Analysis of Ipulsive Natural Phenoena through Finite Difference Methods A MATLAB Coputational Project-Based Learning Nicholas Kuia, Christopher Chariah, Mechatronics Engineering, Vaughn College of Aeronautics and Technology, NY, USA, nicholas.kuia@vaughn.edu, christopher.chariah@vaughn.edu Mentor: Hossein Rahei, Ph.D, Vaughn College of Aeronautics and Technology, NY, USA, hossein.rahei@vaughn.edu Abstract - This paper is concerned with Project Based Learning regarding the nature of the application of finite difference analysis in studying ipulse response and vibration that occur in earthquakes. During any natural phenoena, oscillations occur that can be odeled by a second order differential equation. The finite difference technique has been ipleented to deterine the response of a springass syste, a atheatical odel of one-degree-of-freedo frae structure, subjected to an ipulsive force siilar to one produced by an earthquake ground otion. Soe attention has also been given when considering the application of various finite difference techniques; it is realized that the centered difference ethod is the best approach with the sallest error in analyzing the response of a vibrating syste. Finally, a MATLAB script can be developed to generate ipulsive response based on a finite difference nuerical approach. Keywords: Coputational prograing, Ipulsive response, Finite difference ethods, MATLAB, Earthquake analysis I. INTRODUCTION At its core, engineering has always been about the application of science and atheatics to innovate on century-old solutions, or to solve new age probles. In any cases, atheatical odels have been constructed to describe the phenoena that is occurring, often resulting in coplex second-order differential or partial differential equations. Obtaining eaningful solutions to these equations in closed for solutions, often is not practical, whether it be known phenoena such as the heat equation, the wave equation or even describing a siple haronic oscillator equation analytically. With one of the ore iportant goals of engineers being to ake things faster, ore efficient and ore accurate, it is iportant to have a ediu through which students are exposed to a proble that allows the to ake connections and deonstrate their knowledge while also iproving their critical thinking skills with hands-on experience. With Project Based learning this need is addressed, where coplex real world probles can be tackled by students with oversight but still aintain the independent learning students will encounter in the workplace. For the purposes of this paper, we undertook a Project Based Learning assignent using MATLAB to study vibration, and ipulses occurring in earthquakes, while using finite difference analysis to siplify and study the atheatical description of this phenoenon. Finite difference analysis provides an easy and effective alternative to analytical solutions. Having first odeled a situation with a governing equation and knowing all the variables except for the one which is being solved, the equation can be converted into a finite difference equation. Using siple equations derived fro the Taylor s series involving forward, backward, and centered difference analysis, the governing equation can be converted into one that can be solved nuerically using a MATLAB progra. The siple nature of this conversion is probably one of the greatest advantages of this ethod as the resulting new equation does not need to be reforulated despite certain changes in specification, such as ass. The following is a direct application of finite difference technique to a second order differential equation governing ipulse forces on a 1 D ass which can be considered equivalent to those observed in earthquakes. II. MOTIVATION OF GOVERNING EQUATION Figure 1 Mass - Spring Force Diagra Figure 1 represents a 1 D ass spring syste which consists of, Mass (kg) k Spring contant / Material Stiffness ( kg ) c Daping constant ( N sec ) f(t) Ipulsive Applied force (changes with respect to tie) (N) x(t) Displaceent(changed with respect to tie) () Fro Newton s 2 nd Law of Motion, F = a; Eq. 1 1
Exaining the forces acting on the ass, the following can be observed, F s Force caused by spring F D Force caused by daping F(t) Ipulsive Applied force Considering the x direction to be positive pointing to the right, expansion of Equation 1 can be as follows, F s F D + F(t) = a kx cx + f(t) = x x + cx + kx = f(t) d 2 x dx + c + k dt 2 dt x = f(t) Eq. 2 Equation 2 represents the governing equation that will be analyzed using finite difference ethods to approxiate the solution. Although, the proble is being analyzed in one diension, there is still an inherent coplexity regarding the forces acting on the ass. The ain idea to be gleaned fro this, is that earthquake ground otion produces an ipulsive force on engineering structures siilar to what we ipleented in our analysis of ass-spring syste. Earthquakes result when sections of the earth, that norally ove soothly, stick to each other and cause energy to be stored. When the stored energy is finally great enough to break the two sections of earth free, the energy is released in seisic pulses to the surrounding areas. This energy can be considered coparable to the applied force f(t) in the ass spring syste. The seisic pulse effects of the earthquake are eventually daped out due to the energy dissipation which is analogous to the daping force of the ass spring syste. As a seisic wave propagates through the earth during an earthquake, the wave passes through the crust forcing it to rapidly accelerate and decelerate fro the Priary waves, which cause vertical oveent and Secondary waves that cause horizontal oveent that you see in the Figure 2 [1]. The oscillations caused by these waves can be tracked by a siilar haronic oscillating syste, such as the ass spring syste shown in Figure 3 [1], a fact backed up by the ass springs syste s use as seisographs to easure the intensity of earthquakes. Figure 3 Seisograph Mass Spring syste Figure 4 One Story Frae Structure In order to replicate earthquake ground otion we will analyze the ipulsive response of a one story frae structure seen in Figure 4, which will be subjected to very sharp changes in force over short periods of tie. Fro the graph in Figure 4, a piecewise function can be constructed to represent the ipulsive force present, 17800(0.5 t) for t 0.5 33375(t 0.5) for 0.5 < t 0.7 22250(1 t) for 0.7 < t 1.0 26700(t 1) for 1.0 < t 1.5 f(t) = 13350(2.5 t) for 1.5 < t 2.5 0 for 2.5 < t 10 10000 (10 t) for 10 < t 10.5 10000 (10 t) for 10.5 < t 11 0 for t > 11 Figure 3 Ipulse Graph III. DERIVATION OF FINITE DIFFERENCE APPROXIMATIONS First Order Forward in Velocity and Backward in (FVBR) Figure 2 Earthquake Events Diagra Fro Equation 2, the governing equation can be seen as, d 2 x dx + c + k dt 2 dt x = f(t) 2
Decoposing Equation 2 into a series of first order derivatives, it can be siplified to, d dt (dx i ) + c dx i + k x dt dt i = f(t) Eq. 3 The Forward Difference approxiation of the 1 st Derivative (derived in Appendix A) is, f (x i ) = f(x i+1 ) f(x i ) + O(h) Eq. 4 h Applying Equation 4 to Equation 3, d t (i+1) x t(i) dt (x h Siplification of Equation 5 leads to, dx = h [ f(t i ) dt t (i+1) c ) = f(t i ) c dx t(i) k x dt t (i) Eq. 5 dx t(i) dt k x t (i) ] + dx t (i) dt Eq. 6 The Backward Difference approxiation of the 1 st Derivative (derived in Appendix B) is, x (t i ) = x(t i ) x(t i 1 ) + O(h) Eq. 7 h Applying Equation 7 to the dx in Equation 6, the finite difference dt approxiation is finalized at, x t(i+1) = h 2 [ f(t i ) c v t (i) k x t (i) ] + h v ti + x t(i) Eq. 8 Equation 5 also becoes the function to calculate v ti, v t(i+1) = h [ f(t i ) c v t (i) k x t (i) ] + v t(i) Eq. 9 Because Equation 9 is derived fro the Forward Difference approxiation for the 1 st derivative, it inherently needs an initial condition [2] (v 0 ). However, since Equation 8 contains two finite difference approxiations, it needs two initial conditions. It turns out that one of the initial conditions overlaps because Equation 9 is ibedded within Equation 8. Substituting Equations 10 and 11 into Equation 2, the following relationship is forulated, x t(i+2) = 1 1+ [ (2x hc t (i+1) x t(i) ) kh (x c t (i+1) ) + f(t)h + c hc x t(i+1) ] Eq. 12 As one can see, the final for of the centered difference ethod is copletely unlike that of the forward difference ethod, and backwards difference ethod. The principal difference between the two ethods is that the Forward-Backwards (FVBR) ethod allows for response and velocity to be calculated, whereas the centered difference approxiation only evaluates for response. Due to this fact, it is also known that the initial conditions vary. For the FVBR ethod, an initial position and an initial velocity are needed. In contrast, as seen in Equation 12, the centered difference ethod needs the first two positions, and no velocity condition. APPLICATION OF FINITE DIFFERENCE METHODS Based on Equations 8 and 9, a MATLAB script can be written as follows to calculate the response as well as the velocity of the syste being studied using the Forward in Velocity Backwards in Method, Siply ask the user for the specifications of the syste. Each variable is clearly described by its respective input essage Define the start of the three variables that need to be tracked, x, v and t, response, velocity and tie respectively Centered Difference Method The centered difference ethod is very siilar to the first ethod in that the derivation process is exactly the sae. However, the two ethods vary in their final forulas and their order of error. Starting fro Equation 2, the governing equation is the sae, d 2 x dx + c + k dt 2 dt x = f(t) However, substitute Equation 4 with the centered difference forula for 1 st and 2 nd derivatives (derived in Appendix C), f (x i ) = f(x i+1 ) f(x i 1 ) + O(h) Eq. 10 2h f (x i ) = f(x i+1 ) 2f(x i )+f(x i 1 ) h 2 + O(h 2 ) Eq. 11 Copute the correct forcing ter which is based on tie. Each of the five intervals of the piecewise function is expressed as a condition for a conditional stateent to copute a different forcing ter based on the given ipulsive force Calculate the next ter in each of the response, velocity and tie sequences respectively. As one can see, the finite difference ethod is applied here Display the data to the user and plot tie versus response. Figure 4 Forward in Velocity / Backward in Script 3
Siply ask the user for the specifications of the syste. Sae as Previous Script Define the start of the two variables that need to be tracked, x and t, response and tie respectively (Note: there are two initial positions) MATLAB FINITE DIFFERENCE RESULTS Referring back to the proble shown in the Motivation of the Governing Equation section, let us apply the equations to this situation. Table 1 Initial Conditions Case 2 Average Step Size Case 3 Sallest Step Size Copute the correct forcing ter which is based on tie. Sae as Previous Script Calculate the next ter in each of the response and tie sequences respectively. As one can see, the finite difference ethod is applied here The conditions in Table 1 describe a colun with a stiffness constant of 12,000 kg/ that is initially displaced 0 and at rest. A sudden earthquake occurs with the ipulse force shown in Figure 4. The colun is studied fro the tie of ipact to 20 seconds later. Mixed (Forward in Velocity Backwards in ) Finite Difference Table 2 values for Specific Tie Mixed Difference Display the data to the user and plot tie versus response. Figure 5 Centered Difference Script The FVBR script calculates both the response and velocity of the colun due to the earthquake. The script is driven by the increenting of i to calculate new ters while saving the previous points in a atrix. At the end, each atrix holds data fro t = initial value to t = final value. The finite difference ethod derived in the previous section is defined by lines 34 and 35 of the script. Tie (s) fro 0.001 tie fro 0.0001 tie Velocity fro 0.001 tie Velocity fro 0.0001 tie Based Equation 12, a MATLAB script can be written as follows to calculate the response of the syste being studied using the Centered Difference Method which can be seen in Figure 6. The Centered Difference script calculates only the response of the colun of the one story structure during the earthquake. The script is driven by the increenting of i to calculate new ters while saving the previous points in a atrix. At the end, each atrix holds data fro t = initial value to t = final value. The finite difference ethod derived in the previous section is defined by lines 37 and 38 of the script. Figure 6 FVBR Method (0.001 tie step) Figure 9 FVBR Method (0.0001 tie step) Figure 10 FVBR Method 2 (0.001 tie step) Figure 11 FVBR Method 2 (0.0001 tie step) 4
Centered Finite Difference Table 3 values for Specific Tie Centered Difference proving the data fro the finite difference ethods as accurate representations of the proposed situation. Tie (s) fro 0.01 tie interval () fro 0.001 tie interval () fro 0.0001 tie interval () Figure 16 Typical Earthquake response graph Figure 12 Centered Difference 0.001 tie step Figure 13 Centered Difference 0.0001 tie step ERROR ANALYSIS Fro the derivation of each finite difference ethod, it was shown that each approxiation contains their share of error. Both the Forward and Backwards Difference Methods had an error in the order of h while the Centered Difference has an error in the order of h 2. The response studied in this application contains very sensitive data which changes in the hundredths place. As a result, a sall step size had to be chosen in order to accurately represent the response of the colun to the earthquake. Looking at the difference between the response fro a 0.01 tie step, a 0.001 tie step and a 0.0001 tie step, it can be seen that the data eventually converged to the actual value as a saller tie step was considered. There is no analytical data to copare to the finite difference data. Nevertheless, knowing that the data should represent the actual analytical data, an approxiate error can be calculated to exaine the error between the two finite difference ethods. Approxiate error can be calculated as follows, Figure 14 Centered Difference 2 (0.001 tie step) Figure 15 Centered Difference 2 (0.0001 tie step) This data reveals that the colun would need to be able to sustain a response range of about 18 (given no energy absorption into the colun) in order to withstand an earthquake this size. The graphs in Figures 7, 8, 11 and 12 display the response needed by the colun in the first 20 seconds of the earthquake using two difference finite difference ethods with two step sizes. Figures 9, 10, 13 and 14 exaine the response needed in the first 2.5 seconds of the earthquake. All of the graphs are approxiately the sae which shows the overall accuracy of the finite difference ethods; however, fro the data in Tables 2 and 3, it is evident that there is soe error. Figure 15 shows a typical acceleration vs. tie graph for an earthquake 4. Acceleration is siply the second derivative of response. Since the graph of acceleration can be represented as sine s and cosine s, the for of acceleration and response should follow suit. The graph in Figure 15 has a longer earthquake ipulse force; nevertheless, the response is siilar. Using the 1940 El Centro Earthquake Graph [4] as an experiental reference, the finite difference seisic graphs and the actual seisic graphs can be related % Error = Experiental Value 1 Experiental Value 2 Experiental Value 1 100% Eq. 13 Applying this equation to the 0.0001 tie step data for response for both ethods provides the following results, Table 4 Error Analysis between Step Sizes Tie (s) 1 - Centered Method fro 0.001 tie 2 - Centered Method fro 0.0001 tie Approxiate Error (%) between 1 and 2 00.000-0.787 0.317 1.516-1.797-0.008-1.046 3.041-0.077 5
Table 5 Error Analysis between Finite Difference Methods Tie (s) FVBR Method fro 0.001 tie interval () Centered Method fro 0.0001 tie Approxiate Error (%) 0.000-0.023-0.003 0.101-0.283-0.057-1.469 1.339-0.233 Note: The 0.0001 tie step data was used in error calculated because it ost accurately represents the true data. Both ethods present very accurate data with an error percentage of under 2% for the sallest tie step. Table 4 shows that the 0.01 tie step provided a very large error of 104.3% which is beyond unacceptable, especially for an application such as earthquake. Because the ipulsive force is so short and the oscillations are so close, a very sall tie step needs to be chosen to account for the high frequency. There is a tradeoff between accuracy and efficiency when considering step sizes. For exaple, the 0.01 step size created 2000 data points for 20 seconds, whereas the 0.0001 step size created 200,000 data points. For a ore coplex application, considering all three diensions of otions, such calculations would becoe eory consuing and inefficient for a coputer; thus, the need to choose a larger tie increent. Table 5 deonstrates a sall error proving the accuracy of both ethods. REAL WORLD APPLICATION: GROUND MOTION AND BUILDING FREQUENCIES When studying the effects the produced ipulsive forces have on engineering structures, we see certain characteristics eerge that have a large influence on building response. These characteristics include duration, aplitude (of displaceent, velocity and acceleration) and frequency of ground otion. As it can be iagined, the otions of the building are very coplex, alost as coplicated as the ground otion itself. The priary issue engineers worry about however, is the building's tendency to vibrate around Figure 17 Building Coparison one particular frequency, known as its natural or fundaental frequency. Generally speaking, the frequency buildings vibrate at have a lot to do with their height. The shorter a building is, the higher its natural frequency. The taller the building is, the lower its natural frequency which can be seen in Figure 16. This eans that a short building with a high natural frequency also has short natural period. A very tall building with a low frequency has a long period. However, in reality, these nubers are quite short where a typical one story building has a period of 0.1sec. Although the natural period of buildings tend not to change, other factors such as loose ground soil, and cracking structures can have the effect of aplifying waves. Now you ay be inclined to think that these waves alone result in large oveents that cause huge aounts of daage. This is only partially true. In reality, the oveent of the ground and buildings during an earthquake is not uch regardless of earthquake size. So it is not the distance that the building oves that causes daage, it is the sudden force that causes the building to shift quickly that causes the daage. In other words, the daaging force is acceleration. During an earthquake, the speed at which both the ground and building are oving will reach soe axiu. The ore quickly they reach this axiu, the greater their acceleration, the greater the daage. Figure 18 Dynaic Analysis Dynaically this is related to Newton s second law and D'Alebert's Principle, which states that a ass acted upon by an acceleration tends to oppose that acceleration in an opposite direction and proportionally to the agnitude of the acceleration as shown in Figure 17. This in layan s ters eans the larger the ass and/or the greater the acceleration, the larger the resulting force. When this force F is iposed upon the building's structural eleents, beas, coluns, load-bearing walls, floors, as well as the connecting eleents that tie these various structural eleents together, if they are large enough, cause the building's structural eleents to suffer daage of various kinds. Figure 19 Support Types In Figure 18 we see an exaple of the daage that can result fro the acceleration caused by the earthquake ground otion. Assuing the free standing block is siply sitting on the ground without any attachent to it, the block will ove freely in a direction opposite to the ground otion, with a force 6
proportional to the ass and acceleration of the block [5]. If the sae block, however, is solidly grounded and no longer able to ove freely, it ust in soe way absorb the inertial force internally. In Figure 18, this internal uptake of force is shown to result in cracking near the base of the block. As entioned earlier this has a tendency to aplify waves which increases the severity of the daage, especially when the natural frequency of the building is equivalent to that of the earthquake, which causes resonance to occur along with a continuous absorption of energy until the structure collapses. The reason resonance is so destructive is the transfer of large aounts of energy causes large increases in the aplitude of a given wave, resulting in a structure that is unable to dapen the oscillations effectively and thus vibrating until the energy in the syste forces a collapse. The potential daaging effects of resonance can be seen in the oscillations of the faous Tacoa Bridge collapse. Opened on July 4, 1940 the bridge during its four onths of active life was observed to have any transverse odes of vibration. A axiu crest to trough aplitude was observed to be about 5ft; the frequency of vibration at that tie was 12 vib/in. On the orning of Noveber 7 the wind velocity was 40 to 45 ph, Traffic was shut down and aplitude of about 3ft was observed. At about 10:00 a.., the ain front began to vibrate torsionally in 2 segents with a frequency 14 vib/in. The aplitude of the torsional vibration quickly built up to about 35 vib/in in each direction fro the horizontal peaking, until just after 11 a when the left side of the bridge was 28ft higher than the right. At this point the bridge twisted and broke. In this situation, the applied force, caused by the wind, continually added energy to the syste allowing for zero daping until the structure failed. The net outcoe fro the bridge deonstrates the destruction oscillations can cause [6]. When studying structural natural frequencies, we see that it is iportant that we have a good idea of the regional natural frequencies of earthquakes. As evident fro the Tacoa bridge collapse, knowing the potential oscillation of any natural phenoena, and avoiding potential resonance can not only prevent disasters but save lives. Matheatically, these frequencies can be odeled as eigenvectors and eigenvalues fro second order differential equations using finite difference technique. Although the application of this is beyond the scope of this paper this is just one ore area where finite difference technique can be used to glean useful inforation applicable to earthquake otions effects on buildings. Figure 20 Tacoa Bridge Collapse [7] CONCLUSION Knowing ore about the dangers of earthquakes and the potential hazards that they create for us we now can see why it is iportant for us to have useful ethods for analysis of engineering structures. With the use of finite difference ethods for analysis we are able to replicate the ipulsive forces produced by earthquake ground otion and as a result generate useful data for how they act on engineering structures using a siple ass-spring syste. Although potentially far ore coplex than our analysis seen here for 1D otion, this technique has useful applications in 3D otion as well as in eigenvalues and eigenvectors odeling the frequencies during which destructive resonance can occur. Although those applications are beyond the scope of this paper, it can be seen that finite difference has any useful real world applications allowing otherwise nuerically challenging odels to be accurately approxiated using siple forward, backward and centered differences. The MATLAB code solidifies this theory by allowing for fast, reliable and continual coputations. REFERENCES [1] Wald, Lisa. "The Science of Earthquakes." The Science of Earthquakes. "The Green Frog News" Web. 11 Apr. 2015. <http://earthquake.usgs.gov/learn/kids/ eqscience.php>. [2] DuChateau, Paul and Zachann, David. Auxiliary Conditions..A Schau s Outline of Partial Differential Equations. USA. McGraw-Hill Copanies, 2011. 2 3. Print. [3] Rahei, Hossein and Baksh, Shazi. Finite Difference Nuerical Analysis. Finite Difference Ipulsive Analysis of a Frae Structure A MATLAB Coputational Project Based Learning. New York. Vaughn College of Aeronautics and Technology, 2010. 6 9. Print. [4] Seisic Design Eurocode. Earthquake response. Tie History Record. Eurocode Standards. Web. 11 Apr. 2015. < http://www.eurocode.us/seisic-design-eurocode-8/info-tpl.htl > [5] Nelson, Stephen A. "Earthquakes." Earthquakes. Tulane University. Web. 11 Apr. 2015. < http://www.earthsci.org/processes/geopro/seisic/seisic.htl > [6] "How Earthquakes Affect Buildings." MCEER Inforation Service "" Web. 11 Apr. 2015. < http://ceer.buffalo.edu/infoservice/reference_services/eqaffectbuilding. htl > 7
[7] "Tacoa Bridge ENGINEERING.co." Tacoa Bridge ENGINEERING.co. 24 Oct. 2006. Web. 11 Apr. 2015. <http://www.engineering.co/library/articlespage/ tabid/85/articleid/171/tacoa-bridge.aspx& > [8] Rahei, Hossien. "A Finite Difference Nuerical Analysis." A Finite Difference and Finite Eleent Nuerical Analysis Using Matlab. Vol. 3. New York: Vaughn College of Aeronautics and Technology, 2013. 97-109. Print. 8