## Demystifying Modal Analysis (Part I)

In this article, I will discuss about modal analysis – a topic that is standard, however I’ll strive to demystify it using a simple example and FAQs.

###### Motivation for Modal Analysis

As a mechanical engineer, life is always interesting because I can correlate the knowledge gained from books to real life scenarios. As a student, my professor gave a real example of a bridge failure due to marching soldiers. What followed was a very interesting lecture about dynamics. Until then, I never understood the power of the words such as dynamics, vibration and resonance. Of course, the example provided food for my thoughts to study more about how a bridge could fail due to lesser dynamic load compared to a heavier static load.

For those of you who are curious, the bridge was England’s Broughton Suspension Bridge that failed in 1831 due to the soldiers marching in step. The marching steps of the soldiers resonated with the natural frequency of the bridge. This caused the bridge to break apart and threw dozens of men into the water. Due to this catastrophic effect, the British Army issued orders that soldiers while crossing a suspension bridge must ‘break step’ and not march in unison.

Such failure has given rise to more emphasis on analyzing the structure (mechanical or civil) for dynamic loads if it undergoes any sort of vibrations. Traditionally test equipment have been used to experimentally monitor vibrations in new designs; this is costly however. We apply finite element analysis (FEA) to solve such problems. FEA solvers have evolved and today’s solvers are powerful not only in statics but in dynamics too.

###### Modal Analysis: Getting Down to the Basics

In any dynamic/vibration analysis, the first step is to identify the dynamic characteristics of the structure. This is done through a simple analysis called Modal Analysis. Results from a Modal Analysis give us an insight of how the structure would respond to vibration/dynamic load by identifying the natural frequencies and mode shapes of the structure.

Modal Analysis is based on the reduced form of dynamic equation.

As there is no external force acting and neglected damping, the equation is modified to:

I have skipped the derivation part of natural frequency as it is easily available in textbooks. Natural frequency is substituted back into the equation to find out the respective mode shapes. These natural frequencies are the eigen values whereas the respective mode shapes are its eigen vectors. Natural frequencies & mode shapes in combination are called as modes.

Eigen vectors represent only the shape of deformation, but not the absolute value. That’s the reason it is called as mode shape. It is the shape the structure takes while oscillating at a respective frequency. Important point to remember is the structure has multiple modes and each mode  has a specific mode shape. If any load is applied with same frequency as natural frequency in the same direction as mode shape, then there will be increase in magnitude of oscillation. With no further damping, the scenario will lead to a failure due to a phenomena called resonance. To avoid this phenomena in dynamics, calculating the modes carries great importance.

Having said that, questions will certainly arise. In my opinion, these are the most commonly asked questions in support calls by customers using ANSYS.

• Why do frequencies from simulation don’t match the test results?
• Why are deformation and stresses in modal analysis very high?

From equation (3), it is clear that natural frequency of structure depends on its stiffness and mass. In order to accurately capture frequencies in FEA, the following points are important for you:

• You need to capture mass of the structure and connecting/ignored members accurately.
• Your mesh can be coarse, but enough refinement so that you can accurately capture the stiffness of the structure. If you are interested in the local modes in slender members, then you’ll need to perform local mesh refinement.
• You need to define appropriate boundary conditions in forced modal analysis in order to capture realistic frequencies.
• You need to accurately model the contact between different bodies in an assembly since they affect the stiffness of the structure drastically.

For the second question, a lot of confusion exists when the modes extracted in modal analysis show deformation magnitude. In Equation (2), you will see that no external load is applied on structure. This will make you wonder where these values come from? Let’s have a look with an example of simple cantilever beam.

Fig. 1 shows its extracted mode shape 1 & stress shape 1 from modal analysis. I observe deformation to be as high as 253 mm and stress as 4,914 MPa which is far greater than the ultimate strength of Steel i.e. 500 MPa. You may wonder, why did we get these high values?

This happens because the FEA solver returns the mode shape (not the deformation magnitudes) as output. By this, I mean that magnitude of the mode shape is arbitrary (as seen in Fig. 1). The high value is because of a scale factor that’s chosen for mathematical reasons and does not represent anything real for the model. However this value helps us in relative measurement. Let’s take the example of the first mode. Maximum deformation occurs at the free end compared to any other location. This changes with the change in mode.

Since we have deformation, you can compute corresponding stresses and strains. Once again, these are relative values. If you ask the FEA solver for stresses & strains, it will use the same scaled deformation magnitudes and calculates stresses & strains. They are referred to as stress shape & strain shape (not to be confused with stress state or strain state) because no loads are applied. The magnitude of stresses and strains are useless but their distributions are useful to find hot-spots in the respective modes.

###### Conclusion

Modal analysis offer much more than just the frequencies and mode shapes. This analysis is primarily the stepping stone for linear dynamics studies to calculate the actual deformation due to different kinds of dynamics loads. Modal analysis has many secondary applications which I will discuss in my next blog.

## ANSYS 18.0 – The latest release is here!

The day has arrived!! Most of our customers would’ve received announcement of the latest release – ANSYS 18. It’s time to rejoice and celebrate this new release.

ANSYS 18.0 ushers in the era of pervasive engineering simulation – an era where all types of engineers use simulation throughout the entire product lifecycle. While simulation was once the primary domain of experts and used mainly for verification, it is now moving upfront in the development process to quickly evaluate changes in design. At the same time, it’s also moving downstream of the product lifecycle process to analyze real-time operational data from connected machines in the industrial internet.

By incorporating simulation into all segments of a product’s lifecycle, ANSYS 18.0 adds tremendous value by spurring innovation, reducing development and operational costs, and improving time to market. Whether your field is structures, fluids, electromagnetics, semiconductors, systems, embedded software or some multiphysics combination of these areas, ANSYS 18.0 is the simulation platform to achieve your engineering and business goals.

###### Digital Launch Event

To learn more about pervasive engineering simulation and ANSYS 18.0, we invite you to the digital launch event on January 31. During this event, the ANSYS CEO, Ajei Gopal, and numerous technology leaders from industry will talk about this exciting new release from ANSYS. Here’s the line-up of the speakers from ANSYS:

• Ajei Gopal, President and CEO, ANSYS, Inc.
• Andre Bakker, Senior Director Fluids Development, ANSYS, Inc.
• Dale Ostergaard, Senior Director Software Development, ANSYS, Inc.
• Larry Williams, Senior Director Electronics Products, ANSYS, Inc.
• Sergey Polstyanko, Senior Director, Research and Development, ANSYS, Inc.
• Eric Bantegnie, Vice President and GM Systems Business Unit, ANSYS, Inc.