Measurement of Chladni Mode Shapes with an Optical Lever Method

1. Experimental setup and procedures

NOTE: Set up the experimental system as shown in Figure 2.

1. Preparation of the vibration system
   1. 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.
   2. Attach each plate to the actuate bar of the vibrator with a bolt in the middle point. Drive the vibrator with a sine wave using a waveform generator, and default settings will be enough for the resonance experiment.
      NOTE: The excitation direction of the vibrator is horizontal for the convenience of moving the screen afterwards.
   3. Acquisition of the resonance frequency
      1. Place the laser pen to project the laser beam to the vibrating plate perpendicularly such that the beam is reflected to the light screen in the distance. The distances between the laser pen and the plate and the light screen are 120 mm and 500 mm, respectively.
         NOTE: The farther the distance between the light screen and the vibrating plate, the more obvious the phenomenon appears. It is also noted that the present method can be used to measure either axisymmetric or non-axisymmetric mode shapes. Due to the consideration of simplicity and convenience, the present manuscript only demonstrates the application in determining axisymmetric mode shapes of three circular plates. Then we just need to measure the vibration amplitude along any radial direction to reconstruct the two-dimensional mode shape of the plate.
      2. Move the laser pen along the direction perpendicular to its length direction to make the incident point scan over a diameter while the signal generator changing its frequency continuously. Do it quickly until the spot length is significantly stretched along the diameter when scanning in a certain frequency range, and some spots with almost no expansion appear. For the plate with a diameter of 150 mm, 200 mm and 250 mm, the frequency ranges swept are 200-400 Hz, 100-300 Hz and 50-250 Hz, respectively.
      3. Scan this certain frequency range slowly and pick out the frequency at which the spot expands most obviously. It is found that for the plate with a diameter of 150 mm, 200 mm and 250 mm, the resonance frequencies are 346 Hz, 214 Hz and 150 Hz, respectively.
2. Preparation of the light path and measurement system
   1. Place the light screen parallel to the vibrating plate. Mark the distance with a meter ruler, and use 500 mm as the starting distance.
   2. Place the laser pen to project the beam perpendicularly on the plate such that the beam is reflected to the light screen in the distance. Make sure that the mark made before can be scanned while the laser pen is moving.
      NOTE: The laser beam light must be projected perpendicularly on the plate.
3. Experimental measurement
   1. Turn on the signal generator and set the excitation frequency to be the same as the resonance frequency obtained in step 1.1.3.3. The signal intensity should as small as possible once the light spot on the light screen is large enough to be recorded.
   2. Adjust the laser pen to make the incident point coincide with the first marker, which is the nearest marker to the fixed point of the plate.
   3. Move the screen from a distance D of 500 mm to 1000 mm and measure the spot length L on the screen every 50 mm. Record data in tabular form.
   4. Adjust the laser pen to make the incident point adjacent to the next marker in turn and repeat step 1.3.3 until all the markers have been measured.
      NOTE: Since acrylic plates are easily deformed plastically under excitation, the experimental measurement process of one plate cannot be paused for a long time.
   5. Replace the former plate with the next one and repeat steps 1.3.1 to 1.3.4.

2. Data processing

1. Determine the angle θ between the incident and reflected light with relationship:
   
   where D is the distance between the rest position of the vibrating plate and the light screen, w is vibrating amplitude of the plate, and L is the length of the light spot on the light screen. Several pairs of D and L are obtained in step 1.3.3.
2. Determine the slope of the mode shape by:
   
   NOTE: The obtained slope is always positive with Eqs.(1) and (2).
3. Use a minus sign between two zero points to obtain the true slope distribution.
   NOTE: It does not matter whether the revision begins from the first or the second zero point.
4. Integrate the slope distribution of each plate and determine the integral constant by the nodes to obtain the mode shape with:
   
   NOTE: Nodes correspond to the largest slope of the mode shape. is a constant determined by the location of the nodal lines of the Chladni pattern shown in Figure 2.
5. Compute the uncertainty of the slope¹² with:
   
   NOTE: t_0.95(n – 2) is the t distribution factor with 95% confidence and degrees of freedom n-2, and it is about 2 here. S_r is the standard error of the linear regression with D and L, U_m denotes the uncertainty of the measured distance D_i, and is 0.5 mm here. The average measured distance is defined by , and n denotes the total number of measured D_i.