     

Modelling and characterization of a piezoceramic inertial actuator

An inertial actuator (also known as a proof mass actuator) applies forces to a structure by reacting them against an “external” mass. This approach to actuation may provide some practical benefits in the active control of vibration and structure-borne noise: system reliability may be improved by removing the actuator from a structural load path; effective discrete point-force actuation permits ready attachment to curved surfaces, and an inherent passive vibration absorber effect can reduce power requirements.This paper describes a class of recently developed inertial actuators that is based on mechanical amplification of displacements of an active piezoceramic element. Important actuator characteristics include resonance frequencies, clamped force, and the drive voltage to output the force frequency response function.The paper addresses one particular approach to motion amplification, the “dual unimorph,” in detail. A model of actuator dynamic behavior is developed using an assumed-modes method, treating the piezoelectrically induced stresses as external forces. Predicted actuator characteristics agree well with experimental data obtained for a prototype actuator. The validated actuator dynamic model provides a tool for design improvement.     

Electromagnetic control of the instability in the liquid metal flow over a backward-facing step

The tile-type electromagnetic actuator (TEA) and stripe-type electromagnetic actuator (SEA) are applied to the active control of the perturbation energy in the liquid metal flow over a backward-facing step (BFS). Three control strategies consisting of base flow control (BFC), linear model control (LMC) and combined model control (CMC) are considered to change the amplification rate of the perturbation energy. CMC is the combination of BFC and LMC. SEA is utilized in BFC to produce the streamwise Lorentz force thus adjusting the amplification rate via modifying the flow structures, and the magnitude of the maximum amplification rate could reach to 6 orders. TEA is used in LMC to reduce the magnitude of the amplification rate via the wall-normalwise Lorentz force, and the magnitude could be decreased by 2 orders. Both TEA and SEA are employed in CMC where the magnitude of the amplification rate could be diminished by 3 orders. In other words, the control strategy of CMC could capably alter the flow instability of the liquid metal flow.     

Deterministic-statistical analysis of a structural-acoustic system

The purpose of this paper is to develop an efficient approach for vibro-acoustic analysis. Being simple and representative, an exited plate-acoustic system is selected as a validation case for the vibro-acoustic analysis as the system presents one two-dimensional statistical component (modal dense structure panel—plate) connected to the other component (deterministic acoustic volume—cavity) through the area junction over a surface domain, rather than at a line boundary. Potential industrial applications of the system vibro-acoustic analysis would be in acoustic modelling of vehicle body panels such as the cabin roof panel, and door panels for the boom noise analysis.A new deterministic-statistical analysis approach is proposed from a combination or hybrid of deterministic analysis and statistical energy analysis (SEA) approaches. General theory of the new deterministic-statistical analysis approach is introduced. The main advantage of the new deterministic-statistical analysis approach is its possibility in place of the time consuming Monte Carlo simulation. In order to illustrate and validate the new deterministic-statistical analysis approach, three approaches of the deterministic analysis, the statistical energy analysis and the new deterministic-statistical analysis are then applied to conduct the plate-acoustic system modelling, and their results will be compared. The vibro-acoustic energy coupling characteristic of the plate-acoustic system will be studied. The most suitable frequency range for the new approach will be identified in consideration of computational accuracy, information and speed.     

The design and dynamic performance of a moment actuator

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Active control of high-frequency vibration: Optimisation using the hybrid modelling method

This work presents active control of high-frequency vibration using skyhook dampers. The choice of the damper gain and its optimal location is crucial for the effective implementation of active vibration control. In vibration control, certain sensor/actuator locations are preferable for reducing structural vibration while using minimum control effort. In order to perform optimisation on a general built-up structure to control vibration, it is necessary to have a good modelling technique to predict the performance of the controller. The present work exploits the hybrid modelling approach, which combines the finite element method (FEM) and statistical energy analysis (SEA) to provide efficient response predictions at medium to high frequencies. The hybrid method is implemented here for a general network of plates, coupled via springs, to allow study of a variety of generic control design problems. By combining the hybrid method with numerical optimisation using a genetic algorithm, optimal skyhook damper gains and locations are obtained. The optimal controller gain and location found from the hybrid method are compared with results from a deterministic modelling method. Good agreement between the results is observed, whereas results from the hybrid method are found in a significantly reduced amount of time.     

Input impedance calculation for a cylindrical microstrip antenna fed by a microstrip line

We consider the problem of excitation of a cylindrical microstrip antenna fed by a cylindrical microstrip line. The problem has been solved by the moments method in the spatial domain using a new representation of the Green’s function. In the model proposed, a cylindrical microstrip line of finite length is attached to the patch and is considered as its part. The so-called lumped delta generator is chosen as the excitation source. At the first stage, current distributions on the patch and in the microstrip line are found by solution of the integral equation. Then using the well-known relations of the transmission-line theory and the equivalent-scheme method, we calculate the reflection coefficient in the microstrip line and the antenna input impedance. Patches of different shapes are discussed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 3, pp. 223–233, March 2008.     

SEA and contribution analysis for interior noise of a high speed train

This paper presents a detailed Statistical Energy Analysis (SEA) and contribution analysis of the interior noise of a high-speed train through extensive simulations and measurements. The SEA model was developed based on the actual geometrical parameters of a benchmark high-speed coach. Sound transmission loss levels of the structural components of the car body, which are required in the SEA model, were tested in a dedicated acoustic laboratory following international standard ISO 140-3:1995. Modal densities of these structural components were derived from measured frequency response functions using the modal counting method. Damping loss factors were obtained using the half-power bandwidth method and the vibration attenuation method. By considering the relationship between sound radiation and power transmission, coupling loss factors between structures and cavities were estimated. Source inputs to the SEA model were derived from field experiment data. Interior noise due to those sources was predicted using the SEA model and the prediction was generally in good agreement with measurement. Contribution analysis was then performed using this validated model through parametric study, and this analysis was further examined experimentally. In conclusion, for the coach that was investigated in this paper, the key factors for interior noise are sidewall vibration, bogie area noise, and floor sound transmission loss. Based on this and other engineering considerations, an interior noise control strategy can be defined.     

基于PZT的非球面能动抛光盘设计优化

基于PZT压电陶瓷驱动器的非球面能动抛光盘,能够在PZT驱动器的作用下改变面形,用于中小口径非球面镜加工。为优化设计基于PZT压电陶瓷驱动器的非球面能动抛光盘,利用有限元分析方法,计算各驱动器的影响函数,计算非球面能动抛光盘的输出面形,与理论面形比较得到剩余残差。以优化设计驱动器排布方式和极头直径为例,当非球面能动抛光盘中心到非球面工件中心的距离L为120mm,分别计算比较,极头直径为Φ10mm时,19单元PZT圆形排布与21单元PZT方形排布的剩余残差;以及19单元PZT圆形排布时,极头直径为Φ10mm与Φ14mm的剩余残差。结果表明,非球面能动抛光盘产生变形后的剩余残差RMS相应分别为0.303μm、0.367μm、0.328μm。因此,基于PZT压电陶瓷驱动器的非球面能动抛光盘确定选用19单元PZT圆形排布且极头直径Φ10mm。