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How to Calculate Natural Frequencies and Mode Shapes of a PZT Disc in OnScale

By Cyprien Rusu 18 July 2019

If you are familiar with OnScale, one of the things that you may be interested in learning is how to calculate natural frequencies of your model.

The real world works in a time domain.

You might be used to doing modal analysis, but it’s actually a mathematical abstraction.

With OnScale you’re able to obtain frequency domain results in exactly the same way that they’re obtained experimentally.

Here’s how:

We can calculate the natural frequencies with OnScale using Time Response and Fast Fourier Transformation (FFT) !

How to calculate Natural frequencies and mode shapes of a PZT Disc in OnScale?

In this video, you will learn:

  • How to calculate the natural frequency of a PZT Disc using FFT in OnScale
  • How to view the mode shapes

 The full step by step tutorial to build this model can be found here

Explanation of the process to calculate the Natural frequencies and mode shapes in OnScale

The general process to extract modal behavior is as follows:

Modal –> Dynamic Time Response –> Monitor Acoustic Pressure at Maximum Pressure Point –> FFT of that Time History Acoustic Response Curve –> Frequency Response Curve –> Natural frequencies of vibration


  1. Run time domain simulation generating a time history of the outputs you are interested in
  2. FFT time history to create spectrum
  3. Spectrum allows user to identify, modes of operation
  4. Mode shapes use this same process but apply the FFT across all points in the model, creating magnitude and phase information for each point. These can be viewed as a movie to understand device behavior under CW operation

There are also advantages of doing it this way ‘true’ modal analysis does not handle fluids/absr boundaries very well – if at all.

‘True’ modal analysis also doesn’t consider the effects of damping and mode coupling. It will show where there might be modes, but in practice they might not be accurate, and in some cases are cancelled out.

Here’s an example of a PZT Disc Model in Water (2D Axi-symmetric):


This is a simple a 2D PZT Disc Axi-symmetric model. Our full tutorial on how to build this can be found here.

The full 3D model deformed shape

A PZT Disc is a disc made of lead zirconate titanate which is a piezoelectric material that deforms mechanically under a voltage load. When a forcing function with a certain time distribution is applied to the electrodes of that disc, it deforms according to that voltage and generates acoustic waves into the medium around it.

We applied a voltage forcing function at the top and bottom of the electrodes, which has a time distribution called “Ricker Wavelet”:

This kind of forcing function allow us to analyze the modes which are close to the main drive frequency of 1 MHz within a certain band of frequencies.

The Corresponding frequency forcing function would look like this:

An interesting thing to note is that by adding some subwavelets it would make that forcing function more narrow-band.

There are 2 electrodes around this PZT Disc:

Natural Frequencies

We monitored the acoustic pressure time history at the same position of this model:


This is the response we get after computing the model on the cloud:

Natural Frequencies


This is a Time Response, so we need to do an FFT (Fast Fourier Transformation) to get the frequency response of our result signal.

You can find the FFT function in the OnScale post processing module main menu toolbar:

After clicking on this “FFT Record Button”, the following 2 FFT transformation records will appear in the Results Manager under the new tab “Frequency History”:

Natural Frequencies

Aprs.ma gives the amplitude response
Aprs.ph gives the phase response

In order to visualize correctly those FFT curves, make sure you reset the current viewport:


Then double click on aprs.ma

You will then get the frequency response amplitude curve:

Natural Frequencies


You can zoom in on it to obtain the exact natural frequencies of vibration and their response:

Natural Frequencies


You can then right click and export those data to CSV if you want to process those results in another software such as Excel for example.

About the Mode Shapes

For more details on the mode shapes you can visit our Extracting and viewing mode shapes in OnScale tutorial. 


To extract the natural vibration frequencies from an OnScale simulation, you can post-processing the time response results using the Fast Fourier Transform (FFT) function which is embedded in OnScale Post process environment.

Why not try this process yourself to understand how it works and if you have questions, do not hesitate to leave them in the comments section or contact sales@onscale.com.


Get Started With OnScale Today!

Cyprien Rusu
Cyprien Rusu

Cyprien Rusu is our Director of Engineering for Asia at OnScale. He has a extensive background in FEM, Technical Marketing, Sales and Support. Cyprien recieved his MS in Civil Engineering from Tsinghua University. At OnScale he is a trusted advisor for our client base in Asia, while creating and providing OnScale training.

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