Acoustic Wave Focusing From Reconfigurable Acoustic Arrays Based on a Bricard-Miura Synthesis

The application of origami-inspired structures in science and engineering fields is extensive because it provides new ways of approaching the design of complex three-dimensional (3D) structures that are portable, reconfigurable, and lightweight. Researchers in fields such as space exploration, medicine, and robotics have studied origami-based folding for solar arrays [1–3], satellite antennas [4–6], biomedical devices [7–9], and soft robots [10–12]. Deployable shelters also exploit the portability and reconfigurability of origami folding [13].

Additionally, origami-based structures have been introduced to the field of acoustic wave focusing. Traditional wave focusing techniques adjust the wave phase and amplitude of the transducer elements with digital controls to emulate the physical realignment of transducers for wave convergence [14,15]. An important application of acoustic wave focusing in medicine is the use of high-intensity focused ultrasound (HIFU) to ablate cancerous tumors within the body [16]. Yet, certain cancerous tumors in the body can be inaccessible for HIFU treatment, due to the size of the transducer array in comparison to the tumor location [16]. Recent studies have shown that acoustic transducer arrays inspired from rigid origami such as Miura-ori can focus acoustic waves and be reconfigured to reduce the array size during transport stages [17,18].

To advance these research efforts, Zhao et al. [19] have explored the integration of waterbomb origami or deployable mechanisms such as the Hoberman ring [20] with regular Miura-ori facets to provide curvature for acoustic wave focusing and structural stability for deployment. Furthermore, Lang et al. [21] have proposed the usage of designed-offset linkages with rigid origami to achieve single-degree-of-freedom motion between the unfolded and folded states. Vlachaki and Liapi [22] have also proposed the usage of linkages with rigid origami by coupling scissor linkages to various origami crease patterns to provide single-degree-of-freedom motion and increased structural stability. While these researchers have demonstrated means to achieve low dimensional motion in origami structures via rigid body linkages, only the Hoberman ring [20] has been studied for reconfigurable arrays in acoustic wave focusing applications.

Alternatively, researchers have explored rigid link behavior that is embedded in zero-thickness rigid origami. For instance, Chen et al. [23] and Feng et al. [24] demonstrate that zero-thickness rigid origami patterns are comparable to rigid bar spherical linkages. In particular, rigid origami with a four-crease vertex has the same kinematic behavior as that of spherical linkages with four rotating joints. Moreover, several researchers have shown that the single-degree-of-freedom behavior of zero-thickness rigid origami can be retained for rigid origami with thick panels by replacing the spherical linkage behavior of the vertices with those of single-degree-of-freedom spatial linkages [25–28]. Yet, these research efforts only address the deployment behavior of origami-inspired structures without considering other physical phenomena, such as wave focusing, in the origami pattern design.

Motived by the shortcomings of these two approaches for exploring rigid origami deployment, the objective of this research is to construct and study an acoustic wave focusing array inspired from Miura-ori origami and the kinematic behavior of a modified threefold-symmetric Bricard linkage [29,30]. The array focuses acoustic waves according to the folded configuration resulting in convergent, sound-radiating facets. The crease pattern and geometric design parameters are shown in Fig. 1(a). The proposed array in unfolded, folded, and inverted folded configurations are shown in Fig. 1(b). As shown in Fig. 1(c), the regular Miura-ori facets are treated as vibrating baffled pistons. The selection of which facets to activate is intuitively based on the normal convergence of the facet planes to the central axis, as well as how close these normal vectors converge. Figure 1(d) shows that the rotational symmetry of the Miura-ori unit cells in the proposed acoustic array is inspired from and can incorporate a linkage inspired from the threefold-symmetric Bricard linkage [30] for single-degree-of-freedom motion.

Analytical and experimental efforts are utilized in this research to study the effects of facet normal vector convergence on wave focusing for the proposed acoustic array. Section 2 presents the analytical model used to scrutinize the wave focusing capabilities of the proposed acoustic array. Section 3 communicates the experimental efforts used to validate observations from the analytical model. Section 4 details the influences of the array facet normal vectors on acoustic convergence and wave focusing using theoretical investigations. Section 5 concludes the report by summarizing the new findings with closing remarks.

In this section, the geometric model of a reconfigurable acoustic array is developed. A modified threefold-symmetric Bricard linkage is used to guide the reconfiguration of three regular Miura-ori unit cells. Direct acoustic radiation from the activated, vibrating Miura-ori facets is then analytically computed using Rayleigh’s integral.

Figure 1(a) shows the geometry of a regular Miura-ori unit cell combined with two triangular facets, which make up the Bricard-Miura unit cell. The regular Miura-ori unit cell has four facets governed by edge lengths a and b, facet angle γ, and folding angle θ. The regular, folded Miura-ori unit cell has dimensions H, S, L, and V, with relationships to the facet design parameters given by Schenk and Guest [31], as summarized in Eqs. (1)–(4)

The coordinates for the remaining vertices on the Bricard-Miura array are then defined by Eq. (9)

The p_rms(R, β, φ) is the root-mean-square value of p(R, β, φ), and p_ref = 20 µPa is the reference acoustic pressure for air.

Proof-of-concept specimens are fabricated so that acoustic wave focusing from the analytical models for the proposed array configurations can be experimentally validated. Wave focusing results from both experimental and analytical efforts are gathered and compared in this section.

Experimental models are constructed for both the folded and inverted folded configurations of the proposed acoustic array. The arrays are fabricated with 1.59-mm-thick polypropylene sheets that are laser cut with the crease pattern from Fig. 1(a) and are shown in Figs. 2(a) and 2(b), respectively. The dimensions for the three Miura-ori facets on each configuration are γ = 60 deg and a = b = 40.64 mm. For the inverted folded configuration of the array, the non-activated inner facets are removed to minimize the effects of diffraction on the measured sound pressure (Fig. 2(b)). Activated facets are emulated in the proof-of-concept specimens by circular miniature loudspeakers (Parts Express, Springboro, OH) bonded to centered laser-cut holes of passive panels at the activated facet positions. Each loudspeaker covers approximately 50% of the corresponding facet area. Support fixtures with fixed angles are constructed with a 3D printer (FlashForge Creator Pro) using acrylonitrile butadiene styrene (ABS) and fastened to each array configuration to position the proof-of-concept specimens at the prescribed folding angles. The miniature loudspeakers are driven at the same frequency with an external amplifier and function generator.

A hemi-anechoic acoustic chamber with dimensions 4 m × 4.25 m × 2.75 m is used to measure the near-field acoustic pressure generated by the proof-of-concept specimens. A microphone (PCB Piezotronics 130F20, Depew, NY) is used to measure the acoustic pressure along the central axis of the specimen, starting at the plane of the unfolded array, as shown in Fig. 2(c). A function generator is used to generate a tonal signal that is sent to an amplifier that drives the loudspeakers of the array. Measured acoustic pressure data are post-processed using matlab to determine the sound pressure level along the axial direction.

A comparison between the sound pressure level results from the experimental and analytical models is presented in Fig. 3. The normal surface velocity amplitude is determined to be 0.05 mm/s for the driving frequencies. In Fig. 3, the results are displayed with the horizontal axes representing the locations of the sound pressure levels along the center axis, starting at the plane of the unfolded array. The vertical axes represent the corresponding sound pressure levels for those locations. For each Bricard-Miura array configuration in Fig. 3, there is a peak SPL at an axial location close to the plane of the unfolded array. This axial location represents the acoustic near field where the acoustic pressure from the activated facets constructively converge and produce acoustic pressure focusing as is seen in the SPL peaks. The results in Fig. 3 also show local maxima in both the analytical and experimental models. The local maxima observed around 0.04 m in the analytical models (see Figs. 3(a)–3(c)) are due to constructive interference of the acoustic waves, whereas the local maxima in the experimental results are caused by a combination of the effects from diffraction, reflection, and the directly radiated waves.

In Fig. 3(a), the results for the analytical model of the Bricard-Miura array are presented for three folding angles of 15 deg, 20 deg, and 25 deg. A driving frequency of 19 kHz has been selected for experimental validation of the Bricard-Miura array due to its notable wave focusing capabilities for various folding angles of the array, as seen in Fig. 3. For a folding angle of 25 deg, the peak SPL is around 67 dB at an axial distance of 0.1 m (5.54 wavelengths). As the folding angle reduces to 20 deg and 15 deg, the peak SPL also reduces to around 66 dB and 65 dB with an increasing axial distance around 0.12 m and 0.13 m, respectively. Figure 3(b) displays the corresponding experimental results. For a folding angle of 25 deg and a driving frequency of 19 kHz, the peak SPL reaches 67 dB around an axial distance of 0.06 m. The experimental results also suggest that for folding angles of 20 deg and 15 deg, the peak SPL reaches 63 dB at 0.11 m and 61 dB at 0.12 m, respectively. When compared to Fig. 3(a), there is qualitative agreement with the acoustic focusing trends for increasing folding angles at axial distances less than 0.2 m despite the discrepancies for exact SPL and axial distance values. These discrepancies are due to the diffraction and reflection of acoustic waves from adjoining facets, as well as the assumption that full area activation on a given facet cannot be realized experimentally. For axial distances greater than 0.2 m, the qualitative agreement is not as good between the analytical and experimental results. This is due to a combination of the effects from diffraction, reflection, and the directly radiated waves. It is expected that as the folding angle of the activated facets increases, the peak SPL and corresponding axial location increases and decreases, respectively. This is because increasing the facet folding angle reduces the convergent region for the radiated sound field and positions it closer to the plane of the unfolded array.

Figure 3(c) compares the analytical axial SPL for the Bricard-Miura and inverted Bricard-Miura array configurations for a 25-deg folding angle and 19-kHz driving frequency. The analytical results show that the Bricard-Miura array has a peak SPL around 67 dB at 0.1 m along the center axis, and the inverted Bricard-Miura array has a peak SPL around 62 dB at a 0.19 m axial distance. Corresponding experimental results for both array configurations are shown in Fig. 3(d), where the Bricard-Miura and inverted Bricard-Miura array configurations have SPL peaks reaching 67 dB and 58 dB at axial distances of 0.06 m and 0.21 m, respectively. Both the analytical and experimental results show that the Bricard-Miura array has a higher peak SPL and closer axial distance to the unfolded plane in comparison to the inverted configuration despite the discrepancies in exact SPL and axial distance values. These discrepancies occur for the same reasons that are explained in the results shown in Figs. 3(a) and 3(b). The acoustic focusing trend for the increase in peak SPL and decrease in the axial distance is similar to the previously discussed folding angle influence, whereas now the size and position of the convergent region for the radiated sound field are due to the distance between the activated facets and the central axis of the array. This means that the convergent region reduces and is positioned closer to the unfolded array plane when the distance between activated facets and the center axis is reduced, similar to increasing the folding angle.

One of the objectives of this research is to identify activated facet configurations for the Bricard-Miura array that permit acoustic wave focusing along the central axis of the array. There are three configurations with activated facets that are analyzed. These configurations are the Bricard-Miura array with 12 activated facets (six quadrilateral and six triangular), the Bricard-Miura array with six activated quadrilateral facets, and the inverted Bricard-Miura array with six activated quadrilateral facets. These three activated facet configurations are shown in Fig. 1(c) and in the insets of Fig. 4. Analytical models are determined for each of the candidate configurations at driving frequencies of 10 kHz and 19 kHz and facet folding angles of 15 deg and 25 deg. These driving frequencies have been selected for this study to demonstrate the versatility of the Bricard-Miura array in wave focusing for multiple driving frequencies with the same geometric dimensions. Results from the analytical models are presented for a 15-deg folding angle and 10-kHz driving frequency in Fig. 4(a), a 25-deg folding angle and 10-kHz driving frequency in Fig. 4(b), a 15-deg folding angle and 19-kHz driving frequency in Fig. 4(c), and a 25-deg folding angle and 19-kHz driving frequency in Fig. 4(d). The horizontal axes represent the locations of the sound pressure levels along the center axis in terms of the number of wavelengths, starting at the unfolded array plane.

Figure 4 shows that the Bricard-Miura array with six activated quadrilateral facets has higher SPL peaks and smaller axial distances in comparison to the inverted Bricard-Miura array for the specified driving frequencies and folding angles. This behavior is expected based on the results discussed in Sec. 3. The results in Fig. 4 also show that activating the six triangular facets in the Bricard-Miura array marginally increases the SPL peaks and axial distances. Furthermore, the results in Fig. 4 show that the focal range, defined as the axial distance where the SPL is no more than 6 dB less than the peak SPL, varies for each array configuration, driving frequency, and folding angle.

A comparison of the results from Figs. 4(b) to 4(a) and 4(c) to 4(d) shows that the peak SPL increases and the axial distance decreases as the folding angle increases for the three array configurations, similar to the preliminary results in Fig. 3. The results from Figs. 4(a) and 4(b) show that the focal range decreases from 6.9 to 5.5 wavelengths for the 12 activated facet Bricard-Miura array configuration with an increase in folding angle from 15 deg to 25 deg. The focal ranges for the six activated facet Bricard-Miura and inverted Bricard-Miura array configurations also decrease from 6.5 to 5.1 wavelengths and 11.6 to 9 wavelengths, respectively. A decrease in focal range for the three activated facet configurations is expected due to the increasing folding angle that intuitively reduces the convergent region of the sound field. Comparing the results of Fig. 4(c) to those of Fig. 4(d) shows a decrease in the focal range from 16.1 to 9.7 wavelengths, 14.8 to 8.9 wavelengths, and 23.5 to 16.2 wavelengths, for the 12 activated facet, six activated facet, and inverted Bricard-Miura array configurations, respectively. A decrease in the focal range is also expected at this driving frequency for the same reasons that are previously explained.

A comparison of the results in Fig. 4 shows that increasing the driving frequency increases the peak SPL and corresponding axial distance, as well as the focal range. Increasing the folding angle also increases the peak SPL, yet decreases the axial distance and focal range of the peak. Furthermore, the Fig. 4 results show that the Bricard-Miura array configuration with 12 activated facets has the highest peak SPL in comparison to the other two configurations. This behavior is expected due to this configuration having a greater number of vibrating pistons contributing to the radiated sound field. Yet, the Bricard-Miura array configuration with six activated quadrilateral facets has a smaller focal range, and the resulting peak SPL can be increased to match that of the 12 activated facets configuration simply by increasing the normal surface velocity amplitude of the facets. Either of these activated facet configurations for the Bricard-Miura array is ideal for acoustic wave focusing, with the six activated quadrilateral facets configuration having a marginally smaller focal range.

To further understand how the activated facets affect the acoustic wave focal range, the facet normal convergence area is analyzed for each configuration. The facet normal convergence area is the two-dimensional region where the normal vectors of the facets converge at the axial position of the peak SPL. Figure 5 shows the facet normal convergence areas for the three array configurations at a driving frequency of 19 kHz and a 25-deg folding angle, as well as the influences of the elevation angle, β, and the azimuth angle, φ, on SPL focusing for the Bricard-Miura array with six activated quadrilateral facets. The convergence area for each configuration is where normal vectors for the activated facets are extended to the axial position of the peak SPL. The end coordinates of the normal vectors form a six-sided area. This area is the convergence region. Figure 5(a) shows that the Bricard-Miura array configuration with 12 activated facets has two convergence areas of 480 mm² and 8330 mm². These areas suggest that the quadrilateral facets in red (dark shading) converge toward the center axis, and the triangular facets in blue (light shading) converge toward each other instead of the center axis of the array. Figure 5(b) shows that the Bricard-Miura array configuration with six activated quadrilateral facets has a convergence area of 500 mm². This is marginally greater than that of the quadrilateral convergence area in Fig. 5(a) and is due to the axial distance being smaller (5.7 versus 5.8 wavelengths), as it is seen in Figs. 5(a) and 5(b). Figure 5(c) shows that the inverted Bricard-Miura array configuration has a convergence area of 1460 mm².

Comparing the quadrilateral facet convergence areas for each configuration alongside the focal range results from Fig. 4(d) suggests that the Bricard-Miura array with either activated facet configuration has a smaller wave focal range and higher peak SPL than that of the inverted Bricard-Miura array configuration. This is because the Bricard-Miura array normal vectors converge closer to the central axis than for the inverted array counterpart. A comparison of Figs. 5(a) to 5(b) alongside the results from Fig. 4(d) suggests that the inclusion of the triangular facets increases the peak SPL. Yet, including the triangular facets also increases the focal range because of the increased convergence region. Based on these observations, acoustic arrays with smaller convergence regions have higher SPL peaks and smaller focal ranges in comparison to arrays with larger convergence regions.

Figure 5(d) shows the influences of β and φ on the focused SPL of the Bricard-Miura array with six activated quadrilateral facets. The plot is generated using the same parameters that have been used for Fig. 5(b), with the exception of R being set equal to 5.7 wavelengths. As it is seen in Fig. 5(d), as β increases from 0 deg to 50 deg, the focused SPL of the Bricard-Miura array decreases from 67 dB to 43 dB when φ is equal to 30 deg. Figure 5(d) also shows that the focused SPL of the Bricard-Miura array decreases to 30 dB when β is equal to 30 deg, and φ is equal to 90 deg, 210 deg, and 330 deg. Figure 5(e) shows a few examples of the focused SPL from Fig. 5(d) overlaying the Bricard-Miura array geometry as SPL field points. A comparison of the field points in Figs. 5(d) and 5(e) confirm that the peak SPL focusing occurs at the center axis of the Bricard-Miura array, as it is suggested in Figs. 5(a)–5(c).

Geometric parameters for the rigid Miura-ori facet also influence the wave focusing behavior of the acoustic array. Figure 6 shows the peak SPL and focal range results for the Bricard-Miura array configuration with six quadrilateral activated facets as a function of the facet angle, edge length ratio, and folding angle of the Miura-ori unit cell. The edge length ratio ranges from 0.5 to 2, with edge b = 40.64 mm. The facet folding angle ranges from 5 deg to 30 deg, and a driving frequency of 19 kHz is used to vibrate the activated facets. The facet angle γ is also set to 45 deg in Figs. 6(a) and 6(b), and 60 deg in Figs. 6(c) and 6(d). The insets in the corners of the subfigures for Fig. 6 represent the Bricard-Miura array when a = 0.5b and a = 2b for folding angles of 5 deg and 30 deg. Figure 6(a) shows that for a facet angle γ of 45 deg, the peak SPL increases from 58 to 64 dB as the folding angle and edge length ratio a/b increase. Note that when edge a is smaller than edge b the peak SPL only increases by no more than 1 dB as the folding angle increases. Figure 6(b) shows the corresponding focal ranges for the peak SPL values in Fig. 6(a), where the focal range is defined as being the axial distance range in which the SPL is no more than 6 dB less than the peak SPL. The results show that when a ≥ b, increasing the folding angle decreases the focal range from 69 to 14 wavelengths. For a facet angle of 60 deg, Fig. 6(c) shows that the peak SPL increases from 62 dB to 69 dB as the folding angle increases from 5 deg to 30 deg, and the edge length ratio a/b increases from 0.5 to 2. Figure 6(d) shows the corresponding focal range results for Fig. 6(c). The results show that the focal range decreases from 63 to 8 wavelengths as the ratio a/b decreases from 2 to 0.5, and the folding angle increases from 5 deg to 30 deg.

A comparison of Figs. 6(a) to 6(c) demonstrates that increasing the facet angle from 45 deg to 60 deg increases the peak SPL of the Bricard-Miura array. This is due to larger activated areas contributing to the radiated sound field. Comparing Figs. 6(b) to 6(d) shows that increasing the facet angle from 45 deg to 60 deg also decreases the focal range, with the exception of increasing the focal range for folding angles around 5 deg. Furthermore, the focal range of the array reduces quicker as the folding angle increases, when comparing the results for the 60-deg facet angle to the results for the 45-deg facet angle. A decrease in the focal range from an increase in the facet angle occurs because the larger facet angle aligns the activated facet areas so that the emitted waves converge closer together. Alignment of the activated facet areas means that deviation in the shortest distance to the central axis for each facet is reduced. Improvement in the facet area alignment, for an increase in facet angle from 45 deg to 60 deg, can be seen by comparing the bottom insets in Figs. 6(a) and 6(b) to the corresponding insets Figs. 6(c) and 6(d). Similarly, when edge a is reduced to be smaller than edge b, the focal range of the peak SPL reduces because deviation in the shortest distance of the activated facets to the central axis is reduced. This can be seen by comparing the top insets to the bottom insets in Fig. 6(d).

This research explores a reconfigurable acoustic array inspired from a single-degree-of-freedom linkage and origami structure with shape changes that promote acoustic wave focusing. The analytical results show that acoustic arrays inspired from a Miura-ori and modified threefold-symmetric Bricard linkage synthesis are capable of wave focusing. Experiments are conducted with proof-of-concept models to validate analytical results for wave focusing. The peak SPL and focal range behaviors of the array are scrutinized for varying facet geometric parameters and principles of ray acoustics are leveraged for interpretation. The results show that the relative shortest distance of the facets to the center axis enhances the focal range when decreased. This research motivates the exploration of other reconfigurable acoustic arrays synthesized from origami structures and single-degree-of-freedom linkages for acoustic wave guiding functionality.

This project is supported by the National Science Foundation Faculty Early Career Development Award (Grant No. 2054970).

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.