Power flow analysis of an L-shaped plate structure

In practical applications of the statistical energy analysis (SEA) or other power flow techniques, the calculation of the power flow between subsystems is of fundamental concern. When the subsystems are connected by multimodal structures, for instance, vibration mounts, values for the coupling loss factors as found in the literature are not suitable. In this paper we present a general theoretical framework for calculating the coupling loss factor when the coupler is numerically modeled by classical dynamics, notably by means of a finite element analysis (FEA). After presenting some pertinent results and definitions from SEA, FEA, and structural dynamical theory, an expression for the “coupling” impedance matrix, which relates the forces and free velocities at nodes along the subsystem/coupler boundaries, is derived. This impedance is formed from the admittance matrix of the coupler, calculated using FEA, and input impedances for the systems, analytically calculated. From the coupling impedance, expressions for the coupling loss factors are derived for the cases where the SEA subsystems are plates or beams and include flexural, longitudinal, and torsional forms of energy transmission. Finally, numerical examples are given for the application of this technique to structures consisting of two beams or plates connected by a multimodal coupler.

In the past it has been demonstrated that energy finite element analysis (EFEA) allows computation of high-frequency structural vibration in complex marine structures and produces results similar to statistical energy analysis (SEA) predictions [Vlahopoulos et al., ‘‘Numerical implementation, validation, and marine applications of an energy finite element formulation,’’ accepted by the Journal of Ship Research]. In this paper a basic theoretical development of a hybrid finite element method is presented. Conventional finite element analysis (FEA) is combined with EFEA in order to achieve solution to mid-frequency vibrations. The new formulation is based on deriving appropriate interface conditions at the joints between members modeled by the FEA and the EFEA methods. Effects of strong coupling, power re-radiation, and power re-injection can be accounted in the analysis. Examples of systems of co-linear beams will be presented. Analytical solutions are compared to the hybrid solution in order to validate the theoretical developments. [Research supported by Automated Analysis Corporation.]

Energy finite element analysis (EFEA) has been successfully applied to steady-state response prediction over the past three decades. Compared with other energy-based methods, such as statistical energy analysis (SEA), EFEA can consider more local structural information without increasing the computational consumption too much, which makes it attractive. Inspired by the transient local energy approach (TLEA), a general transient energy balance equation was derived by assuming that the plane wave condition is satisfied. The properties of the energy balance equation were studied, and the analytical solutions with different initial conditions were provided. Utilizing the derived transient energy balance equation, transient EFEA is proposed, which has the same advantages as EFEA. A general formula is presented for the energy transmission coefficients of any number of coupled in-plane beams. The present approach was validated using a single beam and a coupled collinear beam structure under unloading conditions. The coupled collinear beams were also investigated using constant and quasi-static input power. The validation results show that TEFEA can accurately predict the local response of the structure. All of these results were compared with those of finite element analysis (FEA), simplified TEFEA (sTEFEA), transient statistical energy analysis (TSEA), and analytical formulas.

Noise and vibration control are important design features in various industries. Energy Finite Element Analysis (EFEA) and Statistical Energy Analysis (SEA) have been widely used in analyzing realistic structural and acoustic system for mid-to-high frequency regime. The EFEA is derived from the wave equation and uses energy density as its primary variable in its governing equation which is numerically solved in finite element approach. Generally, the EFEA calculates the mean values of given vibro-acoustic systems. This research introduces variance bounds into the EFEA analysis. Specifically, the variance originated from input power and joint matrix is incorporated to the EFEA. The formulation for calculation of the variance in the EFEA will be discussed and followed by several benchmark examples.

<div class="htmlview paragraph">The Energy Flow Method (EFM), which is based on a Finite Element Analysis (FEA) model and its modal frequency response solution is presented in this paper. The energy and power for each subsystem are the primary response and excitation parameters as in the Statistical Energy analysis (SEA) method. This gives a broad-brush prediction by averaging over both frequency and spatial domain. This prediction is useful when uncertainties exist in the model. The FEA model is used to capture the geometry detail, which is critical in mid-frequency vibration. As an example, a five-plate system is studied using various methods, including traditional FEA, SEA and EFM. The last one has been implemented in MSC/NASTRAN. A discussion is given to understand the limitation of SEA and FEA application in mid frequency response.</div>

The use of statistical energy analysis (SEA) for the solution of vibroacoustic problems that occur with the operation of machinery, has only recently become practicable. This paper charts the developments that led from the early ideas of energy accountancy (EA) to the current SEA modeling techniques. The basic concept of EA, which was conceived to show the influence of the overall design parameters on the radiated noise, is explained and its limitations discussed. The paper then goes on to describe how the various practical difficulties preventing the application of SEA were overcome to the point where it could be used as a practical tool. Finally, a case history is presented in which an experimental SEA model is used to determine input (exciting) powers, power flow and sensitivity of the noise to parameter changes. In addition, the use of an optimization algorithm for driving the noise level down to a target spectrum within parameter and weight constraints is also demonstrated.

The Energy Finite Element Analysis (EFEA) offers an improved solution to the established Statistical Energy Analysis (SEA) for simulating the vibration of large-scale structures. A theoretical overview and comparison of the two methods is presented first. The main aspects of a numerical EFEA implementation are discussed. A published marine application is analyzed, and comparison of the results offers an initial validation of the EFEA. Numerical models for a fishing boat are generated for both methods and utilized for further comparison between the two formulations. The applicability of the EFEA in the marine field is demonstrated and its benefits are discussed.

<div class="htmlview paragraph">In this paper, sound transmission characteristics of typical stamped steel automotive body panels are investigated. In particular, focus was placed on the dash panel since it separates the interior space from the engine. Three dash panels were considered with the same geometry but different damping characteristics. These dash panels included a bare steel dash, a bare steel dash with a Patch Constrained Layer (PCL) and a laminated steel dash. Sound transmission loss was measured in a transmission loss suite. It was observed that the measured transmission loss of the laminated steel dash was significantly higher than the other dash panels throughout the mid and high frequency regions. A hybrid approach using both Finite Element Analysis and Statistical Energy Analysis (SEA) was used to predict and correlate the sound transmission loss for these panels. The geometry effects of the dash panels were considered while calculating SEA parameters. Coupling loss factors between the dash panel and acoustic cavity were predicted considering the effects of increased stiffness and the impedance mismatches due to discontinuities along the surface of the dash. The power contributions to the receiving cavity by resonant paths and non-resonant paths were investigated. It was observed that the resonant power contribution was significantly lower for the laminated steel dash as compared to the other dash panels. This resulted in higher transmission loss for laminated steel dash compared to the transmission loss for other two panels.</div>

This article presents a fluid–structure interaction modelling, based on a coupling between component mode synthesis or finite element and statistical energy analysis (SEA). The hybrid strategy is applied on a panel–cavity coupled system using a modal analysis with uncoupled modes of the subsystems and through a finite element model of the coupled system. The determination of the energy transfer parameters is then considered. The hybrid SEA model is then validated in the high-frequency domain by comparison with an SEA model. Finally, a parametric survey is offered through the established modelling and conclusions on its validity domain are drawn.

A bioengineering course is offered to senior students in the Mechanical Engineering major as an elective at Penn State Behrend in the first semester of their senior year. On average around 20 students enroll in this project-based course and 72% of their grade depends on three comprehensive group projects, one of which is to design a bone plate for a fractured femur. 6% of the grade comes from an individual application presentation, discussing different circumstances in which the concepts in mechanical engineering can be applied to solve/analyze biomedical problems. Background of physiology is introduced in the class while the majority of applicable engineering knowledge was learned in prerequisite courses, such as statics, dynamics, strength of materials, system dynamics, etc. Finite Element Analysis (FEA), an elective offered in both semesters of senior year, is not a prerequisite for this bioengineering course. Most of students enrolled in our bioengineering course either do not have FEA course or are taking FEA concurrently. Therefore in spite of its popularity in the areas related to biomechanics, thus far FEA has not been incorporated in this bioengineering course. Previously, the fractured femur project utilized the applicable theoretical analyses including those from statics, strength of materials, materials, and machine design. In Fall 2015, the instructor of bioengineering course made an effort to have an FEA study on femur and ensuing presentation be completed by a willing student. This student was taking FEA simultaneously with bioengineering, and was scheduled to present during the last week of semester. The instructor of FEA wrote an introductory manual of FEA procedures to analyze a healthy femur using a CAD model of the femur. Then the student used the manual to initiate the FEA study of the fractured femur. A major challenge to the student was the complicated loading and assembly feature of the bone plus the addition of bone plate. After a several week struggle and independent learning, while guided by the faculty members of bioengineering and FEA, the student was able to accomplish the FEA analysis and presented the results to the class. The discrepancy between FEA results and theoretical analysis was shown to be less than 5% in most results. The presentation also included FEA results for a diverse realistic loading and boundary conditions. Since all students had finished the fractured femur project, they actively participated in a meaningful discussion after the presentation. The completion of this project demonstrated the feasibility and importance of incorporating FEA in our bioengineering course. In addition, the presenting student was able to complete a working manual for the FEA analysis of the fractured femur, which can be passed to the future students of this course. A plan is to include FEA in the project as a bonus activity, and regarding the formation of project teams, the instructor would appoint a member who is enrolled in the FEA course. An assignment or project, based on the present FEA study of the femur, can also be incorporated in the FEA course offered at our campus.

<div class="htmlview paragraph">Automotive interior noise at mid and high frequencies is typically dominated by the airborne noise from acoustic sources that are spatially distributed around a vehicle. Each source is typically spatially compact (for example, a tire contact patch) but the source radiates sound that then propagates across the entire exterior surface of the vehicle. To characterize a source it is therefore necessary to know both the sound pressure level in the vicinity of the source and also the way in which sound from the source diffracts around the vehicle. The former depends on the details of the source, the latter typically depends on the overall vehicle geometry. When creating Statistical Energy Analysis (SEA) models of interior noise, the diffraction of airborne loads around a vehicle is often measured experimentally. Since SEA deals with averaged quantities, it is typically only necessary to obtain an approximate estimate of the space and frequency averaged sound pressure level across various surface regions of the vehicle (rather than a very detailed narrowband prediction of the diffraction).</div> <div class="htmlview paragraph">This paper provides a numerical investigation of the diffraction of typical acoustic sources around a vehicle using the Fast Multipole Boundary Element Method (FMM BEM). In particular, the paper investigates the sensitivity of the space and frequency averaged diffracted field to various modeling details (for example, geometric detail, mesh density/quality, surface impedance, frequency resolution etc.). The objective of the study is to investigate how much “detail” is needed in an FMM BEM model in order to get accurate predictions of the exterior acoustic loads for use in Statistical Energy Analysis (SEA) models. Significant reductions in computational expense can be obtained by matching the level of detail in the FMM BEM model to the level of detail required in the input results for an SEA model (and therefore avoiding “over computing” results that are subsequently frequency and space averaged).</div>

The theoretical development of a hybrid finite element method is presented. It combines conventional finite element analysis (FEA) with energy FEA (EFEA) to achieve a numerical solution to midfrequency vibrations. In the midfrequency range a system comprises some members that contain several wavelengths and some members with just a few wavelengths within their lengths. The former are considered long members, and they are modeled by the EFEA. The latter are considered short, and they are modeled by the FEA. The new formulation is based on deriving appropriate interface conditions at the joints between sections modeled by the EFEA and the FEA methods. The formulation for one flexural degree of freedom in colinear beams is presented in this fundamental development. The excitation is considered to be applied on a long member, and the response of the entire system is computed. Uncertainty effects are imposed only on the long members of the system. Validation cases for several configurations are presented. They compare closed-form analytical solutions to numerical results produced by the hybrid finite element method. Good correlation is observed for all analyses. The resonant behavior of the short members is captured correctly in the response of the system.

Predicting the vibration of the switched reluctance motor (SRM) is the key to design low vibration motor. In this paper, a comprehensive study of SRM vibration prediction from low frequency to high frequency is presented. For the low frequency vibration, the prediction model based on transfer function is proposed. The transfer function is calculated using modal analysis and mechanical harmonic analysis by 3D FEM. A detailed lookup table of radial force versus phase current and rotor angle is constructed. For the high frequency vibration, the statistical energy analysis (SEA) is employed. The SEA model of SRM including 4 subsystems is built. The internal loss factor and the coupling loss factors are measured through the power injection method; the analytical calculation method of input power is deduced. Both methods are verified by experiments. The comparison between the predicted results and the measured results demonstrates the accuracy of the transfer function based method and the SEA method for SRM low frequency and high frequency vibration prediction, respectively.

This paper describes an approach to control the vibration of flexible structures based on the power flow concept. The μ-synthesis is used to design the controller that optimizes the power flow and retains the robustness for the unmodeled dynamics. The power flow control is based on SEA (Statistical Energy Analysis), and it is able to augment the vibration damping by controlling the power flow that flows into the plant from the outside. The plant model for controller synthesis is found using the identification method called transfer function curve fitting. For identified model, the controller that achieves both performance and robustness is designed using μ-synthesis. Designed controller is evaluated by numerical analysis and experiments. As the results, it is shown that the proposed approach can design the controller with good performance and robustness.

In 1996 a series of standard test structures was conceived and manufactured by members of the Institute of Noise Control Engineering (INCE) [Cuschieri, Burroughs, and Carroll, Evaluation of Structure-Borne Noise Prediction Techniques Review, Proceedings of Noise-Con 98, April 1998, pp. 315–320]. The structures include a Lexan T-shaped beam and two ribbed panels of identical geometries but different materials: aluminum and lexan. In subsequent years, a wide variety of investigators from the U.S. and around the world have conducted experimental and numerical studies on the test structures, particularly on energy flow parameters, such as power input, power dissipation, and power flow. Most of the studies have been performed at low frequencies, and have shown phenomena such as the conversion of flexural wave power to longitudinal wave power at the T-beam joint and the nature of the structural intensity fields in the ribbed panels. Measurements and computations compare well. At high frequencies, Statistical Energy Analysis (SEA) techniques have shown that energy tends to become trapped in the drive leg of the T-beam. SEA studies on the ribbed panels show that the Lexan panel transmits less energy across the ribs than the aluminum panel does due primarily to differences in material loss factor.