August 2011 

NASA/TM–2011-217171 

Statistical Energy Analysis (SEA) and Energy 

Finite Element Analysis (EFEA) Predictions for 

a Floor-Equipped Composite Cylinder 

Ferdinand W. Grosveld 

Lockheed Martin, Hampton, Virginia 

Noah H. Schiller and Randolph H. Cabell 

Langley Research Center, Hampton, Virginia 
NASA STI Program . . . in Profile

 Since its founding, NASA has been dedicated to 

the advancement of aeronautics and space science. 

The NASA scientific and technical information (STI) 

program plays a key part in helping NASA maintain 

this important role. 

 The NASA STI program operates under the 

auspices of the Agency Chief Information Officer. It 

collects, organizes, provides for archiving, and 

disseminates NASA’s STI. The NASA STI program 

provides access to the NASA Aeronautics and Space 

Database and its public interface, the NASA Technical 

Report Server, thus providing one of the largest 

collections of aeronautical and space science STI in 

the world. Results are published in both non-NASA 

channels and by NASA in the NASA STI Report 

Series, which includes the following report types: 

 TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

programs and include extensive data or 

theoretical analysis. Includes compilations of 

significant scientific and technical data and 

information deemed to be of continuing 

reference value. NASA counterpart of peerreviewed formal professional papers, but having 

less stringent limitations on manuscript length 

and extent of graphic presentations. 

 TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or of 

specialized interest, e.g., quick release reports, 

working papers, and bibliographies that contain 

minimal annotation. Does not contain extensive 

analysis. 

 CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees. 

 CONFERENCE PUBLICATION. Collected 

papers from scientific and technical 

conferences, symposia, seminars, or other 

meetings sponsored or co-sponsored by NASA. 

 SPECIAL PUBLICATION. Scientific, 

technical, or historical information from NASA 

programs, projects, and missions, often 

concerned with subjects having substantial 

public interest. 

 TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and 

technical material pertinent to NASA’s mission. 

 Specialized services also include creating custom 

thesauri, building customized databases, and 

organizing and publishing research results. 

 For more information about the NASA STI 

program, see the following: 

 Access the NASA STI program home page at 

http://www.sti.nasa.gov

 E-mail your question via the Internet to 

help@sti.nasa.gov

 Fax your question to the NASA STI Help Desk 

at 443-757-5803 

 Phone the NASA STI Help Desk at 

443-757-5802 

 Write to: 

 NASA STI Help Desk 

 NASA Center for AeroSpace Information 

 7115 Standard Drive 

 Hanover, MD 21076-1320
National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23681-2199 

 

August 2011 

NASA/TM–2011-217171 

Statistical Energy Analysis (SEA) and Energy 

Finite Element Analysis (EFEA) Predictions for 

a Floor-Equipped Composite Cylinder 

Ferdinand W. Grosveld 

Lockheed Martin, Hampton, Virginia 

Noah H. Schiller and Randolph H. Cabell 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an 

official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics 

and Space Administration. 
1

TABLE OF CONTENTS

INTRODUCTION ........................................................................................................................................4

COMPOSITE CYLINDER CONFIGURATION.........................................................................................4

NUMERICAL MODELS OF THE COMPOSITE CYLINDER..................................................................5

Finite Element Model ................................................................................................................. 5

Statistical Energy Analysis Model.............................................................................................. 5

Energy Finite Element Analysis Model...................................................................................... 6

COMPARISON BETWEEN SEA AND EFEA PREDICTIONS ................................................................7

Structural Response Predictions ................................................................................................. 7

Interior Acoustic Sound Pressure Level Predictions .................................................................. 9

CONCLUSIONS.........................................................................................................................................16

REFERENCES ...........................................................................................................................................17

TABLES .....................................................................................................................................................19

FIGURES....................................................................................................................................................22

LIST OF TABLES

Table 1. Volume, bounding area and total perimeter of the acoustic cavities above and below the floor..19

Table 2. Mechanical material properties of the composite cylinder components.......................................19

Table 3. Beam section properties................................................................................................................19

Table 4. Isotropic mechanical material properties for modeling the composite beam components in VA 

One..............................................................................................................................................................19

Table 5. Damping loss factors for the main subsystems of the floor-equipped composite cylinder...........20

Table 6. VA One computed modes in band for a single bay and the sidewall structure (fifteen times the 

area of the single bay), with and without curvature....................................................................................20

Table 7. Length, width, area and perimeter values for a single bay and for the sidewall structure (fifteen 

times the area of the single bay)..................................................................................................................21

Table 8. Flat and curved logarithmic radiation efficiency ratios of a single bay and the sidewall structure 

(fifteen times the bay area). ........................................................................................................................21

LIST OF FIGURES

Figure 1. Composite cylinder supported by two endcaps and held together by wood covered (to provide 

damping) tension rods.................................................................................................................................22

Figure 2. Cross-section of the stiffened, floor-equipped composite cylinder and a detailed view of the 

rivet-bonded frame-stringer-shell intersection............................................................................................22

Figure 3. Wireframe isometric view of the floor-equipped composite cylinder energy finite element model 

with one endcap removed for clarity...........................................................................................................23
2

Figure 4. VA One (SEA) model of the floor-equipped stiffened composite cylinder without the front 

endcap and the acoustic cavity subsystems.................................................................................................23

Figure 5. VA One area, line and point junctions with the area junctions between the acoustic cavities and 

the front endcap removed for clarity...........................................................................................................24

Figure 6. Comet Enflow point, line and area joints of the composite cylinder energy finite element model.

....................................................................................................................................................................24

Figure 7. The unit power excitation bay (X) and the fourteen response bays (1-14)..................................25

Figure 8. Sidewall structure subsystem (area of fifteen bays). ...................................................................25

Figure 9. VA One (SEA) root-mean-square velocity predictions on the sidewall structure (same at all 

locations).....................................................................................................................................................26

Figure 10. Comet Enflow (EFEA) root-mean-square velocity predictions in the imaginary bays (X and 5-

8) of the sidewall structure subsystem. .......................................................................................................26

Figure 11. Comet Enflow (EFEA) predicted root-mean-square velocities at nodes away from the unit 

input power at Node #1538 compared to VA One (SEA) predictions for the sidewall structure subsystem 

and five smaller bay subsystems in the 200 Hz one-third octave band.......................................................27

Figure 12. Comet Enflow (EFEA) predicted root-mean-square velocities at nodes away from the unit 

input power at Node #1538 compared to VA One (SEA) predictions for the sidewall structure subsystem 

and five smaller bay subsystems in the 2000 Hz one-third octave band.....................................................27

Figure 13. VA One (SEA) and Comet Enflow (EFEA) area averaged root-mean-square velocity 

predictions of the sidewall structure ...........................................................................................................28

Figure 14. Sidewall structure divided into fifteen bay subsystems by longitudinal stringers and ring 

frames..........................................................................................................................................................28

Figure 15. VA One (SEA) root-mean-square velocity predictions for the fifteen bay subsystems separated 

by longitudinal stringers and ring frames. ..................................................................................................29

Figure 16. Comet Enflow (EFEA) predicted root-mean-square velocities of the sidewall structure 

including the longitudinal stringers and ring frames...................................................................................29

Figure 17. VA One (SEA) root-mean-square velocity predictions for the fifteen bay subsystems separated 

by longitudinal stringers and ring frames but their responses excluded in the solution matrix. .................30

Figure 18. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem X root-mean-square 

velocities for bay subsystems separated by longitudinal stringers and ring frames....................................30

Figure 19. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 5 root-mean-square 

velocities for the bay subsystems separated by longitudinal stringers and ring frames..............................31

Figure 20. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 6 root-mean-square 

velocities for the bay subsystems separated by longitudinal stringers and ring frames..............................31

Figure 21. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 7 root-mean-square 

velocities for the bay subsystems separated by longitudinal stringers and ring frames..............................32

Figure 22. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 8 root-mean-square 

velocities for the bay subsystems separated by longitudinal stringers and ring frames..............................32

Figure 23. VA One (SEA) predicted root-mean-square velocity responses in bay subsystems X, 5-8 of the 

floor-equipped composite cylinder. ............................................................................................................33

Figure 24. VA One (SEA) predicted root-mean-square velocities in the bay subsystems X, 5-8 of the 

floor-equipped composite cylinder without the beam responses in the SEA solution matrix. ...................33

Figure 25. Comet Enflow (EFEA) predicted root-mean-square velocities in the bay subsystems X, 5-8 of 

the floor-equipped composite cylinder. ......................................................................................................34
3

Figure 26. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the 

excitation bay subsystem X of the floor-equipped composite cylinder. .....................................................34

Figure 27. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the 

excitation bay subsystem 5 of the floor-equipped composite cylinder. ......................................................35

Figure 28. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the 

excitation bay subsystem 6 of the floor-equipped composite cylinder. ......................................................35

Figure 29. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the 

excitation bay subsystem 7 of the floor-equipped composite cylinder. ......................................................36

Figure 30. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the 

excitation bay subsystem 8 of the floor-equipped composite cylinder. ......................................................36

Figure 31. VA One (SEA) and Comet Enflow (EFEA) predicted sound pressure levels in the upper 

acoustic cavity subsystem of the floor-equipped composite cylinder.........................................................37

Figure 32. Two subsystem configuration of a bay and an acoustic cavity excited by unit input power on 

the bay.........................................................................................................................................................37

Figure 33. Predicted sound pressure levels in the acoustic cavity for unit power excitation of a singly 

curved shell (VA One - SEA) or a plate to which curvature was applied (Comet Enflow - EFEA). .........38

Figure 34. Predicted root-mean-square velocities for unit power excitation of a singly curved shell (VA 

One - SEA) or a plate element to which curvature was applied (Comet Enflow - EFEA). ........................38

Figure 35. VA One (SEA) and Comet Enflow (EFEA) predicted sound pressure levels in the acoustic 

cavity for unit power excitation of the flat shell subsystem........................................................................39

Figure 36. VA One (SEA) predicted modal density for a singly curved orthotropic shell and a flat 

orthotropic plate..........................................................................................................................................39

Figure 37. VA One (SEA) predicted radiation efficiency for a singly curved shell and a flat plate. .........40

Figure 38. VA One (SEA) predicted coupling loss factor for a singly curved shell or a flat plate connected 

to the acoustic cavity...................................................................................................................................40

Figure 39. VA One (SEA) predicted coupling loss factor for the acoustic cavity connected to a singly 

curved shell or a flat plate...........................................................................................................................41

Figure 40. VA One (SEA) predicted sound pressure levels in the acoustic cavity for unit power excitation 

of the flat or curved shell bay subsystem....................................................................................................41

Figure 41. Comet Enflow (EFEA) predicted sound pressure levels in the acoustic cavity for unit power 

excitation of the flat or curved shell bay subsystem. ..................................................................................42

Figure 42. Sidewall structure subsystem (area of fifteen bays; no stiffeners) connected to the acoustic 

cavity subsystem. ........................................................................................................................................42

Figure 43. VA One (SEA) and Comet Enflow (EFEA) predicted sound pressure levels in the acoustic 

cavity for unit power excitation applied to a flat or curved sidewall structure with an area equal to fifteen 

single bays...................................................................................................................................................43

Figure 44. Cavity sound pressure level differences due to unit power excitation of the flat or curved 

sidewall structure (fifteen times the area of a single bay) as predicted by VA One (SEA) and Comet 

Enflow (EFEA). ..........................................................................................................................................43

Figure 45. VA One (SEA) predicted radiation efficiency for a flat or curved sidewall structure with a total 

area equal to fifteen single bays..................................................................................................................44

Figure 46. Cavity sound pressure level differences due to unit power excitation of the flat or curved, bay 

or sidewall structure (fifteen times the bay area) as predicted by VA One (SEA) and compared to 10 

log(σbay/σsidewall). ..........................................................................................................................................44
4

INTRODUCTION

Finite Element Analysis (FEA), Boundary Element Analysis (BEA) and Statistical Energy Analysis 

(SEA) have been used for several decades to analyze vibro-acoustic problems and have matured to the 

point that reliable predictions can be expected cognizant of the limitations of each method. These 

prediction tools have been validated for numerous structurally and/or acoustically excited aerospace, 

automotive, shipbuilding, and architectural structures in anechoic, reverberant and in-situ acoustic 

environments. It is generally accepted that the discrete FEA and BEA numerical analyses are applicable 

towards the low end of the frequency scale, where the individual modal response is dominant. SEA is 

applied at frequencies where the modal density and the modal overlap are high. Predictions of the vibroacoustic response made by the FEA and SEA analysis tools were compared with structural and acoustic 

measurements on a floor-equipped, composite cylinder in the 1980s (References 1-6). This composite test 

cylinder was originally developed to resemble a composite aircraft fuselage for impact dynamics and 

acoustic transmission verification (Reference 7). In the last few decades new measurement equipment, 

improved experimental tools, and increased data storage have become available to enhance the 

experimental characterization of a structure such as the composite cylinder. Innovative hybrid analysis 

and prediction tools have emerged to take advantage of both finite element modeling and statistical 

analysis. A new set of measurements and predictions of the vibro-acoustic response of the composite 

cylinder is being pursued to validate these new analysis tools and enhance the physical understanding of 

the problem.

Comet Enflow developed by the Comet Technology Corporation is a commercially available, high 

frequency vibroacoustic analysis software founded on Energy Finite Element Analysis (EFEA) and 

Energy Boundary Element Analysis (EBEA). The EFEA is based on deriving governing differential 

equations in terms of energy density variables and then employing a finite element approach to 

numerically solve them. In the EFEA, both the structural and acoustic domains are modeled using finite 

elements. Joint elements based on power flow continuity across the junction are utilized at the interface of 

geometric and material discontinuities. In this method the same FEA mesh developed for structural and 

acoustic low frequency analyses can be used for the high frequency solutions. In a program managed by 

the NASA Langley Research Center Structural Acoustics Branch, EFEA as implemented in Comet 

Enflow 2009 is being validated on the floor-equipped composite cylinder by comparing EFEA 

vibroacoustic response predictions with SEA and experimental results. The SEA predictions are made 

using the commercial software program VA One 2009 from ESI Group. Comet Enflow predicted 

response for the power flow from an acoustic space to a structural subsystem was investigated in 

Reference 8. The purpose of the current study is to compare VA One and Comet Enflow predictions for 

the floor-equipped composite cylinder with respect to the root-mean-square velocity of the shell and the 

sound pressure levels in the cavity above the floor. The predictions will be compared with experimental 

results in a future effort. The frequency region of interest for this study covers the one-third octave bands 

with center frequencies from 100 Hz to 4000 Hz. 

COMPOSITE CYLINDER CONFIGURATION

The composite cylinder test article (Figure 1) is a 3.658 m long filament-wound, stiffened cylinder 

with a 1.676 m diameter. The composite material of the cylinder shell consists of carbon fibers embedded 

in an epoxy resin. The nine-layer ply-sequence of the cylinder shell is ±45, ±32, 90, 32, 45 for a total 

thickness of 1.7019 mm. The cylinder is stiffened by ten evenly spaced J-section ring frames and twentytwo evenly spaced hat-section longitudinal stringers. A 12.7 mm thick plywood floor is installed 0.544 m 

above the bottom of the cylinder. The floor is carried by ten horizontal aluminum beams at the locations 

of the ring frames. Twenty vertical aluminum beams support the floor, with one pair of supporting beams 
5

in each ring frame cross-section and spaced 1.045 m apart. The shell and stiffener elements of the 

fuselage model are rivet-bonded together. Figure 2 depicts a cross-section of the cylinder showing the 

beam supported floor, a ring frame and the cross-sections of the twenty longitudinal stringers. A support 

structure consisting of two rigid 0.0889 thick medium-density fiberboard (MDF) endcaps provide 

constrained edge conditions to the cylinder. The cylinder rests in grooves carved out in the endcaps and 

the entire structure is held together by four tension rods. Wooden beams are attached to each tension rod 

to increase damping. A hinged door (0.61 m by 0.61 m) was incorporated in the front endcap to provide 

access to the interior of the cylinder. The access door can be locked into place by six over-center latches 

as shown in Figure 1.

NUMERICAL MODELS OF THE COMPOSITE CYLINDER

Finite Element Model

A finite element model of the floor-equipped composite cylinder was developed using the pre/postprocessor MSC.Patran 2007 Rev.2. The model development including the geometry, the element property 

sets, the composite material lay-up, the mechanical material properties, the beam section properties and 

other characteristics of the model is presented in Reference 9. A wireframe isometric view of the 

composite cylinder finite element model is presented in Figure 3 without the front endcap to more clearly 

show the floor and the supporting beams. The model consists of a total 2242 CQUAD4 plate elements for 

the cylinder shell, the floor and the two endcaps. The longitudinal stringers, the ring frames, the floor 

beams and the floor supports were modeled using 1198 CBEAM beam elements. The two acoustic 

cavities above and below the floor consist of 5452 CHEXA three-dimensional elements. The volume, 

bounding area and total perimeter values of the acoustic cavities above and under the floor are presented 

in Table 1.

The mechanical properties of the shell, the ring frames, the longitudinal stringers, the floor, the floor 

support beams and the endcaps are listed in Table 2 and include the elasticity moduli, the shear modulus, 

the density and the Poisson ratios. The beam section properties are listed in Table 3 including the crosssectional area, the moments of inertia, the torsional constant about the centroid, and the shear stiffness 

factors. The cross-sectional dimensions of the ring frames and the longitudinal stiffeners supporting the 

cylinder shell were measured in situ yielding slightly different values for the beam section properties in 

Table 3 than reported in Reference 9, which were obtained from the composite cylinder design report7. 

Statistical Energy Analysis Model

The SEA model depicted in Figure 4 was constructed by importing the geometry and the nodes of the 

floor-equipped composite cylinder finite element model into the VA One prediction program. The bays 

bounded by the ring frames and the longitudinal stringers were modeled as subsystems consisting of 

singly curved shells having a thickness of 1.7 mm and a radius of 0.8382 m. The smaller bays above the 

floor were also modeled using singly curved shells. These bays were bounded by the floor, two ring 

frames and a stringer. The bays adjacent to, but below the floor, were represented by flat plates since 

these bays were small and not enough nodes were available from the finite element model to define the 

curvature of those subsystems. The composite nine-layer shell was represented by the equivalent 

orthotropic material properties of the finite element model (Table 2). The 0-degree fiber orientation was 

defined parallel to the cylinder axis. The composite ring frames and longitudinal stringers were modeled 

with isotropic material properties as this was the only option available for beam sections in VA One 2009. 

The isotropic material properties for the beam subsystems are listed in Table 4. The beam section 

properties including the shear center offsets were taken from the finite element model listed in Table 3. 
6

The horizontal floor beams and the vertical beams supporting the floor were modeled using aluminum 

with section properties derived from the finite element model (Table 3). The plywood floor and the two 

MDF endcaps were represented by flat plate subsystems with finite element model derived mechanical 

material properties (Table 2). The subsystems for the acoustic cavities above and below the floor were 

defined by the geometry of the cylinder finite element model and the volume, area and perimeter 

parameters listed in Table 1. The properties of the air included a mass density of 1.21 kg/m3and a speed 

of sound of 343 m/s.

The flexural damping loss factors of the cylinder shell and floor, and the damping loss factors of the 

acoustic cavity subsystems were estimated experimentally using the impulse response decay method 

(IRDM)10,11,12 and the interrupted noise method, respectively. However since the flexural loss factors for 

the shell and floor were estimated in-situ, the estimates are anticipated to be high since they account for 

losses due to damping as well as coupling. The damping loss factors are tabulated in Table 5. The 

extensional and shear damping loss factors of the shell and the floor, and all the damping loss factors of 

the beams and the endcaps were arbitrarily chosen to equal 0.01 and were included in the SEA matrix 

solution. 

Junctions between subsystems were automatically created using the autoconnect feature in VA One 

resulting in 332 point, 568 line, and 279 area junctions. The junctions are shown in Figure 5. The front 

area junctions between the upper acoustic cavity and the front endcap, and between the lower acoustic 

cavity and the endcap were removed for clarity. Only resonance energy paths were considered.

Energy Finite Element Analysis Model

The finite element bulk data file including the geometry, element properties and material properties 

was imported into Comet Enflow. The program automatically generates junctions at geometric and 

material discontinuities resulting in 324 point joints, 1272 line joints and 2048 area joints. The 324 beambeam point joints are shown in Figure 6a. The line joints consist of 174 plate-plate joints and 1198 platebeam-plate line joints which are depicted in Figure 6b. The 1700 plate-acoustic joints and the 348 plateacoustic-plate joints are shown in Figure 6c. The number of EFEA point joints is about the same as the 

number of SEA point junctions. However, the number of EFEA line and area joints is much larger than 

the number of SEA line and area junctions as the automatic joint generation in EFEA is based on the 

finite elements while the SEA modeling is based on shell subsystems bounded by the ring frames and 

longitudinal stringers. This is illustrated in Figure 5 where a SEA shell subsystem is bounded by four 

stiffener segments and is represented by one area junction and four line junctions, compared to the same 

cylinder shell area in EFEA which consists of six area joints (Figure 6c) and ten line joints (Figure 6b). 

Similarly, the number of EFEA plate-acoustic-plate joints is much higher than the SEA area junctions for 

the cylinder floor. The increased size of the energy balance matrix will increase the computation time to 

obtain a solution for the matrix. The radiation dimensions of the single bay radiating subsystem were 

specified in the input data file along with a curvature of 1.193 for the cylinder shell. The same beam 

section properties were input for the beam stiffeners as were used to define the SEA beam elements. In 

addition, the St. Venant‘s ratio was defined as the square-root of the ratio of the torsion constant and the 

polar moment of inertia, which was equal to 0.049 for the ring frame, 0.0858 for the longitudinal 

stiffeners and 0.0543 for the floor beams and floor supports. The volume, area and perimeter parameters 

of the cavity above and below the floor (Table 1) were entered into the Comet Enflow data file. The same 

damping loss factors as in the SEA formulation (Table 5) were used for the shell, the floor, the stiffeners, 

the endcaps, and the acoustic cavities. Only resonance energy paths were considered. Comet Enflow was 

executed using the default Gaussian Elimination solver.
7

COMPARISON BETWEEN SEA AND EFEA PREDICTIONS

The SEA model was excited by a unit power input at the location of Bay X, which is the shell 

subsystem bounded by two longitudinal stringers and two ring frames (Figure 7). The SEA predictions

were performed with VA One following the recommendations of the User‘s Manual in Reference 13. Bay 

X in the EFEA model consists of six elements and two interior nodes not coinciding with the bay 

boundaries. The unit power input for the EFEA analysis was applied to the bay interior Node 1538 which 

is located closest to the back endcap. The Comet Enflow User‘s Manual14 describes the program 

execution, the input and output file management and the input entries to the code for the EFEA 

predictions. The SEA and EFEA structural response levels of the cylinder shell and the sound pressure 

levels in the acoustic cavities were predicted by the VA One and Comet Enflow software programs and 

are compared in the following sections. 

Structural Response Predictions

Structural response predictions were performed for Bay X and Bays 1-14 of the floor-equipped 

composite cylinder, the locations of which are indicated in Figure 7. This section compares predictions 

from the SEA and EFEA methods for two sidewall configurations and a complete model of the composite 

cylinder. The acoustic cavities below and above the floor were included in the structural analysis of the 

complete model, but their presence had a negligible effect on the structural response of the cylinder 

components. The input to the prediction codes, including mechanical material properties, geometrical 

characteristics and damping loss factors (Table 2 - Table 5), was the same for both energy methods to 

ensure an equitable comparison. Where possible, the energy paths and associated computational 

procedures in the two energy methods were also specified to match one another. 

The first configuration discussed here consists of a portion of the sidewall only, with no stiffeners. 

The area of the sidewall structure covered the fifteen numbered bays of the cylinder (Figure 7) such that 

the total modeled section is the segment shown in Figure 8. The structure was represented in SEA by a 

singly-curved panel subsystem. A unit power excitation was applied to the subsystem and a single, 

spatially averaged root-mean-square velocity was predicted in VA One for the entire subsystem. The 

resulting velocity is shown in Figure 9 as a function of frequency. In EFEA, the sidewall was modeled 

using a relatively coarse finite element mesh which contained six plate elements per subpanel, for a total 

of ninety plate elements for the entire sidewall structure. A finite element approach is employed in Comet 

Enflow to solve the governing differential equation in terms of energy density variables numerically at the 

node locations of the model. This allows evaluating the root-mean-square velocity variations from node to 

node within the sidewall structure subsystem. The excitation corresponded to a unit power input at the 

location of one of the two nodes within the imaginary Bay X as indicated in Figure 8. For the results 

shown here, the nodal velocities were averaged over the nodes inside each imaginary bay, and the 

resulting root-mean-square velocities are shown in Figure 10 for the locations of the imaginary bays X 

and 5-8. Since there are no stringers and ring frames, the velocity response decays when moving away 

from the excitation location due to the structural damping of the sidewall material. The bay-to-bay 

response variation is more obvious when the velocity variation is plotted along a horizontal line parallel to 

the centerline of the sidewall, through the excitation point. This velocity variation for the Comet Enflow 

predictions is shown in Figure 11 for the 200 Hz one-third octave band and compared to the unvarying 

response of the VA One prediction. The EFEA prediction, shown at each node, decays exponentially 

away from the drive point due to damping in the sidewall. A similar comparison for the root-mean-square 

velocity predictions as function of nodal location is presented in Figure 12 for the 2000 Hz one-third 

octave band. Excellent agreement is observed between the SEA and EFEA energy methods when the 

Comet Enflow results are averaged over all nodes of the sidewall structure as evidenced in Figure 13.
8

The second configuration incorporated longitudinal stringers and ring frames, dividing the sidewall 

structure into fifteen individual bays as shown in Figure 14. The longitudinal stringers and ring frames 

were modeled as beam subsystems at the connecting lines between the bay subsystems. The autoconnect 

feature in VA One was used to create the point and line junctions connecting the bay subsystems. The 

SEA root-mean-square velocity predictions for Bay X and Bays 5-8 are shown in Figure 15 while the 

Comet Enflow results are presented in Figure 16. The results in the figures show a large discrepancy 

between the EFEA and SEA models at frequencies below 1000 Hz. After further study, it was determined 

Comet Enflow does not account for energy in the beams. To confirm this finding, the VA One SEA 

model was modified so the flexural, torsional, and extensional beam energies were excluded from the 

solution matrix. The resulting response predictions are shown in Figure 17 showing much better 

agreement with the EFEA results in Figure 16. These results appear to confirm that the Comet Enflow 

EFEA model does not consider beam energies in the solution. 

More detailed comparisons on a bay-by-bay basis of the SEA (with no beam energy) and EFEA 

predictions are presented in Figure 18 through Figure 22. Within the driven Bay X, the higher VA One 

root-mean-square velocity predictions above 315 Hz (Figure 18) may be partially attributed to the 

different formulations defining the SEA coupling loss factor and the EFEA equivalent coupling loss 

factor. The average coupling loss factor ηij for a line junction (Figure 5) between adjacent shell 

subsystems  ̳i‘and  ̳j‘ is defined in SEA by15

2

gi ij

ij

i ij

c L

A





 





(1)

where cgi is the flexural group speed, τij is the power transmission coefficient, ω is the rotational 

frequency, L is the length of the line junction, and Aiis the surface area of the subsystem. In EFEA the 

coupling loss factor of a conservative joint at a line discontinuity between two bay subsystems is 

analogous to the expression for the point coupling of two beams given in Reference 16

gi ij

ij

i ii jj

c L

A r r











(2)

where the reflection coefficients rii and rjj are defined by rii=1-τij and rjj=1-τji resulting in

2

gi ij

ij

i ij ji

c L

A





  



 

(3)

The coupling loss factor at a line junction in VA One is thus proportional to τij/(2-τij) while the 

equivalent coupling loss factor at a line joint in Comet Enflow is proportional to τij/(2-τij- ji). Without the 

additional power transmission coefficient τji in the EFEA formulation the coupling loss factors are the 

same and only valid for weakly coupled systems. The VA One and Comet Enflow predicted velocity 

responses in Bays 5-8 are compared in Figure 19-Figure 22. The energy flow across the ring frames 

decreases more rapidly for the VA One than for the Comet Enflow predictions and is most notable in the 

frequency range from 315 to 1250 Hz.

The final structural configuration studied is a full model of the floor-equipped composite cylinder. 

The VA One predicted root-mean-square velocity responses in bay subsystems X and 5-8, including the 

beam energies, are shown in Figure 23 while the root-mean-square velocities without the beam responses 

included in the solution matrix are presented in Figure 24. The Comet Enflow results for the same bays 
9

are shown in Figure 25. Comparisons of the VA One and Comet Enflow predicted root-mean square 

velocity responses for Bay X and Bays 5-8 in Figure 26 through Figure 30, respectively, show the same 

trends for the floor-equipped cylinder as found for the fifteen bay subsystems separated by the 

longitudinal stiffeners and the ring frames discussed previously.

Interior Acoustic Sound Pressure Level Predictions

The sound pressure levels in the acoustic cavity above the floor of the composite cylinder were 

predicted by VA One and Comet Enflow for unit power input to Bay X. Energy storage and dissipation in 

the stiffeners were not included in the prediction models. The results are depicted in Figure 31 with VA 

One predicting sound pressure levels up to 12.8 dB lower than Comet Enflow below the 250 Hz one-third 

octave band and up to 6.2 dB higher than Comet Enflow above the 250 Hz one-third octave band. 

To investigate the discrepancy between the two prediction methodologies a simplified structuralacoustic model was created consisting of a single non-rigid bay (Bay X), defined as the area between two 

ring frames and two longitudinal stringers, and the acoustic cavity above the floor. Within the VA One 

model of this two subsystem configuration an area junction was created between the shell bay and the 

acoustic cavity. A corresponding finite element model of a single non-rigid bay and interior acoustic 

cavity was imported in Comet Enflow and an area joint was automatically generated between the bay and 

the interior acoustic space. The applicable geometric, material and damping properties from the previous 

full composite cylinder models were preserved. The shell bay area in VA One consisted of a singly 

curved shell subsystem, while the bay area in Comet Enflow was constructed from six plate elements that 

were connected at an appropriate angle to mimic the curvature of the shell. In both models the curved bay 

was excited by unit power input. The results are shown in Figure 33 with VA One predicting sound 

pressure levels up to 5.2 dB lower than Comet Enflow below the 250 Hz one-third octave band and up to 

6.7 dB higher than Comet Enflow above the 250 Hz one-third octave band. The character of these 

discrepancies is similar to the differences in the SEA and EFEA sound pressure level predictions for the 

full composite cylinder model (Figure 31). The root-mean-square velocities predicted by SEA and EFEA 

for the curved bay excited by unit input power were virtually identical as shown in Figure 34, suggesting 

that the coupling between the curved bay and the acoustic cavity is different for the two energy flow 

methods.

The sound pressure level calculations were repeated for the same model configuration but for a nearly 

flat bay area instead of a curved bay. For the VA One model this was achieved by replacing the 0.838 m 

radius of the singly curved shell by a very large number. In EFEA the curvature of the plate group was set 

to zero. The sound pressure levels were predicted by VA One and Comet Enflow in the acoustic cavity 

for excitation of the virtually flat plate by unit power input, resulting in almost identical results as shown 

in Figure 35. The results in Figure 33 and Figure 35 indicate that when curvature is applied to a flat plate 

the different approaches in VA One (SEA) and Comet Enflow (EFEA) yield different sound pressure 

level predictions. The methodology followed in VA One as well as Comet Enflow for predicting sound 

radiation from a curved shell are investigated in the following sections.

VA One curved shell sound radiation prediction

The power П radiated by a flat plate with length a and width b into the cylinder cavity above the floor 

is dependent on the space averaged, mean-square velocity 

2

  v

of the radiating surface, and the radiation 

efficiency σ

17
10 2

0    c ab v

(4)

where ρ0 is the density of the fluid (air) and c is the speed of sound in the fluid. Above the critical 

frequency, the lowest frequency at which the speed of the forced bending waves equal the speed of sound 

in the adjacent fluid, the plate radiates sound from its entire surface into the acoustic far field. At 

frequencies below the critical frequency the plate radiates sound into the far field by means of edge and 

corner modes.18 The critical frequency fcr of an orthotropic plate is defined by19

 

4

2

11 22 3

m

cr

c

f

D D



 

 

  

    

(5)

where ρm is the mass per unit area of the plate. D11 and D22 are the flexural rigidities in two perpendicular 

material directions and are given by19

 

3

11,22

11,22

12 21 12 1

h E

D

 





(6)

where h is the thickness of the plate, E11 and E22 are the elasticity moduli along the two main material 

directions and ν12 and ν21 are the Poisson ratios. The parameter α is defined in Reference 19 by

 

12 66

0.5

11 22

D D

D D







(7)

where D12 and D66 are flexural rigidity matrix values of the orthotropic plate. If α is not known, two 

critical frequencies in the primary directions of the laminate can be defined as

1,2

2

11,22 2

m

cr

c

f

D







(8)

For the cylinder shell material properties listed in Table 2 the two critical frequencies were calculated 

to occur at 6238 Hz and 6983Hz. Since the frequency range of interest is 100 Hz to 4000 Hz the current 

analysis is for sound radiation below the critical frequency.

The total number of modes N(f) of a flat orthotropic plate up to a frequency f is given in Reference 20

by

 

 

/ 2

0

abf h d

N f

D



 

 

 

(9)

where D(θ) is the prime material axis polar coordinate dependent flexural rigidity of the flattened 

orthotropic shell and can be written as20

   

3

' 4 ' 2 2 ' 4

11 12 22 12 22 cos 2 2 cos sin sin

12

h

D E E G E              

(10)
11

where G12 is the shear modulus and

' 11,22

11,22

12 21 1

E

E

 





(11)

Lyon21 simplified the integral in the expression for the number of modes N(f) by taking the average of 

the flexural rigidities along the two prime material directions. The modal density of the flattened 

orthotropic shell nflat=dN(f)/df can thus be expressed as

11 22

1 1

4

flat

ab h

n

D D

  

       

(12)

and is independent of the frequency f. The VA One predicted modal density of the flattened orthotropic 

shell is plotted in Figure 36.

Compared to a flat plate the curved shell is stiffened in the direction of the curvature causing the 

modes of the shell below the breathing cylinder mode at the ring frequency fR to resonate at higher 

frequencies.22 The shell curvature effects are important below the ring frequency fR while the shell will 

behave like a flat plate well above the ring frequency.18 The plane strain ring frequency for a radius of 

curvature r is defined by23

2

2

l

R

c

f

 r



(13)

while the longitudinal group speed in the orthotropic shell material along the two prime axes is given by24

 

1,2

11,22

12 21 1

l

E

c

  





(14)

The ring frequency for the composite cylinder was computed to occur at 1036 Hz. Compared with the 

modes of a flat plate, the modes of the curved shell shift towards its ring frequency resulting in an 

increased modal density in that frequency region. An empirically based curve fitting approximation for 

the curved shell modal density ncurved(f) was proposed by Szechenyi25 for three different one-third octave 

band regions. The regions are defined in terms of ξr, which is the ratio of the frequency f over the plane 

stress ring frequency fr for which the axial stress component is zero

 

 

2.6

r

curved flat

B

n f n





(15)

where

r 2 / 22     rf E

(16)

for 

0.48 r   2.5 B   r r



for 

0.48 0.83   r   3.6 B   r r



for 

0.83 r      

2 2 2 2.9001cos 1.3861 0.8884cos 2.1968 B    r r r    
12

Langley26 evaluated the modal density of a cylinder and obtained a closed form solution in terms of 

elliptic integrals showing that at high frequencies the modal density of the shell approaches the modal 

density of a flat plate as predicted by Szechenyi.25 Lyon and DeJong15 assessed the mode count of a thinwalled cylinder. A smooth curve fit was applied to the solutions below and above the plane strain ring 

frequency to avoid the discontinuity around the ring frequency and to obtain a result valid over the entire 

frequency region. The modal density for a curved shell is expressed by15

 

1/4 4

3.5

1

4 2 R

curved R

R

ab n f

r





   



                  

(17)

where ξR=f/fR and κ is the radius of gyration for bending. Results for the predictions above compare well 

with the computational results from the VA One program, which are shown in Figure 36 for the flat plate 

and the singly curved shell. The VA One computed number of modes in band are listed in Table 6 for the 

flat and the curved bay subsystem showing the shift of modes towards the ring frequency while the 

number of modes well above the ring frequency is about the same for the flat and the curved bays.

The radiated power into the acoustic cavity defined previously is not only dependent on the modal 

density but is also dependent on the radiation efficiency. The radiation efficiency of a flat plate, below the 

critical frequency, is approximated by27

  2 2

1 2 ln

1 1 1 rad cr

a b f f

ab k

 



    

     

      

      

(18)

where k=2πf/c is the wave number and the parameter μ equals (f/fcr)

1/2. Below the critical frequency the 

radiation efficiency decreases with lower frequencies and shorter plate dimensions a and b. The VA One 

predicted radiation efficiency for the shell subsystem without curvature is shown in Figure 37. All 

radiating modes in this region are acoustically slow and only radiate through edge and corner modes. 

Adding curvature to the shell produces the ring frequency. In the region of the ring frequency some of the 

modes become acoustically fast as their wavelengths are longer than the acoustic wavelength which, in 

turn, causes an increase in the radiation efficiency. The VA One predicted radiation efficiency of the 

curved shell is compared with the flat plate radiation efficiency in Figure 37. The SEA coupling loss 

factor ηsa between the structure and the acoustic cavity is related to the radiation efficiency σrad of the 

structure18

0 rad

sa

m

 c







(19)

The VA One predicted coupling loss factors for the flattened and the curved shell are presented in 

Figure 38. The modal density in the acoustic cavity na(ω) is defined as15

 

2

2 3 2 2 2 8 16 a

V S P

n

c c c

  

  

  

(20)

where V is the volume, S is the cavity bounding surface area and P is the perimeter. Applying reciprocity 

for the energy flow to and from the acoustic cavity results in the coupling loss factor ηas from the acoustic 

cavity to the bay subsystem
13 

 

s sa

as

a

n

n

 







(21)

The VA One predicted acoustic cavity to bay subsystem coupling loss factor for the flattened and 

curved shell is shown in Figure 39. 

The ratio of energy level in the acoustic cavity Ea and the energy level in the shell Esis related to the 

coupling loss factors ηsa and ηas, and the damping loss factor ηa of the acoustic cavity by15

a sa

s a as

E

E



 





(22)

where 

60

13.816

a

T







(23)

and T60 is the reverberation time in the acoustic cavity.

The mean square vibrational energy in the shell can be expressed in terms of the mean square velocity

2 E ab v s m

 

(24)

and the energy level in the acoustic cavity is related to the mean-square sound pressure 

2

2

0

a

p V

E

 c



(25)

The sound pressure level in the cavity is then given by

2 2

0

2

10log 10log

m

sa

a as

c ab v

p

V

  

 

   



      

(26)

where

0

4

0

3

60

2

13.816 2 3

sa m

a as

l m

c

ab c

T c h V

 

 

    

  







(27)

The VA One predicted sound pressure level response in the acoustic cavity due to the unit power 

input for the flattened and curved bay subsystem is presented in Figure 40. 
14

Comet Enflow curved shell sound radiation prediction

The EFEA power transmission coefficient τsa from a plate into an acoustic space is defined by28

2 rad

sa

gf rad

h

c h

 



 





(28)

where ηrad is the radiation damping loss factor and cgf is the flexural wave group velocity. The EFEA 

power transmission coefficient from the acoustic space into the structure is discussed in Reference 8

 

2

2

4 rad

as

gf

c

c

 

   

(29)

where the coefficient ζ(ω) is defined by8

 

2

2 3

2

2 3 2 2

2

2 8 16

V

c

V S P

c c c



  

 

  



 

(30)

with V being the volume of the acoustic cavity, S the total surface surrounding the cavity and P the 

perimeter of the acoustic cavity. The rotational frequency ω dependent flexural group velocity for a flat 

orthotropic plate is given by14

2

12 4

gf

m

D

c







(31)

where D12=ν21D11+G12h

3

/12. In Comet Enflow the increased stiffness due to the curvature of the shell 

element is accounted for by adjusting the bending wave speed below the ring frequency.25 The flat plate 

flexural group velocity can be written in terms of the modal density nflat

gf

flat

S

c

n







(32)

Knowing this relationship the same coefficients25 as used for the modal density can be employed for 

the flexural group velocity in Comet Enflow when correcting for the effect of curvature 

for 

0.48 r

1/4

gf r gf c c 







for 

0.48 0.83   r

1/2

c c gf r gf 0.841







for 

0.83 r

no correction was applied to the flexural group speed since in Comet Enflow it was 

assumed that above the ring frequency the curved shell behaves like a flat plate. The radiation damping 

loss factor ηrad for the curved shell was not modified. The effect of curvature on the Comet Enflow 

predicted sound pressure level inside the acoustic cavity is shown in Figure 41.
15

Evaluation of the VA One and Comet Enflow interior sound pressure level predictions

Single bay excitation – In the preceding sections the effects of curvature on the sound radiation into 

the acoustic cavity were analyzed for both the VA One SEA and Comet Enflow EFEA methodologies. 

The differences in interior sound pressure levels for the excitation of a curved single bay (Figure 33) can 

be explained by the different approaches in each energy prediction code. The interior predicted sound 

pressure levels for excitation of a flat single bay result in almost identical predictions (Figure 35).

The trends of the VA One and Comet Enflow sound pressure level predictions for the curved single 

bay subsystem in Figure 33 are similar to the results obtained for single bay excitation of the full, floor 

equipped, composite cylinder model in Figure 31. This suggests that the differences in the VA One and 

Comet Enflow predicted interior sound pressure levels for the floor equipped composite cylinder are 

caused by the different approaches in the VA One and Comet Enflow theories to account for the energy 

flow between a curved structure and the acoustic interior space. 

Sidewall structure excitation - It was shown in Table 6 for the single bay subsystem that only the 

one-third octave frequency bands above 500 Hz contained at least two modes. The 500 Hz and lower 

frequency bands are therefore, strictly, not suitable for energy analysis. To investigate the effect of 

curvature for a larger radiating area which has more modes at lower frequencies a sidewall structure 

subsystem was considered with an area as large as fifteen bays, but without any stiffeners. The sidewall 

structure was connected to the upper acoustic cavity subsystem as shown in Figure 42. The length, the 

width, the perimeter and the area of the sidewall structure are listed along with the dimensions of the 

original single bay in Table 7. Table 6 shows that more than two modes are resonant in the flat sidewall 

structure for all one-third octave bands of 100 Hz and higher while the curved sidewall structure exhibits 

more than two modes in the one-third octave bands at 125 Hz and above. The VA One and Comet Enflow 

predicted sound pressure levels in the acoustic cavity, for the unit power excitation applied to the flat or 

curved sidewall structure, are presented in Figure 43. The VA One and Comet Enflow computed sound 

pressure levels for the flat sidewall are approximately the same, but the VA One predictions for the 

curved sidewall structure are up to 12.5 dB higher than the Comet Enflow predictions.

The differences in the predicted SEA and EFEA sound pressure levels with the excitation applied to a 

single bay or the sidewall structure (flat as well as curved) are shown in Figure 44. The SEA and the 

EFEA predictions for the flat plate and the EFEA predictions for the curved plate are approximately 5.5 

dB higher for the single bay excitation than for the sidewall structure excitation over the entire range of 

100 – 4000 Hz one-third octave bands. However, the approximate 5.5 dB difference only applies to the 

SEA predictions for the curved bay and sidewall excitations above the 1000 Hz one-third octave band. 

Below this frequency the sound pressure level difference varies between 0 and 1.5 dB (Figure 44). 

The differences in interior sound pressure level predicted by Comet Enflow due to the change in area 

of the excited structure can be explained by evaluating the power transmission coefficients. The EFEA 

power transmission coefficient τsa from a plate into an acoustic space can be expressed in terms of the 

radiation efficiency σ

28

2

2

rad

sa

rad











(33)

where β=cρ0/csρsis the ratio of the characteristic impedances of the fluid (air) and the structure. Assuming 

that βσrad≪2 and only the area of the excited structure is changed, then the power transmission coefficient 

is dependent on only the radiation efficiency. Below the critical frequency the radiation efficiency, which 

was defined previously in Equation (18), scales with (a+b)/ab. The change in the predicted interior sound 

pressure levels when exciting the sidewall structure (with an area covering a matrix of five by three bays) 
16

instead of the single bay is then 10log(3+5)/15=5.7 dB. This difference applies to the flat as well as to the 

curved subsystems.

The interior sound pressure levels differences predicted by VA One in Figure 44, due to the different 

areas of the single bay and the sidewall structure, can be explained by evaluating the previously presented 

expression for the interior sound pressure levels in Equation (26). Assuming that the SEA coupling loss 

factor ηas (Figure 39) is much smaller than the damping loss factor ηa of the acoustic cavity (Table 5), the

ratio of ηsa/(ηa+ηas) becomes almost equal to ηsa/ηa. The acoustic cavity sound pressure level then can be 

rewritten as

0

2 3

2 2 60 2

10log 10log

13.816

c T

p ab v

V





 

     

 

(34)

In this equation, for the same unit power excitation of the single bay or the sidewall structure (fifteen 

times the area of the single bay), the product of the area ab and the mean-square velocity 

2

  v

is about 

constant and the sound pressure level in the acoustic cavity scales with approximately ten times the 

logarithm of the radiation efficiency or 10log(σ). The VA One predicted radiation efficiencies for the flat 

and curved sidewall structures are shown in Figure 45. The VA One predicted sound pressure levels for 

the single bay in Figure 40 and for the sidewall structure in Figure 43 can thus be explained by the 

radiation efficiency curves in Figure 37 and Figure 45, respectively. The approximation of ten times the 

logarithm of the radiation efficiency ratio for the sound pressure level differences in the acoustic cavity 

for the single bay or sidewall structure excitation is illustrated in Figure 46 to be very close to the sound 

pressure level differences computed by the VA One prediction program. 

CONCLUSIONS

Statistical Energy Analysis (VA One) and Energy Finite Element Analysis (Comet Enflow) response 

predictions were made for a floor-equipped composite cylinder excited by a unit power input on a bay 

subsystem bounded by two longitudinal stringers and two ring stiffeners. The input to the prediction 

codes, including mechanical material properties, geometrical characteristics and damping loss factors, 

was the same for both energy methods. The forced transmission loss path for the floor was not included. 

and the flexural, extensional and torsional energy storage and dissipation in the stiffeners were not 

modeled. The VA One predicted root-mean-square velocity in the bay excited by the unit power was 

similar to the root-mean-square velocity predicted by Comet Enflow. 

A sidewall structure with an area the size of fifteen bays was initially considered to be a single subsystem 

and stringers and ring frames were not included in the analysis. Root-mean-square velocity responses at 

nodes away from the unit input power node were predicted by Comet Enflow and were compared to the 

VA One predictions in the 200 Hz and 2000 Hz one-third octave bands. While VA One produces a single 

estimate for a subsystem, the root-mean square velocity responses predicted by Comet Enflow vary as 

function of location within the subsystem. 

The structural response of a configuration including fifteen bay subsystems separated by the longitudinal 

stiffeners and the ring frames was evaluated. Comet Enflow considers the beams to not take energy and 

therefore the VA One bay velocities were predicted excluding the beam energies in the solution matrix. 

The energy flow across the ring frames to bays away from the excitation bay decreased more rapidly for 

the VA One than for the Comet Enflow predictions. A possible explanation was offered by the fact that 
17

the coupling loss factor at a line junction in SEA is proportional to τij/(2-τij) while the equivalent coupling 

loss factor at a line joint in EFEA is proportional to τij/(2-τij- ji).

The VA One predicted sound pressure levels inside the upper acoustic cavity of the composite cylinder 

were higher than the Comet Enflow computed levels in one-third octave bands below 250 Hz, while the 

VA One predictions were lower at the higher frequency bands. Similar trends were observed in a simple 

two-subsystem configuration, consisting of a curved single bay shell and the acoustic cavity. The 

differences in the VA One and Comet Enflow interior sound pressure level predictions were shown to be 

caused by the way the curvature of the shell is incorporated in the VA One and Comet Enflow theoretical 

analyses. While the VA One implementation of the curvature is based on the modal density of the 

structure, the curvature in Comet Enflow is addressed by adjusting the bending wave speed in the 

structure below the ring frequency. The different curvature implementations cause large differences (up to 

12.5 dB) in the predicted interior sound pressure levels below and just above the ring frequency. At 

frequencies well above the ring frequency both interior sound pressure level predictions converge as the 

structure can be modeled progressively like a flat plate. 

REFERENCES

1. Grosveld, F. W., and Beyer, T. B., Modal Characteristics of a Stiffened Cylinder with Open and

Closed End Conditions. AIAA 86-1908, AIAA 10th Aeroacoustics Conference, Seattle, 

Washington, July 9-11, 1986.

2. Beyer, T. B. and Grosveld, F. W., Validation of an Interior Noise Prediction Model for a 

Composite Cylinder. AIAA-87-0529, AIAA 25th Aerospace Sciences Meeting, Reno, Nevada, 

January 12-15, 1987.

3. Grosveld, F. W. and Beyer, T. B., Vibratory Response of a Stiffened, Floor-Equipped, Composite 

Cylinder. Proceedings of the 5th International Modal Analysis Conference, London, England, 

April 6-9, 1987, pp. 812-820.

4. Grosveld, F. W., Sullivan, B. and Marulo, F., Aircraft Interior Noise Prediction Using a Structural 

Acoustic Analogy in NASTRAN Modal Synthesis. Proceedings of the 6th International Modal 

Analysis Conference, Orlando, FL, February 1-4, 1988, pp. 1191-1198.

5. Beyer, T. and Silcox, R., Noise Transmission of a Large Scale Composite Fuselage Model. 

AIAA-90-3965, AIAA 13th Aeroacoustics Conference, Tallahassee, Florida, October 22-24, 

1990.

6. Grosveld, Ferdinand W., Noise Reduction of a Composite Cylinder Subjected to Random 

Acoustic Excitation. AIAA-89-1049, AIAA 12th Aeroacoustics Conference, San Antonio, TX, 

April 10-12, 1989.

7. Jackson, A. C., Balena, F. J., Labarge, W. L., Pei, G., Pitman, W. A. and Wittlin, G., Transport 

Composite Fuselage Technology: Impact Dynamics and Acoustic Transmission. NASA-CR4035, December 1986.

8. Grosveld, Ferdinand W., Schiller, Noah H., Cabell, Ran H, ―Comparison of Comet Enflow and 

VA One Acoustic-to-Structure Power Flow Predictions,‖ NASA/TM-2010-216191, February 

2010, 51p.

9. Grosveld, Ferdinand W., Schiller, Noah H., Cabell, Ran H, ―Finite Element Models and 

Properties of a Stiffened Floor-Equipped Composite Cylinder,‖ NASA/TM-2010-216192, 

February 2010, 33p.

10. Cabell, Randolph, Klos, Jake, Buehrle, Ralph and Schiller, Noah, ―Vibroacoustic Response Data 

of Stiffened Panels and Cylinders,‖ Proceedings of Noise-Con 2008, Dearborn, Michigan, July 

28-31, (2008).
18

11. Cabell, Randolph, Schiller, Noah, Allen, Albert and Moeller, Mark, ―Loss Factor Estimation 

Using the Impulse Response Decay Method on a Stiffened Structure,‖ Proceedings of Inter-Noise 

2009, Ottawa, Canada, August 23-26, (2009).

12. Schiller, Noah, Cabell, Randolph and Grosveld, Ferdinand, ―Impact of Damping Uncertainty on 

SEA Model Response Variance,‖ Proceedings of Noise-Con 2010, Baltimore, Maryland, April 

19-21, (2010).

13. VA One User‘s Guide. ESI Group North America, San Diego, 2007.

14. Comet Enflow User‘s Manual. Comet Technology Corporation, Ann Arbor, MI, 2007.

15. Lyon, Richard H. and DeJong, Richard G., ―Theory and Application of Statistical Energy 

Analysis,‖ Second Edition, Butterworth-Heinemann, Newton, Massachusetts, 1995.

16. Wang, S. and Bernhard, R. J., ―Prediction of Averaged Energy for Moderately Damped Systems 

with Strong Coupling,‖ Journal of Sound and Vibration 319(3), 2009, pp. 426-444.

17. Fahy, F. J., ―Response of a Cylinder to Random Sound in the Contained Fluid,‖ Journal of Sound 

and Vibration 13(2), 1970, pp. 171-194.

18. Fahy, Frank and Gardonio, Paolo, ―Sound and Structural Vibration,‖ Second Edition, Elsevier 

Academic Press, Amsterdam, 2007.

19. Renji, K. and Nair, P. S., ―Critical and Coincidence Frequencies of Flat Panels,‖ Journal of Sound 

and Vibration 205(1), 1997, pp. 19-32.

20. Bosmans, I., Mees, P. and Vermeir, G., ―Structure-Borne Sound Transmission Between Thin 

Orthotropic Plates: Analytical Solutions,‖ Journal of Sound and Vibration 191(1), 1996, pp. 75-

90.

21. Lyon, R. H., ―Statistical Energy Analysis of Dynamic Systems,‖ Cambridge, Massachusetts, MIT 

Press, 1975.

22. Lyon, Richard H., ―Designing for Product Sound Quality,‖ CRC Press, Mechanical Engineering 

Series, ISBN 0824704002, June 2000.

23. Manning, Jerome E. and Maidanik, Gideon, ―Radiation Properties of Cylindrical Shells,‖ Journal 

of the Acoustical Society of America, Volume 36, Number 9, September 1964.

24. Spintsis, I. A., ―Determination of the Physicomechanical Characteristics of Viscoelastic Materials 

in Orthotropic and Isotropic Layers from the Propagation Velocities and Damping Constants of 

Longitudinal and Shear Vibrations,‖ Mekhanica Polimerov, Vol. 3, No. 1, 1967, pp. 161-166.

25. Szechenyi, E., ―Modal Densities and Radiation Efficiencies of Unstiffened Cylinders Using 

Statistical Methods,‖ Journal of Sound and Vibration 19(1), 1971, pp. 65-81.

26. Langley, R. S., ―The Modal Density and Mode Count of Thin Cylinders and Curved Panels,‖ 

Journal of Sound and Vibration 169(1), 1994, pp. 43-53.

27. Leppington, F. G., Broadbent, E.G. and Heron, K. H., ―Acoustic Radiation from Rectangular 

Panels with Constrained Edges,‖ Proc. R. Soc. Lond. A 393, 1984, pp.67-84.

28. Bitsie, F., ―The Structural acoustic Energy Finite Element Method and Energy Boundary Element 

Method.
19

TABLES

Table 1. Volume, bounding area and total perimeter of the acoustic cavities above and below the floor.

Acoustic Cavity Volume Area Perimeter

[m

3

] [m

2

] [m]

Above the floor 5.83 20.79 24.26

Below the floor 2.25 14.37 21.82

Table 2. Mechanical material properties of the composite cylinder components.

Mechanical Material Properties E11 E22 G12 ρ ν12 ν21

[GPa] [GPa] [GPa] [kg/m3]

Cylinder shell Composite 40.6 32.4 32.2 1590 0.629 0.502

Ring frame ‘J’ stiffeners Composite 72.7 28.8 18.3 1574 0.753 0.298

Longitudinal ‘Hat’ stringers Composite 95.8 22.9 19.1 1590 0.588 0.141

Aluminum floor beams and floor supports Isotropic 70.0 26.0 2700 0.30

Plywood floor Isotropic 12.4 4.66 711 0.33

Medium-density fiberboard (MDF) endcaps Isotropic 2.8 1.037 730 0.35

Table 3. Beam section properties.

Property Parameter Label Floor Beams Floor Supports Longitudinal Stringers Ring Frames

Shape ‘U’ ‘U’ ‘Hat’ ‘J’

Cross sectional area A [m2] 0.0002607052 0.0002607052 0.0001330128 0.0001864582

Moments of inertia Ixx [m4] 1.515540E-08 1.515540E-8 4.538572E-9 7.771676E-8

Iyy [m4] 1.531684E-07 1.531684E-7 2.126092E-8 1.199627E-8

Torsional constant J [m4] 4.96766E-10 4.96766E-10 1.891524E-10 2.209614E-10

Distance shear center horizontal [m] 1.318182E-8 1.318182E-8 2.855553E-7 -0.008786919

to centroid vertical [m] -0.01378559 -0.01378559 0.007624588 -0.007008278

Shear stiffness factors K1 0.2574783 0.2574783 0.2609327 0.3591745

K2 0.4953249 0.4953249 0.4140227 0.4492481

Table 4. Isotropic mechanical material properties for modeling the composite beam components in VA One.

Mechanical Material Properties E11 G12 ρ ν12

[GPa] [GPa] [kg/m3]

Ring frame ‘J’ stiffeners Isotropic 48.8 16.3 1574 0.499

Longitudinal ‘Hat’ stringers Isotropic 60.3 20.1 1590 0.499
20

Table 5. Damping loss factors for the main subsystems of the floor-equipped composite cylinder.

One-third Octave Band 

Center Frequency

[Hz]

Shell

(flexure)

[-]

Floor

(flexure)

[-]

Upper

Acoustic Cavity

[-]

Lower

Acoustic Cavity

[-]

100 0.079 0.076 0.0226297 0.0226297

125 0.174 0.076 0.0242013 0.0242013

160 0.207 0.076 0.0344726 0.0344726

200 0.187 0.076 0.0264031 0.0264031

250 0.185 0.076 0.0190452 0.0190452

315 0.105 0.076 0.0184962 0.0184962

400 0.038 0.08 0.0160743 0.0160743

500 0.09 0.081 0.0155026 0.0155026

630 0.056 0.066 0.0151923 0.0151923

800 0.064 0.074 0.0126923 0.0126923

1000 0.038 0.074 0.00938272 0.00938272

1250 0.028 0.068 0.007705 0.007705

1600 0.036 0.066 0.00613358 0.00613358

2000 0.043 0.064 0.00435032 0.00435032

2500 0.034 0.07 0.0035221 0.0035221

3150 0.041 0.062 0.00250785 0.00250785

4000 0.041 0.060 0.00227444 0.00227444

Table 6. VA One computed modes in band for a single bay and the sidewall structure (fifteen times the area of the 

single bay), with and without curvature.

One-third Octave Band

Center Frequency

[Hz]

Single Bay

(flat)

[-]

Single Bay

(curved)

[-]

Sidewall

Structure

(flat)

[-]

Sidewall

Structure

(curved)

[-]

100 0.4 0.1 5.5 1.6

125 0.5 0.2 7.0 2.3

160 0.6 0.2 8.9 3.3

200 0.7 0.3 11.1 4.7

250 0.9 0.4 14.0 6.7

315 1.2 0.6 17.6 9.6

400 1.5 0.9 22.2 13.9

500 1.8 1.3 27.7 20.2

630 2.3 2.0 35.2 30.6

800 2.9 3.3 44.1 49.0

1000 3.7 5.4 55.7 81.6

1250 4.7 5.4 70.1 81.7

1600 5.9 6.4 87.9 95.4

2000 7.4 7.8 111.1 116.4

2500 9.3 9.6 139.5 143.5

3150 11.7 11.9 175.9 179.0

4000 14.8 14.9 221.4 223.8
21

Table 7. Length, width, area and perimeter values for a single bay and for the sidewall structure (fifteen times the 

area of the single bay).

Structural Component Length Width Perimeter Area

[m] [m] [m] [m

2

]

Single bay 0.381 0.240 1.241 0.091

Sidewall structure (15x area single bay) 1.905 0.719 5.248 1.367

Table 8. Flat and curved logarithmic radiation efficiency ratios of a single bay and the sidewall structure (fifteen 

times the bay area).

One-third 

Octave Band

Center 

Frequency

[Hz]

Radiation

Efficiency

(flat)

Bay

[-]

Radiation

Efficiency

(flat)

Sidewall

[-]

10log

(Bay/Sidewall)

(flat)

[dB]

Radiation

Efficiency

(curved)

Bay

[-]

Radiation

Efficiency

(curved)

Sidewall

[-]

10log

(Bay/Sidewall)

(curved)

[dB]

100 0.008393 0.002683 5.0 0.005132 0.677799 0.1

125 0.009330 0.003050 4.9 0.006624 0.004981 0.1

160 0.012427 0.003538 5.5 0.012795 0.006483 1.5

200 0.013385 0.004020 5.2 0.015906 0.008983 1.2

250 0.014695 0.004519 5.1 0.020462 0.012000 0.8

315 0.016483 0.005099 5.1 0.027977 0.017214 0.1

400 0.018920 0.005722 5.2 0.041888 0.027562 -0.2

500 0.021918 0.006489 5.3 0.064878 0.044266 0.0

630 0.025951 0.007450 5.4 0.08793 0.064223 -0.4

800 0.031222 0.008610 5.6 0.14944 0.095874 -0.3

1000 0.037065 0.009970 5.7 0.181283 0.158383 4.5

1250 0.043775 0.011695 5.7 0.071405 0.063786 5.2

1600 0.052952 0.014273 5.7 0.066974 0.021734 5.4

2000 0.064075 0.017573 5.6 0.073718 0.019316 5.5

2500 0.080843 0.022492 5.6 0.088578 0.020986 5.5

3150 0.111502 0.031133 5.5 0.118805 0.025143 5.5

4000 0.178355 0.050359 5.5 0.187725 0.033563 5.5
22

FIGURES

Figure 1. Composite cylinder supported by two endcaps and held together by wood covered (to provide damping) 

tension rods.

Figure 2. Cross-section of the stiffened, floor-equipped composite cylinder and a detailed view of the rivet-bonded 

frame-stringer-shell intersection.
23

Figure 3. Wireframe isometric view of the floor-equipped composite cylinder energy finite element model with one 

endcap removed for clarity.

Figure 4. VA One (SEA) model of the floor-equipped stiffened composite cylinder without the front endcap and the 

acoustic cavity subsystems.
24

Figure 5. VA One area, line and point junctions with the area junctions between the acoustic cavities and the front 

endcap removed for clarity.

a) point joints b) line joints c) area joints

Figure 6. Comet Enflow point, line and area joints of the composite cylinder energy finite element model.
25

Figure 7. The unit power excitation bay (X) and the fourteen response bays (1-14).

Figure 8. Sidewall structure subsystem (area of fifteen bays).

X

1

2

3

4

5

6

7

8

9

10

11

12

● Excitation

Location

14

13
26

Figure 9. VA One (SEA) root-mean-square velocity predictions on the sidewall structure (same at all locations).

Figure 10. Comet Enflow (EFEA) root-mean-square velocity predictions in the imaginary bays (X and 5-8) of the

sidewall structure subsystem.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

125 250 500 1000 2000 4000

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

125 250 500 1000 2000 4000

EFEA - Imaginary Bay X

EFEA - Imaginary Bay 5

EFEA - Imaginary Bay 6

EFEA - Imaginary Bay 7

EFEA - Imaginary Bay 8
27

Figure 11. Comet Enflow (EFEA) predicted root-mean-square velocities at nodes away from the unit input power at 

Node #1538 compared to VA One (SEA) predictions for the sidewall structure subsystem and five smaller bay 

subsystems in the 200 Hz one-third octave band.

Figure 12. Comet Enflow (EFEA) predicted root-mean-square velocities at nodes away from the unit input power at 

Node #1538 compared to VA One (SEA) predictions for the sidewall structure subsystem and five smaller bay 

subsystems in the 2000 Hz one-third octave band.

 Node Number

Root-mean-square Velocity, m/s

0.01

0.02

0.03

0.04

0.05

0.06

1527

1538

1549

1560

1571

1582

1593

1604

1615

1626

1637

1648

1659

1670

1681

1692

SEA Sidewall Structure - 200 Hz

EFEA Sidewall Structure - 200 Hz

 Node Number

Root-mean-square Velocity, m/s

0.01

0.02

0.03

0.04

0.05

0.06

1527

1538

1549

1560

1571

1582

1593

1604

1615

1626

1637

1648

1659

1670

1681

1692

SEA Sidewall Structure - 2000 Hz

EFEA Sidewall Structure - 2000 Hz
28

Figure 13. VA One (SEA) and Comet Enflow (EFEA) area averaged root-mean-square velocity predictions of the 

sidewall structure 

Figure 14. Sidewall structure divided into fifteen bay subsystems by longitudinal stringers and ring frames.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

125 250 500 1000 2000 4000

SEA - Average Sidewall Structure

EFEA - Average Sidewall Structure

X

X

5

10

11

12

6

7

3

8

X

4

2

1

14

13

9
29

Figure 15. VA One (SEA) root-mean-square velocity predictions for the fifteen bay subsystems separated by

longitudinal stringers and ring frames.

Figure 16. Comet Enflow (EFEA) predicted root-mean-square velocities of the sidewall structure including the

longitudinal stringers and ring frames.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay X

SEA - Bay 5

SEA - Bay 6

SEA - Bay 7

SEA - Bay 8

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

EFEA - Bay X

EFEA - Bay 5

EFEA - Bay 6

EFEA - Bay 7

EFEA - Bay 8
30

Figure 17. VA One (SEA) root-mean-square velocity predictions for the fifteen bay subsystems separated by

longitudinal stringers and ring frames but their responses excluded in the solution matrix.

Figure 18. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem X root-mean-square velocities for 

bay subsystems separated by longitudinal stringers and ring frames.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay X

SEA - Bay 5

SEA - Bay 6

SEA - Bay 7

SEA - Bay 8

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay X

EFEA - Bay X
31

Figure 19. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 5 root-mean-square velocities for 

the bay subsystems separated by longitudinal stringers and ring frames.

Figure 20. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 6 root-mean-square velocities for 

the bay subsystems separated by longitudinal stringers and ring frames.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay 5

EFEA - Bay 5

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay 6

EFEA - Bay 6
32

Figure 21. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 7 root-mean-square velocities for 

the bay subsystems separated by longitudinal stringers and ring frames.

Figure 22. VA One (SEA) and Comet Enflow (EFEA) predicted bay subsystem 8 root-mean-square velocities for 

the bay subsystems separated by longitudinal stringers and ring frames.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay 7

EFEA - Bay 7

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

125 250 500 1000 2000 4000

SEA - Bay 8

EFEA - Bay 8
33

Figure 23. VA One (SEA) predicted root-mean-square velocity responses in bay subsystems X, 5-8 of the floorequipped composite cylinder.

Figure 24. VA One (SEA) predicted root-mean-square velocities in the bay subsystems X, 5-8 of the floor-equipped 

composite cylinder without the beam responses in the SEA solution matrix.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay X

SEA - Bay 5

SEA - Bay 6

SEA - Bay 7

SEA - Bay 8

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay X

SEA - Bay 5

SEA - Bay 6

SEA - Bay 7

SEA - Bay 8
34

Figure 25. Comet Enflow (EFEA) predicted root-mean-square velocities in the bay subsystems X, 5-8 of the floorequipped composite cylinder.

Figure 26. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the excitation bay 

subsystem X of the floor-equipped composite cylinder.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

EFEA - Bay X

EFEA - Bay 5

EFEA - Bay 6

EFEA - Bay 7

EFEA - Bay 8

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay X

EFEA - Bay X
35

Figure 27. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the excitation bay 

subsystem 5 of the floor-equipped composite cylinder.

Figure 28. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the excitation bay 

subsystem 6 of the floor-equipped composite cylinder.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay 5

EFEA - Bay 5

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay 6

EFEA - Bay 6
36

Figure 29. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the excitation bay

subsystem 7 of the floor-equipped composite cylinder.

Figure 30. VA One (SEA) and Comet Enflow (EFEA) predicted root-mean-square velocities in the excitation bay 

subsystem 8 of the floor-equipped composite cylinder.

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay 7

EFEA - Bay 7

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.02

0.04

0.06

0.08

0.1

0.12

125 250 500 1000 2000 4000

SEA - Bay 8

EFEA - Bay 8
37

Figure 31. VA One (SEA) and Comet Enflow (EFEA) predicted sound pressure levels in the upper acoustic cavity 

subsystem of the floor-equipped composite cylinder.

Figure 32. Two subsystem configuration of a bay and an acoustic cavity excited by unit input power on the 

bay.

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level, dB

85

90

95

100

105

110

115

125 250 500 1000 2000 4000

SEA

EFEA

X
38

Figure 33. Predicted sound pressure levels in the acoustic cavity for unit power excitation of a singly curved shell 

(VA One - SEA) or a plate to which curvature was applied (Comet Enflow - EFEA).

Figure 34. Predicted root-mean-square velocities for unit power excitation of a singly curved shell (VA One - SEA) 

or a plate element to which curvature was applied (Comet Enflow - EFEA).

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level, dB

95

100

105

110

115

125 250 500 1000 2000 4000

SEA - singly curved shell

EFEA - curvature

 One-third Octave Band Center Frequency, Hz

Root-mean-square Velocity, m/s

0

0.05

0.1

0.15

0.2

0.25

0.3

125 250 500 1000 2000 4000

SEA

EFEA
39

Figure 35. VA One (SEA) and Comet Enflow (EFEA) predicted sound pressure levels in the acoustic cavity for unit 

power excitation of the flat shell subsystem.

Figure 36. VA One (SEA) predicted modal density for a singly curved orthotropic shell and a flat orthotropic plate.

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level, dB

95

100

105

110

115

125 250 500 1000 2000 4000

SEA - no curvature

EFEA - no curvature

 One-third Octave Band Center Frequency, Hz

Modal Density, 1/(rad/s)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

125 250 500 1000 2000 4000

SEA - singly curved shell

SEA - flat plate
40

Figure 37. VA One (SEA) predicted radiation efficiency for a singly curved shell and a flat plate.

Figure 38. VA One (SEA) predicted coupling loss factor for a singly curved shell or a flat plate connected to the 

acoustic cavity.

 One-third Octave Band Center Frequency, Hz

Radiation Efficiency

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

125 250 500 1000 2000 4000

SEA - singly curved shell

SEA - flat plate

 One-third Octave Band Center Frequency, Hz

Coupling Loss Factor

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

125 250 500 1000 2000 4000

SEA - singly curved shell

SEA - flat plate
41

Figure 39. VA One (SEA) predicted coupling loss factor for the acoustic cavity connected to a singly curved shell or 

a flat plate.

Figure 40. VA One (SEA) predicted sound pressure levels in the acoustic cavity for unit power excitation of the flat 

or curved shell bay subsystem.

 One-third Octave Band Center Frequency, Hz

Coupling Loss Factor

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

125 250 500 1000 2000 4000

SEA - singly curved shell

SEA - flat plate

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level, dB

95

100

105

110

115

125 250 500 1000 2000 4000

SEA - singly curved shell

SEA - no curvature
42

Figure 41. Comet Enflow (EFEA) predicted sound pressure levels in the acoustic cavity for unit power excitation of

the flat or curved shell bay subsystem.

Figure 42. Sidewall structure subsystem (area of fifteen bays; no stiffeners) connected to the acoustic cavity 

subsystem.

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level, dB

95

100

105

110

115

125 250 500 1000 2000 4000

EFEA - curvature

EFEA - no curvature

X
43

Figure 43. VA One (SEA) and Comet Enflow (EFEA) predicted sound pressure levels in the acoustic cavity for unit 

power excitation applied to a flat or curved sidewall structure with an area equal to fifteen single bays.

Figure 44. Cavity sound pressure level differences due to unit power excitation of the single bay or the sidewall 

structure (flat as well as curved) as predicted by VA One (SEA) and Comet Enflow (EFEA).

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level, dB

95

100

105

110

115

125 250 500 1000 2000 4000

SEA - curved sidewall

SEA - flat sidewall

EFEA - curved sidewall

EFEA - flat sidewall

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level Difference, dB

-5

0

5

10

125 250 500 1000 2000 4000

EFEA flat: SPLbay - SPLsidewall EFEA curved: SPLbay - SPLsidewall SEA flat: SPLbay - SPLsidewall SEA curved: SPLbay - SPLsidewall
44

Figure 45. VA One (SEA) predicted radiation efficiency for a flat or curved sidewall structure with a total area equal 

to fifteen single bays.

Figure 46. Cavity sound pressure level differences due to unit power excitation of the flat or curved, bay or sidewall 

structure (fifteen times the bay area) as predicted by VA One (SEA) and compared to 10 log(σbay/σsidewall).

 One-third Octave Band Center Frequency, Hz

Radiation Efficiency

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

125 250 500 1000 2000 4000

SEA - curved sidewall

SEA - no curvature

 One-third Octave Band Center Frequency, Hz

Sound Pressure Level Difference, dB

-5

0

5

10

125 250 500 1000 2000 4000

Flat: SPLbay - SPLsidewall

Flat: 10 log (bay/sidewall)

Curved: SPLbay - SPLsidewall Curved: 10 log (bay/sidewal
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Statistical Energy Analysis (SEA) and Energy Finite Element Analysis 

(EFEA) Predictions for a Floor-Equipped Composite Cylinder 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Grosveld, Ferdinand W.; Schiller, Noah, H.; Cabell, Randolph H.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20035

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified Unlimited

Subject Category 71 

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

Comet Enflow is a commercially available, high frequency vibroacoustic analysis software founded on Energy Finite Element 

Analysis (EFEA) and Energy Boundary Element Analysis (EBEA). Energy Finite Element Analysis (EFEA) was validated on 

a floor-equipped composite cylinder by comparing EFEA vibroacoustic response predictions with Statistical Energy Analysis 

(SEA) and experimental results. Statistical Energy Analysis (SEA) predictions were made using the commercial software 

program VA One 2009 from ESI Group. The frequency region of interest for this study covers the one-third octave bands with 

center frequencies from 100 Hz to 4000 Hz. 

15. SUBJECT TERMS

Acoustics; Composite cylinder; Energy finite element analysis; Interior noise; Statistical energy analysis; Vibro-acoustics

18. NUMBER

 OF 

 PAGES

49

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39.18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

877868.02.07.07.04.01.01 

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/TM-2011-217171

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and 

Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person 

shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

01- 08 - 2011