Coupling Loss Factors Research Articles

Finite panel sound radiation through an absorbent layer

It is often required to predict the sound radiated from a vibration through an absorbent layer (trim). Statistical energy analysis may be applied in such a case. The required radiation coupling loss factor is estimated from the radiation efficiency. To get the radiation efficiency for the trimmed panel, that of the panel (into air, without trim) is combined with the transmission loss of the absorbent layer to give a corrected radiation efficiency. Although this procedure is quite straightforward, it may produce poor results. An alternative and more accurate procedure for the estimation of the radiation efficiency is presented. The absorbent layer is fully considered in the calculation, which is done using a wave number integration technique. The results from both methods show remarkable differences.

An approach for modeling spot-welded joints in an energy finite element formulation

An emerging simulation technology is the energy finite element analysis (EFEA) [P. Cho, ‘‘Energy Flow Analysis of Coupled Structures,’’ Ph.D. dissertation, Purdue University, 1993; O. M. Bouthier and R. J. Bernhard, ‘‘Models of space averaged energetics of plates,’’ AIAA J. 30(3) (March 1992)]. In order for this method to be applicable to automotive structures it is important that it be suitable for modeling spot-welded joints. This work presents a development effort in computing the energy transmitted through spot-welded joints, and utilizing the information in an EFEA formulation. It is based on principles utilized for deriving coupling loss factors for statistical energy analysis (SEA) from conventional finite element models [C. Simmons, ‘‘Structure-borne sound transmission through plate junctions and estimates of SEA coupling loss factors using the finite element method,’’ J. Sound Vib. 144(2), 215–227 (1991)]. There are two components to the new development: (i) Deriving numerically information about the energy ratio between spot-welded members. (ii) Incorporating information for the energy transferred between members in the EFEA analysis. Technical information and comparison of numerical results to test data will be presented. [Work supported by Automated Analysis Corp. and Ford Motor Co.]

SEA modeling and analysis of stowage bin noise isolation and reduction

The statistical energy analysis method is applied to the noise prediction and analysis of large passenger airplane stowage bin isolation. The purpose is to reduce the cabin noise generated from the external turbulent boundary layer and transmitted from the stowage bins via their supports. AutoSEATM is adopted for the modeling and analysis of the coupled structural-acoustic problem where the connection between the structural fuselage and the stowage bins is represented and implemented by a coupling loss factor (CLF). A significant amount of noise level reduction can be achieved by using elastomeric rod ends to replace the metallic rigid connection between the structural fuselage and the stowage bins. It is also possible to obtain a significant amount of weight reduction using the stowage bin isolators instead of the blanket applied to the cabin crown. Some analysis results have shown good agreement with flight test data. This modeling and analysis provides an effective way of reducing cabin noise for large commercial airplanes without any weight penalty. Economic gains will be obtained from both noise reduction and weight saving. [Support for this effort from B. Cimerman of Vibro-Acoustics Sciences, Inc. and Evan B. Davis of Boeing is gratefully acknowledged.]

SEA analysis of the T-beam and acoustic cavity using alternative SEA approaches

SEA results have been generated for the T-beam structure using two different approaches—a home grown SEA solution and a solution generated using a commercial SEA package. The results generated from these two approaches, while both generally matching the experimental mean results, are different. In this paper, possible reasons for the differences in the results are explored by analyzing the determined coupling loss factors, other components in the solution and the solution in general. The same type of analysis is performed for the acoustic cavity structure. In this case as well, SEA results are generated using the alternative SEA approaches. From a comparison of the full and intermediate SEA results obtained from the different approaches, the possible reasons for the differences are investigated. The two main conclusions of this study are that, (a) when using SEA as a predictive tool, very much depends on the input model and the basis of the solution; (b) the predictive results have to be treated with some caution. [Work sponsored by NASA Langley.]

THE TRANSMISSION OF VIBRATION THROUGH A COUPLED PERIODIC STRUCTURE

Complex built-up structures often consist of structural elements of a periodic nature. Examples of these include plate or shell elements reinforced by stiffeners at regular intervals. This paper describes a theoretical method for evaluating the coupling loss factor (CLF) of a coupled periodic structure for use in the application of statistical energy analysis. The structure considered consists of a plate with periodic stiffeners coupled at right angles to a uniform plate and the method of travelling wave analysis was used for the investigation. An experimental program was conducted to verify the CLF of the structure. Results from the measurements of input power and vibrational energy were used in a numerical procedure to determine the internal loss factors and CLF. The experimental results show good agreement with theoretical predictions at low frequencies. However, some discrepancies between the theoretical and experimental results were observed at high frequencies, possibly due to the frequency dependent nature of the assumptions involved in the theoretical analysis.

AN ALTERNATIVE TO THE SEA COUPLING LOSS FACTOR: RATIONALE AND METHOD FOR EXPERIMENTAL DETERMINATION

For a combination of theoretical and phenomenological reasons, it is argued that the conventional SEA coupling loss factor is not the most appropriate form of coefficient for relating energy transfer between SEA subsystems to their vibrational states. An alternative ‘power transfer coefficient’ is suggested and an experimental method for its determination, which obviates the need to measure input powers, is proposed.

POWER TRANSMISSION BETWEEN TWO FINITE BEAMS AT LOW MODAL OVERLAP

The coupling loss factors frequently used in Statistical Energy Analysis are frequently based on a wave approach that assumes that the transmitted waves returning to the junction are uncorrelated with the directly transmitted waves. This assumption is generally not fulfilled for structures at low frequencies where the response exhibits distinctly well separated resonances. A method is presented here for calculation of an improved coupling loss factor for one-dimensional subsystems. By applying statistical considerations to an analytical expression of the net power transmission it is possible numerically to calculate an average value of the ratio of reflected to incident wave amplitude at the junction. From this ratio an improved coupling loss factor can be calculated which takes into account the re-radiation and re-injection of power which normally occurs between two reverberant fields. This improved coupling loss factor has the following features: it is valid at low modal overlap, it is independent of the exact boundary conditions, it depends upon the loss factor of the receiving subsystem, and its calculation is not dependent on the strength of the coupling. To illustrate the method, a specific beam system is considered in the frequency range where the modal overlap is small. Coupling loss factors are calculated and are employed in a traditional SEA model to predict mean square velocities for each beam element with good accuracy.

THE SEA OF TWO COUPLED PLATES: AN INVESTIGATION INTO THE EFFECTS OF SUBSYSTEM IRREGULARITY

A system comprising two edge-coupled plates is considered. Theoretical predictions of the coupling power and coupling loss factor are made using traditional, “asymptotic” SEA wave theory and an analytical wave solution application to rectangular plates. These are compared to numerical frequency averages found by FEA of the complete system. Results are presented for variously shaped plates and different levels of damping. If the damping is large enough (i.e., for “weak” coupling) the response is independent of the shape of the plates. For lighter damping (i.e., “strong” coupling) the response depends significantly on the specific geometry of each plate: the coupling power is often substantially less than that predicted by traditional SEA theory, which also overpredicts the coupling loss factor. Both the coupling power and coupling loss factor are least for rectangular plates, for which the irregularity of the subsystems is least. The reasons for this behaviour are attributed to wave coherence or, in modal terms, to localisation of the global modes of the structure within one or other subsystem. A parameter γ0is given which provides an estimate of the strength of coupling. This parameter arises in the analysis of coupled rectangular plates and gives a conservative estimate for irregularily shaped plates.

Rudimentary statistical energy analysis and structural fuzzies

Structural fuzzies (Sfs) is a dynamic complex that is attached to a primary structure in order to provide a higher degree of dissipation. The dissipation is with respect to the power that is injected into the driven primary structure. It is argued that care needs to be exercised in the definition of the loss factor that is associated with this dissipation. This definition is examined and discussed; then, the role of the Sfs is analyzed in terms of a rudimentary statistical energy analysis (SEA). It is shown that the effectiveness of the dissipation is enhanced if the coupling of the Sfs to the primary structure is strong enough to approximate equipartitioning of modal energies, if the modal density in the Sfs either approximates or exceeds that in the primary structure and, finally, if the inherent loss factor in the Sfs is sufficiently higher than that in the primary structure. The definitions of strong enough coupling, modal densities, and inherent loss factors will be given. Examples of Sfs will be cited and discussed.

THE DIRECT AND INDIRECT POWER FLOWS OF THREE NON-CONSERVATIVELY SERIES COUPLED OSCILLATORS

The formulae of power flows among three non-conservatively series coupled oscillators excited by random forces have been derived and analyzed. The analysis results show that the theory of the power flows of both three conservatively series coupled oscillators and two non-conservatively coupled oscillators are special examples of the theory of the power flows of three non-conservatively series coupled oscillators, and the reciprocity of the indirect power flows among three non-conservatively series coupled oscillators is untenable. The relationships between the power flows and the parameters of coupling have been studied by means of the numerical method. The numerical results show that the indirect power flows correspond to the resonant transmission between the indirectly coupled oscillators, and the indirect power flows should not be ignored in strong coupling conditions. Using the definition of the direct coupling loss factors (coupling loss factor in classical Statistical Energy Analysis) and the power flows of three non-conservatively series coupled oscillators, the influence of the indirectly coupled oscillators on the direct coupling loss factors has been studied by means of the numerical method. It is shown that the existence of the indirectly coupled oscillators has great influence on the direct coupling loss factors. The direct coupling loss factors in multi-coupled oscillator system are different from the coupling loss factors of the two coupled oscillators, and it is necessary to modify the direct coupling loss factors when the energy balance equations of SEA are modified. Experimental verification has been carried out with a floating raft isolation system. The agreement between the results estimated by SEA and those experimentally measured are good.

A NEW METHOD FOR DETERMINING COUPLING LOSS FACTORS FOR SEA

The objective of this communication is to present a method of determining coupling loss factors by the power injection method which is easy to compute, accurate and which is not affected by the strength of the coupling between subsystems. The method of determining coupling loss factors presented here assumes the possibility of accurately determining the power injected to subsystems

EFFECTIVE INTERNAL LOSS FACTORS AND COUPLING LOSS FACTORS FOR NON-CONSERVATIVELY COUPLED SYSTEMS

What causes negative loss factors in measuring SEA parameters by the power injection method? Analytical results in this paper show that negative loss factors are caused by non-conservative coupling, and non-conservative coupling does not always increase the effective internal loss factors (EILFs). To study the power flow of non-conservatively coupled systems, some new SEA models for non-conservatively coupled systems were suggested by researchers. However, the SEA parameters for non-conservatively coupled systems have not been extensively discussed by using the model of classical Statistical Energy Analysis (CSEA) up to now. The aim of this study is to give insight into the problem of the energy balance mechanism of non-conservatively coupled systems by using both the model and the energy balance equations in CSEA. A method of calculating EILFs and coupling loss factors (CLFs) for non-conservatively coupled machine structures is introduced. The possibility of causing negative loss factor is investigated by analysis of two non-conservatively coupled oscillators. Furthermore, the influences of the coupling stiffness and the coupling damping on EILFs and CLFs are investigated in detail. Finally, an application example is included to demonstrate the accuracy of the method.

The measurement of coupling loss factors using the structural intensity technique

The structural intensity technique can directly measure and distinguish power transmissions between different structural wavetypes across a joint, and its applications are numerous in practical engineering. In this paper, a new method is proposed for the measurement of coupling loss factors (CLFs) using the structural intensity technique. An outline of the method’s theory is also given. Theoretically the proposed method gives an approximate estimation of CLFs. The approximate error depends on the ratios of the effective modal overlap factor of the receiving subsystem to the modal overlap factors of the subsystems coupled through the joint of interest. If the effective modal overlap factor of the receiving subsystem is large compared with the modal overlap factors of the coupled subsystems, the approximate error is small and negligible. A series of measurements were carried out to verify the outlined theory. The results measured on the joints with two subsystems showed that this proposed method can be used to accurately measure CLFs when the modal overlap factor of the receiving subsystem is larger than that of the source subsystem. The approximate error decreases with increasing frequency and the modal overlap factor ratio of the receiving subsystem to the source subsystem.

End correction in vibrational power flow measurement through joints

In the determination of coupling loss factors for complex joints, such as adhesive or bearing joints, analytical methods may not provide satisfactory results. The experimental approaches are normally preferred. In the measurements of the coupling loss factors for the joints, it is assumed that the end reflection can be neglected. However, it is difficult to create such nonreflection termination in the experiments. In this paper, the effects of end reflection in the determination of the coupling loss factors were studied. A new method using the measured reflection coefficients of the termination to correct the measured coupling loss factors was proposed. The experimental study was carried out in the stepped beams and adhesively bonded beams. The experimentally corrected results were compared with the theory.

Diffuse transmission of structure-borne sound at periodic junctions of semi-infinite plates

Vibration attenuation at junctions of point connected plates is an important aspect of structure-borne sound transmission in many realistic structures. The vibrational response of point connected plates can be predicted using statistical energy analysis (SEA), where coupling loss factors are calculated based on mobility formulations. However, in the process of deriving the coupling loss factors for point connected systems, a number of assumptions and simplifications were introduced. In this paper, a more advanced calculation model, based on a wave approach for elastically coupled semi-infinite plates, is presented. Local rigid connections are modeled using an elastic interlayer characterized by a space-dependent stiffness. Periodic boundary conditions are assumed at the junction and the plate response is described using a Fourier decomposition technique. Calculation results obtained for two different junction geometries demonstrate the influence of the distance between the point connections on the structure-borne sound transmission. Finally, a comparison between a number of numerical and experimental results illustrates the ability of the model to predict the influence of the density and the width of the point connections with good accuracy.

DETERMINATION OF STATISTICAL ENERGY ANALYSIS LOSS FACTORS BY MEANS OF AN INPUT POWER MODULATION TECHNIQUE

An input power modulation technique is proposed as an alternative to the widely used steady-state power injection method for thein-situdetermination of dissipation and coupling loss factors of subsystems selected for an SEA model of a physical system. There is no requirement to measure input power and, in principle, response measurements are made at only one point on each subsystem. Theory is developed for a two-subsystem model. An input power modulation device is described, and the loss factor estimates resulting from its application to systems comprising two structurally coupled plates and two acoustically coupled rooms are presented and compared with those from steady state tests. A phenomenon of energy response modulation phase dispersion is observed at modulation frequencies in excess of about 10 Hz.

VIBRATIONAL ENERGY FLOW ANALYSIS USING A SUBSTRUCTURE APPROACH: THE APPLICATION OF RECEPTANCE THEORY TO FEA AND SEA

A method for studying the vibrational energy flows through structures based on receptance theory is presented. The structures are considered to be made up of subsystems, which may, in turn, be substructures modelled by using finite element analysis (FEA), each having been separately analyzed for its eigenvalues and eigenvectors. The method may be classified as a form of substructuring using free{puen}free interface conditions. It differs significantly from traditional substructuring in its use of matrices composed of the substructure Green functions, evaluated as summations over their uncoupled modes, to obtain the displacement contributions of the external and boundary coupling forces; also, the method can readily take into account variations in substructure damping. The proposed method can readily take into account variations in substructure damping. The proposed method additionally calculates the time averaged substructure vibrational energy levels by evaluating the balance between input and dissipated energies and the energy transfers through coupling nodes. It is therefore of particular interest when using FEA substructures to carry out statistical energy analysis (SEA) studies, since the resulting energy data can be readily applied to evaluate SEA parameters such as coupling loss factors.The formulation developed has been implemented as a computer program which uses substructure modal information from a commercial FEA package and then combines this to predict the response of the global model. Two simple examples involving two- and three-dimensional FEA models built from beam elements are presented, which show that there is good agreement between the substructure based predictions and the equivalent global models. Moreover, the method presented is computationally more efficient than using global FEA models, even when all the substructure modes are used. The method is then applied to study the SEA coupling loss factors of two further example structures: first, two thin plates joined along an edge at right angles are examined; and then a second, more complicated structure formed from a section of a large marine vessel is studied. The approach is shown to be applicable to any general finite element model which is considered as an SEA subsystem.

AN INVESTIGATION OF THE COUPLING LOSS FACTOR FOR A CYLINDER/PLATE STRUCTURE

This paper presents a theoretical and experimental study of the transmission of vibration through a two element structure which consists of a cylindrical shell couple to an end plate. The first part of this paper deals with the derivation of coupling loss factor (CLF) for the study of vibration transmission using Statistical Energy Analysis (SEA). The derivation is based on travelling wave analysis and the assumption of equipartition of energy amongst resonant modes. Numerical results for three examples of cylinder/plate structures are presented and compared with those of their equivalent plate/plate structures. The results show that the CLF values of the cylinder/plate structures asymptote to their equivalent plate/plate structures above the ring frequency of the cylinder. In the second part of this paper, an experimental program for measuring the CLF of an example cylinder/plate structure is described. Experimental results of the cylinder/plate structure show good agreement with theoretical predictions and confirm the validity of the present formulation of CLF.

A SEA-LIKE APPROACH FOR THE DERIVATION OF ENERGY FLOW COEFFICIENTS WITH A FINITE ELEMENT MODEL

The Finite Element method is combined with a Statistical Energy Analysis-like (SEAL) energy flow balance to derive the power transmission between two thin plates. A fundamental difference between SEA and the SEAL procedure is that former is applied to ensembles, while the latter addresses the individual case. Energy Flow Coefficients (EFCs) are derived and explicit use of the non-resonant part of the kinetic energy is made. It is demonstrated that the EFC, as opposed to the SEA Coupling Loss Factor (CLF), can become negative at some frequencies. The EFC exhibits characteristics that are individual for each case examined, although it tends towards the CLF at high frequencies. It is also shown that the SEAL energy flow balance can be used for narrow bands, and at any frequency.

INVESTIGATION OF DAMPING EFFECTS ON STATISTICAL ENERGY ANALYSIS OF COUPLED STRUCTURES

The effects of damping on energy sharing in coupled systems are investigated. The approach taken is to compute the forced response patterns of various idealised systems, and from these to calculate the parameters of Statistical Energy Analysis model for the systems using the matrix inversion approach [1]. It is shown that when SEA models are fitted by this procedure, the values of the coupling loss factors are significantly dependent on damping except when it is sufficiently high. For very lightly damped coupled systems, varying the damping causes the values of the coupling loss factor to vary in direct proportion to the internal loss factor. In the limit of zero damping, the coupling loss factors tend to zero. This is a view which contrasts strongly with ‘classical’ SEA, in which coupling loss factors are determined by the nature of the coupling between subsystems, independent of subsystem damping. One implication of the strong damping dependency is that equipartition of modal energy under low damping does not in general occur. This is contrary to the classical SEA prediction that equipartition of modal energy always occurs if the damping can be reduced to a sufficiently small value. It is demonstrated that the use of this classical assumption can lead to gross overestimates of subsystem energy ratios, especially in multi-subsystem structures.