A simple method of measuring the Chladni mode shape on an elastic plate by the principle of an optical lever is proposed.
Quantitatively determining the Chladni pattern of an elastic plate is of great interest in both physical science and engineering applications. In this paper, a method of measuring mode shapes of a vibrating plate based on an optical lever method is proposed. Three circular acrylic plates were employed in the measurement under different center harmonic excitations. Different from a traditional method, only an ordinary laser pen and a light screen made of ground glass are employed in this novel approach. The approach is as follows: the laser pen projects a beam to the vibrating plate perpendicularly, and then the beam is reflected to the light screen in the distance, on which a line segment made of the reflected spot is formed. Due to the principle of vision persistence, the light spot could be read as a bright straight line. The relationship between the slope of the mode shape, length of the light spot and the distance of the vibrating plate and the light screen can be obtained with algebraic operations. Then the mode shape can be determined by integrating the slope distribution with suitable boundary conditions. The full-field mode shapes of Chladni plate could also be determined further in such a simple way.
Chladni mode shapes are of great interest in both science and engineering applications. Chladni patterns are reactions of physical waves, and one can illustrate the wave pattern with various methods. It is a well-known method to show the various modes of vibration on an elastic plate by outlining the nodal lines. Small particles are always employed to show the Chladni patterns, since they can stop at the nodes where the relative vibrating amplitude of the plate is zero, and the positions of the nodes vary with resonant mode to form various Chladni patterns.
Many researchers have paid attention to various Chladni patterns, but they only show the nodal lines of the mode shapes, the mode shapes (i.e., vibration amplitude) between the nodal lines are not illustrated. Waller investigated the free vibrations of a circle1, a square2, an isosceles right angled triangles3, a rectangular4, elliptical5 plates, and different Chladni patterns are illustrated therein. Tuan et al. reconstructed different Chladni patterns through both experimental and theoretical approaches, and the inhomogeneous Helmholtz equation is adopted during the theoretical modeling6,7. It is a popular method of using Laser Doppler Vibrometer (LDV) or Electronic Speckle Pattern Interferometry (ESPI) to quantitatively measure the mode shapes of the Chladni patterns8,9,10. Although LDV enables femtometer amplitude resolution and very high frequency ranges, unfortunately, the price of LDV is also a little expensive for classroom demonstration and/or college physics education. With this consideration, the present paper proposed a simple approach to quantitatively determine the mode shapes of a Chladni pattern with low cost, since only an extra laser pen and a light screen are needed here.
The present measurement method is illustrated in Figure 111. The vibrating plate has three different positions: the rest position, position 1 and position 2. Position 1 and 2 represent the two maximum vibrating places of the plate. A laser pen projects a straight beam on the surface of the plate, and if the plate locates at the rest position, the laser beam will be directly reflected to the light screen. While the plate locates at position 1 and 2, then the laser beam will be reflected to point A and B on the light screen, respectively. Due to the effect of persistence of vision, there will be a bright straight line on the light screen. The length of the bright light L is related to the distance D between the light screen and the location of the laser point. Different points on the plate have different slopes, which could be determined by the relationship between L and D. After obtaining the slope of the mode shape at different points on the plate, the problem turns into a definite integral. With the help of the boundary vibration amplitude of the plate and the discrete slope data, the mode shape of the vibrating plate can be obtained easily. The whole experimental setup is given in Figure 211.
This paper describes the experimental setup and procedure for the optical lever method to measure the Chladni mode shapes. Some typical experimental results are also illustrated.
1. Experimental setup and procedures
NOTE: Set up the experimental system as shown in Figure 2.
2. Data processing
The excitation frequency that can excite axisymmetric Chladni pattern is determined through the frequency sweeping test. Three circular acrylic plates with diameters of 150 mm, 200 mm and 250 mm are tested, and results show that the first order axisymmetric resonance frequencies are 346 Hz, 214 Hz and 150 Hz for the three plates respectively. It is concluded that with larger diameter, the plate is more flexible, and the corresponding resonance frequency will be smaller. The Chladni patterns of the acrylic plate with different diameters are given in Figure 311.
Under the corresponding resonant frequency, the length of the light spot on the light screen of different plates can be measured and recorded. The regression value of mode shape slope can be obtained with Eq.(1), whose distributions along the radial direction of plate A, B and C are given in Table 111, and they are determined by measuring several different light spot lengths L of the specific laser point with different distance D.
Numerical simulation with ANSYS is performed to verify the present experimental results. The script code of APDL (ANSYS Parametric Design Language) is provided as a Supplemental File 1. Figure 411 shows the comparisons of the present experimental results and numerical results on the mode shape of different plates. It is very clear that all results with different conditions compare very well, which prove the feasibility of the present method in measuring the mode shape of plates.
Figure 1: Illustration of the present measurement method.
The basic measurement principal is illustrated in this figure, with an emphasis on the incident and reflect light beam and the relationship of different geometric parameters. Please click here to view a larger version of this figure.
Figure 2: The experimental setup.
The picture of experimental setup is provided for clearly understand and replicate the measurement approach easily. Please click here to view a larger version of this figure.
Please click here to view a larger version of this figure.
Please click here to view a larger version of this figure.
Figure 3: Chladni pattern of different acrylic plates: (a) 150 mm, (b) 200 mm, (c) 250 mm.
The Chladni patterns of three different acrylic circular plates are given respectively. The brown particles are sands and clearly show the nodal line of the Chladni patterns. Please click here to view a larger version of this figure.
Please click here to view a larger version of this figure.
Please click here to view a larger version of this figure.
Figure 4: Comparisons of experimental results and numerical simulation for the mode shapes of different plates: (a) 150 mm, (b) 200 mm, (c) 250 mm.
The numerical results obtained with ANSYS and the present experimental results are compared to verify the reliability of the present experimental method. Please click here to view a larger version of this figure.
Plate A (Diameter=150 mm) |
Plate B (Diameter=200 mm) |
Plate C (Diameter=250 mm) |
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r/mm | Directly calculated slope | Revised slope | r/mm | Directly calculated slope | Revised slope | r/mm | Directly calculated slope | Revised slope |
5 | 0.001913 | 0.001913 | 7 | 0.002668 | 0.002668 | 7 | 0.0013 | 0.0013 |
10 | 0.001478 | 0.001478 | 12 | 0.00269 | 0.00269 | 12 | 0.001613 | 0.001613 |
15 | 0.00144 | 0.00144 | 17 | 0.002785 | 0.02785 | 17 | 0.002055 | 0.002055 |
20 | 0.001088 | 0.001088 | 22 | 0.00269 | 0.00269 | 22 | 0.002283 | 0.002283 |
25 | 0.00061 | 0.00061 | 28 | 0.002543 | 0.002543 | 27 | 0.002618 | 0.002618 |
30 | 0.000388 | 0.000388 | 38 | 0.001858 | 0.001858 | 32 | 0.00256 | 0.00256 |
35 | 0.000883 | -0.000883 | 48 | 0.000748 | 0.000748 | 37 | 0.00209 | 0.00209 |
40 | 0.001733 | -0.001733 | 58 | 0.000668 | 0.000668 | 42 | 0.002128 | 0.002128 |
45 | 0.002478 | -0.002478 | 68 | 0.00082 | -0.00082 | 47 | 0.001723 | 0.001723 |
50 | 0.003433 | -0.003433 | 72 | 0.001583 | -0.001583 | 52 | 0.001568 | 0.001568 |
55 | 0.00389 | -0.00389 | 77 | 0.00241 | -0.00241 | 57 | 0.001 | 0.001 |
60 | 0.002705 | -0.002705 | 82 | 0.002813 | -0.002813 | 62 | 0.004175 | 0.004175 |
65 | 0.002283 | -0.002283 | 87 | 0.0026 | -0.0026 | 67 | 0.001175 | 0.001175 |
70 | 0.002223 | -0.002223 | 97 | 0.002264 | -0.002264 | 72 | 0.002825 | -0.002825 |
77 | 0.000873 | -0.000873 | ||||||
82 | 0.001205 | -0.001205 | ||||||
87 | 0.001538 | -0.001538 | ||||||
92 | 0.00176 | -0.00176 | ||||||
97 | 0.001983 | -0.001983 | ||||||
102 | 0.002278 | -0.002278 | ||||||
107 | 0.002745 | -0.002745 | ||||||
112 | 0.00269 | -0.00269 | ||||||
117 | 0.002783 | -0.002783 | ||||||
122 | 0.002218 | -0.002218 |
Table 1: Slope distribution of the mode shape along radial direction. The calculated slope distribution of the mode shape along the radial direction is provided, and both original and revised slope are given to illustrate the process of revision.
Supplement File 1: ANSYS script for simulating the dynamic response and mode shape of a plate. Please click here to download this file.
The optical lever method is adopted in this paper to determine the mode shape of a plate, since the Chladni pattern can only show the nodal lines of a vibrating plate. To determine the mode shape of the plate, the relationship between the slope and distance of the light screen and spot length should be obtained in advance. Then through definite integration calculation, the mode shape of the Chladni pattern could be quantitatively determined.
Generally, the whole process of the present approach includes the following steps: (1) Perform the forced vibration test to obtain the resonance frequency of the plate. (2) Conduct forced vibration test near the resonance frequency, and record the coordinates of the nodes of Chladni pattern. These data are used for calibrating the absolute mode shape obtained by experimental tests. (3) The laser spot is perpendicularly projected to different radial locations of the plate, and the length of the light spot on the light screen is measured. This test needs to be repeated several times with different distances between the vibrating plate and light screen to obtain the linear regression value of the mode shape slope with Eq. (2). (4) Obtain the experimental mode shape of the Chladni pattern with Eq. (4) through post processing the raw experimental data.
It should be pointed out that, although the present experimental demonstration only shows the measurement of axisymmetric Chladni patterns, it also could be used for the determination of nonaxisymmetric Chladni patterns in a forward manner. Not only circular plates, but also other shapes, such as triangle, rectangular, and even irregular shapes could be employed to show the beauty of Chladni patterns. Furthermore, if the measuring point density, laser source, measuring tool, as well as the integral calculation method are carefully chosen, the accuracy of the proposed method could be adapted to required level.
The authors have nothing to disclose.
This work was supported by National Natural Science Foundation of China (grant no. 11772045) and Education and Teaching Reform Project of University of Science and Technology Beijing (grant no. JG2017M58).
Acrylic plates | Dongguan Jinzhu Lens Products Factory | Three 1.0-mm-thickness mirrored circular acrylic plates with diameter of 150 mm, 200 mm and 250 mm respectively. They are easily deformed. | |
Laser pen | Deli Group | 2802 | Red laser is more friendly to the viewer. The finer the laser beam, the better. |
Light screen | Northern Tempered Glass Custom Taobao Store | Several layers of frosted stickers can be placed on the glass to achieve the effect of frosted glass. | |
Ruler | Deli Group | DL8015 | The length is 1m and the division value is 1mm. |
Signal generator | Dayang Science Education Taobao Store | TFG6920A | Common ones in university laboratories are available. |
Vibrator | Dayang Science Education Taobao Store | The maximum amplitude is 1.5cm.The power is large enough to cause a noticeable phenomenon when the board vibrates. Otherwise, add a power amplifier. |