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What is Modal Analysis and Why is it Necessary?

We often hear the terms “modal analysis,” “eigenvalues,” “eigenmodes” or “eigenfrequencies” and by the time one comprehends a single terminology – new terminology emerges. People always prefer the fancy ways to make things sound more complicated than they need to be. More often it’s out of necessity, but many times it’s to express things with mathematical accuracy. But first let’s explain the difference between quasi-static, dynamic, and modal analysis.

What is modal analysis?

The most common type is quasi-static analysis where the load is applied at a very slow rate such that the acceleration is negligible (or almost zero). Alternatively, dynamic analysis is that where the effects of acceleration cannot be ignored. Both these types of analysis provide a one-to-one relation between a particular input (for example force applied on a system) to its system response (for example displacement of the system due to its load).

In contrast to both, modal analysis provides an overview of the limits of the response of a system. For example, for a particular input (like an applied load of certain amplitude and frequency) what are the limits of the system response (like when and what is the maximum displacement).

Amplitude of system response (y-axis) w.r.t. the frequency of the applied input (x-axis)

Fig 01: Amplitude of system response (y-axis) w.r.t. the frequency of the applied input (x-axis) (Source)

As shown in Fig 01, every object has an internal frequency (or resonant frequency) at which the object can naturally vibrate. It is also the frequency where the object will allow transfer of energy from one form to the other with minimal loss – here vibrational to kinetic. As the frequency increases towards the “resonant frequency,” the amplitude of response asymptotically increases to infinity. In other words, the result of the modal analysis are these frequencies at which the amplitude increases to infinity.

How are eigenvalues / eigenvectors / eigenmodes related here?

Any object can be considered as a connection of complicated springs and then the system response “x” for any applied input “y” could be given using a scaling factor as

k*x = y 

This is similar to the spring equation where “k” is the spring stiffness, “x” is the spring displacement and “y” is the applied force. For any generic system, it can be written as

[K] {x} = {y}

where {x} can be displacements, temperatures etc. while {y} is a force, flux etc. The matrix [K] can be considered as a scaling factor and more commonly known as stiffness matrix. Now, for some response {x} = {a}, if the applied input was {y} = L*{a}, then L are known as the eigenvalues and the response of the system {a} are known as the eigenvectors corresponding to the eigenvalue L.

02

Fig 02: Vector scaling using eigenvalues (Source)

In other words, as shown in Fig 02, the magnitude of the applied input and its response are just a multiple. The eigenfrequencies are those at which this scaling is maximum (i.e. the eigenvalue).

So why are these frequencies important?

Every system can be described in terms of a stiffness matrix that connects the displacements (or system response) and forces (or system inputs). These frequencies are known as natural frequencies of the system and given by the eigenvectors of the stiffness matrix. These frequencies are also known as the resonant frequencies.

These resonant frequencies related to mechanical structures are known as mechanical resonance. Similarly, every system like acoustic, thermal, electromagnetic etc has their own resonant frequencies at which resonance occurs. As shown in Fig. 03, as the frequency of applied load (or input on the x-axis), nears the resonant frequency, the amplitude of response (on the y-axis) nears infinity!

As governed by the first law of thermodynamics, one form of energy is only converted to another. However, energy is neither created nor destroyed. In any mechanical system, when an external time-varying load is applied – it is equivalent to supplying the system with some kinetic or vibrational energy. This is transmitted through the system resulting in a displacement of the structure. Yet, due to the presence of friction, some part is also dissipated as heat.

Amplitude of response as a function of the frequency ratio

Fig 03: Amplitude of response as a function of the frequency ratio (Source)

To understand this process more physically, one can consider that structure is in a constant state of motion sub-atomically. The energy supplied is transported from one part of the structure to the other through energy transfer by atomic processes. However, when the frequency of loading is the same as averaged vibrational frequency of the atoms in the structure, the energy is transferred with minimum loss. In other words, in simple terms, one can think of it as two waves (one being the external load and another being that of the internal atomic structure) that are being superimposed. When the frequencies are the same, they tend to add up.

Thus, it is important to know these frequencies at which the structure can behave erratically.

Practical examples for modal analysis

There are several examples where a prior accurate modal analysis could have prevented loss to lives and property. Some famous ones include:

Tacoma Narrows Bridge Disaster of 1940

Tacoma Narrows bridge was built in the state of Washington (USA). On November 7, 1940, around 11 a.m., the bridge came down instantaneously. Later upon investigation, it was found that the cause of the collapse was aeroelastic flutter.

Collapse of the Tacoma Narrows Bridge

Fig 04: Collapse of the Tacoma Narrows Bridge (Source)

Or in other words, it was a wind-induced collapse. The winds were blowing at a particular frequency that coincided with the resonant frequency of the structure leading to a sudden collapse of the structure.

Mexico City Earthquake of 1985

Another real-life example was the 1985 earthquake in Mexico City. The energy released during this earthquake was equivalent to 1114 nuclear detonations and the earthquake was felt up as far as Los Angeles which is over 800,000 km away. Up to 1950’s, no earthquake codes existed. However, it was in the later 1950’s and 1970’s that codes were introduced for building constructions. Yet, none of these accounted for an event of magnitude 7.0 plus that occurred in 1985.

During the earthquake, most of the 6 to 15-story high-rises collapsed resulting in huge loss of life and property. Buildings with less or more storys were surprisingly not damaged much. Buildings with 9 were completely destroyed and resulted in rubble! Two explanations were reasoned for the earthquakes: the duration of shaking and the resonance with the lakebed frequency. In other words, the resonant frequency of the 6 to 15-story structures coincided nearly with the frequency of the earthquake!

At present, there are several earthquake codes being used by civil engineers especially for structures being built in a zone vulnerable to earthquakes.

Taipei 101 & Burj Khalifa

A real-life example is that of today’s skyscrapers like Taipei 101 in Tokyo (Japan) or Burj Khalifa in Dubai (UAE). These megastructures use tuned mass dampers to absorb the energy and dampen the oscillations of the structures.

Taipei 101 in Tokyo where the local wind systems are so complex that the swirl around the structures. The building acts like a large sail of a boat causing vortex shedding and thus twist / bend the structures in unimaginable ways. Taipei 101 uses a tuned mass damper in the form of a large pendulum between the 88th and 92nd floors. A video of the working on the pendulum is available on YouTube as shown below:


The Burj Khalifa is known to oscillate about 5-6 ft at the top. Such large motion can be felt as creaking and can be significantly uncomfortable for the inhabitants. Thus, Burj Khalifa does not use a damper system. Alternatively, they change the external profile of the building based on the wind system and thus reduce the overall force of the wind.

Conclusions

To summarize, we understood the meaning of modal analysis and various nomenclature. It is very important for a designer to understand the natural vibration frequencies of a system to ensure that they are not the same as excitation frequencies and thus ensuring safety standards.

Modal analysis Truss

Fig 05: Different eigenmodes of a truss bridge with SimScale

This is a key component in many fields like civil, aerospace, automotive engineering where loss of life and property is a major concern. Starting with hand calculations in the 1980s, computer simulations have made a great breakthrough to help improve the quality and robustness of design processes. We look forward to the day when computer simulations can replace engineering codes!

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