## 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.

## Electromagnetic Simulation for Antennas

In this first part of a multi-part series, I will discuss many aspects of antenna design & analysis with the underlying theme of electromagnetic simulation-driven product development. In this part, I will briefly talk about performing stand-alone Antenna Design, Analysis & Optimization using Electromagnetic Simulation.

###### Increasing Importance of Electromagnetic Simulation (EM)

While still in university, I imagined antenna design to be very simple. Based on the given frequency, we will need to calculate dimensions and then fabricate the design. That’s it. A decade ago, I found simulation to appear like dark art or black magic. If the fabricated antenna did not work well, I needed to iterate the physical design till it gave good results.

During the recent years, several EM simulation tools have emerged to evaluate the exact solution of Maxwell Equations for estimating the electromagnetic behavior of the devices. These tools used underlying methods like Finite Element Method (FEM), Method of Moments (MoM) and Finite Difference Time Domain (FDTD). Generally, we can divide the part components of the electronic design into active and passive devices. The modelling of the active devices is based on nonlinear measurement data parameters like S-parameters and X-parameters. When we come across modelling of passive devices, they are very simple because of their linear nature. However, it is important to understand the limitations of those devices.

The main role of the simulation is for to engineers to be able to accurately predict how complex products will behave in real-world environment enabling the complete virtual prototyping. ANSYS HFSS, a state-of-the-art high-frequency electromagnetic simulation, helps to estimate the radiation characteristics of the antenna and optimize the design as per requirement.

###### Parameters To Be Considered For Antenna Simulation

In general, engineers know that dimension can be reduced by increasing the substrate dielectric constant. Using standardized equations, we can estimate the size of the patch. However we cannot estimate radiation characteristics among a few other quantities. Using simulation tools, we can replace physical iterations with virtual iterations; we can identify the optimal design that matches the required specifications.

Why do some engineers get different results? Is there anything else that needs to be considered? Yes, engineers who focus only on model dimensions and not on boundaries and excitation will obtain inaccurate results.

###### Modeling and Setup

Let me consider the example of a GPS antenna that needs to have a gain of 3.5dB. For this gain, we’ll need to identify a antenna design with the smallest possible antenna dimension.

Let’s look at three substrates RT Duriod 5880, FR4 and Alumina. Using ANSYS HFSS, you can model the full antenna by using in-built modeling options or import the design from external CAD software. Initial dimensions of the patch are calculated using standard formulas available in academic literature.

Patch antennas can be fed power by various methods such as microstrip line or coaxial/SMA. While using coaxial input, many don’t consider the dimensions of the coaxial. A good engineer initially checks for the dimensions of the coax in order to get the characteristic impedance, which directly affects the frequency of operation and voltage standing wave ratio or VSWR.

For assignment of different materials for model, HFSS has an inbuilt material library where you can select the required material for substrate, conductors, etc. If you want to use a material which is not in the library or if you want to add some frequency-dependent properties, then you can modify or create a new material.

For antenna design, radiation is another important boundary in order to accurately estimate the EM emission. As a good practice, the distance of at least λ/4 or λ/8 must be maintained between the antenna and the boundary. For example, λ/4 will be a good distance for radiation boundary and λ/8 for PML boundary. This is an important aspect that many engineers fail to consider. Upon completion of the initial setup, I ran the simulation to check for its performance.

###### Parameterization of Antenna

After simulation, check the input electric field in coaxial and the impedance of the transmission line/coax in order to verify the expected excitation. In post-processing, do check important parameters for radiation characteristics like pattern and gain. Even the near field data, which is complex to obtain from measurement, can be estimated with simulation.

Since we are not considering any fringing field and probe effects, there will be variation of results. To further improve the design, I suggest using optimization algorithms such as Optimetrics or ANSYS optiSLang. Such tools also permit sensitivity of the design due to fabrication tolerances.

###### Optimal Design of Antenna

Finally, the best design can be selected after evaluating the gain characteristics of the all variations. For the three substrates, I evaluated the optimized dimensions of the patch using Optimetrics:

• 12.5 x 10 cm² for Duroid
• 9.5 x 7.5 cm² for FR4
• 7 x 5 cm² for Alumina

Per this, antenna with FR4 substrate meets the required gain of 3.5dB with the least possible dimension. Better performance can be obtained by varying other parameters such as height of the substrate, etc.

The next time you perform electromagnetic simulation of antenna, do remember to consider all the boundaries.

This concludes the first part of a multi-part series on antenna design & analysis. In the next part, I will discuss about antenna placement analysis.

If you have any questions, please feel free to comment or fill out the contact form.