Visualization of Chladni Patterns at LowFrequency Resonant and NonResonant Flexural Modes of Vibration
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In this study, Chladni patterns corresponding to resonant and nonresonant vibration modes are visualized on square plates made in steel and aluminum alloys in the low frequency domain of 10–210 Hz. Using a laser sensor, the plate displacement at its central excitation point is measured, and from the obtained frequency response, the resonant and antiresonant vibration modes are identified. Using the qualityfactor method, the damping ratio corresponding to the 1st resonant peak is evaluated. Over a wide range of excitation frequencies, transitions of Chladni figures between resonant patterns via nonresonant patterns could be observed. Such Chladni figures, of the simplest geometrical configuration, can be used to achieve a certain desired movement path of the particles on the vibrating plate by controlling the excitation frequency.
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Introduction
Chladni patterns are the geometrical figures obtained by sprinkling particles on the surface of a vibrating plate [1], [2]. Using violin bows [1], [3], shakers and speakers [4]–[7], laser beams [8], etc., a stationary flexural vibration mode is to be generated into the horizontally supported plate.
Sprinkled particles, such as sand, salt, and sugar grains [9], silica micrograins [8], tea powder [7], green soybeans [6], flour, baking soda, and baby powder [10] are usually agglomerating on the plate in the regions of nil or very small amplitude of vibration. Thus, they visualize the nodal lines and produce the socalled standard Chladni patterns, which are dominated by the gravitational effects.
Yet, flexural vibration produces air currents blowing across the plate from nodes to antinodes and then upward. Finer grains, such as dust particles, might be carried by such currents in the regions of large amplitude of vibration, and, as a result, they might visualize the antinodal lines [10].
In fact, the resulting velocity of a particle bouncing on the surface of a vibrating plate depends on the plate velocity, the particle velocity before the impact, and the restitution coefficient of the kinetic energy after impact, which can be perceived as a viscous damping coefficient [9], [11]. Since the region for particle deposition on the plate depends on the vector of the resultant velocity, the gravitational effect might be diminished.
Similarly, reduction of the gravitational effect on the particle motion can be achieved through buoyancy by submerging the vibrating plate inside a fluidic medium [10], [12]–[15]. Besides, the interaction between a liquid layer and the vibrating plate can be applied to move liquid droplets on it [16], in which case the gravitational effect is diminished by the surface tension contribution. Using these techniques, sophisticated, unconventional Chladni patterns can be easily generated [12]–[16].
The complexity of the Chladni figure depends also on the excitation frequency and the plate shape. Thus, on one hand, frequency augmentation leads to more complex patterns due to the superposition of various degenerate or nearly degenerate eigenmodes [4]–[6]. On the other hand, vibration modemixing phenomenon corresponding to a specific geometry of the plate has been reported. Concretely, the vibration modemixing is different for triangular plates [5], square plates [4]–[7], rectangular plates [17], hexagonal plates [18], circular plates [7], perforated circular plates [19], violinlike shaped plates [1], [3], [20], etc. Concerning the modemixing phenomenon, although the plate is excited by a harmonic central source, there are successive backward wave reflections on the boundaries of the plate, directed towards the source, and therefore, superposing on the main central mode. As a result, a dissonant sound that irritates the ear can be perceived during the experiments [9], [18].
Due to their numerous applications in the fields of Civil, Structural, Mechanical, Aerospace, and Seismic Engineering [9], [21]–[28], Chladni patterns appearing at relatively higher resonant frequencies on centrally excited square plates have been extensively studied [4], [7], [25]. Thus, Table I presents the dimensions, i.e., the side length L and thickness h, of various thin square plates made in aluminum, the frequency interval explored in the Chladni tests, and the values of the resonant frequencies, as reported by [4], [7], and [25]. On the other hand, there is quite a wide spectrum of Chladni configurations corresponding to nonresonant frequencies to be used for motion control of objects on vibrating plates in a broad range of applications connected to drug delivery, tissue engineering, microsurgery, microrobots manipulation, etc. [10], [13]. However, pertinent data necessary for the actual motion control is currently unavailable in the literature.
Reference  Plate side length L (mm)  Plate thickness h (mm)  Searched frequency range (Hz)  Experimentally found resonant frequencies (Hz) 

[4]  240  0.91  1–1000  190, 340, 490, 800, 955 
[7]  160  0.61  300–5000  340, 660, 760, 900, 1000, 1450, 1770, 1940, 2540, 3250, 3360, 3430, 3480, 3910, 4178, 4987 
[25]  320  1.00  200–3000  363, 592, 701, 776, 1257, 1378, 1553, 1659, 1960, 2451 
In this work, an analysis of various flexural vibration modes associated with Chladni patterns occurring on square plates made in aluminum, steel, and stainless steel is performed. In our attempt to visualize the Chladni patterns, appearing at low frequencies, in the range of 10–210 Hz, fine beach sand is uniformly sprinkled on these square plates, which have the same side length and close thicknesses.
Several resonant and nonresonant patterns of simple and symmetrical geometry are identified. One experimentally proves that the shape of the Chladni figure changes at augmentation of the excitation frequency. Thus, the circular shape observed at the lowest frequency, is successively shifting into a ringlike shape, then into an elliptical shape, followed by two ellipticalarcs, two parallel segments, two hyperbolicalarcs, and finally is changing into the plate diagonals. These unsophisticated patterns of the simplest geometrical configuration can be used to achieve a certain desired movement path of particles on the vibrating plate by simply adjusting the excitation frequency.
Test Rig and Experimental Procedure
Visualization Method of the Chladni Patterns
Fig. 1 shows a photo of the test rig used to visualize the Chladni patterns. By using a salt sprinkler, fine beach sand, with a mean particle diameter of 0.2 mm, is uniformly sprinkled on the plate surface before commencing the experiments. Flexural vibration of the plate is achieved by employing a 3BScientific shaker consisting of a function generator and a vibration generator. At the plate center, a small hole of 3 mm diameter is drilled to horizontally fit the plate above the vertical shaking rod via a banana connector.
Fig. 2 illustrates the vibration model associated with the test rig shown by Fig. 1. The vibration generator consists of an elastic element of stiffness k = 2 N/mm and a shaking rod of mass m_{0} = 36 g, which provides the socket necessary to connect the plate. Such a system of two coupled oscillators, consisting of a driving oscillator, i.e., the vibration generator, and a driven oscillator, i.e., the tested elastic plate, is expected to display a complex dynamic behavior with distinctive resonant and antiresonant frequencies [29]. However, since at very low excitation frequencies, the plate behaves as a rigid element, the frequency of the 1^{st} resonant peak, recorded in our experiments, can be computed as:
(1)fn=k/(m0+m)/(2π)where m is the mass of the plate.
Table II shows the dimensions and the material properties, i.e., the side length L, thickness h, mass m, Young modulus of elasticity E, Poisson ratio µ, and density ρ of the square plates used in this work. Tested plates, denoted as P1–P4, are made in carbon steel S45C, stainless steel SUS304, and aluminum alloy A5052. They have the same side length of 180 mm but slightly different thicknesses (see Table II). In order to achieve good contrast with the grey sand particles, all the tested plates are painted black.
Plate type  P1  P2  P3  P4 

Material  Steel S45C  Aluminum alloy, A5052  Stainless steel, SUS304  
Side length, L (mm)  180  180  180  180 
Thickness, h (mm)  1.15  1.05  1.05  1.20 
Mass, m (g)  262.0  91.7  246.6  300.8 
Young modulus of elasticity, E (GPa)  205  70  195  
Poisson ratio, μ (−)  0.3  0.33  0.295  
Density, ρ (kg/m^{3})  7,840  2,660  7,980 
Tests are performed in the low frequency domain, of 10–210 Hz, with a frequency increment of 1 Hz, at the maximal amplitude of excitation, obtained by setting the output signal of the function generator to its maximal value, of 10 V.
A camera, supported by a tripod above the square plate, is used to record the Chladni figure taking form at a certain selected frequency. After each visualization test, conducted at a specific given frequency, the remaining sand is fully removed from the plate surface using a brush.
Recording Method of the Plate Frequency Response
Fig. 3 shows the test rig used to examine the frequency responses of the plates P1−P4, which are excited in the same way as in the visualization tests of the Chladni patterns.
A laser sensor, Keyence LB60, is employed to measure the plate displacement at its central excitation point. The output signal of the sensor is amplified and then inputted into an analyzer, Yokogawa DL850, used to perform a Fast Fourier Transform of the recorded time response.
For each plate, two vibration spectra were obtained. One type gives the variation of the vibration amplitude W, and the other type shows the variation of the vibration level 20·lg(M) [dB] versus the frequency f, at the central point of excitation. Here, M represents the magnification factor [29].
In order to obtain a stationary flexural vibration mode of the tested plate, for each selected frequency in the range of 10–210 Hz, the plate was excited for a time interval of 30 seconds before the signal recording and analysis.
By examining the frequency responses (see Figs. 5–12) of the plates P1–P4, the resonant and antiresonant bending vibration modes were identified (see Tables III–VI).
Parameter  Values/ Range 

Lowest frequency for Chladni pattern formation  100 Hz 
Domain of interest for Chladni patterns control  100–182 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From Fig. 5)  12.9 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From (1))  13.0 Hz 
Left frequency for Qfactor evaluation, f_{L} (Fig. 5)  11.3 Hz 
Right frequency for Qfactor evaluation, f_{R} (Fig. 5)  13.8 Hz 
Qfactor, f_{n}/(f_{R} – f_{L}); Damping ratio, 0.5 (f_{R} – f_{L})/f_{n}  5.160; 0.097 
Main resonant frequencies in the domain of interest for Chladni patterns control (Fig. 6)  100, 112, 127, 156, 164, 171 Hz 
Main antiresonant frequencies in the domain of interest for Chladni patterns control (Fig. 6)  105, 116, 144, 160, 169 Hz 
Domain for the circular Chladni patterns  100–107 Hz 
Domain for the ringlike Chladni patterns  108–144 Hz 
Domain for the elliptical Chladni patterns  145–154 Hz 
Domain for the ellipticalarcs Chladni patterns  155–157 Hz 
Domain for the linear Chladni patterns (II type)  158–168 Hz 
Frequency of the II type resonant peak, f_{II} (Fig. 5)  164 Hz 
Domain for the hyperbolicalarcs Chladni patterns  169–177 Hz 
Domain for the diagonal Chladni patterns (X type)  178–181 Hz 
Frequency of the X type resonant peak, f_{X} (Fig. 5)  181 Hz 
Domain for the recurrent ringlike Chladni patterns  182–210 Hz 
Frequencies for the atypical Chladni patterns  120, 161–167 Hz 
Parameter  Values/ Range 

Lowest frequency for Chladni pattern formation  80 Hz 
Domain of interest for Chladni patterns control  80–178 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From Fig. 7)  19.7 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From (1))  19.9 Hz 
Left frequency for Qfactor evaluation, f_{L} (Fig. 7)  15.95 Hz 
Right frequency for Qfactor evaluation, f_{R} (Fig. 7)  20.6 Hz 
Qfactor, f_{n}/(f_{R} – f_{L}); Damping ratio, 0.5(f_{R} – f_{L})/f_{n}  4.237; 0.118 
Main resonant frequencies in the domain of interest for Chladni patterns control (Fig. 8)  100, 145, 167 Hz 
Main antiresonant frequencies in the domain of interest for Chladni patterns control (Fig. 8)  88, 103, 146 Hz 
Domain for the circular Chladni patterns  80–99 Hz 
Domain for the ringlike Chladni patterns  102–135 Hz 
Domain for the elliptical Chladni patterns  136–137 Hz 
Domain for the ellipticalarcs Chladni patterns  138–142 Hz 
Domain for the linear Chladni patterns (II type)  143–149 Hz 
Frequency of the II type resonant peak, f_{II} (Fig. 7)  145 Hz 
Domain for the hyperbolicalarcs Chladni patterns  150–172 Hz 
Domain for the diagonal Chladni patterns (X type)  173–177 Hz 
Frequency of the X type resonant peak, f_{X} (Fig. 7)  177 Hz 
Domain for the recurrent ringlike Chladni patterns  178–210 Hz 
Frequencies for the atypical Chladni patterns  100, 101 Hz 
Parameter  Values/ Range 

Lowest frequency for Chladni pattern formation  100 Hz 
Domain of interest for Chladni patterns control  100–175 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From Fig. 9)  13.0 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From (1))  13.4 Hz 
Left frequency for Qfactor evaluation, f_{L} (Fig. 9)  11.9 Hz 
Right frequency for Qfactor evaluation, f_{R} (Fig. 9)  14.0 Hz 
Qfactor, f_{n}/(f_{R} – f_{L}); Damping ratio, 0.5(f_{R} – f_{L})/f_{n}  6.190; 0.081 
Main resonant frequencies in the domain of interest for Chladni patterns control (Fig. 10)  151, 163 Hz 
Main antiresonant frequencies in the domain of interest for Chladni patterns control (Fig. 10)  152 Hz 
Domain for the circular Chladni patterns  100–129 Hz 
Domain for the ringlike Chladni patterns  130–144 Hz 
Domain for the elliptical Chladni patterns  145–147 Hz 
Domain for the ellipticalarcs Chladni patterns  148–150 Hz 
Domain for the linear Chladni patterns (II type)  151–153 Hz 
Frequency of the II type resonant peak, f_{II} (Fig. 9)  151 Hz 
Domain for the hyperbolicalarcs Chladni patterns  154–171 Hz 
Domain for the diagonal Chladni patterns (X type)  172–174 Hz 
Frequency of the X type resonant peak, f_{X} (Fig. 9)  174 Hz 
Domain for the recurrent ringlike Chladni patterns  175–210 Hz 
Frequencies for the atypical Chladni patterns  None 
Parameter  Values/ Range 

Lowest frequency for Chladni pattern formation  140 Hz 
Domain of interest for Chladni patterns control  140–208 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From Fig. 11)  12.4 Hz 
Frequency of the 1^{st} resonant peak, f_{n} (From (1))  12.3 Hz 
Left frequency for Qfactor evaluation, f_{L} (Fig. 11)  11.7 Hz 
Right frequency for Qfactor evaluation, f_{R} (Fig. 11)  14.3 Hz 
Qfactor, f_{n}/(f_{R} – f_{L}); Damping ratio, 0.5(f_{R} – f_{L})/f_{n}  4.769; 0.105 
Main resonant frequencies in the domain of interest for Chladni patterns control (Fig. 12)  151, 184, 186, 194, 207 Hz 
Main antiresonant frequencies in the domain of interest for Chladni patterns control (Fig. 12)  185, 188 Hz 
Domain for the circular Chladni patterns  140–155 Hz 
Domain for the ringlike Chladni patterns  156–174 Hz 
Domain for the elliptical Chladni patterns  175–180 Hz 
Domain for the ellipticalarcs Chladni patterns  181–182 Hz 
Domain for the linear Chladni patterns (II type)  183–186 Hz 
Frequency of the II type resonant peak, f_{II} (Fig. 11)  184 Hz 
Domain for the hyperbolicalarcs Chladni patterns  187–196 Hz 
Domain for the diagonal Chladni patterns (X type)  197–207 Hz 
Frequency of the X type resonant peak, f_{X} (Fig. 11)  207 Hz 
Domain for the recurrent ringlike Chladni patterns  208–210 Hz 
Frequencies for the atypical Chladni patterns  None 
Experimental Results and Discussions
Case of Typical Chladni Patterns
Fig. 4 shows the typical Chladni patterns, corresponding to various resonant and nonresonant vibration modes, excited in the range of 10–210 Hz, for all the plates P1–P4.
Figs. 5, 7, 9, and 11 present the variation of the vibration amplitude W versus the frequency f, measured at the central excitation point of plates P1, P2, P3, and P4, respectively.
In order to fully identify the vibration model shown by Fig. 2, damping ratio associated with the 1^{st} resonant peak of the system is clarified, as well. Damping ratio is determined here by using the qualityfactor (Qfactor) method [30]. Hence, after finding the maximal amplitude W_{max} that corresponds to frequency f_{n} (Figs. 5, 7, 9, and 11), at the intersection of the resonant peak with the red horizontal line that corresponds to an amplitude of 0.707·W_{max} one finds the left f_{L} and right f_{R} frequencies (see Fig. 7), which are used in Tables III–VI to evaluate the Qfactor, and then, the damping ratio.
Additionally, Figs. 6, 8, 10, and 12 show the variation of the vibration level 20·lg(M) [dB] versus the frequency f, determined at the central point of excitation of the plates P1, P2, P3, and P4, respectively. Such diagrams were obtained by performing a Fast Fourier Transform of the time responses of all plates for each excitation frequency selected to an integer value in the range of 10–210 Hz.
Tables III–VI present the relevant frequency information concerning the Chladni figures and the vibration spectra of the plates P1, P2, P3, and P4, respectively. The following frequency information was considered as relevant to the present work: the lowest frequency for the Chladni pattern formation; domain of interest to control the shape of Chladni figures of the simplest geometrical configuration; frequency f_{n} of the 1^{st} resonant peak as found from tests, and also, by using (1); left and right frequencies (f_{L}, f_{R}) used to evaluate the Qfactor, and then, the damping ratio; main resonant and antiresonant frequencies, inside the domain of interest for pattern control; frequency domains for typical Chladni patterns, i.e., circular, ringlike, elliptical, ellipticalarcs, parallel segments (linear, IItype), hyperbolicalarcs, diagonal segments (linear, Xtype), and recurrent ringlike; frequencies of the II and Xtype resonant peaks (f_{II}, f_{X}); and finally, the frequencies of some atypical Chladni patterns, which were visualized only on the plates P1 and P2.
From Tables III–VI, one notices good agreement between the experimental and theoretical results for the frequency of the 1^{st} resonant peak. As expected, frequency f_{n} decreases at augmentation of the plate mass m, the highest value of 19.7 Hz being recorded for the lightest plate P2, and the lowest value of 12.4 Hz being obtained for the heaviest plate P4.
Damping ratio varied from the lowest value of 0.081, recorded for plate P3, to the highest value of 0.118, found for plate P2. These values indicate a deviation of about ±19% relative to the average damping ratio, which showed a value of 0.1.
Chladni patterns could be visualized for all the excitation frequencies larger than a certain critical frequency. As shown by the lower value of the frequency span for circular Chladni figures, recorded in Fig. 4, the lowest frequency for pattern formation was found to be 80 Hz for P2, 100 Hz for P1 and P3, and 140 Hz for P4. These values, recorded in the 2^{nd} line of Tables III–VI, are used to define the lower limits for the domains of interest, suggested for Chladni pattern control, as shown by the leftsided red vertical lines in Figs. 5–12.
Fig. 4 provides the experimental evidence that various resonant and nonresonant Chladni patterns of simple and symmetrical geometry, relative to a central Cartesian system of coordinates, taken on the plate surface with the axes parallel to the plate sides, are formed in the range of low frequencies. This result is somewhat unexpected since usually, with very few exceptions, only the resonant Chladni patterns are reported in the literature. Moreover, continuous change of the pattern geometry, gradually leading to shape transitions, was observed at augmentation of the excitation frequency.
Therefore, the visualized Chladni patterns were divided into 8 typical domains. The same sequential transitions from one domain to another, at augmentation of the frequency, have been observed on all four plates, as follows: circular shape at the lowest frequency, changing into a ringlike shape, followed by an elliptical shape, then changing into two ellipticalarcs, followed by two parallel segments, then changing into two hyperbolicalarcs, followed by diagonal segments, and finally changing into a recurrent ringlike shape. For each plate, ranges of variation for the excitation frequency corresponding to all these 8 typical domains are presented in Fig. 4 and also in Tables III–VI.
Resonant flexural modes of the plates P1–P4, indicated in Figs. 5, 7, 9, and 11, as well as in Tables III–VI by the frequency symbols f_{II} and f_{X}, are of particular interest. They correspond to two types of Chladni patterns consisting of straight lines, i.e., parallel and diagonal segments, which were previously reported in the literature [1], [28]. While both theoretical and experimental evidence was provided for the Xtype Chladni pattern, only theoretical evidence has been given for the IItype Chladni figure. A reason for this might be the fact that the IItype Chladni pattern is quite difficult to visualize on a plate made of carbon steel. This is indicated in Table III by the atypical Chladni figures occurring around the resonant frequency of 164 Hz, i.e., in the range of 161–167 Hz.
As specified in the last line of Tables III and IV, other atypical Chladni patterns have been observed both on plate P1, at a frequency of 120 Hz, and on plate P2, at a frequency range of 100–101 Hz. Therefore, a detailed discussion concerning this phenomenon is given below.
Case of Atypical Chladni Patterns
During visualization experiments, some atypical Chladni patterns have been noticed on plates P1 and P2 but not on plates P3 and P4.
For instance, the upper part of Fig. 13 depicts an atypical pattern occurring on plate P1 in the ringlike domain at the excitation frequency of 120 Hz. A similar behavior has been observed on the plate P2 during the transition between the circular and ringlike domains at the excitation frequencies of 100 and 101 Hz. However, since such phenomenon has not been observed during tests carried on plates P3 and P4, it might be regarded as atypical.
Further, Fig. 14 illustrates the atypical Chladni figures visualized around the resonant frequency of f_{II} = 164 Hz, on the plate P1. Thus, the parallel segments, observed for frequencies of 159 Hz and 160 Hz, are firstly changing into two vertical ellipticalarcs in the range of 161–163 Hz, which are then changing into two horizontal ellipticalarcs in the range of 163.33–163.6 Hz, through the transitional state consisted of four small vertical ellipticalarcs, noticed at 163.32 Hz. Next, the horizontal ellipticalarcs are changing via similar transitional states, appearing in the range of 163.65–163.66 Hz, into the resonant pattern (f_{II} = 164 Hz) consisting of four small circles. Then, these circles are gradually degenerating into four small ellipses (see the range of 164.3–165.3 Hz). Next, this configuration evolves into the recurrent patterns composed of two vertical ellipticalarcs in the range of 165.5–167 Hz and, finally, into two parallel segments at a frequency of 168 Hz.
Note that such complex transitions of patterns, occurring at quite small frequency increments, have not been observed during tests carried out on plates P2, P3, and P4. Therefore, this phenomenon might be regarded as atypical. For instance, Fig. 15 provides the experimental evidence that there is no change of the Chladni figures, visualized on the plate P2, in the range of 143–149 Hz, i.e., around the resonant frequency of f_{II} = 145 Hz (see also Figs. 7, 8 and Table IV).
Conclusion
In this work, Chladni patterns associated to resonant and nonresonant flexural vibration modes were visualized on square plates made in steel and aluminum alloys in the low frequency domain of 10–210 Hz.
Using a laser sensor, the plate displacement at its central excitation point was measured, and two types of vibration spectra, one showing the variation of the vibration amplitude and the other presenting the variation of the vibration level versus the excitation frequency, were obtained. Based on such diagrams, the resonant and antiresonant bending vibration modes were clearly identified.
A vibration model associated with the test rig, used to visualize the Chladni patterns, was proposed. Stiffness and damping ratio, corresponding to such a model of the dynamic system, were identified by using the frequency of the 1st resonant peak and the qualityfactor method, respectively.
For excitation frequencies exceeding a certain critical value, not only resonant but also nonresonant Chladni patterns of simple and symmetrical geometry were noticed.
Continuous change of the pattern geometry, gradually leading to shape transitions, was observed at augmentation of the excitation frequency. Therefore, these Chladni figures were divided into 8 typical domains, according to their specific shape, as follows: circular, ringlike, elliptical, two ellipticalarcs, two parallel segments, two hyperbolicalarcs, diagonal segments, and recurrent ringlike.
Control of the microparticle motion on the vibrating plate by adjusting the excitation frequency was suggested as a possible actual application for these Chladni patterns of simple geometry and fair sensitivity at the frequency change.
Atypical Chladni figures were observed on the plate made of carbon steel, especially around the resonant frequency related to the pattern consisting of two parallel segments. One suggested that this might be the reason for the lack of experimental evidence in the surveyed literature for this resonant flexural mode of the lowest frequency.
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