A few months ago, I went to for my (supposedly) annual medical checkup, which I had managed to skip a year earlier, and just to be on the safer side this year, opted for all the tests on list. While going through the test routine, I noticed, that no matter what the doctors were intending to check, be it my eyes, my lungs, or even my teeth, the medical staff will have me do a blood pressure, and a heartbeat rate measurement, to check for the vital signs, and only then the doctor will examine. And it seemed that the most important, and critical test was the “stress test”, where the heart rate was continually monitored and recorded while I ran on treadmill. As the doctor explained, responding to my curiosity, that the heart rate (and the blood pressure) was the key indicator of the basic health status, and indication of any irregularity.
As a structural engineer, it had me thinking, would it not be great to be able to do such a quick check with our structures as well? How can we treat our structures as doctors treat their patients? Can we check their pulse rates? Can we listen to their heartbeats?
And it occurred to me, that we can and probably, many of us already do. Well not literally, but in a manner of speaking, if we think of the natural frequency of structure as its heartbeat. Just like the number of heartbeat per minute is a reliable, and extensively used indicator of human physical fitness, the natural frequency (number of cycles of vibration per second) of a structure is a fairly reliable indicator of its structure’s stiffness, damage state and overall health. For example, like the normal heartrate of a healthy person at rest should be between 60 to 80, a free vibration frequency of a tall buildings of say 30 stories should be about 0.25-0.35 hz or a period of vibration of 2-3 seconds. If the heart rate is too fast for a human being, it may indicate stress or abnormal conditions, and if too low, may indicate health issues. In the same the way, if the frequency of a structure is too high, it may indicate an abnormally stiff structure, and a low frequency may indicate soft or weak state.
Just to be clear, this free vibration response is often obtained using what is called the “Modal Analysis” of a structure. Before we discuss some of the applications where the vibration characteristics (or the results obtained from the modal analysis) can be extremely useful in structural engineering, let’s first take a quick look how at how it all started. The basic idea and the origin of modal analysis can be traced back to the times of Sir Isaac Newton (1643 – 1727) and Joseph Fourier (1768 – 1830). The essential concept is that the dynamic response of physical objects can be described as a sum of few simple waves. Lord Rayleigh (1842 – 1919) developed the concept further and presented in his book “The Theory of Sound”, published in 1877. Later, an Italian structural engineer, Arturo Danusso (1880 – 1968) made key contributions in earlier 20th century to introduce and apply the basic concepts of modal analysis, originally developed in acoustic theory, to describe the dynamic response of building structures against earthquakes.
The free vibration analysis of a structure involves the determination of its natural frequencies, mode shapes, and modal participation factors. Vibration mode shapes are the “shapes” in which the structure would like to oscillate (with corresponding natural frequencies), if allowed to vibrate freely. Each mode shape is an inherent property of the structure and is completely defined by its mass and stiffness. If a linear elastic structure is deformed into a linear combination of few mode shapes and released to oscillate freely, every mode will be independently present (with its own natural period) in the resulting combined time history of deformation. This is the key concept behind the notion that the combined vibrational response of any structure can be decomposed into contributions from few vibration modes, each behaving (and can be solved) independently. The degree of participation of a certain mode in the overall vibration is determined by the characteristics of excitation source(s), and spatial distributions of the mass and stiffness of the system. But I am sure, you already knew that!
You might also know, that the modal analysis has profound applications in diverse areas, covering civil engineering, mechanical and aeronautical engineering, industrial and manufacturing engineering, biomedical engineering, acoustics, space structures, electrical engineering, and so on. The traditional engineering designs require the structures to be lighter in weight, and yet strong-enough to sustain external excitations. These requirements can easily make them susceptible to undesirable vibrations. In such cases, a proper consideration of dynamic response becomes a key factor in design. Nowadays, the use of versatile finite element solvers have opened a whole new paradigm where the modal analysis is finding its innovative applications – from car manufacturing on one side, to the non-destructive testing of high-rise buildings on the other side – from the design of space structures to scientifically evaluating the violins and guitars, and even studding the human heart and the movement of the brain mass within the skull.
Having said that, I would now like to talk a bit more about the significance of modal analysis as applied to the structures, emphasizing the importance of underlying “heartbeat” for the structures. This can help us to:
- First, and foremost, gain an insight into the complex dynamic response of a structure. The experimentally determined natural periods, damping factors and mode shapes can provide an important information regarding the structural characteristics, which can then be used to convert the detailed structural models in to simple modal models.
- And equally important, it helps to provide a quick check on the analytical structural models. For example, if the natural period computed for the model is not what is expected, then the model data can be checked. Most common errors such as incorrect mass density, or wrong elastic modulus (both caused sometimes due to unit confusion, and sometimes wrong material type,) can be detected.
- The plot and animation of mode shapes of modal analysis, in addition to being fun to watch, can help to detect the disconnected members, structural irregularities, eccentric response, torsional modes etc.
(See Ashraf Habibullah, the President of Computers and Structures Inc, CSi, discussing the role of animations in structural engineering on this link. https://www.youtube.com/watch?v=SGa1wzZEUsQ)
- The detailed finite element models of structures can be calibrated with the modal data (natural periods, mode shapes etc.) which is either obtained experimentally or is statistically derived based on some large-scale experimental studies. The modal model is expected to represent the close-to-real dynamic behavior of a structure and therefore, can be used to “correct” the finite element model.
- At the design stage of new building structures, the optimum dynamic response can be easily obtained by altering the mass and stiffness distributions. Various modifications, based on the “what-if” analysis, can be proposed, which can then help in selection of an optimum lateral-load resisting system for a building.
And if that is not enough of reasons to do a modal analysis, either analytically through software, or physically through measurement, we have some more advantages:
- The modal analysis can be used to predict the response of linear elastic structures against a given dynamic loading. Similarly, it can also be used in reverse to predict the dynamic forces against a given measured vibrational response of the structure. With a known dynamic response, the fatigue life of a structure can also be predicted with a reasonable accuracy.
- The modal parameters can be used to detect the damage (cracking or yielding etc.) in structure, thus leading to structural health monitoring (SHM). Any reduction in stiffness because can be easily detected, measured and analyzed by the change in modal properties. The systematic comparison of modal properties before and after structural damage can be used to develop useful relationships for this purpose. An important example of such applications is the regular bridge monitoring and maintenance using ambient vibrations.
- And, perhaps, the most important application of modal analysis in structural engineering is in the design of control mechanisms to suppress unwanted dynamic response of structures. The analysis and application of almost all control devices (dampers, actuators, sensors etc.) require a detailed knowledge of free vibration response and modal characteristics. The engineers can play with natural frequencies, shift them to desired values, change the mode shapes, and relative contributions of different modes, to obtain the optimum structural response.
An interesting video showing various mode shapes of a 40-story RC shear-wall buildings can be seen here https://youtu.be/DnBnckVMztg. This example shows how visualizing the free vibration response can be not only fun to watch (music added!), but also helpful in developing the feel of dynamic behavior and the relative contributions of different vibration modes to the total dynamic response.
We can all appreciate that the ever-growing real-life complexities, and structural demands in buildings and infrastructure are posing new challenges to civil and structural designers. These require developing a feel for the structural response, a greater understanding and expertise in analytical and design procedures and development of out-of-the-box solutions
So, let’s start listening to the “heartbeat” of our structures, and understand what they are saying!