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In this section, first, I will talk about structural vibrations and (acoustic and sound) wave propagation and also explain how all vibrations are actually due to wave propagation. I mentioned about this a little bit on the Structural Health Monitoring section on the home page. Here, I will expand my explanations.
Then, I will talk about the books that I consider the most useful for learning structural dynamics and wave propagation.
I hope you will enjoy :)
In structural dynamic and earthquake engineering courses, we are thought to model the structures using discrete models. As an example, the figure on right (Figure 1) shows the modeling of a building to analyze its lateral vibrations by assigning masses and stiffness to the lateral degree of freedoms at each story, By doing so, the discretized mass and stiffness matrices (Eq. 1 and 2) are formed, and they are used in the equation of motion (Eq.3) to find the eigenvalues (resonance frequencies) and eigenvectors (mode shapes). In most of the cases, we deal with first a few resonance frequencies and we know that these frequencies are usually low (under 10-20 Hz for most of the structures).
Figure 1
Eq.1 Eq.2 Eq.3
However, the low-frequency global vibration parameters of structures (e.g., resonance modes) are actually due to propagating waves. How this happens is as follows: the waves whose half wavelengths are integer fractions of the length of a structure (e.g., simply supported beam) can propagate along the structure with less attenuation (since they satisfy boundary conditions), compared to the waves whose half wavelengths are not integer fractions of the length of the structure. And actually, due to reflections from boundaries, there are waves propagating in two different directions. These waves interact with each other constructively and destructively. Such interactions cause standing waves, which are the resonance modes of structures. The discrete modeling allows for finding the resonance frequencies and mode shapes easily for structures (usually large structure) that can be easily modeled discretely. So it is a good technique for practical purposes, however it masks how these modes are actually formed physically.
The figure below (Figure 2) provides the visualization for the occurrence of resonance frequencies due to wave propagation: Figure 2 (a) shows that some waves are propagating on an infinitely long beam. Consider that these waves are induced on the beam due to an excitation whose frequency content is broadband (see Figure 2 (b)). As a result, waves with different frequencies propagate freely on the infinitely long beam. Accordingly, a frequency spectrum that is similar to the one shown in Figure 2 (b) would be the frequency response of the beam. However, if we have a finite beam as shown in Figure 2 (c), the frequency response of the beam would look similar to the one in Figure 2 (d). This is because only the waves whose half wavelengths are integer fractions of the length of the beam can propagate on the finite beam, leading to resonance frequencies. Figure 2(e) shows the waves propagating in opposite directions in the finite beam due to reflection from boundaries. The constructive and destructive interaction of these waves leads to resonance modes. For example, the 4th resonance mode is shown in Figure 2 (f) at different time instants. Figure 2 (g) shows that the first four resonance modes of the beam. Bending waves create bending natural frequencies, torsional waves create torsional resonance frequencies, and so on.
Now, considering the wave propagation; the number of propagating waves increases as the frequency increases, and the cross-sectional deformations of these waves exhibit more complex shapes. Therefore, in lower frequencies, all the cross-section deforms (e.g., due to bending, torsional, lateral waves), while in higher frequencies, the cross-sectional deformation of the waves can localize at different sections of the cross-section (e.g., surface waves). The dispersion curve (which you start to hear a lot when you start to get into wave propagation) shows the wave speed or wavelength at different frequencies. However, as I explained below, when a structure is restrained at both ends, only the waves having specific wavelengths can propagate easily.
The analytical solution for finding the vibrational parameters when a structure is modeled continuously (with boundary conditions otherwise resonance frequencies and mode shapes do not occur) is based on writing the governing equation of motion using an infinitely small element (see Figure 2 (h) below). After the equation of motion is formulated, the frequency and shapes of waves that can propagate on these structures (which creates resonance frequencies and shapes since standing waves occur) are found using the boundary conditions. If the structure is infinitely long (e.g., plates, beams, or similar structures), the dispersion curves of the waves propagating freely on these structures can be found using the governing equation of motion of an infinitely small element (without considering boundary conditions). Dispersion curves show the speed, wavelength and attenuation of the waves that can propagate freely on an infinitely long structure with respect to different frequencies. When the cross-sectional deformation of the waves is complex (happens in higher frequencies), it is not possible to model such cross-sectional deformations using analytical models since analytical models can have a few degrees of freedom. Therefore, the semi-analytical method is used. The semi-analytical method (at first, I thought it was very complex), is just based on using finite elements to model the cross-sectional deformations and using an analytical term to model the free wave propagating along the longitudinal direction. So, this method is actually simple and straightforward.
Figure 2
I explained above how the wave propagation creates the resonance frequencies and mode shapes on a simply supported beam. Now, what if the structure is a periodically supported infinitely long beam? This is the case when neither the pure structural dynamics approach (based on resonance frequencies and shapes) nor the pure free wave propagation approach is not applicable. This is because structural dynamics (based on resonance frequencies and shapes) and pure wave propagation approaches get combined for this case (in my opinion). Below, I will explain this. Afterward, I will talk about the dynamics of finite multi-span beams. Then, I will use vibrations of rails to wrap up my explanations regarding wave propagation and structural dynamics approaches, as well as the cases in which one of them cannot be solely adopted. Rail is a great example to do so since it is an infinitely long, periodically supported beam with a complex cross-section.
Let's start with a periodically supported beam that is infinitely long. What happens in such beams is that waves having frequencies at specific ranges (i.e., propagation zones) can propagate freely because the wavelength of the waves satisfies all the boundary conditions at the supports. The waves whose frequencies are not in the propagation zones attenuate quickly as they can not satisfy the support conditions. This attenuation is a result of reflection from supports since they can not satisfy the boundary conditions, and thus it is different from attenuation due to damping. Therefore, there is a relation between the bounding frequencies of the propagation zones and the resonance frequencies of finite beams. The relation is that: (i) the beginning frequency of the N’th propagation zone corresponds to the N’th resonance frequency of a finite beam that has the same supports at its two ends as the supports of the periodically supported infinitely long beam, (ii) the ending frequency of the N’th propagation zone corresponds to the N’th resonance frequency of a finite beam that has fixed-fixed supports. As an example, Figure 3 (a) shows two of the waves that can propagate on an infinitely long beam which is periodically supported by pin supports. These two waves actually correspond to the first vibrational mode of a simply supported beam and a fixed-fixed beam, respectively. Figure 3 (b) shows a representative spectrum where the 1st and 2nd propagation zones are displayed. The paper “Wave propagation and natural modes in periodic systems: 1. Monocoupled systems” explains this phenomenon nicely.
In reality, the span length and forces imposed by the connections are usually not identical due to imperfections in manufacturing. Such imperfections create spans with different dynamic properties. As a result, each span has a similar but slightly different propagation zones. Therefore, several studies examined the dynamic behavior of infinitely-long nearly periodic beams. To explain what happens in this case, let’s consider three consecutive spans with different lengths in an infinitely long nearly periodic beam. Denote these spans as span number #1, #2, and #3, as shown in Figure 3 (c). The figure also shows their 1st propagation zones. As the figure shows, each span has different propagation zones. Therefore, only the waves whose frequency corresponds to the frequency zone common in the propagation zones of these three spans can propagate freely along these three spans. The waves whose frequency does not lie in the common frequency zone attenuate due to reflections from supports.
Now, let's consider a multi-span beam with 3 identical spans. What happens, in this case, is that the first 3 resonance frequencies of the multi-span beam are within the first propagation zone of a periodically supported infinitely-long beam that has the same span length. The next 3 resonance frequencies of the multi-span beam are within the second propagation zone and so on. The paper “Natural flexural waves and the normal modes of periodically supported beams and plates free wave propagation in periodically supported infinite beams” explains this phenomenon very nicely. However, if the span lengths deviate, things get interesting. I will explain this over the figure below (Figure 3 (d) and (e)). Let’s consider a two-span beam where two spans have similar lengths. Let’s consider the first two frequencies of each span (if they were not connected together) to be 1 and 5 Hz and 1.2 and 5.7 Hz, respectively. When they are connected, they will have different frequencies due to coupling. Let’s say the first 4 frequencies of the multi-span beam is 1.3, 1.45, 5.2, and 5.92 Hz (guessed based on my experience), respectively. Figure 3 (d) shows how the mode shapes of this multi-span beam will be. Now, let’s consider another two-span beam (with very different span lengths) where the first two frequencies of each span (if they were not connected together) are 1 and 5 Hz and 4.6 and 21.3 Hz, respectively. When they are connected, they will have different frequencies due to coupling. Let’s say the first 4 frequencies of this multi-span beam is 1.23, 5.42, 5.8, and 22.90 Hz (guessed based on my experience), respectively. Figure 3 (e) shows how the mode shapes of this multi-span beam will be. Now, as you can see from Figures 3 (d) and (e), the first 4 frequencies and shapes of the two-span beams are governed by the modes of each beam within them. You can also see that the shapes over the spans which did not govern the vibration mode of the two-span beams approximately look like the mode shape of that span (the span that did not govern) at that frequency. Therefore, the resonance frequencies and shapes of finite length multi-span beams are also governed by wave propagation (as expected since all vibrations are due to wave propagation). The paper “Vibration characteristics analysis of disordered two-span beams with numerical and experimental methods” provides vibrational analysis of a two-span beam.
Figure 3
Now, I will use rail as an example to help you visualize all the things that I explained above. The figure on left (below) shows the dispersion curve of rail (infinitely long, no supports). As you see, as the number of frequencies increases, the number of propagating waves increases. Up to around 25 kHz, the cross-sectional deformation of the propagating waves are global (i.e., all the parts of cross-section of the rail deforms together). This dispersion curve was obtained using the semi-analytical method since analytical methods are capable of modeling dispersion curves of rails only up to 4-5 kHz (this is because analytical models have a few degrees of freedom, so they can model only a few waves having simple cross-sectional deformations). The figure on right shows the bending waves propagating in a rail that is periodically supported by fasteners. Therefore, this wave can propagate at specific propagation zones (as I explained above). If the rail was supported by 2 fasteners at its ends (instead of being periodically supported and infinitely long), only the waves that create the resonance frequencies could propagate, and resonance modes would occur as the rail would be supported only by at two ends. The spectrum below (a representative spectrum, not obtained based on computations) shows the propagation zones for the bending waves for the rail that is infinitely long and periodically supported by fasteners.
Note: Here, I only showed the bending waves. There are many different waves that can propagate on the rail as you can see from the dispersion curve. Approximately after 2-3 kHz, it is not possible to call the waves with simple names such as bending, torsional, longitudinal. This is because the waves exhibit more complex cross-sectional deformations as the frequency increases.
Figure 4
When we consider the waves having frequencies higher than 25 kHz, waves start to localize at different sections of the cross-section of rail (this happens in other types of beams and plates as well). Waves localized in the head and web of the rail can propagate freely as they do not interact with fastener connections that are located at the foot of the rail. However, the waves propagating at rail foot interact with fasteners, so they can not propagate freely.
Figure 5
Now, I am showing here how low frequency waves that create resonance modes in structures can be seen during an earthquake. Therefore: I am showing the recorded acceleration signals from different stories of a tall building during a small earthquake (see Figure 3). I got this figure from my M.Sc. thesis. That time, I did not notice the time delay in the recorded signals (see the dashed line in the figure which I added now). This is because I had no idea that the earthquake generated waves that propagated along the structure. Actually, there are studies that use wave propagation in structures for damage detection purposes by modeling the structures continuously or periodically.
Figure 6
Now, I am explaining two different damage detection approaches performed using wave propagation (i.e., non-destructive testing). Figure 4 (a) shows that a transducer is used to generate waves within a structure so that the reflected waves from a defect (if exist) can indicate the existence of a defect. These waves have very high frequencies and thus very short wavelengths (they are in the ultrasonic range). These waves are called bulk waves, they only have longitudinal and shear deformations, and they are used to inspect the areas located near the transducers. Figure 4 (b) shows that a transducer generates a wave that can propagate along the structures for long distances. This kind of wave can be used for damage detection where the damage is located away from the transducer. The waves that can propagate along the structures are called guided waves since wave propagation is guided by the boundaries of the structure. However, since these guided waves are the waves that can propagate along the structure, they are the waves that create resonance frequencies or propagation zones in structures (depending on the boundary and support conditions as well as length). Therefore, (in my opinion), these are the natural waves that can occur in a structure as a result of excitations such as random excitations (e.g., wind), earthquakes, and impact excitations. For example, all the waves in the rails that I showed above are guided waves. So, guided waves are not necessarily ultrasonic waves as they start from 0 Hz. The ultrasonic range is considered to be above 20 kHz. However, when these waves are used in the ultrasonic range, their wavelengths get very small, so they can reflect from small defects such as cracks. So, guided waves can be adopted in ultrasonic damage detection applications. But, again, (in my opinion), they are not necessary ultrasonic waves unless they are used in higher frequencies. To sum up, any vibration in structures is due to waves. The ones at higher frequencies are usually adopted in non-destructive testing as their wavelengths get very small (they can also localize at different sections of the cross-section) so that these waves become sensitive the very small defects such as cracks.
Figure 7
Note: my Ph.D. research is about defect detection of rails. Therefore, my Ph.D. helped me to understand everything that I am mentioning in this section of the webpage. It was difficult at the beginning since I was a structural dynamics person, and research articles about rails usually focused on wave propagation while some of them focused on structural dynamics since rail behaves differently at different frequency ranges. Therefore, It took me a while to get this “A-ha” moment and understand everything that I am mentioning in this section. However, so far (up to my knowledge), I could not find a brief document explaining (i) how structures behave at different frequencies in terms of wave propagation, (ii) when resonance frequencies can occur or not, (iii) when the structural dynamics and wave propagation approaches are combined (i.e., periodic structures), (iv) what waves are used in non-destructive testing. Therefore, I wanted to create this section to help you see the big picture before you get into structural dynamics and wave propagation. I hope that you enjoyed reading this section.
The Books
For learning structural dynamics, reading the famous book of Chopra: “Dynamics of Structures” is enough (in my opinion).
For learning wave propagation, since modeling of waves is based on analytical methods in most cases, except the semi-analytical method, a good understanding of elasticity and continuum mechanics (especially for the notations used in formulations) is required before getting into wave propagation. The books “Elasticity, Theory, Applications and Numerics” and “Continuum Mechanics” are very good books to get into these topics. Then, to learn about wave propagation, the books “Wave Motion in Elastic Solids” and “Wave Propagation in Elastic Solids” are considered to be very good books.
As I mentioned on the home page, when learning wave propagation, it is better to focus on the structure that you are dealing with. The wave propagation books have chapters for different applications. When I was getting into wave propagation, I thought that I needed to know all for my research. However, I realized that this was not the case. For example, if you are working with beams, or pipes, or plates, or large mediums (i.e., for seismicity), it is better to study the wave motion for the type of structure that you are working on. However, to start with wave motion in strings and beams is not a bad idea regardless of the type of structure that you are working on.
Lastly, “Ultrasonic Guided Waves in Solid Media” and “Ultrasonic Waves in Solid Media” are very good books to get into ultrasonic waves and non-destructive testing.
I hope that you will find the books that I listed useful when you are learning wave propagation and structural dynamics.
