Visualization of Chladni Patterns at Low-Frequency Resonant and Non-Resonant Flexural Modes of Vibration

In this study, Chladni patterns corresponding to resonant and non-resonant 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 anti-resonant vibration modes are identified. Using the quality-factor method, the damping ratio corresponding to the 1 st resonant peak is evaluated. Over a wide range of excitation frequencies, transitions of Chladni figures between resonant patterns via non-resonant 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.

Sprinkled particles, such as sand, salt, and sugar grains [9], silica micro-grains [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 so-called 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 mode-mixing phenomenon corresponding to a specific geometry of the plate has been reported.Concretely, the vibration mode-mixing is different for triangular plates [5], square plates [4]- [7], rectangular plates [17], hexagonal plates [18], circular plates [7], perforated circular plates [19], violin-like shaped plates [1], [3], [20], etc. Concerning the mode-mixing 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 non-resonant 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, micro-robots manipulation, etc. [10], [13].However, pertinent data necessary for the actual motion control is currently unavailable in the literature.
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 non-resonant 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 ring-like shape, then into an elliptical shape, followed by two elliptical-arcs, two parallel segments, two hyperbolical-arcs, 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.

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 3B-Scientific 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 anti-resonant 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:    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.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.A laser sensor, Keyence LB-60, 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.

Recording Method of the Plate Frequency 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 anti-resonant bending vibration modes were identified (see Tables III-VI).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 quality-factor (Q-factor) 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 Q-factor, and then, the damping ratio.

Case of Typical Chladni Patterns
Additionally, Figs.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 Q-factor evaluation, f L (Fig. 5) 11.3 Hz Right frequency for Q-factor evaluation, f R (Fig. 5) 13.8 Hz Q-factor, 19.9 Hz Left frequency for Q-factor evaluation, f L (Fig. 7) 15.95 Hz Right frequency for Q-factor evaluation, f R (Fig. 7 (From Fig. 9) 13.0 Hz Frequency of the 1 st resonant peak, f n (From (1)) 13.4 Hz Left frequency for Q-factor evaluation, f L (Fig. 9) 11.9 Hz Right frequency for Q-factor evaluation, f R (Fig. 9 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 Q-factor, and then, the damping ratio; main resonant and anti-resonant frequencies, inside the domain of interest for pattern control; frequency domains for typical Chladni patterns, i.e., circular, ring-like, elliptical, elliptical-arcs, parallel segments (linear, II-type), hyperbolical-arcs, diagonal segments (linear, X-type), and recurrent ring-like; frequencies of the II-and X-type 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 left-sided red vertical lines in Figs.5-12.Fig. 4 provides the experimental evidence that various resonant and non-resonant 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 elliptical-arcs, followed by two parallel segments, then changing into two hyperbolical-arcs, followed by diagonal segments, and finally changing into a recurrent ring-like 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 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 X-type Chladni pattern, only theoretical evidence has been given for the II-type Chladni figure.A reason for this might be the fact that the II-type 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 ring-like 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 ring-like 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 elliptical-arcs in the range of 161-163 Hz, which are then changing into two horizontal elliptical-arcs in the range of 163.33-163.6Hz, through the transitional state consisted of four small vertical elliptical-arcs, noticed at 163.32 Hz.Next, the horizontal elliptical-arcs are changing via similar transitional states, appearing in the range of 163.65-163.66Hz, 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 IV).

Conclusion
In this work, Chladni patterns associated to resonant and non-resonant 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 anti-resonant 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 quality-factor method, respectively.

Fig. 1 .
Fig. 1.Photo showing the test rig used to visualize the Chladni patterns.

Fig. 2 .
Fig. 2. Vibration model associated to the test rig shown by Fig. 1.

Fig. 3 .
Fig. 3. Photo showing the test rig used to examine the frequency responses (amplitude and vibration level) of the square plates P1-P4.

Fig. 6 .
Fig. 6.Variation of the vibration level, determined at the central point of excitation of the plate P1, versus the frequency f.

Fig. 7 .
Fig. 7. Variation of the vibration amplitude W , measured at the central point of excitation of the plate P2, versus the frequency f.

Fig. 8 .
Fig. 8. Variation of the vibration level, measured at the central point of excitation of the plate P2, versus the frequency f.

Fig. 9 .
Fig. 9. Variation of the vibration amplitude W , measured at the central point of excitation of the plate P3, versus the frequency f.

Fig. 10 .
Fig. 10.Variation of the vibration level, measured at the central point of excitation of the plate P3, versus the frequency f.

Fig. 11 .
Fig. 11.Variation of the vibration amplitude W , measured at the central point of excitation of the plate P4, versus the frequency f.

Fig. 12 .
Fig. 12. Variation of the vibration level, measured at the central point of excitation of the plate P4, versus the frequency f.

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.

Fig. 4
shows the typical Chladni patterns, corresponding to various resonant and non-resonant 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.

) 20. 6
Hz Q-factor, f n /(f Rf L ); Damping ratio, 0.5(f Rf 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 anti-resonant 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 ring-like Chladni patterns 102-135 Hz Domain for the elliptical Chladni patterns 136-137 Hz Domain for the elliptical-arcs 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 hyperbolical-arcs 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 ring-like Chladni patterns 178-210 Hz Frequencies for the atypical Chladni patterns 100, 101 Hz 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.Visualization of Chladni Patterns at Low-Frequency Resonant and Non-Resonant Flexural Modes of Vibration Suciu and Karimine ) 14.0 Hz Q-factor, f n /(f Rf L ); Damping ratio, 0.5(f Rf 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 anti-resonant 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 ring-like Chladni patterns 130-144 Hz Domain for the elliptical Chladni patterns 145-147 Hz Domain for the elliptical-arcs 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 hyperbolical-arcs 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 ring-like Chladni patterns 175-210 Hz Frequencies for the atypical Chladni patterns None TABLE VI: Relevant Frequency Information Concerning the Chladni Patterns and the Vibration Spectra of the Plate P4 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 Q-factor evaluation, f L (Fig. 11) 11.7 Hz Right frequency for Q-factor evaluation, f R (Fig. 11) 14.3 Hz Q-factor, f n /(f Rf L ); Damping ratio, 0.5(f Rf 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 anti-resonant frequencies in the domain of interest for Chladni patterns control (Fig. 12)

Fig. 13 .
Fig. 13.Atypical Chladni patterns observed inside the ring-like domain of the plate P1, and also, at the transition between the circular and ring-like domains of the plate P2.

Fig. 14 .
Fig. 14.Atypical Chladni patterns observed around the resonant frequency of f II = 164 Hz, on the plate P1.
Visualization of Chladni Patterns at Low-Frequency Resonant and Non-Resonant Flexural Modes of Vibration

TABLE II :
Dimensions and Material Properties of the Plates used in the Visualization Tests of the Chladni Patterns 6, 8, 10, and 12 show the variation of the vibration level 20•lg(M) [dB] versus the frequency

TABLE III :
Relevant Frequency Information Concerning the Chladni Patterns and the Vibration Spectra of the Plate P1

TABLE IV :
Relevant Frequency Information Concerning the Chladni Patterns and the Vibration Spectra of the Plate P2

TABLE V :
Relevant Frequency Information Concerning the Chladni Patterns and the Vibration Spectra of the Plate P3 -165.3 Hz).Next, this configuration evolves into the