Assessing the similitude of vibrating plates
Christian ADAMS(1) , Joachim BÖS(1) , Tobias MELZ(1),(2)(1) Technische Universität Darmstadt, research group System Reliability, Adaptive Structures, and Machine Acoustics SAM,

Darmstadt, Germany, adams@sam.tu-darmstadt.de(2) Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany

Abstract
The similitude of vibrating structures allows for transferring vibration responses from a scaled structure toan original structure using scaling laws. Numerical calculations and experimental measurements show thatcomplete similitude conditions such as equal damping and geometrically complete similitude are essential toaccurately transfer vibration responses from small-scaled structures to the original structure or vice versa. However, complete similitude conditions can hardly be achieved in practice. A different damping or geometrical
distortions cause the mode shape order to change, which leads to errors when using scaling laws. Nevertheless,the scaling laws can transfer the vibration responses from the scaled structure to the original structure with asufficient accuracy in some cases. This paper analyzes vibration responses of rectangular plates with various
sizes. The vibration responses are obtained from laser D OPPLER vibrometry, and the loss factors are obtained
from transfer function measurements. The vibration responses of an original rectangular plate is transferred toseveral scaled rectangular plates. The accuracy of the transferred vibration responses is assessed by comparing
them with the measured vibration responses of the scaled rectangular plates, where several measures are used inorder to assess how well the vibration responses agree.

Keywords: similitude, vibrations, rectangular plates, laser D OPPLER vibrometry

1

INTRODUCTION

Similitude analyses allow for deriving scaling laws, which are used to transfer vibration responses from scaled
structures to an original structure or vice versa. Scaling laws for vibrating structures have been introduced inseveral papers. A review paper written by C ASABURO [1] summarizes scaling methods for structural design
engineering, where similitude analysis is reviewed with respect to methods to derive scaling laws, structures
used in similitude analyses, and applications of similitude analysis methods in various engineering subjects. It
is shown that – despite the fact that similitude analysis has a long history in engineering – incomplete similitudeconditions still limit the application of scaling laws in practice. In vibroacoustics, incomplete similitude conditions arise from a different damping of the original structure and of the scaled ones and from geometrically
incomplete similitude. Thus, the limitations of scaling laws need to be elaborated, which requires appropriate
measures to assess the accuracy of scaled vibration responses.
This paper shows how incomplete similitude conditions due to the damping and due to geometrically incomplete similitude affects the accuracy of scaled vibration responses. A vibrating rectangular plate is used as arepresentative example. A frequency response analysis is carried out for each rectangular plate using a scanninglaser D OPPLER vibrometer (SLDV), and the loss factors are obtained from frequency response function measurements. The vibration responses of an original rectangular plate are transferred to several scaled rectangularplates using scaling laws, which are derived from numerical calculations using the method proposed in [2, 3].
The similitude of the rectangular plates is assessed by several measures that describe how well the transferredvibration responses agree with the measured ones.

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2

FUNDAMENTALSThis section introduces the scaling laws that are used to transfer (hereafter referred to as to replicate) thevibration responses of the original rectangular plate to the scaled rectangular plates. The measures that are used
to assess the similitude are introduced as well.
2.1 Scaling laws of vibrating rectangular plates
The scaling laws are derived provided that the rectangular plates are in complete similitude. In a next step,
the scaling laws are enhanced towards geometrically incomplete similitude. In order to clearly identify therectangular plates, the following notation is used:• parent: original rectangular plate with known vibration responses. These vibration responses are used toreplicate the vibration responses of the scaled rectangular plates.
• replica: scaled rectangular plate in complete similitude. Its vibration responses are estimated using scalinglaws.
• avatar: scaled rectangular plate in incomplete similitude. Its vibration responses are estimated using scaling laws.
The scaling laws are derived for the rectangular plates’ mean squared transfer admittance (MSTA), which isdefined asṽ2 ( f ).
(1)
Sh2T ( f ) = S 2
F̃ ( f )S, ṽ2 ( f ), and F̃ 2 ( f ) denote the vibrating surface area, the surface-averaged squared RMS1 vibration velocity,and the squared RMS excitation force, respectively. f denotes the frequency. It is assumed that the rectangularplates are scaled in their geometrical dimensions, i.e., length a, width b, and thickness t. Next, the scaling
factors
a(r)
b(r)
t (r)
φa = (p) = φb = (p) = φl and φt = (p)
(2)
a
b
t
for length, width, and thickness are introduced, respectively. The superscript (p) denotes the parent and thesuperscript (r) a replica. Complete similitude conditions require φa and φb to be equal. Applying the scaling
method from [2, 3] yields
2 (r)ShT ( f (r) )= φSh = φl2 φt−4
(3)2 (p) (p)ShT ( f )for the MSTA, where the frequencies are scaled byf (r)= φ f = φl−2 φt1 .
f (p)

(4)

In order to enhance the scaling laws to geometrically incomplete similitude, the scaling factor φl in Eqs. (3)and (4) is substituted by1
φl = (φa + φb ) .
(5)
2
Thus, the vibration responses of avatars in geometrically incomplete similitude are obtained from another replicathat is as close as possible to the avatar [5].
1 RMS: root mean square4250

2.2 Similitude measures
Three measures are introduced that are used to asses how well the replicated MSTA matches the measured
MSTA of replicas (and avatars):• the H AUSDORFF distance [8], which is used in [9] to assess the similitude of vibrating rectangular plates,• the M AHALANOBIS distance [10], which is introduced in [3] to assess the similitude of vibrating rectangular plates, and• the frequency response function similarity metric (FRFSM), which is introduced by L EE [11].
The H AUSDORFF distance of two (arbitrary) frequency response functions A = {a1 , a2 , . . . , aN } andB = {b1 , b2 , . . . , bM } is defined asdH = max dH0 (A, B) , dH0 (B, A) ,(6)

where dH0 (A, B) and dH0 (B, A) are the unidirectional H AUSDORFF distances
dH0 (A, B) = max min ka − bk2
aε A bε B

and

dH0 (B, A) = max min ka − bk2 ,bε B aε A(7)of these frequency response functions. k·k2 denotes the E UCLIDEAN norm [8]. The H AUSDORFF distance canbe seen as the maximum distance of minimum distances between the two frequency response functions, thus,
describing the maximum mismatch between A and B [8].
The M AHALANOBIS distance is similar to a E UCLIDEAN distance that is weighted by the inverse of the covariance matrix Σ−1 . Taking again (arbitrary) frequency response functions A and B, the M AHALANOBIS distanceyieldsqdM (A, B) =(A − B)T Σ−1 (A − B).

(8)

−1

Due to the inverse of the covariance matrix Σ , the M AHALANOBIS distance is weighted by the amount ofcorrelation between A and B. In this paper, the mean M AHALANOBIS distance
dM =

1 M
1 N
dM (A, B) +∑∑ dM (B, A)
2N i=1
2M i=1(9)

is used, which is the mean of the bidirectionally calculated M AHALANOBIS distances dM (A, B) and dM (B, A).
A zero H AUSDORFF distance or mean M AHALANOBIS distance indicates a perfect agreement, while small distances indicate a high degree of agreement. Higher distances indicate deviations between the measured and thereplicated MSTA. However, absolute values for small and high distances depend on the investigated structures.
Thus, the distances only describe the relative degree of agreement of several replicas (or avatars) to each other.
In contrast, the FRFSM measures the degree of agreement on an absolute scale between 0 (no agreement) and1 (perfect agreement). L EE [11] shows that the FRFSM correlates with the subjective judgment of experts, who
rated the degree of agreement of frequency response functions obtained from vibration measurements of passenger cars. In this paper, it is investigated whether the FRFSM can be used to assess the degree of agreement of
the replicated MSTA and the measured MSTA as well.

3

EXPERIMENTAL SETUP

Table 1 lists the geometrical dimensions of the rectangular plates that are subject to the experimental investigations of this paper. All plates are manufactured three times and are made from aluminum. The material4251

properties have been measured for each plate using the method in [7]. The averaged material properties are:YOUNG ’ S modulus E = 70.59 · 109 N m−2 , P OISSON ’ S ratio µ = 0.33, and mass density ρ = 2 676 kg m−3 . A
simply supported boundary condition of the rectangular plates is obtained by bonding blades with 0.5 mm thickness to the plates’ edges. The blades are clamped to brackets, which are mounted on the test stand. Further
details on the design of the simple supported boundary conditions can be found in [4, 6].
Table 1. geometrical dimensions and scaling factors of the rectangular plates; ∗) : calculated according to Eq. (5)
plate
p
r1
r2
a1
a2

parentreplica
replicaavataravatar

a

b

h

φa

φb

φh

φl

870 mm
435 mm
435 mm
375 mm
375 mm

620 mm
310 mm
310 mm
310 mm
310 mm

5.0 mm
5.0 mm
2.5 mm
5.0 mm
2.5 mm

1.0
0.5
0.5
0.43
0.43

1.0
0.5
0.5
0.5
0.5

1.0
1.0
0.5
1.0
0.5

1.0
0.5
0.5
0.466∗)
0.466∗)The vibration velocities are obtained from SLDV measurements in the frequency range of 0 . . . 5 kHz for one
plate of each size as described in [4], and the MSTA is calculated from Eq. (1). The loss factors are measured
for all three plates of each size in order to assess the similitude of the damping. Each rectangular plate isexcited by an impact and the time-decaying vibrations are measured by an accelerometer. The loss factors ofeach rectangular plate are obtained at 30 peaks in the vibration spectrum using the 3 dB bandwidth. Further
details on the measurement procedure can be found in [3]. The loss factor measurements are performed atthree different states: (1) rectangular plates with free boundary conditions, (2) rectangular plates with bladesbonded to the edges and free boundary conditions, and (3) rectangular plates with simply supported boundary
conditions. This allows for determining the effect of the boundary conditions on the loss factor. In order toanalyze the significance of the similitude of the loss factors, an analysis of variance (ANOVA) is performed.
Besides the significance of the boundary conditions, the significance of the plates’ sizes is analyzed within the
ANOVA as well.

4

RESULTSFirst, the replicated MSTA is analyzed for all replicas and avatars and the accuracy of the replicated MSTA
is qualitatively assessed by a subjective judgment. Second, the results of the loss factor measurements are
discussed. Third, the measures defined in Section 2.2 are calculated and the results are discussed.
4.1 Replicating the MSTA
Figure 1 shows the replicated and measured MSTA of the replicas r1 and r2. The replicated MSTA levels agree
sufficiently well with the measured ones up to a frequency of approximately 1 kHz. Above 1 kHz, the frequencies with locally increased MSTA levels (peaks) are still accurately replicated, but the replicated MSTA levels
themselves tend to be higher than the measured MSTA levels. Since the replicas r1 and r2 are in geometricallycomplete similitude to the parent, the deviation above 1 kHz is rather caused by an incomplete similitude of thedamping than by a geometrically incomplete similitude.
Figure 2 shows the replicated and measured MSTA of the avatars. The MSTA levels agree well at somefrequencies, e.g., at approximately 0.9 kHz for avatar a1 or at approximately 0.3 kHz for avatar a2. Due togeometrically incomplete similitude conditions, the vibration behavior of the avatars differs from that of thereplicas and, thus, from that of the parent. For example, the replicated MSTA levels of avatar a1 show threepeaks between 1.5 kHz and 2.0 kHz that are not shown by the measured MSTA levels.

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level of ShT

2 (r2)

in dB

level of ShT

2 (r1)

in dB

measured

replicated

40
30
20
10
0
−10
−20
−30
40
30
20
10
0
−10
−20
−30

102

103

104

frequency in Hz

Figure 1. MSTA levels (re 6.25 · 10−6 m4 N−2 s−2 ) versus frequency of the replicas r1 (top) and r2 (bottom),frequency axis applies to both plots, figure obtained from [3]level of ShT

2 (a2)

in dB

level of ShT

2 (a1)

in dB

measured

replicated

40
30
20
10
0
−10
−20
−30
40
30
20
10
0
−10
−20
−30

102

103

104

frequency in Hz

Figure 2. MSTA levels (re 6.25 · 10−6 m4 N−2 s−2 ) versus frequency of the avatars a1 (top) and a2 (bottom)
The previous analyses of the agreement of the measured and replicated MSTA levels is obtained from visuallyinspecting Figures 1 and 2, which is a subjective judgment made by the authors of this paper. Other individuals may obtain a different view when analyzing the figures. As a consequence, more objective measures arenecessary to assess the agreement between the replicated MSTA and the measured MSTA. Section 4.3 proposessuch measures. These measures are analyzed in order to fund out whether they match the subjective judgmentof Figures 1 and 2 presented in this section.

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4.2 Similitude of the loss factor
The results of the ANOVA are presented as box plots. A detailed analysis can be found in [3]. In this paper,
only the main findings are presented. The loss factors are found to be log-normally distributed, which requires
the logarithmized loss factors to be used within the ANOVA. It is found that the loss factor significantly increases on a 95% confidence level if the blades are bonded to the plates’ edges, which can be seen by the factthat the notches of the boxplot are not overlapping, see Figure 3. The simply supported boundary condition

logarithmized
loss factor ln (η)

−2
−4
−6
−8
−10
−12

free

bonded blades, free simply supported
boundary condition

Figure 3. box plot of the logarithmized loss factors subdivided by plates’ boundary conditions
yields not only another significant increase of the loss factor on a 95% confidence level, but also causes thevariance to increase, which is illustrated by the box’ size. As a consequence, the similitude of the loss factors depends on the boundary conditions. Particularly, the simply supported boundary conditions increase thevariance of the loss factors, which may lead to an incomplete similitude of the damping and, thus, limits theapplication of scaling laws in practice. It is shown in [4] that the MSTA levels of rectangular plates in geometrically complete similitude can be replicated sufficiently well if the loss factor is in complete similitude.
Figure 4 shows the box plot of the logarithmized loss factors subdivided by the different plates’ sizes. The
loss factors significantly differ on a 95% confidence level, while the variance is approximately equal. Thus,
the loss factors depend on the plates’ sizes as well. Note that this effect is independent from the effect of
the boundary conditions, which has previously been described. As a consequence, an incomplete similitudeof the loss factors leads to incomplete similitude conditions – even for the rectangular plates in geometricallycomplete similitude. Thus, the application of scaling laws is limited in practice and incomplete similitude causes
the replicated MSTA levels to differ from the measured ones as can be seen in Figures 1 and 2. In order tofind the limits of the application of scaling laws, it is necessary to assess the agreement of the measured andthe replicated MSTA, which requires appropriate measures. Such measures are analyzed in Section 4.3.

logarithmized
loss factor ln (η)

−2
−4
−6
−8
−10
−12

p

r1

r2

r3

r4

rectangular plate

Figure 4. box plot of the logarithmized loss factors subdivided by plates’ sizes

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4.3 Assessing the similitude
The results in Section 4 show that
1. analyzing the agreement between replicated MSTA and measured MSTA is somewhat subjective, and2. limitations of the application of scaling laws arise from the fact that complete similitude conditions are
hardly fulfilled in practice.
However, both findings lead to the fact that the degree of agreement between replicated MSTA and measured
MSTA needs to be assessed. Therefore, dH , d M , and the FRFSM are calculated using the measured and thereplicated MSTA of each replica and avatar. As discussed in Section 4.1, the MSTA of the replicas r1 and r2
subjectively agree well below 1 kHz. Thus, dH , d M , and the FRFSM are not only calculated over the entire
frequency range 0 . . . 5 kHz, but also for 0 . . . 1 kHz and > 1 . . . 5 kHz.
Table 2 shows the results. Considering the entire frequency range the mean M AHALANOBIS distance d M andthe FRFSM match the subjective judgment by the authors for the replicas, since the mean M AHALANOBIS
distance d M (the FRFSM) of replica r1 is higher (lower) than that of replica r2. Based on the subjective
judgment it has been concluded that the replicated MSTA agrees well with the measured MSTA below 1 kHz
for the replicas r1 and r2 (see Section 4.1), while the MSTA agrees to a lesser extent above 1 kHz. The
H AUSDORFF distance and the FRFSM reflect this subjective judgment, since the H AUSDORFF distances and the
FRFSM in the frequency range 0 . . . 1 kHz of the replicas r1 and r2 are smaller than those in the frequency
range > 1 . . . 5 kHz. In case of the avatars, the H AUSDORFF distance, the mean M AHALANOBIS distance, andthe FRFSM do not generally show that the MSTA agrees to a lesser extent than the MSTA of the replicas.
For example, the mean M AHALANOBIS distance equals 4.36 · 10−3 m4 N−2 s−2 and 0.44 · 10−3 m4 N−2 s−2 in thefrequency range 0 . . . 5 kHz for the replica r2 and the avatar a2, respectively, which indicates that the replicatedand measured MSTA agree well for avatar a2, while they (somewhat) deviate for replica r2. However, thesubjective judgment (see Section 4.1) yields a different result.
Table 2. measures to assess the degree of agreement of the MSTA, dH and d M are given in m4 N−2 s−2 , while
FRFSM is dimensionlessdH

0 . . . 5 kHz
dM

FRFSM

dH

dM

FRFSM

dH

> 1 . . . 5 kHz
dM

FRFSM

r1
r2

0.15 · 10−3
6.60 · 10−3

11.91 · 10−3
4.36 · 10−3

0.18
0.32

0.06 · 10−3
10.34 · 10−3

10.50 · 10−3
238.16 · 10−3

0.54
0.43

0.15 · 10−3
1.16 · 10−3

18.72 · 10−3
138.52 · 10−3

0.11
0.30

a1
a2

0.15 · 10−3
2.50 · 10−3

107.24 · 10−30.44 · 10−3

0.21
0.30

0.02 · 10−30.77 · 10−3

2.39 · 10−3
12.74 · 10−3

0.45
0.37

0.25 · 10−3
2.50 · 10−3

1661.40 · 10−30.42 · 10−3

0.16
0.28

plate

5

0 . . . 1 kHz

CONCLUSIONS

Scaling laws derived from numerical calculations are applied in order to replicate the measured MSTA of simply supported rectangular plates in geometrically complete and incomplete similitude. An ANOVA of the loss
factors shows that rectangular plates are not in complete similitude regarding the damping, which is caused by
the simply supported boundary conditions and by the plates’ sizes themselves. Both incomplete similitude dueto the damping and geometrically incomplete similitude cause the replicated MSTA to deviate from the measured MSTA.
The H AUSDORFF distance, the mean M AHALANOBIS distance, and the FRFSM are used to assess how well
the scaling laws replicate the measured MSTA. These measures match the subjective judgment of the agreementof the replicated and measured MSTA in some cases, while they are contradictory to the subjective judgment4255

in other cases. Further research will, thus, focus on developing an improved measure to determine the degreeof agreement. Such an improved measure should reliably turn the subjective judgment of individuals into an
objective figure. Thus, judging of the agreement of the MSTA (or frequency response functions in general) by
individuals needs to be understood in more detail.

ACKNOWLEDGEMENTS
The authors acknowledge the financial support for the scanning laser D OPPLER vibrometer from Deutsche
Forschungsgemeinschaft (DFG) under grant number INST 163/520-1.

REFERENCES[1] Casaburo, A.; Petrone, G.; Franco, F.; De Rosa, S. A review of similitude methods for structural engineering. Applied Mechanics Reviews, 71(3), 2019, 030802-1–030802-32.
[2] Adams, C.; Bös, J.; Slomski, E.M.; Melz, T. Scaling laws obtained from a sensitivity analysis and applied
to thin vibrating structures. Mechanical Systems and Signal Processing, 110, 2018, 590–610.
[3] Adams, C. Similitudes and sensitivities as contributions to scaling laws in machine acoustics. PhD dissertation, Technische Universität Darmstadt, Darmstadt, Germany, 2019.
[4] Adams, C.; Bös, J.; Melz, T. An experimental investigation of vibrating plates in similitude and the possibility to replicate the responses using sensitivity-based scaling laws. In 6th NOVEM – Noise and Vibration
Emerging Methods, Ibiza, Spain, May 07–09, 2018.
[5] De Rosa, S.; Franco, F. Analytical similitudes applied to thin cylindrical shells. Advances in Aircraft andSpacecraft Science, 2(4), 2015, 403–425.
[6] Robin, O.; Chazot, J.D.; Boulandet, R.; Michau, M.; Berry, A.; Atalla, N. A plane and thin panel withrepresentative simply supported boundary conditions for laboratory vibroacoustic tests. Acta Acustica united
with Acustica, 102(1), 2016, 170–182.
[7] Langer, P.; Sepahvand, K.; Guist, C.; Bär, J.; Peplow, A.; Marburg, S. Matching experimental and three
dimensional numerical models for structural vibration problems with uncertainties. Journal of Sound andVibration, 417, 2018, 294–305.
[8] Huttenlocher, D.P.; Klanderman, G.A.; Rucklidge, W.J. Comparing images using the Hausdorff distance.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(9), 1993, 850–863.
[9] Meruane, V.; De Rosa, S.; Franco, F. Numerical and experimental results for the frequency response ofplates in similitude. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of MechanicalEngineering Science, 230(18), 2016, 3212–3221.
[10] Mahalanobis, P.C. On the generalized distance in statistics. Proceedings of the National Institute of Sciences of India, 2(1), 1936, 49–55.
[11] Lee, D.; Ahn, T.S.; Kim, H.S. A metric on the similarity between two frequency response functions.
Journal of Sound and Vibration 436, 2018, 32–45.

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