Multifrequency ultrasonic transducers with spatially distributed
stop band material
Johannes Henneberg(1,2) , Simone Preuss(1,2) , André Gerlach(2) , Steffen Marburg(1)(1) Technical University of Munich, Chair of Vibroacoustics of Vehicles and Machines, Germany, simone.preuss@tum.de(2) Robert Bosch GmbH, Corporate Research, Germany, johannes.henneberg@de.bosch.com

Abstract
Ultrasonic transducers are widely used for surround sensing applications in the automotive industry. A major
challenge for today’s mobility solutions consists in the development of autonomous driving functions. Highly
automated vehicle concepts require a comprehensive surround sensing performance, which leads to an increasing number of sensors. The applied ultrasonic sensors are desired to operate in parallel or in shorter intervals.
Hence, transducers with two or more well discriminated operating frequencies are of great interest. Known
multifrequency ultrasonic transducers use multiple electromechanical coupling elements resulting in more complex sensor electronics. To overcome this issue, the authors investigate a method for controlling the operationaldeﬂection shapes. In ﬁnite element simulation, a generic plate-like model for the ultrasonic transducer is employed and different resonator conﬁgurations are studied. Spatially distributed stop band material is used toachieve two well-separated operating frequencies with similar and appropriate sound radiation. Furthermore, it
is investigated whether the shape modiﬁcation is caused by the resonant behavior or the added resonator mass.
Based on the results, locally distributed resonators are identiﬁed as a suitable solution to modify the operationaldeﬂection shapes and the corresponding sound radiation of the generic model. Thus, a multifrequency ultrasonicsensor with a single coupling element can be realized.
Keywords: ultrasonic sensor, stop band material, acoustic metamaterial, periodic structure

1

INTRODUCTION

Owing to the recently increased demand for autonomous driving functions, high potential surround sensing applications are required. For decades, ultrasonic sensors provide a reliable environment analysis as part of thedriver assistance systems. In order to measure the distance between the sensor and the surrounded objects, thepulse-echo method is used. The sensor, working in its resonant operation point, evaluates the object location bymeasuring the time difference between emitted sound signal and detected echo [1]. To achieve a detailed resolution of the car’s environment, more and more ultrasonic sensors are applied. The time efﬁcient operation ofvarious sensors leads to the desire for multiple operating points. A design with two or more well discriminatedoperating frequencies and appropriate deﬂection shapes beneﬁt parallel sensor operation and therefore, shorter
operation intervals.
In literature, a few approaches for multifrequency ultrasonic sensors can be found. In [2], a stepped-frequency
transducer consisting of loosely coupled magnetostrictive elements is presented. Simultaneous excitation of allelements of a stack leads to a broadband transducer. Although this concept could be used as a multifrequency
transducer, it suffers from a lack of efﬁciency compared to single-frequency equivalents. A split-mode ultrasonictransducer is investigated by [3]. In order to obtain a transducer with multiple acousto-electric resonances, astructure of periodically poled domains in a ferroelectric wafer with free surfaces is used. [4] introduces apiezoelectric micromachined ultrasonic transducer (PMUT). By using multiple electrodes, the device provides
high sensitivity at the ﬁrst, third, and ﬁfth fundamental frequency of the transducer. Further investigations
focus on a capacitive micromachined ultrasonic transducer (CMUT) [5]. The authors employ a CMUT arrayfor multiband operation with acoustic elements of different dimensions. For medical applications, especially forinterstitial thermal therapy, a transducer with a single electromechanical coupling element but multiple operating962

points has been designed. [6] introduces this new approach with matching front layers being able to transmit
ultrasound at multiple frequencies in the range of several MHz. Another recently developed sensor array for
medical purposes works with a zirconate titanate (PCT) transducer in dualfrequency mode (4.1 MHz and 13.3
MHz) [7]. [8] and [9] suggest hybrid solutions for multifrequency sensor usage. By combining different electromechanical coupling mechanisms, e.g. a CMUT and PMUT in one element, multiple resonance frequencies withsuitable sound radiation properties can be obtained.
However, the aforementioned approaches show some disadvantages regarding the implementation in driver assistance systems. One lack consists in the usage of multiple electromechanical coupling elements. This leads tomore complex electronics and therefore, more expensive sensors. Furthermore, the sound radiation characterized
by the directivity pattern should be appropriate, meaning that radiation similar to monopole or dipole behavior without distinctive side lobes is required. The last problem goes with the frequency dependent damping
characteristics of air. Operational frequencies in the high ultrasonic range lead to low echo amplitudes due tosigniﬁcant airborne sound damping, while in the lower range various sound sources from the car’s environment
interact. Thus, the operating frequency has to be in a certain range. For ultrasonic-based parking assistancesystems, a operation point between 40 and 50 kHz turned out to be the best compromise [10].
To overcome this disadvantages, the authors investigate a novel approach employing locally distributed resonators in order to obtain a multifrequency transducer with a single electromechanical coupling element andsuitable sound radiation at two distinctive resonance frequencies. By adding structures of resonators to a geometry, it is possible to prevent free wave propagation in certain frequency ranges. Thus, so called stop band
materials can be formed [11, 12]. Recent publications in this research area demonstrate that depending on thefrequency range, stop band material can decrease both, the acoustical [13] and the vibrational [14] responseof dynamic structures. [13] presents a promising outlook for periodic resonant structures causing desired bandgaps for low frequencies. Moreover, the periodic resonators represent an easy tuning possibility for the band
gap behavior. Another research work by [15] investigates the inﬂuence of the resonator spacing. It is shownthat random arrangement can also lead to appropriate band gaps but the distance in between the resonators is
required to be at a sub scale of the corresponding wave-length. Due to their ability to generate frequency ranges
without free wave propagation, stop band materials are predestined to increase sound transmission loss of structures [14]. For this purpose, [16] considers rubber coated lead spheres, which are embedded in epoxy, in orderto create a stop band material. In [17], the authors use stop band material to reduce mechanical cross-couplingin phased-array transducers.
In this study, stop band material is applied to modify the operational deﬂection shape (ODS) of a generic transducer model. As stop band material is only effective within a certain frequency range while outside this range
there is only marginal inﬂuence on the dynamic behavior, certain operating frequencies can be tuned mainly independently of each other. These characteristics open up the possibility for a multifrequency transducer design.
The recently published work of [18] provides a more detailed insight into the topic of this paper. While in
the present paper, numerical studies are carried out to investigate the inﬂuence of the particular dynamic behavior of the resonators, [18] extends the frequency range to ultrasound and further includes experimental results.
Moreover, different boundary conditions are considered and the sound radiation behavior of the transducer test
cases is investigated.

2

MODEL DESCRIPTIONIn this section, the investigated models are presented. First, a simpliﬁed model representing the ultrasonictransducer is discussed. Thereafter, the according ﬁnite element models of all test cases are presented.
2.1 Test cases
To analyze the potential of stop band material for multifrequency approaches, a generic model is employed. The
ultrasonic transducer is reduced to a rectangular plate with ﬁxed boundary conditions on all sides. Assuming
the transducer to be in sound emission mode, the electromechanical coupling element can be substituted by963

a centric dynamic load. The top of the plate, made of epoxy, represents the sound radiating surface. The
described model forms the reference case for further modiﬁcations. One design variation consists in adding
spatially distributed resonators to the plate. The resonators are attached to the downside of the model and are
distributed as shown in Figure 1. A middle section without any resonators (I) and two border sections witheach containing 72 resonators (II) are formed.

    


 





 





Figure 1. Generic model of the ultrasonic transducer. Plane epoxy plate as reference case (ref) and modiﬁcation
with spatially distributed resonators (CON). Excitation is realized through a dynamic load.
Furthermore, two conﬁgurations of the resonators are investigated. As shown in Figure 2, each resonator consistsof a beam like shaft and a tip mass. While the ﬁrst conﬁguration (CON1) is entirely made of epoxy material,
similar to the plate, conﬁguration 2 (CON2) has an additional middle section where another type of material isassigned. In here, an artiﬁcial epoxy equivalent with the same density and Poisson’s ratio but only a tenth of theYoung’s modulus is used. However, introducing this more ﬂexible material offers the possibility to separate theeffects caused by the dynamic behavior of the stop band material from additional mass effects. With regard toautomotive applications, smaller and more complex transducer geometries are relevant. The authors presented astudy with such geometries in [19]. But once proved on the generic model, the principle of spatially distributed
stop band material can be transferred without difﬁculty.



Figure 2. Two different test cases of the resonator design. CON1: Resonator entirely made of epoxy. CON2:
Resonator made of epoxy and a more ﬂexible epoxy equivalent in the middle shaft section.

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2.2 Finite element model
The dynamic behavior of the above shown test cases is investigated in a ﬁnite element (FE) simulation. The
simulations are carried out with the commercial software Simulia Abaqus 2017. The employed FE model is set
up of 20 quadratic node-brick elements (element type C3D20) [20]. In order to avoid mesh dependencies, amesh convergence study following the guidelines of [21] is carried out. All applied materials are considered tobe linear elastic, therefore, fulﬁlling Hooke’s law. The contact between the resonators and the plate is modeled
as a tie constraint with a surface-to-surface formulation. In a harmonic analysis, the forced response function
and the ODS is analyzed in the frequency range from 3 kHz to 25 kHz with a step size of 55 Hz.

3RESULTS

The sound radiation behavior of transducers during operation is indicated by the deﬂection shape in the corresponding operation point (OP). The desired radiation properties for industrial applications, characterized by anhomogeneous directivity pattern, is achieved for the ﬁrst normal mode. Therefore, the new design should show
two ODS, which are similar to latter. The ﬁrst way of realizing this approach is through setting the operating
frequency of the system to the ﬁrst eigenfrequency of the plate-like structure. As shown in Figure 3 (a), thereference test case has three resonance peaks in the analyzed frequency range. The calculated mode shapes are
similar to the reference cases from literature, following the rules of Kirchhoff’s plate theory [22, 23]. Hence,
the ﬁrst peak at 7.8 kHz provides the desired ODS. Figure 3 (b) shows the frequency response function (FRF)
for the modiﬁed test cases. In comparison to the reference model, the additional resonator mass causes a frequency drop of 1.4 kHz. On the other hand, the stiffening effect caused by the attached resonators is negligible.
However, the stiffening effect on the plate-like structure due to attachement of the resonators is present for both
conﬁgurations CON1 and CON2 in the same way. Therefore, the ﬁrst resonance peak occurs at 6.4 kHz forboth conﬁgurations.
(a)

(b)
1

1

normalized displacement [-]normalized displacement [-]

ref

10-1

10-2

CON1
CON2
-1

10

-2

10

10-3

10-3
5

10

15
frequency [kHz]

20

25

5

10

15
frequency [kHz]20

25

Figure 3. FRF of the sound radiating surface evaluated at the center point. (a) FRF of the reference model.
(b) FRF of the conﬁgurations with locally distributed resonators. Resonance frequencies with ODS suitable forsound radiation marked with ∗ for 1st operating point and with  /  for 2nd operating point
Another manner of designing suitable directivity patterns consists in identifying an ODS where one part of thestructure behaves similar to the ﬁrst normal mode while the rest remains in equilibrium position. The FRF
plots in Figure 3 (b) depict more resonance peaks in the higher frequency range. At 18.3 kHz and 13.3 kHz in
conﬁgurations CON1 and CON2, respectively, the middle section II shows free wave propagation with all nodes

965

moving with the same phase. In contrary to that in the outer sections I with added stop band material, thewave propagation is attenuated. As a consequence, the nodes remain almost without displacement. Due to theimplementation of a less stiff epoxy equivalent in CON2, the resonance frequency is reduced. As the mass andgeometry remain the same for both conﬁgurations, the particular ODS is mostly correlated with the dynamic
behavior of the spatially distributed stop band material.
In Figure 4, the ODS corresponding to the above discussed operating points are presented. The plots show
the normalized proﬁles along the x-axis of the plate. To evaluate the amplitude and phase both, the real andimaginary part of the displacement is presented. The ﬁrst graph (a) depicts the ODS of the plain plate for theﬁrst resonant operating point. Compared to the reference case, the geometries equipped with stop band material,
cf. Figure 4 (b), show a more stretched but for both test cases equal proﬁle at OP1. This can be ascribed tothe marginal inﬂuence of the resonator dynamics. Moreover, the additional mass plays the more important role
for the investigated ODS stretching.

1
D

0.5
0

OP 1

1
CON1
CON2

0.5

E

0

1
OP 2

QRUPDOL]HGGLVSODFHPHQW>@

OP 1

ref

F

0.5
0
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
normalized plate length [-]

0.8

0.9

1

Figure 4. Normalized displacement along the x-axis through the center of the plate for the reference test caseat OP1 (a) and the conﬁgurations at OP1 (b) and OP2 (c). The solid lines indicate the imaginary amplitude,
the dotted lines the real values.
Considering the second operating point, a bigger difference between the two test cases is analyzed. Figure 4 (c)shows the normalized proﬁles of CON1 and CON2 at 18.3 kHz and 13.3 kHz, respectively. Both ODS have
all nodes on one side of the equilibrium position and further only slight changes of the phase. By that, an
appropriate sound radiation can be achieved. The resonance frequency and the ODS of this second appropriate
operation point is strongly inﬂuenced by the particular dynamic behavior and the dimension of the resonator.

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4

CONCLUSIONS

The authors present a study on multifrequency ultrasonic transducers with spatially distributed stop band material. Finite element simulation is employed for a generic transducer model. A reference model, consisting of aplain plate-like structure, and two further test cases with different, locally distributed resonators are investigated.
Two operating frequencies with suitable ODS are identiﬁed within a harmonic analysis. While the ﬁrst operating frequency is related to the ﬁrst normal mode of the plain system, the second one strictly depends on thestop band material. By introducing a more ﬂexible epoxy equivalent in the middle section of the resonator, it
could be ruled out that the effect is mainly caused by the mass or the inertia of the resonator. Furthermore, thedistinctive operating frequencies can be tuned mainly independently from each other. Based on the results, spatially distributed stop band material is identiﬁed as a possibility to design multifrequency ultrasonic transducers
with only one electromechanical coupling element and appropriate and designable sound radiation.

ACKNOWLEDGMENTS
The authors gratefully acknowledge Robert Bosch GmbH Germany for the funding of this research.

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