Numerical investigation of the reverberation method for measuring the total
loss factor of plate-like structures
Yasutomo YAMASAKI(1) , Naohisa INOUE(2) , Tetsuya SAKUMA(3)(1) Graduate School of Frontier Sciences, The University of Tokyo, Japan, 2835299483@edu.k.u-tokyo.ac.jp(2) ditto, Japan, n-inoue@edu.k.u-tokyo.ac.jp(3) ditto, Japan, sakuma@e.u-tokyo.ac.jp

Abstract
The total loss factor measurement by the reverberation method is often performed to estimate the boundary
conditions of vibration fields of plate-like structures. In order to clarify how measured values involve discrepancies from the true values, this paper presents a numerical model that simulates the reverberation method todetermine the loss factor in a bending vibration system of a glass plate supported with an elastic material. As areference, the total loss factor of a finite plate system is theoretically calculated based on the diffuse vibration
field assumption, where the random-incidence vibration absorption coefficient on the support edge is given forthe semi-infinite plate terminated by the mechanical and moment impedances. Subsequently, the finite element
analysis is performed for a rectangular plate with the impedance boundary, and impulse responses are calculated with different arrangements of excitation and receiving points. Comparing the theoretical and numericalresults, the validity of the measurement procedures is generally confirmed, however, it is demonstrated that themeasured loss factors tend to become slightly higher than theoretical ones. Additionally, measured results onreal window systems are discussed briefly.
Keywords: Thin plate, Finite element method, Loss factor

1

INTRODUCTION

Understanding vibro-acoustical behaviors of plate-like structures are of great interest in many fields of noise
control engineering. Regarding the acoustic radiation from a rectangular plate, Berry et al. have made a significant remark that the radiation mechanism strongly depends on the boundary condition of the plate [1]. Anotherimportant aspect is that energy loss occurs in the reflection of bending waves at the elastic supported edges,
which reduces the radiation from modal vibrations of the plate [2]. These are why modeling of the elastic
boundary support is crucial for simulating accurate sound transmission loss of plate-like structures.
The elastic boundary support has been usually modeled with mechanical and moment impedances, but it is not
yet well established how to determine the impedances. Several papers have dedicated to investigate the effectof translational and rotational restraint on natural frequencies of finite plates (e.g.[3]), where the analysis was
performed with assuming lumped constants of stiffness, inertance and resistance for the impedance boundary
condition. However, it is not clear to what extent this lumped model is applicable. Besides, input parametersof the boundary impedances are often experimentally determined by excitation tests. The total loss factor (TLF)
measurement by reverberation method is one of the most common measurement [4]. However, the measured
value has not been investigated in terms of the discrepancy from the theoretical value.
Focusing on a thin plate supported by an elastic material with a rectangular cross section, this paper is dedicatedto improve the usability of the impedance boundary modeling. In Sect. 2, the random-incidence vibration
absorption coefficient on the support edge is theoretically formed under the semi-infinite plate terminated by theequivalent mechanical and moment impedances. Subsequently, in Sect. 3, finite element analysis is performedon the impedance model and a precise support material model in order to clarify the reliable condition of the
impedance boundary modeling. Finally, the numerical and experimental measurement of the total loss factor isconducted in Sect. 4 and Sect. 5.

226

2

THEORICAL ANALYSIS

2.1 Governing Equation
A flat plate is assumed to lie on the x − y plane of the Cartesian coordinate. e jω t is assumed as the time
convention throughout this paper. The time-harmonic equation of the Kirchhoff-Love thin plate vibration theoryis given asB∇2 ∇2 w − ρp ω 2 w = fz + z

∂ fy
∂ fx
+z
∂x
∂y

(1)

where ∇2 is the Laplace operator, w is the out-of-plane displacement, B and ρ p are the flexural rigidity and thearea density of the plate. B is given by B = Eptp3 /[12(1 − µ 2 )], where Ep , µ and tp are the Young’s modulus,
the Poisson’s ratio and the thickness of the plate, respectively. fx , fy and fz are the external stress acting on
the plate surface in each direction. z is the signed distance from the mid-plane of the plate, which is tp /2 on
the upper face and −tp /2 on the bottom face, respectively. The relation between the bending-torsional moments
and the displacement is described as follows.
[
]
∂ 2w
2
Mαβ = −B (1 − ν )+ δα ,β ν ∇ w(2)∂ α∂ βwhere α and β take x or y. δα ,β is the component of the unit tensor. Throughout this paper, the internal loss
factor of the plate, ηp , is set as zero in order to focus on the effect of the edge damping.
2.2 Impedance boundary conditions
As Eq. (1) is the partial differential equation of fourth order, two conditions should be defined at a boundary:
one is for translational motion and the other is for rotational motion. Assuming the local reactive boundary,these conditions can be generally described by using the mechanical and moment impedances, ZQ and ZM , asfollows.
)
(
∂ Mn
∂ Ms
∂w
Q̃ =
+2
= − jω ZQ w,
Mn = jω ZM
,
(3)
∂n
∂s
∂n
where Q̃, Mn , and Ms are the effective shear force, normal and torsional moments along the boundary, respectively. ∂ /∂ n and ∂ /∂ s are the normal and tangential directional derivative along the boundary, respectively.
2.3 Oblique-incidence reflection coefficient
As depicted in Fig. 1, let us consider a situation where the plane propagative bending wave impinges to theboundary of x = 0 at an incidence angle of θ . In this semi-near field, general solution of the Eq. (1) is givenC+A–(a) translational motion(b) rotational motionzzEvanescent
Impedance Boundary

Regressive
θ

0

θ

0

y
Progressive

A+

x

O
O

x

O

x

/2

/2

/2

/2

Figure 2. Assumed deformations of a rectangular supFigure 1. Problem setting for the analysis of the plate
port material for (a) translational and (b) rotational
bending wave reflection from the impedance boundary
motions.

227

as [5]w(x, y) = (A+ e− jω kBx x + A− e jω kBx x +C+ e−ω kBx x )e− jkBy y ,(4)where kBx = kB cos θ , kBy = kB sin θ and kEx = kB (1 + sin 2θ )1/2 . kB is the bending wave number on the plate
defined as kB = ω 1/2 (B/ρp )1/4 . Substituting Eqs. (2) and (4) into the former or latter equation of Eq. (3), thefollowing relations are obtained.
(γ− β+ − zq )A+ − (γ− β+ + zq )A− − ( jγ+ β− + zq )C+(β− − zm γ− )A+ + (β− + zm γ− )A− − (β+ + jzm γ+ )C+

= 0,

(5)

= 0,

(6)

where zq , zm are the normalized mechanical impedance and moment impedance defined as zq = ω ZQ /(kB3 B), zm =
ω ZM /(kB B), respectively. And the following values are introduced, β± = 1 ± (1 − ν ) sin 2θ and γ± = (1 ±sin 2θ )1/2 . Combining Eqs.(5) and (6), the oblique-incidence reflection coefficient is obtained asr(θ ) =

γ− β+2 − 2z+ γ− zq zm − jγ+ (β−2 + zq zm − 2γ− zm )
A−
=
.
+
A
γ− β+2 + 2zq + γ− zq zm + jγ+ (β−2 + zq zm + 2γ− zm )(7)

Furthermore, oblique-incidence vibration absorption coefficient is given as α (θ ) = 1 − |r(θ )|2 .
2.4 Equivalent impedances of a rectangular elastic support material
Let us consider deformations of a rectangular elastic support material as depicted in Fig. 2. The threedimensional displacements of the support material are continuous to those of the plate on the joining face
and fixed at the opposite face. Furthermore, the other faces are under the free support. In order to derive theimpedances in the closed form, the following assumptions are introduced:
1. only the one-dimensional longitudinal vibration is excited in the thickness direction of the support material,
2. the translational out-of-plane displacement and the rotational slope of the plate are uniform over the supporting depth.
The validity of the first assumptions is numerically investigated in Sect. 3. The second assumption is considered to be valid when the bending wavelength is sufficiently larger than the supporting depth. Under the first
assumption, the longitudinal modal stress in the support material of the lower side is expressed asσz (x, z) = k1 Ẽs w(x)cos(k1 z)
sin(k1 hs )(8)where w(x) is the displacement on the joining face. The second assumption states that the displacement is
w(x) = w0 + θ0 x with the constant translational displacement w0 and the constant rotational slope θ0 . hs is the
thickness of the support material, k1 is the wave number of the longitudinal wave in the support material, andẼs is the complex Young’s modulus defined as Ẽs = Es (1 + jηs ) with the loss factor ηs . For the translationalmotion, the force on the joining face, z = hs , is obtained by integrating σ (x, hs ) over the supporting depth ds ,whereas for the rotational motion, the moment on the joining face is obtained by integrating σ (x, hs ) × x overthe supporting depth. Considering the reaction forces of the support materials on both sides, the mechanicaland moment impedance are obtained as follows.
∫ d /2ZQ =2 −ds s /2 σz (x, hs )dxjω w02ρs c̃1 ds=,
j tan(ω hs /c̃1 )∫ d /2ZM =2 −ds s /2 σz (x, hs )xdxjωθ0=

ρs c̃1 ds3
6 j tan(ω hs /c̃1 )(9)where c̃1 is the speed of longitudinal wave defined as c̃1 = (Ẽs /ρs )1/2 and ρs is the material density of the
support material. Under the preceding assumptions, Eq. (9) is equivalent mechanical and moment impedance ofthe rectangular elastic support material.

228

model (I) : Precise Model(3-Dimensional Elastic Material Model)
ForcedΩE

FixedTable 1. Physical properties of the plate and supporting material.

hs
npE hs10 mm

x4 x3 x2 x1

Γp

ΓpE

Fixed

100 mm

300 mm

y

x

0
∂Γp
s

Δx = 2.5 mmnΓpForcedmodel (II) : Impedance Boundary Model

Figure 3. Problem settings and domain notation for calculating mechanical and moment impedances.

3

property

plate (glass)Young’s modulus [N/m2 ]Ep = 7.5 × 1010

Poisson’s Ratio []Loss Factor
Material Density [kg/m3 ]Thickness [m]Depth [m]

ν = 0.22
ηp = 0ρp /tp = 2, 500
tp = 0.01
N/A

ds/2

support (putty)
Es = 1.0 × 106 ,1.0 × 108N/Aηs = 0.5
ρs = 1, 000
hs = 0.005
ds = 0.015

NUMERICAL INVESTIGATIONThis section discusses the validity of the mechanical and moment impedances given in the previous section
through the finite element analysis (FEA). Figure 3 shows the problem setting and the domain notation in thissection. The boundary condition of the plate and the elastic material is free support unless otherwise indicated.
The physical properties for the calculation are listed in Table 1.
3.1 Analysis procedure of normal-incidence vibration indicators
The bending vibration field becomes one-dimensional in the strip plate, which is described asw(x) = A+ e− jkB x + A− e jkB x +C+ e−kB x +C− ekB x(10)

By observing the displacements at four points, x1 to x4 , the following matrix equation can be set according to
Eq. (10).
 − jk x  + e B 1 e jkB x1 e−kB x1 ekB x1 w(x1 )
A 





e− jkB x2 e jkB x2 e−kB x2 ekB x2  A−  w(x2 )

(11)e− jkB x3 e jkB x3 e−kB x3 ekB x3  C+  = w(x3 )








w(x4 )
C−e− jkB x4 e jkB x4 e−kB x4 ekB x4
Then, the unknown amplitudes of propagative and evanescent waves, {A+ , A− ,C+ ,C− }T , are obtained by solving
the above equation. It was confirmed by a preliminary study that the theoretical impedances best approximate
those of the precise model (I) just at the middle point of the joining depth: the middle point is set as x = 0
as depicted in Fig. 5. Then, normalized mechanical and moment impedances at x = 0 and normal-incidence
vibration absorption coefficient are calculated as follows:z′q =A+ − A− + jC− jC−,
A+ − A− +C+ +C−z′m =A+ − A− −C−C−,
A+ − A− − jC+ + jC−

αn = 1 −

A−
A+

2

.

3.2 Results and discussions
FEA for the model (I) is performed under shear limp and elastic conditions for the support material. The
former condition corresponds to the presented impedance model, and the shear stress is neglected in the FEA.
The default physical properties are the same as those investigated in the previous section. Calculation is doneat the 1/12 octave center frequency from 16 to 4,000 Hz. The wavelength of the bending wave on the plate at229

Ratio of Absolute impedances []26
1.8 (a) Es = 10
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
10

fq
zM = ∞
zM = 0210Frequency [Hz]

3

10

2
8
1.8 (b) Es = 10
1.6
fm
1.4
zQ = 0
1.2
1
0.8
0.6
0.4
0.2
0 1
10

Shear Limp

Shear Elastic

2

10Frequency [Hz]3

10

Figure 4. Ratio of the absolute mechanical (moment) impedance of the model (II) to that of the theoreticalmodel. The Young’s moduli of the support material are (a) 106 and (b) 108 .
4,000 Hz is 0.16 m, and enough larger than ds . Then, the second assumption stated in Sect. 2.4 is acceptablein this point.
Figure 4 shows calculation results: ratio of the absolute impedance of the model (II) to that of the model(I). For the shear limp condition, the theoretical model presented in Sect. 2 well approximates the mechanicaland moment impedances around and below fq , the translational mass-spring resonance frequency. Above fq ,the support material can no longer be considered as a lumped constant system, which causes the pronounced
discrepancy. Regarding the shear elastic model, the mechanical impedance is almost the same as those of
the shear limp model. However, the moment impedance is underestimated in the entire frequency range byneglecting the shear stress of the support material. In particular, the ratio is constant below fq . Note that thesetendencies are also observed when changing plate thickness and the support material thickness and depth.

4

NUMERICAL MODELING OF A TOTAL LOSS FACTOR MEASUREMENTThe TLF measurement by the reverberation method is often performed to get or to estimate input values fortheoretical or numerical calculations. However, it is not clear how measured values involve discrepancies fromthe theoretical values for the diffuse field, which increases the uncertainty of subsequent calculations. In thissection, the TLF measurement is numerically modeled to understand the behavior of measured values. This
practical information is valuable for experimental verification of the proposed impedance model in future work.
4.1 Theoretical foundation
When the 60 dB decay time, T60 , is measured, a total loss factor ηtot is obtained asηtot =

6 ln 10
,
ω T60

(12)

because of the definition, γ = ηtot ω , where γ is the exponential decay rate.
In the two-dimensional diffuse vibration field, the mean-free path is given as π S/ltot with the plate area S andthe total perimeter length ltot . Then the exponential decay rate, γDF , is defined by γDF = cg ltot αr /π S, where
cg is the group velocity of the bending wave defined as cg = d ω /dkB , and αr is the vibration absorption
coefficient for 2-D random-incidence. Based on these equations, the total loss factor in the diffuse vibration
field is theoretically given asηtot =

cg ltot αr
.
πω S

(13)

4.2 Numerical analysis conditions
Following the reference [6], calculation arrangement is set as illustrated in Fig. 5. In the FEA, all perimeters
are set as impedance boundaries because this measurement is usually performed for specimens in normal service

230

10 0

10 0

(a) Es = 106(b) Es = 108

0.75 × 0.9

1500
300

10−1

1250

1250
625

Receiving points
10
559

354

Total loss factor

1.25 × 1.5

Excitation point

50 100300

Unit: [mm]10−21.75 × 2.1 [m2]

10− 1

Calc.
Theoretical

10− 2

100 125 160 200 250 315 400 500 630 800 10001250 1600

100 125 160 200 250 315 400 500 630 800 10001250 1600

Frequency [Hz]Frequency [Hz]Figure 5. A default geometry for the Figure 6. Loss factors calculated for the plate with different sizes andcalculation of the total loss factor.
the same aspect ratio. The Young’s moduli of the support material are(a) 106 and (b) 108 .
condition. The impedance values are given by Eq. (9). Calculation is executed in 0.5 Hz intervals from 0 to2000 Hz. Subsequently, the transfer function of the acceleration response is converted to the transient responseby the inverse Fourier transform. The reverberation time, T60 , is determined by the least square regression of theenergy decay curve obtained by the backward integration of the filtered transient response. The total loss factor
is determined by Eq. (12) and the 5-point-average of T60 . It is well known that the reverberation of the bandpass filter (BPF) itself affects the reverberation time (RT) of the filtered response. In order to approximately
equalize the cut-off characteristics among the different bands, the order of the 1/3-octave band FIR filter of thecenter frequency fc is set as N × 2M , where N is the order at 1 kHz and M is calculated by
M = ⌊log2 (1000/ fc ) + 0.5⌋,
and in this case, N = 64 is appropriate by a preliminary study.
4.3 Results and discussions
4.3.1 Effect of the plate size
It is obvious from Eq. (13) that TLF of a vibration system depends on the area and the total perimeter length.
The TLF are calculated for three sizes of the plates: 40 % smaller and larger plates than the default size.
Figure 6 compares calculated results with the theoretical values for the diffuse vibration field. The calculated
results appear to capture the frequency trends of the theoretical values. Furthermore, TLFs for the smaller platefluctuate more than those for larger plates do. However, the calculated values are two to four times larger
than the theoretical values in the entire frequency range. In general, the measurement of the sound absorption
coefficient by reverberation room method involves two main error factors that cause the discrepancy from thetheoretical value under the diffuse field assumption. One is the lack of the diffusivity, which usually appears
as the underestimation of the sound absorption coefficient. The other is the diffraction at the edge of thefinite specimen, which usually appears as the overestimation of the sound absorption coefficient. Similarly, theTLF measurement on the rectangular plate involves the non-diffuseness and the diffraction effects, and the latter
seems to be predominant. Although all perimeters are impedance boundaries in the present study, the diffraction
effect is considered to occur at the plate’s corners.
4.3.2 Effect of the support material’s properties
Figure 7(a) shows the calculated TLF with changing the support material’s loss factor and fixing the Young’s
modulus to 1.0 × 106 N/m2 . Figure 7(b) shows the calculated TLF with changing the Young’s modulus of
the support material and fixing the loss factor to 0.5. As discussed in the above, the discrepancy between
calculated and theoretical values appears to be large around the mass-spring resonance frequency at which the
TLF becomes high. Inverse estimation of the support material’s physical parameters or equivalent resistance,
stiffness and inertance constants is often performed in order to obtain the input parameters for theoretical and231

(a) Loss factor

0.1

0.25

125

(b) Young’ s Modulus

0.5

1.0 107

Total loss factor

1.0 106150

125

1.0 108 [N/m2]

300

390

10−1

150
300

385

770

10−1

Receiving points

10−2

200

Exitation points

100

Calc.
Theoretical−3
10
100 125160

200 250

315

200

10−2

400 500

630

800 1000 1250 1600

100 125

75

100
160

200 250315

400 500

630

800 1000 1250 1600

Frequency [Hz]Frequency [Hz]

75

780

Unit: [mm]Figure 7. Loss factors calculated with changing the physical property of the Figure 8. A geometry for theseal: (a) loss factor and (b) Young’s modulus.
measurement of the total loss
factor.
100

glass thickness [m]front sideback side
thickness of air layer [m]sash material

0.004
0.006
0.002
aluminum

Indivisual value
Average value
Total loss factor []Table 2. The types of the measured window.

10-1

-2

10

100

125

160

200

250

315

400

500

630

800 1000 1250 1600 2000 2500 3150 4000 5000

Frequency [Hz]Figure 9. Measured Loss factors on individual excitation
and receiving points
numerical calculations [7]. However, from the above observations, the support material’s damping parametersuch as the loss factor and equivalent resistance constant can be overestimated.

5

EXPERIMENTAL MEASUREMENT OF THE TOTAL LOSS FACTOR

In the previous sections, it is clarified how different the measured value is from the theoretical one. Following
that, this section reports the results of TLF measurement of a real window system.
5.1 Measurement conditions
Figure 8 and Table 2 demonstrate the set-up of the measurement. We arranged double pane windows for certain
reasons, although the glass plate with supporting putty is investigated assuming a single pane window in theprevious section. Also, the physical property of the windows is not clear as the reason for the difficulty ofmeasurement. Therefore, the purpose of this section is following the tendency of measured values. In thismeasurement, two excitation points and five receiving points are set, and each measurement is repeated threetimes. The windows are excited by dropping a steel ball of 20 mm diameter 100 mm above the window, andthis vibration is measured by an acceleration pickup at five receiving points. Finally, the total loss factor ofwindows is determined by measured T60 by averaging the measurements of 30 times and Eq. (12). Note thatthe taps of the FIR filter are set as 128 × 2M by preliminary analysis. Measurement is analyzed at the 1/3 octave
bands from 100 to 5000 Hz.

2325.2 results and discussions
Figure 9 shows measured TLF by individual measurement and its averaging value. Results on same excitation
and receiving points are illustrated by the same color. While the measured values of TLF at the same conditionare very close in the entire frequency range, the ones on different conditions vary widely in low frequencies.
However, estimating the equivalent impedance can be enable based on this result, and it can lead to accurate
simulation of the sound transmission loss.

6

CONCLUSION

In this paper, mechanical and moment impedances of a rectangular supporting elastic material were derived
in closed form with the some assumptions. Theoretical analysis was provided to investigate the behaviors ofbending wave absorption at the impedance boundary. Incidence angle dependency of the vibration absorption
coefficient was confirmed to be weak until about 60-degree-incidence. The proposed impedances were comparedwith those of the precise support material model by the finite element analysis. This study confirmed thatthe presented mechanical impedance agreed well with the precise model around and below the translational
mass-spring resonance frequency. On the other hand, the presented moment impedance was underestimatedin entire frequency range due to neglecting the shear reaction of the support material. Furthermore, above
the translational mass-spring resonance frequency, the support material can no longer be replaced as a lumped
constant system. Thus the impedance boundary model does not sufficiently simulate the behavior of the precise
model in particular at high frequencies.
Numerical modeling of the loss factor measurement was performed in order to investigate the difference between
the theoretical and measured values. Compared with the theoretical TLF under the diffuse field assumption, thecalculated values were overestimated due to the diffraction effect and the self-reverberation of the band passfilter. This tendency should be kept in mind when conducting inverse estimation of the support material’s
physical parameters or equivalent resistance, stiffness and inertance constants.
Finally, experimental measurement is executed. Although this condition widely differs from numerical measurement in Sect 4, the result implies the numerical measurement corresponds to the experimental ones.

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