Journal Vibration The potential of stop band material in multi-frequency ultrasonic transducers

Ultrasonic transducers play a major role in surround sensing for automotive and industrial applications. The development of autonomous driving functions is one of the top challenges for mobility solutions of the 21st century. In this context, surround sensing systemswith high performance serve as a key enabler. This leads to an increasing number of sensors which are desired to operate in parallel or in shorter intervals. Hence, measurements should be conducted with two different frequencies to discriminate the sensor signals. Known multifrequency ultrasonic transducers mostly employ multiple electro-mechanical coupling ele- ments which require more complex sensor electronics. To overcome this issue, the authors present a study on multi-frequency ultrasonic transduc- ers using only one electro-mechanical coupling element. In order to achieve suitable sound radiation properties at two well separated operating frequencies, spatially distributed stop band material is employed. As a result, the operational deﬂection shape can be controlled at a certain frequency. In ﬁnite element simulation, the relation between spatially distributed stop band material and the resulting operational deﬂection shapes is investigated. The sound radiation behavior is estimated using the numerical results from a harmonic analysis as input for the Rayleigh integral. In experimental investigations, the presented approach is validated. element simulation and experimental testing show good accordance. Based on the results of the presented study, it is possible to realize a multi-frequency ultrasonic sensor with one electro-mechanical transducer element only and suitable sound radiation behavior at multiple operating frequencies. CMUT PMUT electro-mechanical transducer can be at two operating frequencies. Suitable sound radiation properties for surround sensing be with the obtained mode shapes at both structure rather than by the boundary conditions. These ﬁndings could also be transferred to other applications where certain dynamic behavior is desired independently from the boundary conditions. Hence, the same design could be used in various applications without inﬂuencing the dynamic behavior. In the present test case, this effect leads to a suitable sound radiation for both conﬁgurations. The results of the directivity pattern in this study must be seen in the context of the generic model. As the directivity pattern strictly depends on the operating frequency and the dimensions of the radiator, the results of the generic model can not be transferred directly to real transducer applications. However, the results are suitable to estimate the potential of spatially distributed stop band material. Due to the additional boundary conditions in


Introduction
In recent years, the development of automatic and autonomous driving function gained rising attention. This requires a more detailed and reliable surround sensing of the environment than for manual driving. Since decades, ultrasonic sensors are widely used in surround sensing applications in automotive and industrial sectors. The distance between the sensor and the object is detected by pulse-echo operation [1]. In order to get a comprehensive resolution of the environment, the number of applied ultrasonic sensors increases continuously. To operate the sensors in shorter intervals or in parallel, it is desired to use multiple operating frequencies. Thus, a signal discrimination is possible by frequency filtering.
In literature, a couple of approaches to realize ultrasonic sensors with multiple operating frequencies are presented. In 1948, a stepped-frequency transducer consisting of multiple magnetostrictive nickel elements is presented by Ref. [2]. The single elements are loosely coupled to form a stack. Via a common electrical winding, the simultaneous excitation of the elements is possible. The aim of that study is to attain transducers with a broader frequency response. However, depending on the damping of the system, the approach would offer also a possibility to form multi-frequency transducers. It is found that the steppedfrequency transducers with multiple elements suffer from a lack of efficiency compared to single-frequency equivalents. In Ref. [3], a split-mode ultrasonic transducer is investigated. A structure with a two-dimensional periodicity of poled domains is created in a ferroelectric wafer with free surfaces. As a result, an ultrasonic transducer with multiple acousto-electric resonances is obtained. A piezoelectric micromachined ultrasonic transducer (PMUT) with multiple electrodes is presented in Ref. [4]. Due to multiple electrodes, the device provides high sensitivity at multiple frequencies coinciding with the first, third and fifth fundamental frequency of the transducer. Capacitive micromachined ultrasonic transducers (CMUT) are employed in Ref. [5]. A CMUT array with acoustic elements of different dimensions is designed having multiple operating frequencies. An approach using only one electro-mechanical coupling element is presented in Ref. [6]. The transmission of ultrasound is possible at multiple frequencies in the range of several MHz by adding an appropriate front layer. It is used for interstitial ultrasound thermal therapy. In the same context of medical ultrasonic therapy, a dual-frequency lead zirconate titanate (PZT)-transducer array with fundamental frequency (4.1 MHz) and third harmonic (13.3 MHz) is characterized in Ref. [7]. Ref. [8] presents a hybrid multifrequency ultrasonic transducer. It consists of different electro-mechanical coupling elements. One of which is a piezoelectric ceramic while the other one is made of PVDF. By hybridization of both characteristic properties, the transducers enable the usage of multiple resonance frequencies. In Ref. [9], also two electro-mechanical coupling mechanisms are combined. Thus, a CMUT as well as a PMUT electro-mechanical coupling is realized in one element. The transducer can be operated at two operating frequencies. Suitable sound radiation properties for surround sensing can be expected with the obtained mode shapes at both frequencies.
However, all presented approaches suffer at least from one of the following lacks: • The approach is not suitable for surround sensing applications. Especially the sound radiation behavior is not appropriate at various operating frequencies. It should show directivity pattern without distinctive side lobes. Hence, they are similar to monopole or dipole behavior at all operating frequencies.
• Multiple electro-mechanical coupling elements are used to realize multi-frequency transducers. This leads to more complex electronics required for surround sensing and is mostly undesired.
To overcome these issues, the present study investigates a novel approach employing spatially distributed stop band material in order to obtain a multi-frequency transducer with only one electro-mechanical coupling element and suitable sound radiation properties at two distinctive operating frequencies. Stop band materials attract attention within vibroacoustics recently [10]. Basically, it is possible to attain structures which do not allow free wave propagation in certain frequency ranges. These frequency ranges are called stop bands or band gaps [11,12]. In 2000, a phononic crystal with particular high sound transmission loss at certain frequencies is presented by Ref. [13]. The phononic crystal consists of rubber coated lead spheres which are embedded in epoxy. In Ref. [14], it is shown that stop bands caused by periodic, resonant structures show good prospect for low frequencies. Furthermore, the band gap behavior can be simply tuned by the periodic resonator. In Ref. [15], it is shown that stop bands can also be attained with randomly arranged resonant structures. However, the spacing between the resonators is required to be at a subscale of the wavelength corresponding to the desired band gap. Due to their ability to generate frequency ranges without wave propagation, stop band materials are predestined to increase sound transmission loss of structures. This is investigated in several studies, e.g. Refs. [16][17][18][19][20][21][22]. The possibility of waveguiding Lamb modes within a certain frequency is studied numerically and experimentally in Ref. [23]. In Ref. [24], the propagation of flexural waves in thin plates is investigated. It is shown that wave propagation can be attenuated by locally attached spring-mass resonators. The behavior of different excitations is investigated with regard to the vibration transmission. In Ref. [25], similar structures are investigated with regard to the sound transmission loss. In Ref. [26], the authors show the good prospect of stop band material in order to reduce mechanical cross-coupling in phased-array transducers.
In this study, stop band material is investigated with a rather different purpose. The operational deflection shape of a plate like structure representing a generic model of an ultrasonic transducer is modified within a certain frequency range. Therefore, sections with and without stop band behavior are spatially distributed. Stop band material shows the advantage that only a certain frequency range is effected by the particular dynamic behavior. Outside of the stop band, the influence on the dynamic behavior of the plate like structure is only small. Hence, certain operating frequencies can be tuned mainly independently from each other. This approach opens up the possibility to achieve a multi-frequency transducer having only one electro-mechanical coupling element and suitable sound radiating behavior at multiple operating frequencies. Thus, the drawback of known multifrequency ultrasonic transducer concepts can be overcome.
This work is structured as follows: initially, we focus on the methods applied in this study. The generic model to investigate the dynamic behavior is presented in Section 3. The approach is investigated numerically with a finite element model as well as in experiments with a physical model. The results are presented in Section 4. In Section 5, the approach is transferred to the ultrasonic frequency range. Finally, a conclusion is drawn.

Methods
In this section, the employed methods are presented briefly. First, the Rayleigh integral is discussed in order to estimate the radiated sound pressure field generated by the plate vibration. Afterwards, the wave finite element method (WFEM) applying periodicity conditions is presented. With this approach, it is possible to characterize the wave propagation behavior in periodic structures. Finally, the experimental validation using laser Doppler vibrometer is described.

Sound radiation
The sound radiation behavior of sound transducers is usually characterized by the directivity pattern. This can be calculated in the far-field employing the Rayleigh integral [27,28]. It provides an exact solution under the following conditions: • the radiating surface is flat, • the radiator is embedded in a flat, infinite, and rigid baffle.
These assumptions are suitable in order to obtain a reasonable approximation of the directivity pattern for ultrasonic transducers embedded in a mounting structure like a car bumper. This is discussed in several studies like Refs. [29,30]. However, in Ref. [31] it is shown that the Rayleigh integral produces reasonable results even for more complex structures. The assumption of a baffled plate holds for the investigated structures and the ultrasonic transducers for surround sensing applications in automotive and robotic industries.
In order to fit the far-field assumption, three conditions must be fulfilled [32]. Firstly, the distance (R) between the radiator and the field point has to be much greater than the dimension (l) of the radiator Secondly, the condition must be fulfilled. Consequently, the error caused by phase shifts is small [33]. Finally, the wavelength ( ) should be small compared to the distance between the radiator and field point For e i t harmonic time dependence, the Rayleigh integral for a plane surface can be expressed by Ref. [34]. with where Γ, k, , f , f , i denote the radiating surface, wavenumber, circular frequency, fluid particle velocity, fluid density, and imaginary unit. The particle velocity distribution on the surface of the structure can be used to estimate the radiated far-field sound pressure as shown in Ref. [35]. In spherical coordinates (R, , ), the Rayleigh integral can be approximated for the sound pressure in the far-field by Ref. [32].
The transformation from Cartesian to spherical coordinates is realized by Analyzing the directivity pattern in polar plots, it is possible to identify the sound radiation behavior of the investigated generic model.

Dispersion in periodic infinite structures
The wave propagation behavior in a periodic structure can be characterized by the dispersion relations in a unit cell. The Floquet Bloch theorem offers a possibility to handle periodic structure and to determine the wave propagation properties [36]. A freely propagating wave can be described for e i t harmonic time dependence with angular frequency by = We i( t−k x x−k y y) , (8) with W describing the wave mode through the thickness of the structure, being a response variable, and k x , k y being the components of the wavenumber in x− and y−direction, respectively. This can be obtained by employing finite element methods. The motion is described by a finite number of generalized displacements q.
In case of two-dimensional periodicity in x− and y−direction, q = [ ] T contains the generalized displacements at nodes 1 to 4, cf. Fig. 1. The Bloch theorem states that the relation between the generalized displacements can be described by Ref. [37].
where r x,y and x,y represent the length of the periodic lattice and the propagation constant in the direction of periodicity, respectively. Introducing the reduction matrix by the generalized displacement vector can be described dependent on q 1 by In the absence of damping, the generalized displacements are related to the generalized forces by with K, M, and f being the stiffness matrix, mass matrix, and force vector, respectively. Inserting Eq. (12) into Eq. (13) leads to Assuming no external forces, the sum of nodal forces connected to node 1 is zero. Thus, with can be written. Premultiplying Eq. (14) with Λ L leads to a reduced eigenvalue problem with The solution of the eigenvalue problem from Eq. (17) is a function of the propagation constants x , y . ( x , y ) characterizes the free wave propagation in the structure. It has to be solved for any combination of the propagation constants within the first Brillouin zone (BZ) with x , y = [− , ] [11]. Usually, the results are depicted in three-dimensional plots showing the dispersion surfaces. Thus, it is possible to identify omni-directional band gaps where no free wave propagation is possible. In case of symmetric unit cells, the calculation can be reduced to the irreducible Brillouin zone (IBZ) or to the contour of the IBZ. The three-dimensional plot of dispersion surfaces is then reduced to a two-dimensional plot of dispersion curves. This is discussed in detail in Ref. [10]. A detailed elaboration of the theory, known as wave finite element method, can be found in Refs. [37][38][39].

Experimental investigation
Experimental investigation is employed to validate the results of the generic model obtained from finite element simulations. In order to achieve a generic model which is comparable to a real ultrasonic transducer, fixed boundary conditions are required, cf. Section 3. These are realized by jointing the investigated structure to a backing with high stiffness and large mass compared to the structure. The velocity perpendicular to the sound radiating surface is measured at 351 scanning points with a noncontact measurement using a scanning laser Doppler vibrometer. The excitation is realized with an alternating current electric voltage of 10 V which is applied between the electrodes of the piezoelectric ceramic. Using a so called pseudo-random signal, all frequencies in the range from 0.3 kHz to 25 kHz are excited simultaneously. Furthermore, no leakage effects are introduced by this signal [40]. The obtained resonance frequencies and the related operational deflection shape (ODS) from experimental testing are then employed to validate the numerical simulations.

Models
In this section, the investigated models are presented. First, the generic model representing the ultrasonic transducer is discussed. The used finite element model is presented afterwards. Finally, the physical model used for experimental validation of the numerical results is described.

Generic model
A generic model is used in order to investigate the approach of using spatially distributed stop band material to attain multifrequency transducers. It consists of a rectangular plate like structure with a sound radiating surface, a piezoelectric ceramic realizing the electro-mechanical coupling, and 112 resonators forming two sections of stop band material, cf. Fig. 2 (a) (II). The plate like structure and the resonators are made of plastics. Different configurations of the plate boundary conditions are realized with blocks of aluminum. In the first configuration CON1 only the shorter edges of the plate like structure are fixed, cf. Fig. 2 (a). This configuration is comparable to ultrasound transducers presented in Ref. [41]. A second configuration CON2 is achieved by additional, partial fixing of the long edges of the plate like structure, cf. Fig. 2 (b). Using two different configurations of the boundary conditions, it is possible to ensure that the behavior at the second operating frequency is caused by the band gap behavior of the structure. If this is the case, the second operating frequency should be mainly independent of the plate like structure boundary conditions. With regard to automotive applications, smaller and more complex transducer geometries are relevant. The authors presented a study with such geometries in Ref. [29]. However, the investigation of the generic model offers the possibility to separate the effects introduced by the stop band material from other geometrical effects. Once proofed, the principle of spatially distributed stop band material can be transferred without difficulty.

Finite element model
The dynamic behavior of the generic model is investigated in a finite element (FE) simulation. The employed model, cf. Fig. 3, consists of 20-node brick elements with quadratic interpolation functions (Element type in Abaqus: C3D20) [42]. In order to avoid mesh dependencies, the guidelines given in Ref. [43] are applied. The calculations are carried out with the commercial software Simulia Abaqus 2017. The contact between different parts is modeled as tie constraint with a surface-to-surface formulation. All materials are assumed to have linear elastic behavior according to Hooke's law. Table 1 shows the material properties applied in the FE-simulation. The piezoelectric ceramic is similar to PZT-5a [44]. Hence, the material properties are comparable to that presented in Ref. [45]. In a harmonic analysis, the forced response function is calculated. As a result, resonance frequencies can be identified. The excitation is realized by applying an electric voltage to the piezoelectric ceramic. Afterwards, the calculated surface velocity of the plate like structure is used to obtain the radiated sound field using the Rayleigh integral.

Physical model setup
The experimental validation of the approach requires a physical model, cf. Fig. 4. The plate like structure and the resonators are manufactured as a monolithic using additive manufacturing techniques. PolyJet [47] technology offers the possibility to produce fully filled, solid structures. In a following step, the piezoelectric ceramic is jointed to the downside of the plate like  structure. The assembly is finished by jointing the aluminum blocks according to configuration CON1, cf. Fig. 2 (a), to the plate like structure. In order to realize the fixed boundaries, the entire assembly is jointed to a backing having a large mass and high stiffness compared to the plate like structure. After the experimental investigation of the test structure, additional aluminum blocks are added to the plate like structure. Hence, configuration CON2, cf. Fig. 2 (b), is realized using the same specimen. Testing one and the same structure with both configurations, allows the comparison of the results without additional influences from manufacturing and assembling of the plate like structure and the piezoelectric ceramic.

Results
In this section, the results obtained from the numerical simulations and experiments are presented. First, the operational deflection shapes of the two configurations are compared. The sound radiation behavior at certain resonance frequencies is depicted using polar plots. Finally, the numerical results are validated by experimental results. Therefore, the ODSs between

Operational deflection shape and stop band behavior
The operational deflection shape of the plate surface characterizes the sound radiation behavior. In order to generate the desired directivity pattern with small side lobes, the ODS should be similar to the first normal mode. This could be either the Fig. 5. Frequency response function of normalized velocity at the center point of the sound radiating surface. First and second operating frequencies are identified by × and •, respectively. The first operating frequency is influenced by the boundary conditions. In contrary, the second one is mainly independent of the additional boundary conditions in configuration CON2. Other peaks in the frequency response function correspond to ODSs which do not show suitable sound radiation behavior for surround sensing purposes. entire structure or only a part of the structure while the rest remains in equilibrium position. Achieving the first case is rather trivial. The operating frequency coincides with the first eigenfrequency of the plate like structure. Due to the different boundary conditions in configurations CON1 and CON2, the lowest operating frequencies are obtained at 0.83 kHz and 2.9 kHz, respectively. These can be identified in the frequency response function (FRF) shown in Fig. 5 highlighted by × . The corresponding ODSs are depicted in Fig. 6.
An ODS where a part of the structure behaves similar to the first normal mode while the rest remains in equilibrium position is achieved at higher frequencies. This frequency is highlighted by • in Fig. 5 for each configuration. At 17.5 kHz and 17.7 kHz in configurations CON1 and CON2, respectively, the middle section of the plate like structure shows an ODS where almost all nodes move with the same phase. The outer sections, consisting of stop band material, remain almost without displacement, cf. Fig. 6. The additional boundary conditions in configuration CON2 have only small influence on the resulting operating frequency. Hence, the operating frequency OF2 is mainly independent of the additional boundary conditions. This ODS seems suitable to serve as a second operating point. The ODS of both configurations are pretty similar. However, at the boundary of the plate like  structure a difference in the ODS can be found. This is due to the missing boundary conditions in configuration CON1.
In order to characterize the band gap behavior of the outer plate sections, the dispersion curves are investigated. Fig. 7 shows the unit cell consisting of a plate like structure and a beam with quadratic cross-section. In order to identify band gaps, the propagation constants are set to fit the contour of the IBZ ΓXM [11]. This is suitable due to the symmetry of unit cell as shown in Ref. [10]. The dispersion diagram shows that a partial stop band is achieved with this type of band gap material. In Ref. [26], a similar unit cell is investigated. It is shown that a band gap is obtained for shear ( ) and bending waves ( ) while longitudinal waves ( ) can pass the unit cell at all frequencies, cf. Fig. 7. It is found that the second operating frequency of the transducer emerges within the stop band where bending waves can not propagate freely through the outer section of the plate like structure. Consequently, the behavior of the plate like structure at the second operating frequency can be interpreted as follows: A harmonic dynamic excitation of the plate like structure is introduced in the center using the piezoelectric ceramic. The excitation frequency coincides with the stop band frequency of outer section. Due to the particular dynamic behavior of this section, the bending wave is reflected at the boundary of the stop band material. Thus, the dynamic behavior mainly depends on the stop band material.

Directivity patterns
The sound radiation behavior is a criterion to assess the usability of the presented approach to achieve multi-frequency transducers. The directivity pattern should exhibit an even form without distinctive side lobes. Hence, they are desired to behave similar to monopole or dipole characteristics. The directivity pattern is obtained applying the Rayleigh integral to the sound radiating surface of the generic model. The visualization is usually done in polar plots, cf. Fig. 8. At the first operating frequency, both configurations radiate sound without any side lobes. The directivity patterns at the second operating frequency differ slightly between the two configurations. In case of configuration CON1, the sound pressure level shows a drop to −6 dB in x−direction at the center point. In configuration CON2, this drop is reduced to −3 dB. This can be ascribed to lower displacements of the structure at the outer sections due to the additional boundary conditions. In y−direction, configuration CON1 shows monopole behavior. In contrast, a characteristic similar to a dipole is found in configuration CON2 in y−direction. In this case, the sound pressure levels shows a drop to −14 dB at an angle of ±71 • . However, in test both cases the directivity pattern does not show distinctive side lobes.

Experimental results
From experimental testing, the resonance frequencies and the related ODS are obtained. Comparing these results with results from numerical simulation, it is possible to validate the employed FE-model. Table 2 shows the operating frequency from FE-simulation and experimental testing. Regarding the first operating frequency, the deviation between simulation and experiment is less than 4% in both configurations. Thus, a suitable accuracy of the applied finite element model is validated.
Considering the second operating frequency, the deviation from numerical results increases to almost 15%. However, the ODSs of the second operating frequency exhibit equal behavior in FE-simulation and experimental testing, cf. Fig. 9. As the second operating frequency is equal for both configurations CON1 and CON2, it is experimentally proved that the additional fixation of the plate like structure has no influence on the second operating frequency for in the investigated configurations.

Discussion
Summarizing the results, the dynamic behavior can be divided into two categories. In the first category, the excitation frequency does not coincide with the frequency of the stop band. In this case, the vibrational behavior of the plate like structure is similar to that known from classical plate theory. Thus, the boundary conditions have significant influence on the resonance frequencies and the corresponding ODSs. In the second category, the excitation frequency is in the range of the stop band. In this case, a particular dynamic behavior of the system is found. This leads to ODSs where a part of the structure behaves similar to the first normal mode while the rest remains in equilibrium position. The behavior can be interpreted as follows. The wave is excited by the piezoelectric ceramic. In section 1, the wave is freely propagated. In contrast, no free wave propagation is possible in section 2 due to the coincidence of the excitation frequency and the frequency range of the stop band. Thus, the wave gets reflected at the edge of section 2. Hence, it becomes clear that the operating frequency OF2 is found within the frequency range of the stop band for bending waves. By the geometrical design of zone I, the resulting ODS as well as the operating frequency laying within the stop band can be adjusted.
Investigating two different configurations, it is shown that the effect is caused by the stop band behavior of the structure rather than by the boundary conditions. These findings could also be transferred to other applications where certain dynamic behavior is desired independently from the boundary conditions. Hence, the same design could be used in various applications without influencing the dynamic behavior. In the present test case, this effect leads to a suitable sound radiation for both configurations. The results of the directivity pattern in this study must be seen in the context of the generic model. As the directivity pattern strictly depends on the operating frequency and the dimensions of the radiator, the results of the generic model can not be transferred directly to real transducer applications. However, the results are suitable to estimate the potential of spatially distributed stop band material. Due to the additional boundary conditions in configuration CON2, a more even directivity pattern is obtained. Hence, this configuration is more suitable to build multifrequency ultrasonic transducers. In order to achieve more even directivity patterns at the second operating frequency, fixed boundary conditions along all edges should be considered. The investigation shows that the operating frequencies are not coupled to each other. Hence, it is possible to tune the operating frequencies mainly independently from each other.
Furthermore, the piezoelectric ceramic could be surrounded entirely by stop band material. In this case, the waves would be reflected as well when traveling in y−direction which would reduce the displacement of the outer section of the plate like structure. As a result, this would generate a more even directivity pattern. Employing the presented approach, it would be possible to achieve ultrasonic transducers with more than two operating frequencies. Therefore, various types of stop band material should be spatially distributed. Exhibiting stop bands at different frequency bands, multiple operating frequencies could be achieved with suitable sound radiation behavior. In experimental investigation, the results from numerical simulation are validated. Good accordance is found for the first operating frequency in both configurations. The second operating frequency obtained in experimental testing shows higher deviation from the simulation results. This can be ascribed to a lack of accuracy in additive manufacturing processes resulting in resonators with a larger cross-section at the plate side and smaller cross-section at the free end, cf. Fig. 4 (c). Consequently, the center frequency of the stop band is increased in experimental testing in comparison to numerical simulation. Finally, this results in an increased second operating frequency. However, the comparison of the ODSs from numerical simulation and experimental testing shows good accordance for both configurations. It is shown that spatially distributed stop band material is a suitable approach to achieve multi-frequency ultrasonic transducers. Thus, it is possible to overcome the drawbacks of known multi-frequency ultrasonic transducer concepts.

Transfer to ultrasonic frequency range
For practical use, the previously presented results need to be transferred to the ultrasonic frequency range. This can be achieved by changing the material as well as by adjusting the geometry of the transducer. In this section, two additional configurations are presented with both operating frequencies in the ultrasonic range (> 16 kHz [48]). Configurations CON3 and CON4 are introduced in Fig. 10.
In contrast to CON1 and CON2, the configurations for the ultrasonic frequency range are made of aluminum. Due to the high ratio from Young's modulus to mass density, the operating frequency increases compared to the previous configurations which were made of plastics. In order to realize proper boundary conditions of the plate like structure, the blocks are considered to consist of ceramic material. The geometrical design of configuration CON3 is similar to the one of CON2. Only the plate width in y−direction is reduced and the corresponding resonators are removed. This is necessary to ensure that the ODS of a part of the structure behaves similar to the first normal mode if the OF2 is increased. In configuration CON4, the length of the plate in x−direction is reduced additionally and the corresponding resonators are removed. Thus, the OF1 is increased.
The operational deflection shapes of both operating frequencies are shown in Fig. 11. OF1 arises at 18.9 kHz and 23.7 kHz for CON3 and CON4, respectively. The second operating frequency OF2 is achieved at 55.2 kHz and 58.5 kHz for CON3 and CON4, respectively. Thus, the results found for the generic model in the previous sections can be reproduced numerically also in the ultrasonic range. Fig. 12 shows the dispersion curves for the configurations CON3 and CON4. OF2 coincides with the stop band for bending waves in a similar way as found in the generic model. The dimensions of zone I are different in configurations CON3 and CON4. As a result, the second operating frequency OF2 differs between configurations CON3 and CON4. However, both are found in the frequency range of the stop band for the bending waves. The resulting directivity patterns show suitable sound radiation behavior for the use in surround sensing applications, cf. Fig. 13. Thus, it is shown, that spatially distributed stop band material is a suitable approach to achieve multi-frequency ultrasonic transducers.

Conclusion
A study on multi-frequency ultrasonic transducers with spatially distributed stop band material has been presented. The desired ultrasonic transducer is required to work at different operating frequencies using only one electro-mechanical coupling element. The directivity pattern of the radiated sound field needs to be even and should not show distinctive side lobes. In a generic model, the effect of spatially distributed stop band material is investigated in order to fulfill these requirements. The approach is studied with different boundary conditions of the generic model. Finite element simulation is employed to identify the operating frequencies and the belonging ODSs. The results are used to calculate the directivity pattern of the radiated sound field using the Rayleigh integral. In experimental testing, the results of the numerical simulation are validated. The obtained ODSs show good accordance between simulation and experiment.
The authors conclude that: (i) spatially distributed stop band material is identified as a suitable solution to achieve multi-frequency ultrasonic transducers using only one electro-mechanical coupling element, (ii) the second operating frequency is related to the frequency of the stop band, (iii) the second operating frequency achieved by the introduction of spatially distributed stop band material and the belonging ODSs are mainly independent of additional boundary conditions in configurations CON1 and CON2, (iv) suitable sound radiation behavior is achieved at two well separated operating frequencies, and (v) distinctive operating frequencies can be tuned mainly independently from each other.
The presented approach offers a possibility to design multi-frequency ultrasonic transducers with only one electromechanical coupling element and suitable sound radiation properties at the same time. Thus, the previously discussed drawbacks of known concepts can be overcome. This is a major contribution to increase the surround sensing performance and reliability of ultrasonic transducers.