Use of Mathematica in Some Studies about Structural Dynamics

Good morning in the States and here in Europe. Good afternoon. We are going to talk about the use of Mathematica in some studies about structural dynamics. Okay. The structure of our presentation will be this. After showing you a brief abstract and talking slightly about us, I will present which studies do we have in progress, the basic of dynamic of structures, why we use Mathematica. And we will show you a short example of what we do. The generation of a theoretical signal, identification of natural frequencies, filtering, and how do we calculate the model damping ratio. After that we will show how to apply this to a real structure and the future studies that we are doing. So, let me show this abstract and as I told you, this presentation tried to summarize the content of the different modules that we are implementing in Warframe Mathematica and that we use as support in the academic studies that we are carving out in the field of structural dynamics. What's the most usual? Well, the most usual case is to have the record of several accelerometers of structure and it allows us to have immediately the maximum absolute values of the recorded acceleration. But the mathematical treatment of samples, such data can provide much more information. And this is the content of our presentation. This is because the vibration of the structure can be decomposed into the sum of various vibrations, which correspond to what we call the vibration modes of the structure. These natural frequencies of vibration are determined from the analysis of the spectrum of the accelerogram, which is obtained using the Fourier transform. And besides, when the structure is in a free vibration region, we can get the model damping ratio. And it's also interesting what happened with the different calculation methods. What's the most important of this? That the symbolic calculation software Mathematica have facilitated us the implementation of these modules and it has allowed the analysis of the registered data for your structure, making possible to advance the knowledge of structural dynamics and giving rise to several scientific publications. Well, something about us, because it's obvious that it's important to see where we work, if you are interested. Miguel Angela Steed is Professor Dr. Civil Engineer and he's a prominent professional in the field of structural engineering, quantum dynamics and finite element analysis, among other areas. Until his retirement, he was full professor of Brick Engineering at the Madrid Polytechnic University upm. And now he continues his collaboration at Carlos Fernandez Casado, one of the most important Spanish project engineering film specializing rich and large scale structure. And I am Juan Antonio Bope Zaragon, civil engineer, a member of the Court of Civil Engineers of the Spanish Ministry for Transport, Mobility and Urban Agenda. And I combined my professional work as senior advisors in the Ministry with my studies for finishing the PhD in the Polytechnic University of Madrid, under the direction of Megan. Of course. If you have any doubt, you can show this in the Q and A or in the chat. We will try to answer this in the end of the presentation or if it's not possible, in the email that I will show you at the end of the presentation. Well, what about the studies that we have in college, where we are in a very interesting general context? We have the Sustainable Development Goals set by the United nations for 2030, the technological evolution, the Internet of Things and Big Data. What happened with civil engineering that now conservation is specialized in the advanced countries is more important than the construction of new infrastructure because we have the need of preventive maintenance and inspections due to the critical character of bridging. We have here two examples that happened in the last year. The Morandi bridge in Italy in 2018 and the collapse of the Lane 12 in Mexico in 2021, with a lot of fatalities. So it shows the importance of the maintenance of bridges. And the monitoring of bridges allows us to detect incidents at an early stage and saving on eventual repair cost. So our work try to analyze the behavior of similar bridges. In order to know how can vary dynamic parameters. We try to answer to this question, which is the behavior of free air structures, which dynamic parameters are stable? How affects the structural typology in variation because of damage? Do we have to consider ambient variables can be modeled or in the case of the model dampen ratio, which is the influence of the calculation methods and if they can be improved. So the dynamic parameters that we obtain is thanks to the dynamic of the structures that it's in this. In this other slide, as it is well known, the equation that models the response of a linear viscoulidamp in single degree of freedom system a spring and the mechanical excitation is this one. We have the mass, the second derivative of the displacement. So the acceleration, the damping, the first derivative, which is the velocity, the stiffness, the displacement and all the zoom is equal to the mechanical excitation or load. So what happened with pre array bridges and buildings that they are multiple degree of freedom systems? Fortunately, to everyone that has to solve this equation, the response of a free restructure is equivalent to the sum of various simple degree of freedom systems. Normally, if you make a test, you only take acceleration thanks to special devices which are called accelerometers and the records are named accelero gans and the waves under free force Vibrations can be modeled as the acceleration is equal to an initial amplitude A sub 0. An exponential function with the damping, the circular frequency and the time. An a cosim function with an argument of the circular frequency, the time and the phase difference, the initial phase difference, which is our objective. Our objective to obtain the dynamic parameters of the system from the accelerograms. And obviously it can be complicated in the case of natural frequency of vibration and model damping ratios. And it's what we are trying to solve. Why Mathematica? As I told initially, obviously we use spreadsheets and all normal software, but we need something more than a common software for this. And besides, we wanted to compare the behavior of different calculation methods. So we need to implement and control the code. We wanted to know exactly what's happening with the code. Since we were accustomed to Mathematica, we considered the possibility of using this software. We are tremendous happy with the decision, as we will see in this presentation, because it allows us to to arise to several presentations. Well, as an example, I will show you what's happened with a theoretical signal. So we will generate here a signal of 15 seconds length, which is the sum of few waves. As a random noise. We have wave number one, two or three with these amplitudes, these frequencies, these model damping ratios and this phase difference. And we will add a random noise slightly higher than the real resolution of accelerometers, in order to simulate most unfavorable conditions to solve this problem. So the function that we are referring is the following, where we will clear the variables. And we have here. So I will execute. Okay, and we have here that the plot of this function. If you have an accelerometer, really you will have a data point to point, point by point. So we will generate a table similarly to this previous plot. If you see here slight changes in the numbers, it is because of that we have add a random noise. So we will generate the data and it's here with the result and we can plot again this. So we have here more or less the same plot that we had before, but with point by point data. Okay, what's the next step that we have to do? Once we have the accelerogram, we have to identify natural frequencies, which is the most often used tool the Fourier transform. Among other options, Mathematica bring us the opportunity of using the two Fourier to obtain the discrete furine transform the DFT from a list. So. So it saves us a lot of computational cost and programming cost. This is an extract of the code for that Sorry, because some of the name of the variables are in Spanish, but it is easy for us. But here is the step. The channel, the parameters of the Fourier transform, the limits of the accelerograms, and all the manipulation, the management that we have to do with the vectors and the matrices. So if we execute this, it is here, if we plot the results of the calculation. The graphic of the plot that we obtain is called the spectrum of the signal, where the peaks correspond with the natural frequencies. If everyone goes right 1, 2 and 4 Earth, as we indicate in the simulate wave. So let's plot the wave and here we can see in respect to the peaks at 1, 2 and 4 hertz, as it was expected. Okay, the next step. The next step is to make it a filter of the signal. Because as we said before, we have to decompose the signal into simple degree of freedom signals. There are a lot of literature about filters with the properties. And here I put one of the easiest one, the passband filter, the rectangle, because it's the easiest one. One of the handicap of this kind of filter is the introduction of some distortion in the boundaries. But we can cut the signal in the boundaries, because it's enough with the central part of the central section of the accelerogram. And of course, the application for that type of filters is possible. And in fact, it will be interesting to study what happened with these filters. Well, so after defining the filter and the cut length of the filter signal, the extract of the code will be this. Here we define that if we want to get the frequency of 1 Hz, we will filter between 0.8 and 1.2. If we want 2.0 Hz, we will filter between 1.8 and 2.2. And if we want to filter around 4.0 Hz, between 3.8 and 4.2. And the cut of the wave will be 20 seconds. Remember that we have a 50 second length wave, so we will get the 10 seconds in the center. I will execute this data and work this manipulation with the vectors and the matrices. And we will apply the filters here. Okay, so we obtain here, for each natural frequency here, natural frequency number one, the plot of the either passband filter between 0.8 and 1.2 hertz, the filter accelerogram without any cut. So from 0 to 50 seconds. And we will get the central part here, from 20 to 30 seconds in the Acceleroca. And the same for natural frequency number two and the same for natural frequency number three. Remember that we obtain field graphic for each natural Frequency filter, filter signal and cut and filter signal. Okay, and the next step is to get the model damping ratio. We can obtain the model damping ratio thanks to different calculation methods. We will see one of the most common in detail, because we have not many times and this is the most often method used method. If the logarithmic decrement, which is the classic procedure per excellence and it's directly applicable to single degree of freedom systems undergoing free vibrations. So the model damping ratio is obtained thanks to the ratio between two maximum displacements. Here we have the Nepalian logarithmic between these two displacements over n consecutive cycles. So this is the expression. And alternatively we can obtain the model damping ratios not only thanks to the displacement, but also the acceleration, which is really the data that we have record. Once the signal is isolated for a given frequency, as we've done before, we have to choose a criterion because we will have the extremes and. And we have to choose with which criteria we will get the model damping. Because we can select the ratio between the first and the last one extreme, or between the first and the second, or the third, the first and the fourth. And making an average or adjusting this curve or with this square adjustments or imagination is free. So I prefer, as we will see here, getting the the extremes and after getting the extreme, doing an arrowhead. So the code for. For this is this one. And I say that I want that the extreme will be separated five cycles, which is these parameters. So, so if I execute this, and again I execute this, what I have for each natural frequency, for instance, in this case, natural frequency number one, the extreme values which is here and the average model damping ratio 0.0.22%. In the case of the natural frequency number 2, 1% and in the case of the natural frequency number 3, 0.5%. Obviously the obtained result are very, very approximately to the theoretical values 2, 1 and 0.5%. In the. In the example, remember that we have add random noise. So it is clear that because of the noise and also all the procedure, the filter and all the numerical approximations and it's very difficult to get the exact value, but it's very, very, very good values. And I said before, there are other calculation methods, the bandwidth method, the cure fitting or the random decrement technique, we have implemented all of them in order to see what happens with one and another and another method. Well, we have applied this to real structures. This is one real structure in Spain. It's called the road viaduct Arcoastral Connector. It's a really high bridge. It is in the river Tagus Tahoe. It's with a span of more than 200 meters. So it's a really big extraction. And thanks to this work. This is the computational model of Teprovich. And we have compared the experimentally registered natural frequencies and modes of vibration with the computational model. Here we can see that the experimental frequencies it's in the first lane, 0.396 Hz and the theoretical one is 0.303 or 0.600 0.639. So it's very interesting. Also we calculate the model damping ratios and we study the evolution of the dynamic parameters over time. And we analyze the relationship between dynamic parameters and ambient variables like temperature, wind or the maximum acceleration, because the acceleration are the inter load measurement in this case. Here we have an example the correlation between the ambient pressure and the natural frequency of vibrations over the time. We can see very, very good relationships with a coefficient of determination of almost 0.82. Remember that the maximum is one if it was a perfect correlation. The other ones are not so good, but 0.6, 0.5 or 0.5 are very, very good data for a real structure. And we have presented our conclusions in different papers and communication. If you are interested. This one about the experimental analysis of of the evolution of the dynamic parameters of a long span metal arc. This other one about the reaching experiences in structure and monitoring of bridges. Or this another one, the singularities observed in the calculation of the model damping ratio of a long span metal arc that we will present in Coimbra in the following days, in a congress that will be celebrated in Portugal in the following days and of course future studies, because we are working in other papers like this one, dedicated to the influence of the model damper patient calculation method that I said before, which have been submitted to a special site publication, now it's under revision. And we are also studying the dynamic behavior of other typology of railway bridges. These two bridges, Piyadut Ahoyo del Valle viaduct and Arrio de las Piercas in Spain. And it will be possible to carry out more studies, for instance, to extend this kind of study to more viaducts with other typologies locations, to analyze the influence of the type of filters, deepening the physical modeling of the relationship that we have observed and to increase the studies about the effect of structural damage over the dynamic parameters, both in computational models and in full scale. Best. Okay, so as conclusion, throughout the presentation we have seen the sense of the Academic studies that we are carrying out in the field of structural dynamics. We have seen how the use of the symbolic calculation software Mathematica has facilitated us the implementation of the different modules and has allowed the analysis of the registered data of real structures. And this made possible to advance the knowledge of structural dynamics, giving rise to several scientific publications. We want to express our gratitude to Wolfram for the organization of the virtual technology conference where we are sharing knowledge and experience with our colleagues, to Universidad Politanica de Madrid, upn, the university where we are developing these studies, to mitma, the Ministry for Transport Mobility and Urban Agenda of Spain, for the access to the real data of bridges and of course to the audience of this presentation for their integral. Thank you very much. And if you are interested in these topics, we have here with several references that you could see if you want. And now if you have any question and answer, I will have a look to the chat. But if you have any doubt or comment in the future, we will please to answer if possible to my mail, which. Which is here. Don't hesitate to contact us in the mail. So thank you very much again and I will have a look because with this window I cannot see properly the chat. If there are any question and answer. Okay, I have here one question from Danny Lich Blau who says besides up and down, does a bridge also have issues with torsional vibration around the length axis or lateral to the span from wind? Yes, we have. In this deep bridge we have several vertical accelerometers and some transversal accelerometers in order to see what happened with. With the wind. Because in fact this. This ridge is in a. In a. Bailey. Great Bailey, as you could see with a lot of wind. So we could see not torsional vibration, but we could see lateral vibration in. In some cases. So we get some values for the. For the natural frequencies in vertical and in lateral mode of vibration. And really it is a very singular case because it is a very. It's a metal bridge and it is known as a singular bridge here in Spain. So it's very interesting being said that if I will share this presentation. Yes, I said before to the organization that I will prepare because I made the presentation with the notebook format. It's complicated to generate a good PDF. So I will prepare a PDF and it will share in the system. I will send this to the organization to be there. Okay. I don't know if there is any other question. So anyway, if you have any question, don't hesitate to ask us. Thank you very much again here. Sharing the notebook can also be very useful. Yes, yes, yes, don't worry, I will. So I will share the notebook also. It can be interesting for everyone. So of course I will share it. So any for inclusion of thermal effect? Well, Thomas Branham told us about the thermal effect. We have studied. Certainly that natural frequency vary varies slightly with. With temperature. It is not something new because late papers talk about this and in fact for our intention, which is to settle, I mean structural health monitoring systems. It's very interesting to know how are these operations because we have to correct the values that we have in the measurements in order to avoid false positive in the SHM system. So in fact we have detect the relationship with the natural frequency, but not with the model dampen ratio. It could be something that you could imagine beforehand because it's a metal structure, so you know that the property will change with the temperature. Perhaps the damping could vary with this, but not in the range. In this range we didn't see any relationship. In fact the coefficient of the determination was almost zero between the model damping ratios and the temperature. By the way, we have data of the temperature both from the ambient temperature and from the arch. Because sometimes it depends on whether the sun. Whether is the sun. You can have some difference. So it. It can be some. Some. It could be interesting. So. Okay, so if you have anything else, I insist, don't hesitate to send an email to me and I will try to answer this. Okay, so thank you very much for everyone and to all the audience, Bye bye.