Modal Frequency Prediction of Chladni Patterns Using Machine Learning ()
1. Introduction
Ernst Florens Friedrich Chladni was a Creat pioneer in the science of acoustics. His studies explored torsional vibrations of pendulums and longitudinal vibrations of strings and rods, and the latter was applied to determining solid-state sound velocity. Additionally, he filled an organ pipe with media other than air to measure sound velocity in those media and taking the pitch of the note produced. He used brass plates clamped at the center and excited by bowing at one end, and sprinkling sand on the vibrating plates, he obtained “Chladni’s figures” [1]. For several centuries, study of acoustics has been conducted. Simultaneously, studies have been conducted to find out correlation between mechanical, harmonical sound wave oscillation and appearance of visible patterns on physical objects has been carried out. Therefore, Cymatics was created. Cymatics deals with studies of visible sound, vibration and physical phenomena including visible sound [2]. Application of Chladni plate is to demonstrating nodes and antinodes point at various frequencies of vibrations. In response to vibrations, the sand on the plate produces specific patterns when it is excited by a vibrating source. These patterns are belonged to the excitation frequency. The sand on the plate moves away from antinode and move towards nodal lines. Node is the point where amplitude of vibration is maximum. Antinode is the point where amplitude of vibration is minimum or zero. Sands on plate create specific patterns. These patterns are known as Chladni figures. Mode shapes are dependent on material of plate, dimension and boundary condition [3]. As a technique for analyzing structures, force hammer is also used as an excitation source to generate vibration modes [4]. Chladni experimented with square and round plates with other loose materials. A visible figure appears when touching the plates with a bow and then touching them in another place with a finger. When sound waves propagate across the plate, loose material on the surface starts moving. Chladni figures are characterized by where the bow and finger touch the plate, as well as the size and shape of the plate. The figures were named after Ernst Chladni, who conducted much of the research on their formation. It was Ernst Chladni who introduced the term “sound figures” [5]. In similar conditions, square or round plates produce different results. Considering the physical properties of figures, the plate begins to vibrate as the sound waves spread due to the action of the bow. Sound waves cause the sand to move from places where the oscillations are more pronounced and then collect in nodal lines. A process by which waves are added, called interference [6]. It is possible to observe what effect standing waves (interference) have on different materials using Chladni figures. As a result, a visible pattern of the energy of wave oscillations can be analyzed [7]. Image is one of the most important information sources of human intelligence activities, as it is the primary medium of human communication and understanding of the world. With the time, the Scope of image processing technology is increasing day by day. In image segmentation, images are divided into
regions that have different characteristics and then useful targets are extracted from each region. The application of machine learning algorithm in image processing can obtain better image processing effect. Based on this background, the image segmentation is implemented by ant colony algorithm. The reason for this is that image segmentation is a combination of optimization problems, and ant colony algorithms have good application optimization solutions [8]. Analytical and quantitative comparisons can be performed using this algorithm at low computational costs (real-time) and high efficiency. The parameters used for comparison are the degree of blurriness and the amount of noise associated with the surface image. A heuristic analysis of these parameters is used to evaluate the quality of the surface image by characterizing the major aspects of human vision. It is the purpose of this research to provide image processing tools for comparing and assessing surfaces processed under different grades of a manufacturing process all the way to optimal [9]. In order to achieve the aim of identifying images, image processing and recognition is done on the actual image transformation. Because of the characteristic of the image information is that it is a two-dimensional space, so the amount of information it contains is very large. The neural network image recognition technology is a new kind of image recognition technology made possible by the combination of modern computer technology, image processing, artificial intelligence, and pattern recognition theory. Before the image recognition need to use digital image processing techniques for image preprocessing and feature extraction. The use of neural networks in image pattern recognition research has become increasingly active because of artificial intelligence theory and computer technology development [10]. Object classification is the critical task in computer vision applications. It is basically a combination of Image Classification and Object Localization. Traditional ways of object classification extract features or descriptors for different classes first and then use classifiers to classify objects. There are several deep-learning models which can be deployed to classify the objects. This includes techniques like R-CNN, Fast R-CNN, Faster R-CNN, YOLO, YOLOv2 and YOLOv3 [11]. The R-CNN deep-learning model is a relatively simple and straight forward application of CNN. .This method combines classical tools from computer vision and deep learning (bottom Using this method, you can combine classic computer vision methods and deep learning techniques (top-down region proposals and convolutional neural networks) and achieve a 30% relative improvement over traditional methods. This performance was achieved through two insights. The first thing that needs to be done is to apply high-capacity convolutional neural networks to bottom-up region proposals in order to localize and segment objects. Secondly, there is a paradigm for training large CNNs in the absence of labeled training data. The network is highly effective when pre-trained with supervision for an auxiliary task where the data is abundant (image classification), and then tuned for the target task where the data is scarce (detection) [12]. Fast R-CNN was proposed as an
extension to the R-CNN model. The research work addresses the speed issues of R-CNN, and summarizes the limitations as follows: It is a multistep pipeline training that is time-consuming and expensive to train, as well as slow in terms of object detection. Make predictions using a deep CNN on so many region proposals is very slow. The model architecture passes the image through a Deep CNN. A pre-trained CNN, such as a VGG-16, is used for feature extraction. The final layer of this Deep CNN network is a custom layer called Region of Interest (ROI) Pooling Layer. This helps in reducing the computation time of the system [13]. The Regional Proposal Network, or RPN, is a faster R-CNN architecture designed to refine region proposals as part of the training process. Improvements reduce the number of region proposals and speed up the model’s test-time operation to near real-time, achieving state-of-the-art performance. This obstacle detection system, called Faster R-CNN, is composed of two modules. In the first module, there is a deep fully convolutional network that proposes regions, and in the second module, there is a Fast R-CNN detector that makes use of those regions. The entire system works as a unified network to detect objects [14]. Generally, the R-CNN models are more accurate. However, for real-time applications, faster models are preferred. This is where YOLO proves to be the favorable option. In this model, each cell predicts a bounding box if the center of a bounding box lies within it based on the input image grid. Each grid cells predict bounding boxes based on their x, y coordinates, widths, heights, and confidence margins [15]. The current work involves the experimentation of modal analysis of Chladni plate and extracted mode shapes in FEA for validation. Further, captured the large number of images with camera for machine learning to predict the modal frequency based on pattern formed on plate.
2. Concept of Image Processing
2.1. Image Processing and Data Directory
High-level image processing is the act of modifying or extracting information from digital images. Depending on the application, this technique can be used to simply adjust contrast, remove noise, or find edges. With the combination of multiple techniques, one can design algorithms which perform increasingly complex tasks. By processing all steps like Import, Pre-process, Segment, Post-process, Classify, the entire image classification process can be obtained.
2.1.1. Import
For image processing, the first step is to import images into the workspace memory, so that you can use those images to achieve your desired outcome. To develop a robust classification algorithm, we need a large number of images to analyze. As shown in Figure 1, Chladni pattern at frequency 900 Hz is imported for machine learning algorithm.
Figure 1. Chladni pattern at frequency 900 Hz.
2.1.2. Preprocessing
When we store image in memory it stored as numeric array. Numeric array is stored in pixels and intensity values. A grayscale image occupies one third the memory space of an RGB image once loaded into memory. Due to the fact that greyscale images contain only one third of the data, they can be processed more quickly and require less computational power. Image processing algorithms are easier to develop with grayscale images than with RGB images. In Chladni pattern images shown below, little information is lost when they are converted to grayscale. There is no loss of essential characteristics, such as the dark line and bright steel plate. As shown in Figure 2, Gray scale image for chladni pattern at frequency 900 Hz.
Figure 2. Gray scale image of 900 Hz frequency pattern.
2.1.3 Segments
One method is to separate the pattern lines from the background. An image can be segmented by separating it into different parts. It can be by many ways like bounding boundary, colour, planes, patterns. The final goal of segmentation is to identify regions of interest. Here we can neglect background by masking for E.g. Indicate area of interest as 1 and other as 0. We provide us only selected view of image in same height and width of pixels. By doing this, we will be able to calculate, change the background, and identify patterns for frequency identification in our case. As shown in Figure 3, Canny edge image for Chladni pattern at frequency 900 Hz.
Figure 3. Canny edge detected image of 900 Hz frequency.
Intensity Thresholding—By thresholding the intensity values of a grayscale image, you can convert it to a binary black and white image. In the case of values below the cut-off, the value is 0, while in the case of values above the cut-off, the value is 1. A grayscale image was segmented using a threshold of certain value of the maximum possible intensity of 255.
2.1.4. Post Processing
Noise Removal—Smooth pixel intensity values to reduce the impact of variation on binarization. To avoid the effect of mismatching pixel intensity with neighbor pixels to smoothen this effect technique is used called spatial Filtering. Useful for blurring edges Background Isolation and Subtraction, Isolate and remove the background of an image before binarizing.
2.1.5. Classify and Batch Processing Data Store
When we have huge data base then it is difficult to process each image, so data store is helpful.
2.2. Training of Data
For machine learning, a large number of data sets are required. Captured around 200 photos of each modal frequency and captured over all-around 3000 photos with a camera of all 14 Chladni patterns for supervised learning.
2.2.1. Neural Network Layers
To create agents, we need to create neural networks to represent actors and critics. Build these networks by specifying the layers that comprise them. A deep layer is made up of various layers in which each layer receives input from previous layer which performs an operation on input and pass on output to next layer for actor or critic networks which are value-based networks, typically the hidden layers are fully connected layers or activation function layers (as shown in Figure 4). A fully connected layer consists of several neurons. Each neuron is a mathematical operation multiplies the input by a constant (called the weight) and adds another constant (called the bias). The resulting value is then passed to next layer.
Figure 4. Neural network layers.
In a fully connected layer, each neuron takes input from every neuron in the previous layer and passes the output to every neuron in the next layer. Each neuron has many weights as well as activation function layers that introduce nonlinearity [16].
Some commonly used activation functions are the rectified linear unit (“ReLU”), in which positive inputs are left unchanged, whereas negative inputs are set to zero, and the hyperbolic tangent is used to smoothly map all inputs into the range [−1, 1].
2.2.2. Network for 1D Neural Network
In this activity, to make a neural network to represent a function of a single scalar value. With one scalar input and one scalar output, you can visualize the behavior of this network like any function y = f(x). This network takes a single value (x) as input.
This value is passed to the two neurons in the second layer. Each neuron multiplies x by a weight and adds a bias. The two output values are then passed to the activation function layer that applies the tanh function to them. These two values then become the inputs to the last fully connected layer. This layer has a single neuron which takes the two inputs, multiplies each by a weight and adds the results plus a bias. This value is the output of the network. This network therefore represents a function f(x) that is defined by seven parameters (four weights and three biases). When you first create the network, these parameters are undefined. When used as the representation of an actor or critic in an RL agent, the parameters are determined by training the agent. However, in this activity, you will set the parameters manually to see how their values determine the shape of the function f(x). A network to represent a stochastic actor takes the observation as input. Toward the end of the hidden layer, there is a fully connected layer that contains one neuron for every possible action. The final layer is softmax function that turns the raw values from the previous layer into probabilities. Usually action is equal to number of neurons [13] (as shown in Figure 5).
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Figure 5. Input, Output and Hidden layers.
3. Methodology
3.1 Experimental Setup
Figure 6. Experimental setup for modal analysis.
As shown in Figure 6, performed modal analysis on experimental setup which consists of sine wave generator (Ningbo Hema Scientific), mechanical oscillator (GelsonLab HSPW 003), amplifier, rectangular plate and camera with tripod. After performing modal analysis, obtained 14 mode shapes for changing frequencies from 0 to 4000 Hz. These mode shapes are known as Chladni patterns.
3.2. Simulation Setup
To valid the experimental result, performed simulation work as modal analysis in Ansys software. First created solid plate as per given dimension in SpaceClaim as shown in below figure. Then performed hexahedral meshing on chladni plate. Used of ‘O’ grade approach to create high quality mesh in SpaceClaim meshing. Element quality of all generated elements are greater than 0.20. As per mesh quality standard acceptable range for element quality should be lie between 0.20 to 1.00 (as shown in Figure 7).
Figure 7. SpaceClaim Cad Model, Meshing model & Element quality.
For performing modal analysis, requirement of material properties in Ansys mechanical. Material properties are as shown in Table 1.
Table 1. Material properties for aluminum.
Material: Aluminum |
Property |
Value |
Youngs Modulus |
71,000 (MPa) |
Poisson Ratio |
0.33 (Unitless) |
Shear Modulus |
26,692 (MPa) |
Bulk Modulus |
69,608 (MPa) |
Density |
2770 (kg/m3) |
In modal analysis, extracted sufficient number of mode shapes corresponding natural frequencies of rectangular Chladni plate. Here 14 mode shapes from modal analysis are relevant to experimental mode shapes for validation purpose.
3.3. Supervised Machine Learning
Performed modal analysis experiments on Chladni plate setup and obtained various mode shapes at various frequencies. For machine learning, Need of a large number of data sets. Captured around 200 photos of each frequency and captured over all-around 3000 photos with a camera of all 14 Chladni patterns for supervised learning.
Our data having special relationship between frequency and modal pattern so, for this problem statement CNN is suitable. The algorithm used for this is CNN type. A convolution neural network algorithm is designed for recognizing two-dimensional image information as a multilayer perception. The input layer, the convolution layer, the sample layer, and the output layer are always present. Furthermore, the convolution layer and sample layer can have multiple layers in a deep network architecture. There are two processes involved in CNN algorithms: convolution and sampling. Convolution process: trainable filter Fx, deconvolution of the input image (the input of the convolution after deconvolution is each layer’s feature image, namely Feature Map), then add a bi as bx, we get the convolution layer Cx. Three main steps are involved in image processing: Importing the image via image acquisition tools, Analyzing and manipulating the image, and Predicting the result based on analysis of the image.
In starting, every modal pattern image of particular frequency is taken from camera is acquired after getting image cropping and other parameters like pixels are adjusted so that all sample images will be in same pattern. Now, our data directory is ready we need to organize it by frequency make each frequency as one class put all images in that in this manner, we have 14 classes as we are working on 14 frequencies. We have done all coding and analysis in jupyter using python in anaconda. After completing the prerequisite of data directory the code to form data directory which is as follow, first import all required directories from it imported libraries are numpy to convert image into arrays ans matplotlib to plot arrays ,os to get access of device os, cv is computer vision to read all images, tqdm used perform over larger loops. Now specify each frequency category and index it. Each image is iterated over each category and converted into an arrays and grayscale each image is converted into grayscale to large avoid space consumption. Now all new images are stored in data directory by appending them. To avoid overfitting, we need to shuffle images. Last stage is pickling, here we need to dump images so, every time when we call directory whole data will show up to avoid this pickle dump is used. After this our directory is formed. After this we need to train and test the model so it will fulfill our desired output of predicting frequency of particular images pattern. For this we need to import new libraries like tensorflow and keras this are open neural network platform which helps to form layers which we have discussed in CNN part. Now after this need to set gpu speed so it won’t consume all our gpu memory. Previously we have dumped images now we need them so pickle load is used so, we will get all our data. Sequential function used to activate the layers like input hidden layers and output layer. Here we can set convolution filters as per our requirement we have used conv2D 64 and kernel size 3 × 3 which specify the height and width of tupel. ReLu is used to read small gradient and number of hidden layers used is two, now all layers are flattened and pooled together using maxpooling for this softmax is used is probability function to smoothen the model. We are forming such a layers there is always some data loss to compensate that adam optimizer is used, to improve accuracy canny edge detection is used. Now at the end provide predication function so, it will predict the resemblance of testing image with reference images and it will provide prediction probability for each class. Now which class having highest resemblance, we can say that is frequency of that test image.
4. Results and Discussion
Validation of experimental work is done with simulation work which is performed in Ansys Mechanical. Meshing on CAD model is performed in SpaceClaim meshing. 14 mode shapes are formed on Chladni plate with the experiment. To validate experimental work, extracted 14 mode shapes in modal analysis using Ansys Mechanical. For supervised machine learning, collected a large number of data sets. Chladni patterns, which are captured with a camera, are converted into grayscale images and canny edge detected images for image recognition. Training of data and testing of data is done for modal frequency prediction of chladni patterns.
In supervised machine learning, created data set and labeled. We have to train the model and testing of data is required. Finally, prediction of frequency is done based on input shape of chladni pattern. As shown in below figure, image processing basically includes the following three steps: Importing the image via image acquisition tools, Analyzing and manipulating the image and Prediction in which result can be altered image or report that is based on image analysis. Accuracy of machine learning program drastically increases by adding edge detection algorithm after grey scale algorithm. For machine learning program required a greater number of data as we captured around 3000 images in 14 classes for machine learning. For training of each frequency, captured around 200 images. For testing of each frequency, no. of images is required 5 as shown in Table 2. Comparative studied among experimental, simulated, canny edge and gray scale chladni images as shown in Figure 8.
Table 2. Data sets for training and testing of machine learning algorithms..
S.N. |
Class (Frequency) |
No. of Images for Training of Data |
No. of Images for Testing of Data |
1. |
340 |
200 |
5 |
2. |
660 |
200 |
5 |
3. |
760 |
200 |
5 |
4. |
900 |
200 |
5 |
5. |
1000 |
200 |
5 |
6. |
1450 |
200 |
5 |
7. |
1770 |
200 |
5 |
8. |
1940 |
200 |
5 |
9. |
2540 |
200 |
5 |
10. |
3250 |
200 |
5 |
11. |
3360 |
200 |
5 |
12. |
3430 |
200 |
5 |
13. |
3480 |
200 |
5 |
14. |
3910 |
200 |
5 |
Figure 8. Experimental, Simulation and Machine learning mode shapes.
Figure 9. Flow chart for supervised machine learning.
Table 3. Test frequency Vs Probability of frequency predication.
Modal Frequency for Chladni plate |
Probability of Frequency Prediction (%) |
330 Hz |
0.3342 |
660 Hz |
0.0966 |
760 Hz |
0.3922 |
900 Hz |
0.1494 |
(1000 Hz) |
(95.333) |
1450 Hz |
0.0728 |
1770 Hz |
0.00002 |
1940 Hz |
1.0264 |
2540 Hz |
0.0549 |
3250 Hz |
0.0267 |
3360 Hz |
2.1363 |
3430 Hz |
0.0281 |
3480 Hz |
0.000006 |
3910 Hz |
0.3467 |
As shown in Figure 9, 14 classes of Chladni patterns were captured and the frequency for each class was labeled. Around 3000 images were captured for data training, and 200 images were captured per frequency. For testing, any random image will be imported which is not used during training of machine learning algorithm. Finally, Machine learning algorithm will predict the resonant frequency corresponding Chladni pattern. Frequency prediction will be in the term of probability as shown in Table 3.
As shown in Table 3, we can say that frequency of imported chladni pattern will be 1000 Hz as the probability for frequency 1000 Hz is so much higher than other frequencies, i.e., the probability of frequency prediction is 95.333% for frequency 1000 Hz.
As shown in Table 4, for checking the accuracy of machine learning program, inserted 5 random images of particular frequency and check the accuracy. After that took the average of accuracy for 5 random images of particular frequency, as shown in Table 2. Same sequence followed for 14 frequencies. Draw the graph between modal frequency and accuracy of machine learning algorithm as shown in Figure 10.
Table 4. Modal frequency Vs Accuracy of machine learning algorithm.
Modal Frequency (Hz) |
Accuracy of Machine learning Algorithm (%) |
330 Hz |
85.78 |
660 Hz |
96.62 |
760 Hz |
84.76 |
900 Hz |
95.34 |
1000 Hz |
95.33 |
1450 Hz |
84.11 |
1770 Hz |
88.11 |
1940 Hz |
94.96 |
2540 Hz |
83.87 |
3250 Hz |
83.82 |
3360 Hz |
87.98 |
3430 Hz |
86.21 |
3480 Hz |
92.89 |
3910 Hz |
97.02 |
Figure 10. Graph b/w frequencies of mode shapes Vs Accuracy of machine learning.
Calculated the accuracy of machine learning algorithm for all 14 modal frequencies.
Achieved the accuracy of machine learning algorithm more than 80 percentage for all 14 modal frequencies as shown in above graph. Highest machine learning algorithm accuracy for modal frequency 3910 Hz is 97.02% and minimum machine learning algorithm accuracy for modal frequency 3250 Hz is 83.82 percentage.
5. Conclusion
In this paper, Chladni Patterns are nodal patterns that are formed on plates when an external source causes to plate vibrate. Chladni patterns are used to visualize the sound. There are two key parameters of standing waves: nodes and antinodes. During oscillation, fixed points are referred to as nodes of the standing wave and correlate with the antinodes, which are the highest amplitudes of the oscillation. For a rectangular Chladni plate, 14 experimental mode shapes are obtained within the frequency range of 0 to 4000 Hz. As the frequency value increases, the complexity of mode shapes increases. A large data set is needed to improve the accuracy of machine learning algorithm, as approximately 3000 photos in 14 classes were collected. Further accuracy of machine learning algorithm increased with adding canny edge detection algorithm after gray scale algorithm. In all 14 modal frequencies, the machine learning algorithm’s accuracy exceeds 80%. On the basis of the patterns formed on the Chladni plate, a machine learning algorithm can be used to predict the modal frequency. This is the inverse approach or reverse engineering in the field of cymatics.
Acknowledgements
The authors would like to sincerely thank to Dr. Sanjay Patil (ARAI, Pune) for providing valuable support.