PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Analysis of displacement amplification characteristics of class IV
flextensional shell based on elliptic perimeter approximation
formulaHoulin FANG1; Difeng SUN; Fang ZHANG; Yang LIU; Tianqing ZHAO; Zhang CHENG; Liangyong
ZHANG; Xubin LIANG; Deyu SUN
1

Northwest Institute of Nuclear Technology, China

ABSTRACTAs the most widely used underwater acoustic transducer, class IV flextensional transducer can get larger
amplitude flexural vibration from longitudinal vibration of driven stack by the displacement amplification
effect, thereby radiating more energy. In order to further quantitatively understand the displacement
amplification characteristics of elliptical tube shell, this paper proposed to calculate the displacement
magnification using the ellipse perimeter approximation formula, and the variation law of the displacement
amplification with the compression coefficient and the deformation rate was obtained, which was analysedand compared with the results by the method of finite element analysis. The empirical formula for calculating
the finite element results by the results of the perimeter approximation formula was got by fitting curves.
When the compression coefficient is 0.24~0.80, the error of the calculation results was not more than 6.11%.
The research of this paper has certain reference significance for the estimation of the radiation performanceof low frequency transducers.
Keywords: Displacement magnification, Class IV flextensional shell, Elliptic perimeter approximation
formula

1. INTRODUCTION
Underwater acoustic transducers which plays an important role on sound information translation
are widely used in ocean monitoring, underwater communication, marine resources development andso on(1). As a member of underwater acoustic transducers, class IV flextensional transducer can getlarger amplitude flexural vibration from longitudinal vibration of driven stack by the displacement
amplification effect, thereby radiating more energy. Due to advantages such as low frequency, highpower and small size, class IV flextensional transducer has always been a resear ch hotspot, so that is
the most widely studied and analyzed.
The research of class IV flextensional transducers is mainly reflected in the two aspects ofmaterials and structures. The materials are mainly driven materials research, supplemented by shellmaterials research. The traditional driven material is piezoelectric ceramics. In recent years, rare earth
giant magnetostrictive materials(Terfenol-D) have been applied to a large extent, and some newly
developed materials such as antiferroelectric ceramics and single crystal materials are also applied inclass IV flextensional transducers. Transducer structure is rich in research. In recent years, transducers
with low frequency, wideband and high power have been the main research directions. Fish -lipflextensional transducer invented by Mo Xiping have adopted a variable-height elliptical shell, whichmakes the shell have both a double-amplification effect (2).The frequency can be effectively widenedby slitting the radiating surface of the shell. The Institute of Acoustics of the Chinese Academy of
Sciences and the Harbin Engineering University have in-depth research (3-5).
In order to maximize the radiated sound power while ensuring the broadband performance of thetransducer, Lan Yu and Mo Xiping have developed a double-shell flextensional transducer (6), which
maximizes the volume displacement output of the transducer to achieve low frequency and high power
acoustic radiation. In 2011, Indian scholars (7) proposed a flextensional transducer with a parabolicreflector with a gain of 1.7-8dB. Multi-mode coupling through the shell is also an effective means of1

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fanghoulin@nint.ac.cn

expanding the frequency band and many scholars have done a lot of work (8, 9).In 2017, Wang Yue
studied waterdrop type flextensional transducer based on antiferroelectric ceramics, which effectively
expanded the bandwidth by first asymmetric bending mode and first bending mode coupling (10). Zhou
Tianfang(11) proposed a transducer structure that is conformally driven with shell, which caneffectively avoid the change of vibration mode caused by the traditional longitudinal drive crystal
stack, and can obtain a lower frequency.
In terms of driving materials, the erbium-doped antiferroelectric ceramics(PLZST) of the Xi'an
Jiaotong University was first applied in class IV flextensional transducer (12).Li Kuan et al. studied aClass IV rare-earth flextensional transducer(13).
Although research on class IV flextensional transducer is so adequate there are few stu dies on thedisplacement amplification characteristics. In the literatures(14,15) only the displacement
amplification schematic diagram is given, and the displacement amplification effect is briefly
described. However, there are few quantitative studies on the displacement amplification
characteristics of the flextensional shell, especially the displacement magnification. Therefore, forthe need of for class IV flextensional transducer in air, this paper proposed a method to estimate thedisplacement magnification using the approximate perimeter formula, and the variation law of thedisplacement magnification with the compression coefficient and deformation rate was obtained,
which was analyzed and compared with FEA results. The work of this paper has certai n reference
significance for the estimation of radiation performance of low frequency transducers.

2. THEORETICAL CALCULATION
2.1 Derivation of Displacement Magnification
The shell of class IV flextensional transducer is an elliptic cylindrical shell, which is simplified toa neutral plane ellipse with long-axis 2a, short-axis 2b (a>b) , focal length 2c ( c  a 2  b2 ) andeccentricity e (e=c/a<1) for convenience. When the load (force or displacement) of stack acts on theellipse, the long-axis deformation is Δa, and the short-axis deformation is Δb (Figure 1).

Figure 1 –Schematic diagram of neutral plane ellipse deformation
Two assumptions are made before and after ellipse deformation:(1) After deformation, especiallywhen the deformation is relatively small, the structure is still approximate ellipse.(2) Before andafter deformation, the deformation of elliptic perimeter is much smaller than that of long - andshort-axis, and the elliptic perimeter is approximately unchanged. So the following derivation ismade. Elliptic perimeter approximation formula isL  π 1.5  a  b   ab (1)Thus, elliptic perimeter of before deformation is equal to that of after deformation as following
function.
1.5  a  b   ab  1.5  a  a  b  b   a  a b  b 

(2)

Let Δb/Δa=α, defined as displacement magnification, and is substituted into above formula to get
the quadratic equation for α.
 a  3 ab
3 ab  b
2.25 2  
 3.5   
 2.25  0
(3)
a
 a


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The solutions is2

 a  3 ab
 3 ab  b 3.5   4  2.25  
 2.25 

(4)
 a

 a


4.5
It can be seen that displacement magnification α is related to half-long-axis a, half-short-axis b
and long-axis deformation Δa. Let b/a=B, defined as compression coefficient in mathematics. Let
Δa/a=A, defined as deformation rate. So the displacement magnification expression can be
simplified to(considering the actual situation, the symbol “±” should be taken as “-”)a  3 ab
3.5 

a

2

1 3 B
3 B B
1 3 B3.5   3.5   9 
 2.25 
A
A
A





4.5
where A and B meet the following inequality group0  A  1;
0  B  1;
A  1 3 B ;

3.5

2
2
8 A  7  6 B  9 B A  1  3 B  0.




 

(5)

(6)



2.2 Law of Change
Based on the above formula (5), a group of curves (Figure 2 and Figure 3) showing the relationshipbetween displacement magnification and compression coefficient b/a and deformation rate Δa/a isobtained. In Figure 2, as the compression coefficient increasing the displacement magnification
decreases and the decrease rate gradually decreases. When compression coefficient is same, the larger
deformation rate, the larger the displacement magnification. And the difference of curves decreases
with the increase of compression coefficient. The minimum compression coe fficient that can be solved
by perimeter approximation formula is related to the deformation rate. The larger the deformation rate,
the larger the minimum compression coefficient. Figure 3 gives a series of similar results, but the trendof curves is opposite. It can be known that if the perimeter approximation formula is solved,
compression coefficient should not be too small and deformation rate should not be too large.
Compression coefficient has a greater influence on the displacement magnification than deformation
rate.

Displacement magnification

6
5.5
5
4.5

a/a=0.002
a/a=0.01
a/a=0.02

4
3.5
3
2.5

a/a=0.05
a/a=0.10
a/a=0.20

2
1.5
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Compression coefficient1

Figure 2 – Displacement magnification versus compression coefficient with different deformation rates

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Displacement magnification

5

4

3

b/a=0.20
b/a=0.25
b/a=0.30
b/a=0.35
b/a=0.40

b/a=0.50210b/a=0.60
b/a=0.70
b/a=0.90
0.05

0.1
0.15
Deformation rate

0.2

0.25

Figure 3 –Displacement magnification versus deformation rate with different compression coefficients

3. FINITE ELEMENT ANALYSIS
3.1 Modeling
The structural parameters of the class IV flextensional shell mainly include long-axis(2a),short-axis(2b), wall-thickness(t) and width(B). Combined with the analysis above, the displacement
magnification is mainly related to the long-axis and the short-axis. Therefore, the long-axis and theshort-axis are mainly changed, and the wall-thickness and width are only selected a value
appropriately.
The FEA modeling firstly establishes the calculation model according to the parameters, thendivides the grid according to the calculation requirements, sets the model load, and finally calculates
the structural deformation nephogram. Further, the displacement value can be extracted to obtain thedisplacement magnification.

Beforedeformation(a) Parameterized model(b) Meshing(c) Load setting(d) Deformation result

Figure 4 –FEA modeling
3.2 Law of Change
On the basis of theoretical calculation, a set of shells with different structural parameters arecorrespondingly designed for FEA. Figure 5 shows the displacement magnification versus
compression coefficient comparing between theoretical calculation and FEA. Note that power
exponent lines fit these data quite well, FEA results are higher at all compression coefficients. The
difference decreases as compression coefficient increasing, when the value tends to 1, namely ellipse
tends to circle, the difference tends to be zero. When the compression coefficient reaches 0.122, there
is no solution by using the perimeter approximation formula, but the FEA results are still consistent
with the overall law.

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Displacement magnification

9

Δa/a=0.0976%_FEA
Δa/a=0.0976%_Peri-appr-f
Δa/a=0.488%_FEA
Δa/a=0.488%_Peri-appr-f
Δa/a=1.0%_FEA
Δa/a=1.0%_Peri-appr-f

8
7
6
5
4
3

2
1
0
0.1

0.2

0.3

0.4
0.5
0.6
0.7
Compression coefficient

0.8

0.9

Figure 5 –Perimeter approximation formula and FEA results of displacement magnification versus
compression coefficient with deformation rate is 0.0976%, 0.488% and 1.0%(which are written as %conveniently). Perimeter approximation formula is replaced by “Peri-appr-f” in the figure.
In figure 6, the curve with a compression coefficient of 0.3 is basically larger than the curve witha compression coefficient of 0.4. FEA results are generally larger than the perimeter approximation
formula results with different trends. Especially, the curve with a compression coefficient of 0.3, theresults of the perimeter approximation formula increases faster with the increase of deformation rateuntil no solution. However, the curve of FEA increases with the deformation rate, of which the rate is
gradually slowed, and then the displacement magnification reaches the maximum and then begins tofall. The curve maximum point with a compression coefficient of 0.3 is smaller than the curve with acompression coefficient of 0.4.

Displacement magnification

4.2
3.7
3.2

b/a=0.3_FEA
b/a=0.3_peri-appr-fb/a=0.4_FEA
b/a=0.4_peri-appr-fb/a=0.6_FEA
b/a=0.6_Peri-appr-f

2.7
2.2
1.7
1.2

0 0.010.020.030.040.050.060.070.080.09 0.1 0.110.12
Deformation rate
Figure 6 –Perimeter approximation formula and FEA results of displacement magnification versus
deformation rate with compression coefficient is 0.3, 0.4 and 0.6. Perimeter approximation formula isreplaced by “Peri-appr-f” in the figure.
In summary, the results indicates that the compression coefficient has a relatively great influenceon the displacement magnification. With the chosen range of compression coefficient and1952

deformation rate, the maximum variation of the perimeter approximation formula results with thedeformation rate is 48.4%, and the FEA results is up to 33.8%. However, the corresponding
maximum variation with the compression coefficient is 392.9% and 587.1%.

4. COMPARATIVE ANALYSIS
4.1 Difference Analysis between Perimeter Approximation Formula and FEA
The displacement nephogram of FEA showing in figure 7 compares the change of the elliptical shellon which a force loaded. For convenience of analysis, draw the long - and short-axes before and after
the deformation and the corresponding ellipse.
It can be seen that when the deformation displacement is large enough (for example Δa=5mm), theneutral surface of the structure is obviously no longer elliptical, the deformation in the long -axis
direction is Δa, and the displacement in the short-axis direction is Δb. If the ellipse is still consideredafter the deformation, as shown by the black dot-dash line, the perimeter will inevitably decrease. If
the perimeter is to be kept constant, the deformed ellipse should be as shown by the pink dash line
ellipse. The displacement deformation in the short-axis direction is Δb’ which is smaller than Δb, so
the displacement magnification obtained by theoretical calculation is smaller than the displacement
magnification calculated by FEA, and the larger the displacement deformation in the long-axis
direction, the larger the difference between the deformed structure and the ellipse, which will lead tothe larger the difference of displacement magnification between theoretical calculation and FEA. Thisis the same conclusion as the previous analysis.

∆b’

∆b

Ellipse with same perimeter (L)Ellipse with same a and b∆a

No deformation

Actual deformation

Figure 7 –Schematic diagram of before and after deformation with corresponding ellipses
4.2 FEA Curve of Deformation Rate Analysis
For the phenomenon in FEA that the displacement magnification reaches the maximum value andthen falls is further analyzed. It can be seen from the lateral deformation nephogram of the shell that
before the displacement magnification reaching the maximum value, the curvature direction of theshort-axis end point remains unchanged after deformation (Namely the structure at the end of theshort-axis is convex), but the curvature direction becomes changed (That is, the structure at the end of
the short-axis becomes inwardly concave), which conforms to the actual deformation characteristics,
and the structural stress also reaches the strength limit of the material. By analyzing the structures withcompression coefficients of 0.3 and 0.4, the relative deformation rate of the short-axis direction when
the displacement magnification reaches the maximum value is 39%-45%.
4.3 Fitting Curve Analysis
In the calculations above, in order to observe the overall trend and law of variation the range ofchosen compression coefficient and deformation rate is large, and the difference between perimeter
approximation formula results and FEA results is large, especially law of change with deformation
rate. Considering the actual situation, the compression coefficient and deformation r ate should be
limited as follow:(1) Strain of driven materials. The strain of driven materials such as piezoelectric ceramics,

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rare-earth materials and antiferroelectric materials directly determines the deformation rate of shellin the long-axis direction. The strain of this kind of material is generally a few thousandths (themaximum strain of Terfenol-D is 1.5‰(1)), and the maximum strain of relaxor ferroelectric single
crystal is up to 1.7%, so the maximum deformation rate is recommended as 2% (0.02).
(2) Using the approximation formula for calculation, the difference will increase when there is no
solution. At the same time, the compression coefficient will be generally less than 0.2 with which theshell is extremely narrow this structure and is rarely used. Therefore, the recommended compression
coefficient is chosen to be a minimum of 0.2.
Based on two points above, fitting curves of displacement magnification versus compression
coefficient gets the relation of displacement magnification between perimeter approximation formulaand FEA:
0.355b peri-appr-f(7)
 
A set of flextensional shells are designed, first estimated by empirical formula (7), and thencalculated by FEA to verify the accuracy of the estimation. Table 1 lists the calculation results,
which shows that the empirical formula can be used to calculate the FEA results from the results ofthe perimeter approximation formula, and the calculation error is not more than 6.11% when th ecompression coefficient is 0.24~0.80.
Table 1 –Results of estimation by empirical formula FEA  1.053  aPerimeter

Estimation

Appro-fof FEA

1

4.090

0.01

1

24

0.01

100

25

0.60

100

0.80

100

NO.

b/a

a/mm

b/mm

Δa/a

Δa/mm

FEA

Error

1

0.20

100

20

0.01

7.626

5.500

38.65%

2

0.22

100

22

3.285

5.921

5.089

16.35%

3

0.24

100

1

2.841

4.965

4.679

6.11%

4

0.25

0.01

1

2.681

4.618

4.486

2.94%

5

60

0.02

2

1.326

1.674

1.709

2.05%

6

80

0.02

2

1.133

1.290

1.226

5.22%

5. PRELIMINARY CONCLUSIONS
According to the calculation of the perimeter approximation formula and FEA results, the influenceof the compression coefficient on the displacement magnification is greater than the deformation rate.
FEA results are larger than the results of perimeter approximation formula.
With the increase of compression coefficient the displacement magnification decreases as power
exponential. The same compression coefficient, the larger the deformation rate, the larger the
displacement magnification, but the difference is not obvious. With the increase of compression
coefficient, the difference between the FEA results and perimeter approximation formula results
decreases.
Law of change with deformation rate, the difference between the FEA results and perimeter
approximation formula results is large, especially with smaller compression coefficient. Firstly thedisplacement magnification increases with the deformation rate, the growth rate is gradually slowed,
and then began to fall after reaching the maximum value. The curve maximum point with acompression coefficient of 0.3 is smaller than the curve with a compression coefficient of 0.4.
When the compression coefficient is 0.24~0.80 and the deformation rate is less than 2%, theempirical formula can be used to calculate the FEA results from the results of the perimeter
approximation formula, and the calculation error is not more than 6.11%.

ACKNOWLEDGEMENTS
The authors would like to thank the Acoustic Engineering Innovation Group Fund for funding thisproject, and also would like to thank every member of the project team for their con tribution to thework of this paper. This work would not have been possible without your help.

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