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).

1. Experimental setup and procedures
NOTE: Set up the experimental system as shown in Figure 2.
<ol>
	<li>Preparation of the vibration system
	<ol>
		<li>Prepare three 1.0-mm-thickness mirrored circular acrylic plates with diameter of 150 mm, 200 mm and 250 mm respectively. Drill a hole of 3 mm in diameter at the center of each plate. Mark several black points every 5 mm along an arbitrary radius.</li>
		<li>Attach each plate to the actuate bar of the vibrator w...

<ol>
	<li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+free+circular+plates.+Part+1:+Normal+modes.">Vibrations of free circular plates. Part 1: Normal modes.</a> Proceedings of the Physical Society. 50 (1), 70-76 (1938).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+free+square+plates:+part+I.+Normal+vibrating+modes.">Vibrations of free square plates: part I. Normal vibrating modes.</a> Proceedings of the Physical Society. 51 (5), 831-844 (1939).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+free+plates:+isosceles+right-angled+triangles.">Vibrations of free plates: isosceles right-angled triangles.</a> Proceedings of the Physical Society. 53 (1), 35-39 (1941).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+Free+Rectangular+Plates.">Vibrations of Free Rectangular Plates.</a> Proceedings of the Physical Society Section B. 62 (5), 277-285 (1949).</li><li>Waller, M. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Vibrations+of+Free+Elliptical+Plates.">Vibrations of Free Elliptical Plates.</a> Proceedings of the Physical Society Section B. 63 (6), 451-455 (1950).</li><li>Tuan, P. H., Wen, C. P., Chiang, P. Y., Yu, Y. T., Liang, H. C., Huang, K. F., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Exploring+the+resonant+vibration+of+thin+plates:+Reconstruction+of+Chladni+patterns+and+determination+of+resonant+wave+numbers.">Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers.</a> The Journal of the Acoustical Society of America. 137 (4), 2113-2123 (2015).</li><li>Tuan, P. H., Lai, Y. H., Wen, C. P., Huang, K. F., Chen, Y. F. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Point-driven+modern+Chladni+figures+with+symmetry+breaking.">Point-driven modern Chladni figures with symmetry breaking.</a> Scientific Reports. 8 (1), 10844 (2018).</li><li>Castellini, P., Martarelli, M., Tomasini, E. P. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Laser+Doppler+Vibrometry:+Development+of+advanced+solutions+answering+to+technology's+needs.">Laser Doppler Vibrometry: Development of advanced solutions answering to technology's needs.</a> Mechanical Systems and Signal Processing. 20 (6), 1265-1285 (2006).</li><li>Sels, S., Vanlanduit, S., Bogaerts, B., Penne, R. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Three-dimensional+full-field+vibration+measurements+using+a+handheld+single-point+laser+Doppler+vibrometer.">Three-dimensional full-field vibration measurements using a handheld single-point laser Doppler vibrometer.</a> Mechanical Systems and Signal Processing. 126, 427-438 (2019).</li><li>Georgas, P. J., Schajer, G. S. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Simultaneous+Measurement+of+Plate+Natural+Frequencies+and+Vibration+Mode+Shapes+Using+ESPI.">Simultaneous Measurement of Plate Natural Frequencies and Vibration Mode Shapes Using ESPI.</a> Experimental Mechanics. 53 (8), 1461-1466 (2013).</li><li>Luo, Y., Feng, R., Li, X. D., Liu, D. H. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=A+simple+approach+to+determine+the+mode+shapes+of+Chladni+plates+based+on+the+optical+lever+method.">A simple approach to determine the mode shapes of Chladni plates based on the optical lever method.</a> European Journal of Physics. 40, 065001 (2019).</li><li>Coleman, H. W., Steele, W. G. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=.">.</a> Experimentation and uncertainty analysis for engineer. , (1999).</li></ol>

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...

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.

<table>
	<tbody>
		<tr>
			<td>Acrylic plates</td>
			<td>Dongguan Jinzhu Lens Products Factory</td>
			<td></td>
			<td>Three 1.0-mm-thickness mirrored circular acrylic plates with diameter of 150 mm, 200 mm and 250 mm respectively. They are easily deformed.</td>
		</tr>
		<tr>
			<td>Laser pen</td>
			<td>Deli Group</td>
			<td>2802</td>
			<td>Red laser is more friendly to the viewer. The finer the laser beam, the better.</td>
		</tr>
		<tr>
			<td>Light screen</td>
			<td>Northern Tempered Glass Custom Taobao Store</td>
			<td></td>
			<td>Several layers of frosted stickers can be placed on the glass to achieve the effect of frosted glass.</td>
		</tr>
		<tr>
			<td>Ruler</td>
			<td>Deli Group</td>
			<td>DL8015</td>
			<td>The length is 1m and the division value is 1mm.</td>
		</tr>
		<tr>
			<td>Signal generator</td>
			<td>Dayang Science Education Taobao Store</td>
			<td>TFG6920A</td>
			<td>Common ones in university laboratories are available.</td>
		</tr>
		<tr>
			<td>Vibrator</td>
			<td>Dayang Science Education Taobao Store</td>
			<td></td>
			<td>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.</td>
		</tr>
	</tbody>
</table>

measurement of chladni mode shapes with an optical lever method

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.

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 diff...

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Measurement of Chladni Mode Shapes with an Optical Lever Method

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 ...

This procedure can provide a clear glance for determining the Chladni mode shape using the optical lever method. The proposed method is a rather simple approach to quantitatively determine the Chladni mode shape at a very low cost. Knowledge of the partial differential equation and the finite element method is important for successful execution of the procedure.To set up the vibration system, acquire two 1mm thick mirrored circular acrylic plates with 150mm diameter and 200mm diameters respectively. Drill a 3mm diameter hole in the center of each plate, and mark several points every 5mm along an arbitrary radius. Attach each plate to the actuate bar of the vibrator with a bolt in the middle point, and use a waveform generator to drive the vibrator with a sine wave.To acquire the resonance frequency, use the laser pen to perpendicularly project the laser beam from a 120mm distance onto the vibrating plate, such that the beam is reflected to the light screen 500mm in the distance. Quickly move the laser pen along the direction perpendicular to its length to scan over the diameter, while the signal generator continuously changes its frequency. When the spot length is significantly stretched along the diameter, and some spots with almost no expansion appear, begin scanning the certain frequency range slowly, and determine the frequency at which the spot expands most obviously.To prepare the light path and measurement system, place the light screen parallel to the vibrating plate, and mark the distance with a meter ruler. Then, using 500mm as the starting distance, place the laser pen to project the beam perpendicularly onto the plate, such that the beam is reflected to the light screen in the distance, making sure that the previously made mark can be scanned while the laser pen is moving. To obtain an experimental measurement, turn on the signal generator, and set the excitation frequency to the resonance frequency at which the spot expanded most obviously.The signal intensity should be as small as possible once the light spot on the light screen is large enough to be recorded. Adjust the laser pen such that the incident point coincides with the first marker, that is, with the nearest marker to the fixed point of the plate, and move the screen from 500mm to 1000mm, measuring the spot length from the screen every 50mm. Then adjust the laser pen to make the incident point adjacent to the next marker, and repeat the measurements from 500mm to 1000mm as just demonstrated.When all of the incident points have been measured, repeat the measurement with the acrylic plate of the next largest diameter. The excitation frequency that can excite an axisymmetric Chladni pattern is determined through the frequency sweeping test. A larger plate diameter correlates with a higher plate flexibility.Under the corresponding resonant frequency, the length of the light spot on the light screen of different plates can be measured and recorded. Plotting this data allows comparison of the mode shapes of different plates. The most important aspect of this procedure is obtaining accurate data when measuring the spot length, as this data determines the accuracy of the entire experiment.This method can also be used to determine non-axisymmetric mode shapes in word minor, to diamonds trace the beauty of Chladni patterns.

Measurement of Chladni Mode Shapes with an Optical Lever Method

Abstract