Estimation of coupling loss factors for rectangular plates with different materials and junctions

Abstract

In statistical energy analysis, coupling loss factor is the essential parameter for vibro-acoustic analysis of complicated structures. The coupling loss factors have been estimated using energy-level difference method. The tightening torque applied at structural junction has been varied. Higher values of coupling loss factor have been observed for higher tightening torque on bolted junction. The coupling loss factors have been determined for various structural junctions of rectangular composite plates. The riveted and bolted junctions have been examined for composite plates in same plane and size. The coupling loss factors for bolted junction are relatively higher than that for riveted junction of composite plates. The values of coupling loss factors are found to increase with increasing tightening torque applied at structural junctions of composite plates. It is also noted that the experimental results of coupling loss factors for point junctions vary with changes in fiber orientations of composite plates. It is firmly believed that the various findings of the coupling loss factors in this article help for vibro-acoustic analysis of complicated structures.

Keywords

Statistical energy analysis coupling loss factor structural junctions power flow equation rectangular plates

Introduction

Statistical energy analysis (SEA) was evolved in the early 1960s to model high-frequency vibro-acoustic interaction in the rocket launch vehicle structures. The term SEA was coined due to the subsequent reasons. “Statistical” notes that the systems under study are assumed to be drawn from populations of similar construction with a known distribution of their dynamic parameters. This enables one to account for the manufacturing tolerance that exists in practical systems. “Energy” is the primary variable considered in SEA. Displacement, velocity, acceleration, and sound pressure are all can be derived from energy. SEA basically consists of computing the storage and flow of energy. The term “analysis” is used to underline that SEA is a framework of study and not a particular technique.

SEA parameters, such as modal density, damping loss factor, input power, and coupling loss factor (CLF), plays an important role for energy flow and path analysis of coupled subsystems. CLFs depend on type of junctions and junction variables between two coupled subsystems.¹ There are several methods, namely wave approach, modal approach,² power coefficient method,³ power injection method,^4,5 intensity technique,⁶ energy-level difference method,⁷ power spectral element method,⁸ statistical modal energy distribution analysis method,⁹ and receptance method,¹⁰ for the estimation of CLFs. Sablik¹¹ obtained transmission coefficient and CLF for an L-joint between two beams. Clarkson and Ranky¹² measured the dissipation loss factor and CLF of two plates at individual level and connected to each other, respectively. Manik¹³ presented the drawbacks for determining CLF of currently available methods. A new numerical method has been suggested by him for determining CLFs. Ming and Pan¹⁴ developed the formulae to predict the approximation error resulted from the use of the energy-level difference method. Improved energy ratio method was proposed by Gu and Sheng¹⁵ for the estimation of CLF. Wilson and Hopkins¹⁶ investigated errors that may occur with SEA and focused on the potential of advanced SEA for predicting vibro-acoustic analysis in the low- and mid-frequency ranges. Díaz-Cereceda et al.¹⁷ presented an automatic methodology for the identification of SEA subsystems within a vibro-acoustic system. Lightweight structures are demanded in today’s era due to its low weight and more rigidity. The inertia forces of lightweight structures are low which results in more vibration levels and noise levels. Täger et al.¹⁸ developed vibro-acoustic analytical simulation models that allow structural and sound radiation analysis adapted to the material of anisotropic multilayer composite plates. Average modal spacing, transmission coefficient, input power, and CLF were determined for symmetrically coupled laminated composite plates by Secgin.¹⁹

There are physical interconnections between subsystems through joints in machines such as automobiles, aerospace, and ships. To predict the response of such systems, determination of coupling loss factor for various systems with bolted, riveted, and screwed joints becomes necessary. This article focuses on estimation of CLF for idealized subsystems with different junctions and materials.

Theoretical computations of CLFs

The SEA method consists of first dividing a structure along its natural boundaries to form a set of “n” structural elements. The important equation used in the SEA is the power balance equation between coupled subsystems. For a subsystem “p” connected to many subsystems “s,” with “s” varying, as shown in Figure 1, the power balance equation can be written as

Π_{i n}^{p} = Π_{d i s s}^{p} + \sum_{s \neq p} Π_{c o u p}^{p s}

(1)

Figure 1.

Power flow between two systems.

The structure is considered under steady-state conditions, and all the power components stated are time-averaged quantities. The transmitted power and internal dissipated power are usually calculated as

\begin{array}{l} Π_{d i s s}^{p} = ω η_{p} E_{p} \\ Π_{c o u p}^{p} = ω η_{p s} E_{p} \end{array}

(2)

The power balance equations can be rearranged as

\begin{array}{l} Π_{i n}^{p} = ω η_{p} E_{p} + ω \sum_{s \neq p} η_{p s} E_{p} - ω \sum_{s \neq p} η_{s p} E_{s} \\ where p = 1, 2, 3, \dots, n \end{array}

(3)

When the number of systems are connected to each other, then the equations are written in matrix form and the total loss factor appears on the diagonal elements

[\begin{matrix} η_{11} & \dots & - η_{n 1} \\ ⋮ & ⋱ & ⋮ \\ - η_{1 n} & \dots & η_{n n} \end{matrix}] {\begin{matrix} ω E_{1} \\ ⋮ \\ ω E_{n} \end{matrix}} = {\begin{matrix} Π_{i n}^{1} \\ ⋮ \\ Π_{i n}^{n} \end{matrix}}

(4)

If the loss factors and power inputs are known as a function of frequency, one can solve for the steady-state energies by standard matrix techniques. Usually only one of the $\prod_{i n}^{p}$ is non-zero. Also, many of the $η_{p s}$ are zero because element “p” is not directly coupled to all other SEA elements. To produce frequency response curves for the various energies $E_{p}$ , equation (4) must be solved repeatedly for each frequency band in the range of frequencies under consideration.

Energy-level difference method

CLFs for coupled plates with different structural junctions were estimated by energy-level difference method. It is assumed that out of “N” subsystems, first subsystem is excited by impact hammer. First subsystem is directly connected to all other subsystems. For source subsystem to receiving subsystems, the CLF can be estimated as

η_{12} = [\begin{array}{l} (η_{22} + η_{21} + \dots + η_{2 N}) - η_{32} \frac{E_{3}}{E_{2}} \\ - η_{42} \frac{E_{4}}{E_{2}} - \dots - η_{N 2} \frac{E_{N}}{E_{2}} \end{array}] \frac{E_{2}}{E_{1}}

(5)

η_{12} = [η_{2} - η_{32} \frac{E_{3}}{E_{2}} - η_{42} \frac{E_{4}}{E_{2}} - \dots - η_{N 2} \frac{E_{N}}{E_{2}}] \frac{E_{2}}{E_{1}}

(6)

where $η_{12}$ is the CLF from subsystem 1 to 2, $E_{1}$ is the vibrational energy of subsystem 1, and $η_{2}$ is the damping loss factor of subsystem 2.

After neglecting the effect of non-source subsystems, equation (6) becomes

η_{12} = η_{2} \frac{E_{2}}{E_{1}}

(7)

Experimental arrangement for estimating SEA parameter

The experimental setup for estimating SEA parameters is shown in Figure 2. The test structure has been excited using impact hammer to induce transient signals. Impact testing has been performed by fixed hammer method. In this method, hammer is applied at a single point and response is measured at various locations on the structure. The impact hammer has been directly connected to data acquisition systems. Piezoelectric accelerometer has been mounted on the structure based on the IS 14883:2000/ISO 5348:1998 standard to measure the acceleration of that structure. The mounting position of the piezoelectric accelerometer on the structure has changed. Piezoelectric accelerometer has been mounted at nine locations on the structure. The electrical signal from piezoelectric accelerometer is passed to data acquisition system. B and K data acquisition system has been used to collect, record, and analyze the data. The sampling rate of the data acquisition system was 65.5 kSamples/s. The Windows-based computer with pulse software has been connected to data acquisition system. Pulse is a useful task-oriented vibration and sound analysis system. It offers the platform a range of PC-based measurement solutions from Bruel and Kjaer. Pulse Type 3560-B compact data acquisition unit up to four input channels has been used. Data acquisition system with pulse software has been used to post-process the measured data for estimating SEA parameters.

Figure 2.

Experimental setup for statistical energy analysis parameter estimation—transient excitation.

Experimental procedure of estimating CLFs for plates in same plane

Thin rectangular plates made up of aluminum and composites with similar dimensions have been joined with bolted junctions. The details of composite plates are given in Table 1. These plates have been connected to each other in same plane. For determining CLF, arrangement of instrumentation is shown in Figure 3. Joined plates have been hung by nylon threads. A rigid frame has been used for connecting one end of nylon thread. The impact hammer has been used to apply the force on one of the plates. The accelerometers have been mounted on the plates to capture the vibration signals of plates. Measurements have been taken by changing the position of the accelerometers on the plates. The accelerometers have been connected to data acquisition hardware. The data have been processed by Fast Fourier Transform (FFT) analysis software, and spectrums have been observed in frequency domain and time domain. For estimating CLFs of plates in same plane, the energy-level difference method was used. The vibration energy (E) was calculated corresponding to selected frequency from the mass of plate and velocity of plate. The values of velocity were taken from autospectrum of accelerometers 1 and 2 corresponding to selected frequency. The torque range has been used to apply the tightening torque at junction of the plates. The bolted and riveted junctions of composite plates are shown in Figure 4.

Table 1.

Material properties of composite plates (Dimensions-(400 × 300 × 2) mm³ and 12 layers).

S.No	Composition of material	Fiber orientation
1	E-glass fiber 10%, LY-556 epoxy resin 90%	Unidirectional, quasi-isotropic, and cross-ply
2	E-glass fiber 10%, LY-556 epoxy resin 90%, and filler 0.83%	Unidirectional and quasi-isotropic

Figure 3.

Arrangement of instrumentation for plates in same plane.

Figure 4.

Bolted and riveted junctions of composite plates.

Results and discussion

Reliability of SEA models depends on good estimate of CLFs. In SEA, subsystems coupling between them are represented by CLFs. CLF is a unique parameter at SEA and is connected with a central SEA result. To predict noise and vibration levels of a product at the design phase, CLF plays an important role. For some simple cases, CLF can be theoretically calculated. However, for most practical situations, mainly for those including different structural junctions such as screwed, bolted, and riveted, they cannot be calculated theoretically and must be obtained from experimental measurements.

In Figure 5, CLF result for point-connected plate is shown. The two plates connected to each other are in same plane. It has been observed from Figure 5 that CLF values are more for composite plates as compared with elastomer, thermoplastic, and non-ferrous metals.

Figure 5.

Coupling loss factors for point-connected plates.

CLF values for two connected plates of steel in same plane are shown in Figure 6. The joining lengths of two steel plates are 0.02, 0.2, and 0.9 m. It has been observed from Figure 6 that CLFs are more for 0.9 m compared with 0.2 and 0.02 m. Also, with the increase in frequency, CLF values decrease for all joining lengths (0.02, 0.2, and 0.9 m) of two steel plates having same dimensions.

Figure 6.

Coupling loss factors for point- and line-connected steel plates.

Screwed and bolted junctions of aluminum plates

It has been observed that CLFs for line-connected plates are more than that for point-connected plates. But for point connections, there is use of bolts, screws, rivets, hinges, and spot welding. It is difficult to predict CLFs of structures joined with different point connections. Two aluminum plates connected by screw and bolt junctions have been experimentally tested. In Figure 7, it has been observed that theoretical values of CLF for point-connected plates and experimental values of CLF for bolted and screwed connected plates are close to each other. Two plates connected in same plane show that screwed junction has lower values of CLF than bolted junction. A reason for this difference is the tightening force effect between two plates at junctions.

Figure 7.

Coupling loss factors for aluminum plates in same plane.

Effect of tightening torque on bolted junction of aluminum plates

The plates of aluminum have been joined with bolted junctions in same plane. Experimentally estimated CLFs for bolted junctions with different torques are shown in Figure 8. The CLFs for 0.9 kgf-m torque are relatively higher than that for 0.3 kgf-m torque for bolted junction of plates. Interfacial pressure increases at structural junctions due to increase in tightening torque applied at bolt. Because of this, connection between two plates becomes stiffer at higher tightening torque. Thus, two subsystems indicate strong coupling rather than weak coupling which gives maximum values of CLF.

Figure 8.

Comparison of experimental results of CLFs for different torques at junction of aluminum plates.

Bolted and riveted junctions of composite plates

Plates made up of composite materials with different fiber orientations have been experimentally tested. Results shown in Figure 9 reveal that the values of CLF for bolted junction (E-1B) are more than that for riveted junction (E-1R). The CLFs of composite plates with different fiber orientations are compared in Figure 10. It is found that the tightening torque applied at structural junctions affects the values of CLF. It increases with increase in tightening torque applied at bolted junction for plates connected in same plane. It has been observed from Figure 11 that at higher tightening torque value of 0.7 kgf-m, the values of CLF are more compared with 0.3 kgf-m torque. Due to the addition of graphene material in resin, there is variation in shrinkage and warpage of resin. It changes the modulus of elasticity and Poisson’s ratio values in x and y directions of composite plates, due to which CLF values relatively decrease as shown in Figure 12. The examined connection shows lower values of CLF for riveted junction as compared with bolted junction because of less transmission efficiency in riveted junction. In bolted junction, two plates are connected with nut and bolt arrangements, which results in stiffer junction rather than the riveted junction between two plates. The materials for rivets and bolts were different because of which the values of CLF are different.

Figure 9.

Coupling loss factors for bolted (E-1B) and riveted (E-1R) junctions of composite plate.

Figure 10.

Coupling loss factors for different fiber orientations of composite plates with bolted junction.

Figure 11.

Effect of tightening torque on coupling loss factors for unidirectional composite plates.

Figure 12.

Effect of graphene on coupling loss factors for unidirectional composite plates.

Conclusion

The SEA approach has been used for dynamic analysis of different structural elements. Estimation of the CLFs is the most important ladder in SEA. The experimental results obtained for CLFs have been validated by the analytical results. Using same experimental methodology, around 19 experiments have been conducted using different structural elements. Following conclusions are drawn based on the results obtained.

For point junctions, the CLFs significantly depend on type of junctions. The effects of bolted, screwed, riveted, and hinged junctions on CLFs have been experimentally verified. The values of CLF are higher for bolted junctions as compared with screwed junctions for aluminum rectangular plates connected in same plane. This can be attributed to the fact that bolted junctions have high transmission efficiency.

It can be concluded that higher values of CLF have been observed at higher values of tightening torques of bolted junctions for thin rectangular aluminum plates and the plates of composite materials connected in same plane. Interfacial pressure increases due to increase in preload of bolted junctions which results in the increase in CLF values.

As compared with riveted junctions, higher values of CLFs have been observed for bolted junctions for composite rectangular plates connected in same plane. When maximum vibro-acoustic energy is required to transmit from one subsystem to another subsystem, bolted junctions are preferably used rather than riveted junctions.

CLF values change with fiber orientation in composite plates. These values are significant at low- and mid-frequency ranges, whereas it is negligible at high-frequency range.

Based on experimentations with composite plates, it is observed that the CLF is higher for cross-ply fiber orientation composite plates in comparison with unidirectional fiber orientation composite plates. It can be concluded that designer can choose either cross-ply fiber orientation or unidirectional fiber orientation based on desired CLF with all other considerations.

CLF values for unidirectional fiber orientation composite plate without graphene are higher than unidirectional fiber orientation composite plate with graphene. Thus, it is possible to use estimated CLFs for solving power flow equations to understand vibro-acoustic energy flow in different structures. This knowledge helps for effective design of structures in vibro-acoustic environment.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mandale Maruti Bhagwan

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