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In this study, Chladni patterns are visualized on square plates made of various metallic materials in the domain of low frequencies around the first flexural resonant peak. Using the membrane and the thin plate models, the undamped and damped natural frequency of the square panel is explicitly expressed versus the mean wavelength of the bending vibration. The wavelength of the idealized Chladni figures is evaluated for linear and curvilinear nodal lines relevant to our experiments. Significant discrepancy between the computed and measured frequencies is corrected based on the wavelength evaluation of the actual Chladni figures, done by considering the cutting and rounding of the nodal lines near the plate edges and near the central point of excitation. Such model able to link the shape of the actual nodal lines with the excitation frequency is useful for the motion control of micro-particles on the vibrating plate.

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Introduction

In our previous work [1], Chladni patterns associated with resonant and non-resonant flexural vibration modes were visualized on square plates made of steel and aluminum alloys in the low-frequency domain. Continuous change of the pattern geometry, gradually leading to shape transitions, was observed at augmentation of the excitation frequency. According to their specific shape, the visualized nodal lines were divided into groups, such as circular, ring-like, elliptical, elliptical arcs, parallel segments, hyperbolical arcs, diagonal segments, etc. Control of the micro-particle’s motion on the vibrating plate by adjusting the excitation frequency was suggested as a possible actual application for these patterns of simple geometry and fair sensitivity at frequency change.

Unfortunately, in the range of low excitation frequencies, predicted values of the natural frequency for square panels, by using the elastic membrane and the thin plate models [2]–[16], are considerably higher than those estimated from the modal testing and analysis [17].

In order to successfully control the movement of micro-particles on the vibrating plate by adjusting the frequency, a reliable model able to correlate the excitation frequency with the shape of the actual nodal lines, is required.

In this investigation, firstly, vibration tests are performed for square plates, which all have the same thickness and side length, but are made in carbon and stainless steels, copper and bronze, as well as aluminum and magnesium alloys. Photos of the nodal lines forming on all these plates in the range of low frequencies, around the first flexural resonant peak, are shown in correlation with the recorded vibration spectra, and then, damping ratio at the resonant frequency is evaluated by using the Q-factor technique.

By using the elastic membrane model and the thin plate model, the undamped and damped natural frequencies of the panel are explicitly expressed versus the wavelength of the bending oscillation. The condition under which both models predict the same value for the frequency is specified.

Then, the wavelength of the idealized Chladni figures is evaluated for linear and curvilinear nodal lines consisting of parallel segments, plate diagonals, four segments forming a rotated square, as well as elliptical and hyperbolical arcs. In order to correct the significant discrepancy between the computed and the measured frequencies, the theoretical approach is revised based on the wavelength evaluation of the actual Chladni figures. By considering the cutting and rounding of the nodal lines near the plate edges and in the vicinity of the central excitation point of the plate, the mean wavelength is revised for the second and third modes of vibration as an example of the application of the suggested geometrical method.

The frequency of the third mode of vibration is computed by using the proposed approach and the equations reported by several studies [2]–[4], [12]. A correction factor multiplying the core of the frequency expression is used to precisely fit the measured values of the frequency. Finally, the influence of the material properties, damping of the resonant peak, shape of the actual nodal lines, vibration type (free or forced), and the boundary conditions attached to the thin plate model upon the value of the correction factor are discussed.

Chladni Figures Observed at Frequencies Around the First Flexural Resonant Peak

Nodal lines, forming in the range of low frequencies, are visualized here on square plates, which all have the same dimensions but are made in various metallic materials, such as carbon steel, stainless steel, copper, bronze, aluminum, and magnesium alloys. Thus, Table I shows the actual type of materials, and Table II presents the dimensions as well as the material properties, i.e., the side length L, thickness h, ratio L/h, mass Mp, Young modulus E, Poisson ratio μ, and density ρ of the plates P1–P6, which are tested in this work.

Plate type Material
P1 Carbon steel, S45C
P2 Aluminum alloy, A5052
P3 Stainless steel, SUS304
P4 Copper, C1100P
P5 Bronze, C2801P
P6 Magnesium alloy, AZ31
Table I. Actual Materials Used for the Tested Square Plates
Plate type P1 P2 P3 P4 P5 P6
Side length, L [mm] 180 180 180 180 180 180
Thickness, h [mm] 1 1 1 1 1 1
Ratio L/h [-] 180 180 180 180 180 180
Mass, Mp [g] 262 92 247 294 279 62
Young modulus, E [GPa] 203 67 193 115 101 43
Poisson ratio, μ [-] 0.30 0.33 0.30 0.33 0.35 0.35
Density, ρ [kg/m3] 7870 2860 8010 8910 8510 1810
Table II. Dimensions and Material Properties of the Plates Used in the Visualization Tests of the Chladni Patterns

Vibration experiments are conducted with the plates horizontally supported and centrally excited by a vertical rod. The actual methods used to excite the plate, visualize the nodal lines, record the plate frequency response, and evaluate the damping ratio are described in detail by Suciu and Karimine [1]. Therefore, this information is omitted here. The recorded vibration spectra illustrate the variation of the vibration level 20·lg (M) [dB] versus the frequency f, where M is the magnification factor [1]. Excitation frequencies are taken around the first flexural resonant peak, and as argued later, they all correspond to a wavelength Λ, which nearly equals the side length L of the plate. The experimental results are illustrated in Figs. 112, as follows.

Fig. 1. Nodal lines forming on the plate P1 in the frequency range of 158–182 Hz, around the resonant frequency of 164 Hz.

Fig. 2. Vibration spectrum of the plate P1, measured around the resonant frequency of 164 Hz.

Fig. 3. Nodal lines visualized on the plate P2 in the frequency range of 139–173 Hz, around the resonant frequency of 145 Hz.

Fig. 4. Vibration spectrum of the plate P2, measured around the resonant frequency of 145 Hz.

Fig. 5. Nodal lines occurring on the plate P3 in the frequency range of 149–175 Hz, around the resonant frequency of 151 Hz.

Fig. 6. Vibration spectrum of the plate P3, measured around the resonant frequency of 151 Hz.

Fig. 7. Nodal lines observed on the plate P4 in the frequency range of 104–143 Hz, around the resonant frequency of 137 Hz.

Fig. 8. Vibration spectrum of the plate P4, measured around the resonant frequency of 137 Hz.

Fig. 9. Nodal lines visualized on the plate P5 in the frequency range of 97–136 Hz, around the resonant frequency of 131 Hz.

Fig. 10. Vibration spectrum of the plate P5, measured around the resonant frequency of 131 Hz.

Fig. 11. Nodal lines observed on the plate P6 in the frequency range of 130–174 Hz, around the resonant frequency of 137 Hz.

Fig. 12. Vibration spectrum of the plate P6, measured around the resonant frequency of 137 Hz.

Fig. 1 presents the nodal lines forming on the plate P1 in the frequency range of 158–182 Hz, around the resonant frequency of 164 Hz (see frm in Fig. 2).

Fig. 3 shows the nodal lines visualized on the plate P2 in the frequency range of 139–173 Hz, around the resonant frequency of 145 Hz (see frm in Fig. 4).

Fig. 5 displays the nodal lines occurring on plate P3 in the frequency range of 149–175 Hz, around the resonant frequency of 151 Hz (see frm in Fig. 6).

Fig. 7 presents the nodal lines observed on the plate P4 in the frequency range of 104 Hz–143 Hz, around the resonant frequency of 137 Hz (see frm in Fig. 8).

Fig. 9 displays the nodal lines forming on the plate P5 in the frequency range of 97–136 Hz, around the resonant frequency of 131 Hz (see frm in Fig. 10).

Fig. 11 shows the nodal lines visualized on the plate P6 in the frequency range of 130–174 Hz, around the resonant frequency of 137 Hz (see frm in Fig. 12).

Figs. 1, 3, 5, 7, 9, and 11 illustrate a continuous change of the nodal line geometry at augmentation of the excitation frequency. Similar shape transitions can be observed versus frequency for all tested plates, but the frequency interval of manifestation for each pattern depends on the plate material. Thus, in agreement with the results reported by Suciu and Karimine [1], two nodal lines of elliptical shape successively shift into two parallel segments (symbol fII is used here for the related frequency), next into two hyperbolical arcs (symbol f)( is used here for the related frequency), and then suddenly change into a square with rounded corners, which is rotated with 45 degrees relative to the plate sides. This square-like pattern is known in the literature as the third mode of vibration of the plate [2]–[4], and its corresponding frequency is denoted here by the symbol f. In some cases, the hyperbolical arcs change their orientation (see Fig. 9 for the frequency range of 116–119 Hz) or degenerate into the plate diagonals (see Fig. 1 for the frequency range of 181–182 Hz and Fig. 5 for the frequency range of 174–175 Hz) prior to the sudden change into the rounded square. The diagonal-like pattern is known in literature as the second mode of vibration of the plate [2]–[4], and the related frequency is denoted here by the symbol fX.

Figs. 112 also indicates two patterns of the dynamic behavior of the vibrating plates. Thus, for plates P4 and P5, made in copper and bronze, a single distinct resonant peak can be observed, which coincides with the third vibration mode of the plate (see frm = f = 137 and 131 Hz on Figs. 8 and 10). On the other hand, for plates P2, P3, and P6, made in aluminum alloy, stainless steel, and magnesium alloy, the main resonant peak (see frm = fII = 145, 151, and 137 Hz on Figs. 4, 6, and 12) is accompanied by a secondary peak of lower height (see frs = f)( = 167, 163, and 155 Hz on Figs. 4, 6, and 12). Note that the nodal lines related to the main resonant peak consist of two parallel segments and the nodal lines corresponding to the secondary peak consist of two hyperbolical arcs.

Remark that plate P1 dynamically behaves in the same manner as the plates P2, P3, and P6, displaying a main resonant peak (see frm = 164 Hz on Fig. 2) accompanied by a secondary peak of lower height (see frs = f)( = 171 Hz on Fig. 2). However, in some visualization trials at the frequency of 164 Hz, a superposition of four elliptical arcs was observed, producing an intermittent switching of shapes between two arcs aligned to the x-axis, and two arcs aligned to the y-axis of the plate (see Fig. 13 for the system of coordinates attached to the plate). In some cases, at the same resonant frequency of 164 Hz, the sand grains agglomerated at the intersection points of these four arcs, i.e., at the points of coordinates (x = L/4, y = L/4), (x = L/4, y = –L/4), (x = –L/4, y = L/4), and (x = –L/4, y = –L/4), to form small ellipses and then small circles (see the frequency of 164 Hz on Fig. 1). Since this atypical pattern could be observed only on the plate P1, it is not covered by the following theoretical study.

Fig. 13. Cartesian system of coordinates (x, y, z) attached to the square plate of side length L and thickness h.

Next, Table III illustrates the values of the frequency and the damping ratio corresponding to the main resonant peak, as well as the frequency f of the third mode of vibration, which can be distinguished on all plates P1–P6.

Plate type P1 P2 P3 P4 P5 P6
Frequency frm [Hz] 164 145 151 137 131 137
Damping ratio, ζ 0.003 0.010 0.003 0.024 0.024 0.014
Frequency, f [Hz] 182 173 175 137 131 174
Table III. Frequency and Damping Ratio Associated to the Main Resonant Peak, and Frequency of the Third Mode of Vibration

The damping ratio was evaluated by using the Q-factor technique [1] (see the red cutting lines of the resonant peaks on the vibration spectra from Figs. 2, 4, 6, 8, 10, and 12).

Theoretical Models of the Vibrating Plate

Relative to the ratio of the side length L to the thickness h of the plate, mainly three types of theoretical models have been proposed: thick plate model, appropriate for L/h < 8; thin plate model, suitable for 8 < L/h < 80; and membrane model, adequate for L/h > 80 [5]. From Table II, one notes that for all the tested plates, the ratio L/h = 180 is larger than 80, and hence, the membrane with free edges is likely to be a suitable model for our analysis.

During visualization tests of the nodal lines, sound is produced by the vibrating plate, and it can be perceived either softer or louder, as the plate excitation frequency is changed. This can be explained by the superposition of waves meeting at one spot, where they interfere [6].

Thus, 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 [7]–[11]. When the waves arrive in phase, constructive interference causes the sound to become louder. Oppositely, when the waves arrive out of phase, destructive interference causes the sound to become softer.

A standing wave occurs when two waves with identical frequency travel in opposite directions, e.g., the wave produced at the central point of excitation of the plate, and the wave reflected by the plate edges. Such interference creates nodes, in the case of a string, and nodal lines, in the case of a plate, where the deflection is nil; it also creates anti-nodes, in the case of a string, and anti-nodal lines, in the case of a plate, where the deflection is maximal. The nodes and anti-nodes occur at fixed locations on the string, and the anti-nodes appear midway between the nodes [6].

In the case of a string or a plate with fixed boundaries, the amplitude of the bending wave is forced to zero at the edges, i.e., the boundary becomes a node or a nodal line. Since other nodes occur as multiples of the half wavelength, in the case of a string, nodes appear at the following spots: 0, λ/2, λ, 3λ/2, 2λ, etc. On the other hand, for a string or a plate with free edges, the amplitude of the bending wave is maximal at the boundary, which becomes an anti-node or an anti-nodal line. Since the first node occurs at quarter wavelength from the boundary, in the case of a string, nodes occur at the following locations: λ/4, 3λ/4, 5λ/4, 7λ/4, etc.

In conclusion, nodes of a string are half wavelength inter-distanced, regardless of the boundary condition. This peculiar feature is considered here relative to the nodal lines, and it is used in the following analysis to find the mean wavelength Λ of the Chladni figure. Thus, depending on the shape of the nodal lines, the wavelength λ, which is constant in the case of a string, should be regarded as variable, i.e., dependent on the actual location on the surface of the vibrating plate.

Incidentally note that the sand grains used in this work have sizes larger than 0.1 mm. According to [12], these relatively large particles agglomerate on the vibrating plate in regions of nil or very small amplitude of vibration, and therefore, they visualize the nodal lines.

Square Plate Modeled as an Elastic Membrane

Since the flexural rigidity D of a plate can be computed as [13]–[16]:

D = E h 3 12 ( 1 μ 2 ) ,

when the plate thickness h tends to zero, i.e., when the thin plate can be modeled as an elastic membrane, the flexural rigidity approaches zero. Therefore, membranes do not have flexural stiffness, and they cannot resist bending loads. In fact, similar to wires and ropes, membranes can only sustain tensile loads, and their inability to sustain compressive loads leads to the well-known wrinkling phenomenon [5].

However, under the conditions that h > 0 and L/h > 80, non-zero flexural rigidity can be forcibly attributed to the membrane, and in these circumstances, the free vibration of small amplitude for the homogeneous thin square plate can be described by the following wave equation [13]–[16]:

2 w t 2 c 2 ( 2 w x 2 + 2 w y 2 ) = 0

where t is the time, (x, y) are the Cartesian coordinates taken in the central plane of the plate (Fig. 13), w is the transversal deflection along the z coordinate, and c is the speed of the wave traveling inside the plate, which for waves with the wavelength λ much longer than the plate thickness h, can be computed as [15]:

c 2 = 4 π 2 D ρ h λ 2 = π 2 3 E ρ ( 1 μ 2 ) ( h λ ) 2 .

Since the nodal lines visualized in this work are symmetrical relative to the Ox and Oy axes of coordinates, the plate deflection w should satisfy the following condition:

w ( t , x , y ) = w ( t , x , y ) = w ( t , x , y ) ,

and in these circumstances, the deflection can be written as:

w = W c o s ( 2 π x Λ C x ) c o s ( 2 π y Λ C y ) c o s ( Ω t Φ ) ,

where Cx and Cy are constants, Λ is the mean wavelength, which is constant for a certain visualized configuration of nodal lines, W is the amplitude, Ω is the angular frequency, and Φ is the phase angle.

By substituting the derivatives (6) of the deflection w:

{ 2 w x 2 = 4 π 2 Λ 2 C x 2 w ; 2 w y 2 = 4 π 2 Λ 2 C y 2 w 2 w t 2 = Ω 2 w

into (2), the angular frequency Ω can be obtained as:

Ω = 2 π Λ c C x 2 + C y 2 .

By imposing into (3) the obvious condition: λ=Λ, and then, by substituting the result into (7), one achieves the following expression for the frequency f of the vibrating membrane:

f = Ω 2 π = π h Λ 2 E ( C x 2 + C y 2 ) 3 ρ ( 1 μ 2 ) .

For a plate with free edges, deflection w should satisfy the following boundary conditions [13]–[16]:

w x ( t , x = ± L 2 , y ) = 0 ; w y ( t , x , y = ± L 2 ) = 0 ,

and in these circumstances, Cx and Cy can be computed as:

C x = Λ L m ; C y = Λ L n ,

where m and n are integers. Additionally, as already stressed, the standing waves are forming when the side length L of the plate is a multiple of the half wavelength. Such condition can be expressed as:

L = k Λ / 2

where k is an integer. This allows us to rewritten (10) as:

C x = 2 m k ; C y = 2 n k .

Then, by substituting (12) into (8), one obtains the following revised expression for the frequency f of the vibrating plate:

f = 2 π h k Λ 2 E ( m 2 + n 2 ) 3 ρ ( 1 μ 2 ) .

Next, note that the lower part of (6) can be rewritten as:

2 w t 2 + Ω 2 w = 0.

In this form one recognizes the equation describing the free vibration of an undamped one-degree of freedom system [6], [17]. Thus, f given by (13) represents the natural frequency of the vibrating plate.

Under the reasonable assumption that some damping is always present in the system, such as damping produced by the air drag force opposing the plate movement, (14) can be rewritten for the free vibration of a damped one-degree of freedom system, as:

2 w t 2 + 2 ζ Ω w t + Ω 2 w = 0 ; ζ = C 2 M P Ω ,

where C is the damping coefficient and ζ is the damping ratio. In these circumstances, the natural frequency f should be replaced by the damped natural frequency fd, which can be computed as:

f d = f 1 ζ 2 = 2 π h k Λ 2 E ( m 2 + n 2 ) 3 ρ ( 1 μ 2 ) ( 1 ζ 2 ) .

Additionally, one observes that, under the condition (10), the solution (5) of the wave (2) can be rewritten as:

w = W cos ( 2 π m x L ) cos ( 2 π n y L ) cos ( Ω t Φ ) .

Moreover, (17) can be generalized to the following form:

w = W [ C 1 cos ( 2 π m x L ) cos ( 2 π n y L ) + C 2 cos ( 2 π n x L ) cos ( 2 π m y L ) ] cos ( Ω t Φ ) ,

without affecting the frequencies given by (13) and (16). Here, C1 and C2 are unrestricted real constants.

Model of the Thin Elastic Plate

Since it is somewhat questionable how flexural rigidity can be attributed to a membrane, which by definition is able to sustain only tensile loads, the model of the thin elastic plate is alternatively considered in our study. Thus, the free vibration of small amplitude of homogeneous square panels, modeled as thin elastic plates, can be described by the Kirchhoff equation [13]–[16]:

2 w t 2 + E h 2 12 ρ ( 1 μ 2 ) ( 4 w x 4 + 2 4 w x 2 y 2 + 4 w y 4 ) = 0 ,

which, for a plate with free edges, should be solved under the following boundary conditions [13]–[16]:

{ 3 w x 3 + ( 2 μ ) 3 w x y 2 = 0 ; x = ± L 2 3 w y 3 + ( 2 μ ) 3 w x 2 y = 0 ; y = ± L 2 2 w x y = 0 ; ( x , y ) = ( ± L 2 , ± L 2 ) 2 w x 2 + μ 2 w y 2 = 0 ; x = ± L 2 2 w y 2 + μ 2 w x 2 = 0 ; y = ± L 2 .

Although the solutions (17), (18) for plate deflection w, found by using the membrane model, satisfy both (19) and the first three conditions of (20), they do not satisfy the last two requirements. In fact, an exact solution to (19), able to satisfy all conditions (20), has not yet been found [16].

In these circumstances, under the assumption that (18) is close enough to an exact solution for (19), after substituting the terms containing the partial derivatives (21) of the plate deflection w:

{ 4 w x 4 + 2 4 w x 2 y 2 + 4 w y 4 = ( 2 π L ) 4 ( m 2 + n 2 ) 2 w 2 w t 2 = Ω 2 w

into (19), the angular frequency Ω can be obtained as:

Ω = 2 π 2 h L 2 E ( m 2 + n 2 ) 2 3 ρ ( 1 μ 2 ) ,

and then, the frequency f can be found as:

f = Ω 2 π = π h L 2 E ( m 2 + n 2 ) 2 3 ρ ( 1 μ 2 ) .

Next, by substituting into (23) the plate side length as given by (11), the frequency f can be rewritten as:

f = 4 π h k 2 Λ 2 E ( m 2 + n 2 ) 2 3 ρ ( 1 μ 2 ) .

Again, by considering the influence of damping, similar to (16), the damped natural frequency fd can be computed as:

f d = f 1 ζ 2 = 4 π h k 2 Λ 2 E ( m 2 + n 2 ) 2 3 ρ ( 1 μ 2 ) ( 1 ζ 2 ) .

Upon the Selection of the Numbers m and n

Note that the membrane and the thin plate models predict the same value of the frequency (see (16) and (25)) only if the following condition is satisfied:

m 2 + n 2 = k 2 / 4.

Table IV illustrates the first ten possible combinations of integer values for m and n, producing gradually increased integer even values for k. The related mean wavelength can be computed as Λ = 2L/k (see (11)).

Combination m n k Λ = 2L/k
1 0 0 0 Λ = 2L/0 = ∞
2 1 0 2 Λ = L/1 = 2L/2
3 2 0 4 Λ = L/2 = 2L/4
4 3 0 6 Λ = L/3 = 2L/6
5 4 0 8 Λ = L/4 = 2L/8
6 5 0 10 Λ = L/5 = 2L/10
7 4 3 10 Λ = L/5 = 2L/10
8 6 0 12 Λ = L/6 = 2L/12
9 7 0 14 Λ = L/7 = 2L/14
10 8 0 16 Λ = L/8 = 2L/16
Table IV. Possible Combinations of Numbers which Satisfy (26) and Lead to Even Values for the Integer k

One observes that the case m = n = 0 corresponds to a non-vibrating plate, which appears as fully covered by the sand particles. Since the vibration frequency is nil, a finite value for the speed of the wave traveling inside the plate can be achieved only if the apparent wavelength tends to infinity.

On the other hand, in order to gain integer odd values of k, one should accept rational numbers for m or n. Thus, Table V shows the first ten possible combinations of numbers m and n, leading to gradually increased integer odd values of k. One observes that by taking fractional values for m, the left side of (9) cannot be satisfied since the plate appears to have two free edges at y = ± L/2 and two fixed edges at x = ± L/2. Consequently, only the combinations of integer values for m and n, shown by Table IV, are relevant to the present study of a square plate with four free edges.

Combination m n k Λ = 2L/k
I 1/2 0 1 Λ = 2L/1
II 3/2 0 3 Λ = 2L/3
III 5/2 0 5 Λ = 2L/5
IV 3/2 2 5 Λ = 2L/5
V 7/2 0 7 Λ = 2L/7
VI 9/2 0 9 Λ = 2L/9
VII 11/2 0 11 Λ = 2L/11
VIII 13/2 0 13 Λ = 2L/13
IX 15/2 0 15 Λ = 2L/15
X 9/2 6 15 Λ = 2L/15
Table V. Possible Combinations of Numbers which Satisfy (26) and Lead to Odd Values for the Integer k

Additionally, one observes that the values of m and n can be interchanged for any line from Tables IV and V, without affecting the results obtained for k and Λ.

In conclusion, the nodal lines visualized on square plates with free edges at low-frequencies, around the first flexural resonant peak, appear to be related to the lowest vibration mode shown by Table IV that correspond to (m, n) = (1, 0). Besides, for such combination, the frequencies predicted by the membrane model and the thin plate model are the same.

Therefore, by substituting m = 1 and n = 0 in (16) or (25), the damped natural frequency of the first resonant peak can be written as below, where the mean wavelength Λ should be taken as equal to the side length L of the plate (Table IV):

f d ( m = 1 , n = 0 ) = π h Λ 2 E ( 1 ζ 2 ) 3 ρ ( 1 μ 2 ) .

Next, by substituting the values of the material properties and dimensions of the plates P1–P6 (see Table II) into (27), one computes the undamped (ζ = 0) and the damped (ζ ≠ 0) natural frequencies, where the damping ratio is taken from Table III. One notes that the undamped and damped natural frequencies display the same values for all the plates P1–P6 (see the results shown by Table VI). This can be explained by the small value of the damping ratio that corresponds to the main resonant peak. Since the term (1 – ζ2)1/2 ≈ 1 can be neglected in (27), the frequency can be simply computed as:

f ( m = 1 , n = 0 ) = π h Λ 2 E 3 ρ ( 1 μ 2 ) .
Plate denomination P1 P2 P3 P4 P5 P6
Undamped natural frequency [Hz] 298 287 288 213 206 291
Damped natural frequency [Hz] 298 287 288 213 206 291
Table VI. Values of the Undamped and Damped Natural Frequencies, Computed by Using (27) for Λ = L

On the other hand, significant discrepancy can be noted between the experimental (see Table III) and the computed (see Table VI) values of the frequency corresponding to the main resonant peak. Therefore, the objective of this work is to suggest possible reasons for this discrepancy, and then, to revise the theoretical approach.

Similar Equations Proposed for Frequency Evaluation

A thoughtful literature survey was performed to search for models able to accurately predict the experimentally determined frequencies for the Chladni patterns visualized in this work. The pertinent information is summarized below:

1) Since the Chladni figures are generated by a complex superposition of waves, many models apply the variational method of Ritz to a mixture of a finite number of waves, which are described by various types of functions, such as, beam functions, Legendre functions, etc. [2]–[4].

Accordingly, for vibrating square plates the frequency can be computed as [2]–[4]:

f = F F π h L 2 E 3 ρ ( 1 μ 2 ) ,

where FF is a numerically determined frequency factor, which depends on the vibration mode under consideration, on the Poisson ratio μ, and on the number of mixed waves used in the numerical procedure.

Thus, by taking the Poisson ratio as μ = 0.3, the frequency factor was estimated to a value of FF = 0.6147725 for the third mode of vibration, and a value of FFX = 0.496375 for the second mode of vibration of the square plate [3].

2) In a novel recently proposed approach, firstly, the plate response was obtained as a function of the excitation wave number by employing the inhomogeneous wave equation, and then, the resonant wave number was identified by using the principles of maximal coupling efficiency and maximal entropy [12]. Furthermore, the visualized nodal lines were successfully reconstructed by substituting the theoretical resonant wave number into the experimental response function. Based on this approach, the following relationship connecting the frequency f with the wave number K = 2π/Λ was proposed [12]:

f = C F 2 π E h 2 12 ρ ( 1 μ 2 ) K 2 = C F π h Λ 2 E 3 ρ ( 1 μ 2 ) .

where CF = 0.9 is a correction factor to account for the slight difference which occurred between the experimental and the theoretical values of the frequency [12].

Note that the proposed relationship (28) for the frequency computation is quite similar to (29) and (30), except for the frequency factor FF, which is predicted relative to the side length L of the plate, or except for the correction factor CF, which is predicted relative to the wavelength Λ.

Since the correction factor CF is not depending on the mode of vibration under consideration, compared to (29), (30) appears to be a more general expression to be used for frequency evaluation. Such conclusion is supported also by the fact that the vibration mode and the wavelength are related. Thus, the mean wavelength Λ of a certain Chladni pattern should be accurately evaluated, as attempted below.

Wavelength of the Idealized Chladni Figures

From a mathematical standpoint, nodal lines can be found by imposing the condition of nil plate deflection at any time instant:

w ( t , x , y ) = 0 ; ( ) t ,

which, when applied to (18), leads to the following equation, to be solved analytically or numerically in order to generate the contour of the corresponding idealized Chladni figure:

C 1 cos ( 2 π m x L ) cos ( 2 π n y L ) + C 2 cos ( 2 π n x L ) cos ( 2 π m y L ) = 0.

For excitation frequencies around the first flexural resonant peak, which corresponds to (m, n) = (1, 0), (32) reduces to:

C 1 cos ( 2 π x / L ) + C 2 cos ( 2 π y / L ) = 0.

Chladni Figure Consisted of Two Parallel Segments

For C1 ≠ 0 and C2 = 0, the contour line (33) reduces to:

cos ( 2 π x / L ) = 0 x = ± L / 4 ; ( ) y ,

which depicts two parallel segments, symmetrically aligned to the y-axis (see the left-side of Fig. 14). Such pattern was visualized on the plates P2, P3, and P6 at and near the main resonant peak. Concretely, this pattern was observed on P2 in the range of 143–149 Hz, i.e., around frm = fII = 145 Hz (Fig. 3), on P3 in the range of 151–153 Hz, i.e., near frm = fII = 151 Hz (Fig. 5), and on P6 in the range of 137–141 Hz, i.e., in the proximity of frm = fII = 137 Hz (Fig. 11).

Fig. 14. Theoretically obtained Chladni figures for the case of two parallel segments aligned to y-axis (left-side) or x-axis (right-side).

Alternatively, for C1 = 0 and C2 ≠ 0, the contour line (33) reduces to:

cos ( 2 π y / L ) = 0 y = ± L / 4 ; ( ) x ,

which similarly depicts two parallel segments, aligned to the x-axis (see the right-side of Fig. 14).

One observes that the inter-distance between the parallel nodal lines equals the half wavelength λ/2, and also equals the half side length of the plate L/2 (see Fig. 14). Since the wavelength is constant, obviously, for this pattern the mean wavelength equals the side length, i.e., λII = ΛII = L.

Chladni Figure Consisted of the Plate Diagonals

For C1 = –C2 ≠ 0, the contour line (33) reduces to:

cos ( 2 π x / L ) cos ( 2 π y / L ) = 0 y = ± x ,

which depicts the plate diagonals (see the left-side of Fig. 15). Such pattern was visualized on the plates P1 and P3 at frequencies of 181 Hz (Fig. 1) and 174 Hz (Fig. 5).

Fig. 15. Theoretically obtained Chladni figures consisted of diagonal segments (left-side), and four inclined segments (right-side).

Compared with the case of two parallel segments, where the wavelength is constant (Fig. 14), here, as indicated in the left-side of Fig. 15, for each value of the y coordinate, the corresponding wavelength λ depends on the x coordinate, and implicitly on the y coordinate (see the case of y = x, in (36)), as follows:

λ = 4 x = 4 y .

However, the mean wavelength ΛX that can be computed as:

Λ × = 2 L 0 L / 2 λ ( y ) d y = 8 L 0 L / 2 y d y = L ,

equals the side length of the plate.

Chladni Figure Consisted of Four Inclined Segments

For C1 = C2 ≠ 0, the contour line (33) reduces to:

cos ( 2 π x / L ) + cos ( 2 π y / L ) = 0 ,

which admits the following solutions:

{ y = ± ( x + L / 2 ) ; x [ L / 2 ; 0 ] y = ± ( x L / 2 ) ; x [ 0 ; L / 2 ]

Note that (40) depicts four inclined segments that produce a square with a side length of 0.707 L, which is rotated with 45 degrees relative to the plate axes (see the right-side of Fig. 15). This idealized figure with orthogonal corners differs from the actual square with rounded corners, which was visualized on the plates P1–P6. The phenomenological cause of this geometrical alteration is explained in Section 5.

As indicated in the right-side of Fig. 15, for each value of the y coordinate, the corresponding wavelength λ depends on the x coordinate, and implicitly on the y coordinate (see the case y = L/2 – x, in (40)), as follows:

λ = 4 x = 2 L 4 y .

However, the mean wavelength Λ that can be computed as:

Λ = 2 L 0 L / 2 λ ( y ) d y = 4 L 0 L / 2 ( L 2 y ) d y = L

equals the side length of the plate.

Curvilinear Chladni Figures of General Shape

The idealized Chladni figures, discussed in Paragraphs 4.1., 4.2., and 4.3. consist of segments and they are well-known in the literature [1]–[4]. Here, one considers curvilinear Chladni figures consisted of elliptical-like and hyperbolical-like arcs (see Fig. 16). Such nodal lines were observed on all the plates P1–P6, in the range of frequencies taken around the first flexural resonant peak.

Fig. 16. Schematic view of the Chladni figures of general shape consisting of elliptical-like and hyperbolical-like arcs.

In order to reduce the number of parameters describing these curvilinear figures, (33) is rewritten as:

cos ( 2 π x / L ) + Γ cos ( 2 π y / L ) = 0 ; Γ = C 2 / C 1

Note that the previously analyzed linear patterns can be regarded as degenerate shapes of these curvilinear figures. Thus, by imposing into (43) the following conditions: Γ = 0, Γ → ∞, Γ = –1, and respectively Γ = 1, the case of two parallel segments aligned to y-axis (34) or aligned to x-axis (35), the case of diagonal segments (36), and respectively the case of four inclined segments forming a rotated square (39), can be easily retrieved.

Due to the double symmetry relative to the (x, y) system of coordinates of the curvilinear patterns depicted by Fig. 16, the following analysis is performed by using only the first quadrant (x, y > 0). Thus, Fig. 17 presents the elliptical-like and hyperbolical-like arcs represented in the first quadrant, and their relevant geometrical features.

Fig. 17. Elliptical-like and hyperbolical-like arcs represented in the first quadrant, and their relevant geometrical features.

Remark that the elliptical-like and hyperbolical-like arcs intersect the x-axis in a point of coordinates (x0, 0), and the plate boundary in a point of coordinates (xb, L/2). From (43), the coordinates x0 and xb can be derived as follows:

x 0 = L 2 π cos 1 ( Γ ) ; x b = L 2 π cos 1 ( Γ )

and they exist if the expressions cos-1(±Г) can be computed. This means that the parameter Г should be restricted to the range [−1; 1], from which the values corresponding to linear patterns, i.e., Г = −1, 0, and 1, should be removed. Note that the elliptical-like arcs are obtained by taking the parameter Г in the range (0, 1), and in this case the coordinates (44) satisfy the following inequality:

x b = L 2 π cos 1 ( Γ ) < x 0 = L 2 L 2 π cos 1 ( Γ )

On the other hand, the hyperbolical-like arcs are obtained by replacing Г with –Г, which falls in the interval (−1; 0), and in this case the coordinates (44) should fulfill the inequality:

x 0 = L 2 π cos 1 ( Γ ) < x b = L 2 L 2 π cos 1 ( Γ )

Fig. 17 shows that regardless the value of Г, all these arcs pass through the fixed point of coordinates (x = L/4, y = L/4), which is also an inflection point. Additionally, these curves are asymmetrical relative to both the vertical line x = L/4 and the horizontal line y = L/4. In order to prove this, note that for any increment ∆y taken in the range of [0; L/4], the horizontal lines y = L/4 ± ∆y intersect the elliptical-like and hyperbolical-like arcs in points of coordinates x = L/4 ± ∆x, where the increment ∆x can be computed as:

Δ x = L 4 L 2 π cos 1 [ Γ sin ( 2 π Δ y L ) ]

For the sake of simplicity, the hyperbolical-like arc is used in the following analysis, but similar results can be obtained by taking into consideration the elliptical-like arc.

Firstly, from Fig. 17 one observes that the area depicted with black dots under the hyperbolical-like arc, and also the area depicted with blue horizontal lines in the left-side of the hyperbolical-like arc have the same surface area S. On the other hand, S equals the half surface area of the rectangle with the width of xbx0 and the height of L/2. Consequently, the surface area S can be computed as:

S = 1 2 ( x b x 0 ) L 2 = L 2 4 [ 1 2 1 π cos 1 ( Γ ) ]

Next, the mean wavelength Λ)( of the hyperbolical-like Chladni figure can be evaluated, relative to either y or x axis, as follows. Thus, relative to y-axis, for each selected value of the y coordinate, by using (43) the wavelength λ can be expressed as:

λ ( y ) = 4 x = 2 L π cos 1 [ Γ cos ( 2 π y L ) ]

and then, the mean wavelength Λ)( can be computed as:

Λ ) ( = 2 L 0 L / 2 λ ( y ) d y = 8 L [ L x 0 2 + S ] = 2 ( x 0 + x b ) = L .

On the other hand, relative to x-axis, for each selected value of the x coordinate, by using (43) the wavelength λ can be written as:

λ ( x ) = 4 y = 2 L π cos 1 [ 1 Γ cos ( 2 π x L ) ]

and then, the mean wavelength Λ)( can be computed as:

Λ ) ( = x 0 x b λ ( x ) d x x b x 0 = 4 S x b x 0 = 4 L 4 ( x b x 0 ) x b x 0 = L

Thus, regardless the value of the parameter Γ, and regard-less the axis of the system of coordinates relative to which the evaluation process is performed, the mean wavelength λ)( of the hyperbolical-like figures equals the side length L of the plate. Obviously, due to the peculiar symmetry of the elliptical-like and hyperbolical-like arcs mentioned above in relation with Fig. 17, the mean wavelength of the elliptical-like figures also equals the side length L of the plate.

In conclusion, all the linear and curvilinear nodal lines, visualized for various frequencies around the first flexural resonant peak, can be ideally described by (33). Since for all these idealized figures, the computed mean wavelength equals the side length of the plate (Λ = L), according to (28) and (30), all these linear and curvilinear patterns should manifest at the same excitation frequency. Unfortunately, this result is in complete disagreement with the experimental evidence. Hence, the geometrical discrepancy between the ideal and actual Chladni figure, which presumably affects the value of the mean wavelength, seems to be the cause of this contradiction between the theoretical and experimental values of the frequency. Therefore, the theoretical approach should be revised, as reported below.

Wavelength of the Actual Chladni Figures

Firstly, note that on the free boundary of the plate, which behaves as an anti-nodal line, and also at the central excitation point of the plate, which appears as an anti-nodal point, the plate displacement cannot be zero.

On the other hand, concerning for example the second mode of vibration of the square plate, the theoretical pattern consisted of diagonal segments (see the left-side of Fig. 15) leads to nil displacement at the corners and also at the central point of excitation. However, since all these points are anti-nodes, such idealized figure cannot be actually achieved. For this reason, the experimentally recorded pattern slightly differs from the theoretical one. Concretely, cutting of the diagonal lines in the proximity of the corners, as well as cutting and/or rounding of the diagonal lines in the vicinity of the excitation point can be observed on the plates P1 and P3 at the frequencies of 181 Hz (Fig. 1) and 174 Hz (Fig. 3).

Similarly, concerning the third mode of vibration of the square plate, the theoretical pattern consisted of a rotated square with orthogonal corners (see the right-side of Fig. 15) leads to nil displacement at the corners of coordinates (x = 0, y = L/2), (x = 0, y = –L/2), (x = –L/2, y = 0), (x = L/2, y = 0). However, since these points are on the free edges of the plate, they are anti-nodes. For this reason the experimental pattern slightly differs from the theoretical one. Concretely, rounding of the corners, through which space is created between the free edges of the plate and the corners of the rotated square, can be observed on all the plates P1–P6.

Wavelength of the Second Mode of Vibration

Fig. 18 presents the geometrical model used to find the influence of the cutting distance and the influence of the rounding radius on the mean wavelength of the second mode of vibration, i.e., the wavelength of the actual Chladni figure consisted of altered plate diagonals. Here, δb is the cutting distance at the edges, δc is the cutting distance at the plate center, and R is the rounding radius at the plate center.

Fig. 18. Geometrical model used to find the influence of the cutting distance and the rounding radius on the wavelength of the second mode of vibration (actual Chladni figure consisted of altered plate diagonals).

For diagonals shortened both at the edges and plate center (see the left-side of Fig. 18), the local wavelength λ is still given by (37). However, the mean wavelength found for the idealized pattern (ΛX = L, see (38)), should be revised for the actual geometry of the nodal lines, as follows:

Λ × = 1 L / 2 δ b δ c δ c L / 2 δ b λ ( y ) d y = L 2 δ b + 2 δ c

Note that for equal cuts, i.e., δb = δc, the mean wavelength still equals the plate side length L, but it decreases (ΛX < L) for δb > δc, and increases (ΛX > L) for δb < δc. Thus, based on (28) and (30) one concludes that the corresponding frequency increases for δb > δc, and decreases for δb < δc.

By referring to the nodal lines visualized on P1 at 181 Hz, it seems that the boundary cut is shorter than the central cut. Since this frequency is smaller than the value of 298 Hz presented by Table VI, it appears that (53) is in agreement with the experimental evidence.

Similarly, for diagonals shortened at the boundary, but rounded at the plate center (see the right-side of Fig. 18), the local wavelength λ can be written as:

λ ( y ) = { 4 ( R 2 R 2 y 2 ) ; y [ 0 ; R 2 / 2 ] 4 y ; y [ R 2 / 2 ; L / 2 δ b ]

and then, the mean wavelength ΛX for the actual geometry of the nodal lines can be revised as follows:

Λ × = 1 L / 2 δ b 0 L / 2 δ b λ ( y ) d y = L 2 δ b + ( 4 π ) R 2 L 2 δ b

Note that the rounding radius R augments, but the edge cut δb reduces the mean wavelength ΛX. In order to gain a lower frequency than the value presented by Table VI, the actual mean wavelength ΛX should be larger than the side length L of the plate, and this can be achieved if the next geometrical condition is satisfied:

R > 2 δ b ( L 2 δ b ) 4 π

Wavelength of the Third Mode of Vibration

Fig. 19 illustrates the geometrical model used to find the influence of the rounding radius R on the mean wavelength of the third mode of vibration, i.e., the mean wavelength of the actual Chladni figure consisted of a rotated square with rounded corners. Since the experimental evidence indicates that the side length of such rounded square, denoted as Lr, might be different than the value of 0.707L obtained for the idealized Chladni figure, parameter Lr is also considered in our geometrical analysis.

Fig. 19. Geometrical model used to find the influence of the rounding radius on the wavelength of the third mode of vibration (actual Chladni figure consisted of a rotated square with rounded corners).

The approach explained in detail by Paragraph 5.1., is also applied here for the geometrical pattern shown in Fig. 19.

Accordingly, the mean wavelength of the idealized figure (Λ = L, see (42)), should be revised for the actual geometry of the nodal lines, as follows:

Λ = 1 L / 2 δ 0 L / 2 δ λ ( y ) d y = 2 L r 2 ( 4 π ) R 2 L r 2 2 ( 2 1 ) R

where the upper limit of the integral is given by:

L / 2 δ = L r 2 / 2 R ( 2 1 )

Next, for this pattern of nodal lines, which is common to all plates P1–P6, the mean wavelength Λ is computed by using (57) and the obtained results are shown in Table VII. In order to accurately evaluate the rounding radius R and the side length Lr photos were magnified to scale of 1:1. Thus, Table VII presents the values of the experimentally found frequency f, side length of the rounded square Lr, rounding radius R, mean wavelength Λ, and ratio Λ/L of the mean wavelength to the side length of the plate. Observe that the mean wavelength Λ of the actual Chladni figure, predicted by (57), exceeds with about 20% the wavelength of the idealized Chladni figure (Λ = L, see (42)).

Plate type P1 P2 P3 P4 P5 P6
Frequency, f [Hz] 182 173 175 137 131 174
Side length, Lr [mm] 137 138 137.5 134 135 138.5
Radius R, [mm] 50 52 51 46 45.5 52
Wavelength Λ [mm] 218.3 219.9 219.1 213.2 214.7 220.7
Ratio Λ/L [-] 1.213 1.222 1.217 1.184 1.193 1.226
Table VII. Experimentally Found Frequency, Side Length of the Rounded Square, Rounding Radius, Mean Wavelength, and Ratio Λ/L Associated to the Third Mode of Vibration of the Plate

Frequency of the Third Mode of Vibration

Table VIII presents a comparison of the experimental and computed values of the frequency f associated to the third vibration mode of the plate. The mean wavelength Λ, as given by Table VII, is substituted in (28), (30) to compute the corresponding frequencies. Note that the results predicted by (29) are acceptable, but the results predicted by (30) are in excellent agreement with the experimental results. On the other hand, in absence of the correction factor CF = 0.9, introduced to account for the difference between the experimental and the theoretical values of the frequency [12], results predicted by (28) are unacceptable.

Plate type P1 P2 P3 P4 P5 P6
Frequency, f [Hz] (Experimental) 182 173 175 137 131 174
Frequency, f [Hz] (Computed, (28)) 202.7 192.3 194.4 151.8 144.8 193.6
Frequency, f [Hz] (Computed, (29)) 183.2 176.5 177.1 131 126.6 179.1
Frequency, f [Hz] (Computed, (30)) 182.4 173 175 136.6 130.3 174.2
Table VIII. Comparison of the Experimental and Computed Values of the Frequency for the Third Mode of Vibration

Note that our approach, based on the mean wavelength computation, and the approach based on the principles of maximal coupling efficiency and maximal entropy [12] lead to the same structure for the frequency equation. However, a correction factor multiplying the core of the frequency expression is necessary to be introduced in order to precisely fit the experimentally determined frequencies. The value of the correction factor CF = 0.9, proposed in [12] for a plate made in aluminum alloy, was found here to be appropriate for various metallic materials. Thus, this factor appears to be uninfluenced by the material properties, by the damping ratio associated to the resonant peak, by the shape of the actual nodal lines, and by the vibration type, which was modeled here as free vibration, but as forced vibration in [12]. In such circumstances, one reasonably argues that the need of a correction factor is the consequence of the fact that, although solutions (5), (17), (18) satisfy both (19) and the first three conditions of (20), they do not satisfy the last two requirements.

Following such rationale, a closer look at conditions (20) might be insightful. For instance, by substituting (5) into the first two requirements of (20), they can be rewritten as:

{ [ C x 2 + ( 2 μ ) C y 2 ] w x ( t , ± L 2 , y ) = 0 [ ( 2 μ ) C x 2 + C y 2 ] w y ( t , x , ± L 2 ) = 0 ,

and similarly, by substituting (5) into the last two conditions of (20), they can be rewritten as:

{ ( C x 2 + μ C y 2 ) w ( t , ± L 2 , y ) = 0 ( μ C x 2 + C y 2 ) w ( t , x , ± L 2 ) = 0

Since for a vibrating plate all the following terms, which are included into expressions (59) and (60), are positive:

{ C x 2 + ( 2 μ ) C y 2 > 0 ; ( 2 μ ) C x 2 + C y 2 > 0 C x 2 + μ C y 2 > 0 ; μ C x 2 + C y 2 > 0

the requirements (59) reduce to (9), which describe the boundary conditions for a plate with free edges, but, on the other hand, the requirements (60) reduce to:

w ( t , ± L 2 , y ) = 0 ; w ( t , x , ± L 2 ) = 0

which describe the boundary conditions for fixed edges.

Solving this contradiction, either by finding a different solution to (19), or by partially revising the conditions (20), might reveal the physical meaning of the correction factor.

Conclusion

In this work, nodal lines occurring in the domain of low frequencies around the first flexural resonant peak were visualized on square plates made in various alloys, such as carbon steel, stainless steel, copper, bronze, aluminum, and magnesium alloys. Correlation between the nodal lines and the vibration spectra revealed two main patterns of dynamic behavior:

1. For plates made in copper and bronze, a single distinct resonant peak, coinciding with the third vibration mode, was observed. Nodal lines formed a rotated square with rounded corners.

2. For plates made in aluminum alloy, stainless steel, and magnesium alloy, the main resonant peak with nodal lines consisted of two parallel segments was accompanied by a secondary peak of lower height, with nodal lines consisting of two hyperbolical arcs.

By employing the elastic membrane and the thin plate models, the undamped and damped natural frequencies of the panel were explicitly expressed versus the wavelength of the flexural vibration. A condition under which both models predict the same value for the frequency was specified.

Such theoretical analysis revealed the following aspects:

3. The undamped and damped natural frequencies shown the same values for all the plates, and this was explained by the small damping ratio associated to the main resonant peak.

4. Significant discrepancy between the computed and the measured values of the frequency was observed. Therefore, a thoughtful literature survey was performed to search for models able to accurately predict the frequency.

In order to achieve good agreement between the predicted and measured frequencies, the theoretical approach was revised based on the precise wavelength evaluation.

5. Firstly, the mean wavelength of the idealized Chladni figures was evaluated for the nodal lines relevant to our tests, i.e., for linear and curvilinear lines consisting of parallel segments, plate diagonals, four segments forming a rotated square, as well as elliptical and hyperbolical arcs. Since the mean wavelength of all these idealized patterns equaled the side length of the plate, from a theoretical standpoint, they were not scattered over a frequency distribution, but shared a common frequency value. However, such a conclusion was found to be in complete disagreement with the experimental evidence provided by the recorded vibration spectra.

6. By considering the cutting and rounding of the nodal lines near the plate edges and in the vicinity of the central excitation point of the plate, the mean wavelength was revised for the actual Chladni patterns, corresponding to the second and third modes of vibration.

7. Frequency of the third vibration mode was computed by using the proposed procedure and the equations reported by several studies [2]–[4], [12]. A correction factor multiplying the core of the frequency expression was used in order to precisely fit the experimentally found values of the frequency. In the end, the influence of the material properties, damping related to the resonant peak, the shape of the actual nodal lines, vibration type (free or forced), and the boundary conditions attached to the thin plate model upon the value of the correction factor was discussed.

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