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- \headline={\tenrm\hfil Koshelev$\ \ $\folio}
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- \centerline{\bf EARTHQUAKE DAMAGE PREVENTION}
-
- \medskip
- \centerline{\bf 1. Introduction}
-
- The goal of this project is to compare several methods of earthquake
- damage prevention. In order to compare them, I simulated the effects of
- an earthquake on a building with different earthquake damage prevention
- methods.
-
- \noindent{This project consists of two programs:}
-
- \parskip 0 pt
- \item{$\bullet$} A program that simulates an earthquake
-
- \item{$\bullet$} A program that simulates how a building moves during an
- earthquake.
- \parskip 8 pt plus 4 pt minus 4pt
-
- \medskip
- \centerline{\bf 2. Description of the earthquake simulator program}
-
- \medskip
- \centerline{\bf 2.1 General Idea}
-
- The main problem with earthquakes is that they are very unpredictable.
- There is no known formula to describe an earthquake. Therefore, we
- cannot simulate an earthquake by simply programming in a known formula
- with a regular, predictable behavior.
-
- Irregular, unpredictable phenomena are very common: weather, the
- economy, etc. To simulate these phenomena Professor Mandlebrot of Yale
- University invented fractals. Here is a description of what a fractals are.
-
- \centerline{\bf 2.2 Fractals}
-
- The word {\it fractal} comes from the word fraction, and it means a set of
- points, whose dimension is not a whole number. Let's describe what a
- dimension is.
-
- From the physical viewpoint, a dimension sort of describes how ``big'' a
- set is. For example: pasta can be in different shapes: it can be long
- and thin (one-dimensional) in spaghetti, it can be thin and
- two-dimensional as in fettucini, and it can be three-dimensional as in
- thick pasta shells. If their sizes are equal, then there is more dough
- in shells than in fettucini, and there is more dough in fettucini than
- in spaghetti.
-
- To describe how to find which sets have more elements and which sets
- have less elements let's consider the following approach.
-
- We have a set. To describe how many elements are in this set we pick a
- distance $D$ and ask a fast food company, for example Subway, to build
- its restaurants so that wherever you are in that set the nearest Subway
- will be no farther than $D$ yds. The company wants to save money and
- therefore, it wants to build as few Subways as possible. The more
- restaurants are needed, the bigger is the set.
-
- First let's consider a one-dimensional case: We want to build Subways on
- a road whose length is $L$. The Subways must serve all of the points
- including the endpoints. Therefore there must be a Subway at a distance
- not farther than $D$ yds from each endpoint. It makes no sense to build
- it closer than $D$ yds from the endpoint because then we would lose
- money. Therefore, we build the first Subway $D$ yds from the endpoint.
-
- This Subway serves the area from the endpoint to $2D$ yds The second
- one must be at the distance $D$ from the first point not covered by the
- first Subway, i.e. at a distance of $2D + D = 3D$ yds from the endpoint.
- And so on.
-
- Here is how to find out how many Subways we need. Every Subway serves
- and area of $2D$ yds in length. The total length is $L$. Therefore, the
- total number $N$ of Subways is $$N \approx {L \over 2D}.$$
-
- Now let's say that we have a two dimensional set, and the company is
- trying to build Subways with the same condition as in the previous
- example. In a one-dimensional set the area served by each Subway
- is a line segment, in a two-dimensional set it is a circle. So now the
- number of Subways would be the area $A$ of the set divided by the area
- of each circle, i.e. by $\pi D^2$: $$N \approx {A \over \pi D^2}.$$
-
- Now let's assume the set is three-dimensional and the
- conditions are still the same. Now the area served by each Subway
- is a sphere. So the number of Subways would be the volume $V$ of the
- set divided by the volume of each sphere, i.e. by $(4/3) \pi D^3$:
- $$N \approx {V \over (4/3) \pi D^3}.$$
-
- If we look at these three cases, we see a pattern. In the 1-D case, $N
- \sim c/D^1$ for a constant $c$ ($c = L/2$); In the 2-D case, $N \sim
- c/D^2$; In the 3-D case, $N \sim c/D^3$. In all three cases, the
- exponent is the same as the dimension. So in a general case the
- dimension is defined as follows: If for a set, $N \sim c/D^\alpha$, then
- $\alpha$ is the dimension. Sets for which $\alpha$ is a fraction are
- called fractals.
-
- \medskip
- \centerline{\bf 2.3 Waves}
-
- A wave is a real-life pattern that repeats itself again and again. Waves have
- several characteristics that describe them. One is
- amplitude which is the distance from the middle of the wave to the crest
- (the top) of the wave. Another one is frequency, which is the number of
- repetitive segments in a second. And the last one is phase,
- which is how far the initial position of the wave is from the middle.
-
- Light from any source can be separated into several basic colors, as in a
- prism. Any sound can be simulated by a piano. This means that the sound
- can be played as a chord, and thus, it can be separated into several
- different sounds. Light and sound are waves. They can be separated into several
- monochromatic waves, or waves with a constant amplitude, frequency and phase.
-
- If you get the sine of several increasing numbers and you graph the
- results you will see that the graph is a monochromatic wave. Therefore the sine
- function is used to make monochromatic waves. If we have a time $t$, an
- amplitude $A$, a frequency $\omega$, and a phase $\phi$ of a point on a
- monochromatic wave, then that point is: $$A \sin (\omega t + \phi).$$
-
- A wave consists of several monochromatic waves. To make a wave out of
- several monochromatic waves you simply add the monochromatic waves
- together If you have a time $t$, a
- set of amplitudes $A_n$, a set of frequencies $\omega_n$, and a set of
- phases $\phi_n$ of a point on $N$ monochromatic waves, then the wave can
- be figured out like this: $$\sum_{n=0}^N A_n \sin (\omega_n t + \phi_n).$$
-
- \medskip
- \centerline{\bf 2.4 The earthquake simulation}
-
- It is known that earthquake can be described by fractals. It is also
- known that when we represent a fractal as a sum of monochromatic waves,
- then the amplitude decreases with frequency $\omega$ approximately as:
- $$A \approx {I \over \omega^\alpha}$$ for constants $I$ and $\alpha >
- 0$. So to simulate a random process, we take: $$A(\omega_n) = R_n \cdot {I
- \over \omega_n^\alpha},$$ where $R_n$ is a random number simulated by the
- computer's random number generator, and we also take a random phase. The
- random phase $\phi_n$ goes from $0$ to $2 \pi$ (360 degrees in radians).
-
- So the earthquake simulation runs as follows:
-
- We fix a value $\Delta\omega$, and choose $\omega_n = n \Delta\omega$
- (i.e., $\omega_0 = 0, \omega_1 = \Delta\omega, \ldots$). Then we make
- two random sequences: $R_0, \ldots , R_n$, and $\phi_0, \ldots, \phi_n$.
- To generate the earthquake force at time $t$, with $N$ monochromatic
- waves, we will use the following
- formula: $$f_{external}(t) = \sum_{n=0}^N A(\omega_n) \sin (\omega_n t +
- \phi_n).$$
-
- \medskip
- \centerline{\bf 3. Description of the building movement program}
-
- \centerline{\bf 3.1 General Idea}
-
- In order to simulate the effect of different controls on the building,
- we use Newton's second law. According to this law, $$f(t) = Ma(t),$$
- where $f(t)$ is a force, and $a(t)$ is an acceleration. From this
- formula we can derive the formula for acceleration: $$a(t) = {f(t) \over
- M}$$
-
- \noindent To use this formula we need to know two things:
-
- \parskip 0 pt
-
- \item{$\bullet$} How force depends on time.
-
- \item{$\bullet$} How to simulate the motion if we know the acceleration.
-
- \parskip 8pt plus 4pt minus 4pt
-
- \medskip
- \centerline{\bf 3.2 How does force depend on time? }
-
- \noindent Force that acts on any object consists of the following components:
-
- \parskip 0pt
-
- \item{$\bullet$} External forces that make it move in the first place.
-
- \item{$\bullet$} Elastic force, this is most visible in a rubber band.
- This force tries to return the object to its initial position. The
- farther we bend the object, the stronger is the elastic force. Therefore
- it is natural to assume that this force is proportional to the
- displacement: $f_{elastic}(t) = -kx(t)$, where $k$ is a coefficient.
-
- \item{$\bullet$} Friction force, that slows down all of the movement.
- The faster the movement the bigger the friction force. Therefore it is
- natural to assume that this force is proportional to the velocity:
- $f_{friction}(t) = -nv(t)$, where $n$ is a coefficient.
-
- \noindent In out case, the external force consists of two components:
-
- \item{$\bullet$} The earthquake force.
-
- \item{$\bullet$} The control force. We will consider three cases.
-
- \itemitem{$\bullet$} No control, i.e. $f_{control}(t) = 0$.
-
- \itemitem{$\bullet$} Active control. The main idea of active control is
- that if the earthquake pulls the building in one direction active
- control must pull it in the other direction. The ideal active control
- is: $f_{control}(t) = -hx(t)$, but we can't do it like that because the
- earthquake is very fast and the machine can't react immediately.
- Therefore there is a delay $T$: $f_{control}(t) = -hx(t - T)$.
-
- \itemitem{$\bullet$} Hybrid control. The main idea of hybrid control is
- this: there is an extra beam in the building that is not attached to
- anything on one end. When the building moves more than a certain limit $a$,
- the beam connects itself to the building. It acts as an extra elastic
- force. Therefore the formula is as follows: if $x(t - T) < a$ then
- $f_{control}(t) = 0$ else $f_{control}(t) = -hx(t)$ where $a$ is a
- coefficient.
-
- \parskip 8pt plus 4pt minus 4pt
-
- \noindent As a result we get the following formula for force: $$f(t) =
- f_{external}(t) + f_{elastic}(t) + f_{friction}(t) + f_{control}(t).$$
-
- \medskip
- \centerline{\bf 3.3 How to simulate the motion if we know the
- acceleration.}
-
- By definition, velocity is distance over time. If we simulate the
- movement every $\Delta t$ seconds, then time is $\Delta t$. Distance is
- the difference between the position of the building in the consequent
- moments of time $t$ and $t + \Delta t$, i.e. distance is $x(t + \Delta
- t) - x(t)$. Therefore the velocity is $$v(t) = {x(t + \Delta t) - x(t)
- \over \Delta t}$$. Let's use this formula to find the position at the
- next point in time based on the position at the current point at time.
- By using Cross Products we derive the following formula: $v(t) \cdot \Delta
- t = x(t + \Delta t) - x(t)$. Moving $x(t)$ to the other side we get:
- $$x(t + \Delta t) = v(t) \cdot \Delta t + x(t).$$
-
- Similarly, acceleration $a(t)$ is defined as velocity over time, i.e.
- $$a(t) = {v(t + \Delta t) - v(t) \over \Delta t}.$$ Using the same
- method as in the previous derivation we get the following: $$v(t +
- \Delta t) = a(t) \cdot \Delta t + v(t).$$
-
- So the algorithm for simulating $x(t)$ is as follows: We start at the
- moment $t=0$. We set $x(t) = 0$ and $v(t) = 0$. To get the coordinate
- and the velocity at the next moment in time we use the following
- formulas: $$x(t + \Delta t) = v(t) \cdot \Delta t + x(t),$$ $$v(t + \Delta t)
- = a(t) \cdot \Delta t + v(t),$$ where $$a(t) = {f(t) \over M}$$ and $$f(t) =
- f_{external}(t) + f_{elastic}(t) + f_{friction}(t) + f_{control}(t).$$
-
- \centerline{\bf 3.4 How to compute energy}
-
- Hybrid control does not use any energy. Active control basically eats
- up lots of energy because it moves the building. Here is how to
- calculate the energy used.
-
- The bigger the force we apply, the more energy we use. Therefore, it is
- reasonable to assume that the energy is proportional to force. But if we
- don't move anything then we don't use any energy. The more we move the
- building, the more energy we use. So the energy is also proportional to
- the displacement $\Delta x(t) = x(t + \Delta t) - x(t).$ Therefore $e(t)
- = f(t) \cdot \Delta x(t).$ To derive how much energy we use altogether we
- use the following formula: $$E = \sum_{t = 0}^T e(t).$$
-
- \centerline{\bf 4. The Results}
-
- \centerline{\bf 4.1 Conclusion}
-
- I used the following algorithms for earthquake intensities $0.1$ to
- $9.9$ on the Richter scale to find out the maximum movement, total
- movements, and total energy used for no control, active control and
- hybrid control.
-
- Active control got the smallest total movements on intensities $0.1$ thru
- $0.9$. Hybrid control got the smallest total movements on the rest of the
- intensities. So basicaly hybrid control won because earthquakes with
- intensities so small do not cause almost any damage.
-
- Active control got the smallest maximum movement on intensities $0.1$
- thru $1.9$. Hybrid control got the smallest maximum movement on the
- rest of the intensities. So bascially hybrid control won for the same
- reason as with the total movements.
-
- And of course hybrid control and no control won the total energy used
- because they do not use any energy and active control uses a lot.
-
- My conclusion is that hybrid control is the best earthquake damage
- prevention method.
-
- \vfill\eject
-
- \centerline{\bf Reference List}
-
- \noindent{}Andrews, D.J. ``A Stochastic Fault Model 1. Static Case.''
- Journal of Geophysical Research Jul. 10, 1980: 3867--3877.
-
- \noindent{}Andrews, D.J. ``A Stochastic Fault Model 2. Time-Dependent
- Case.'' Journal of Geouphysical Research Nov. 10, 1981: 10821--10834
-
- \noindent{}Ingard, K. Uno. Fundamentals of waves \& oscillations.
- Cambridge, New York: Cambridge University Press, 1988.
-
- \noindent{}Kagan, Y. Y. and Knopoff, L. ``Spatial distribution of
- earthquakes: the two-point correlation function.'' Geophysical Journal
- of the Royal Astronomical Society Vol. 62, 1980: 303--320.
-
- \noindent{}Kagan, Yan and Knopoff, Leon. ``Statistical study of
- occuurrence of shallow earthquakes.'' Geophysical Journal of the Royal
- Astronomical Society Vol. 55, 1978: 67--86.
-
- \noindent{}Kagan, Y. Y. and Knopoff, L. ``Stochastic Synthesis of
- Earthquake Catalogs.'' Journal of Geophysical Research April 10,
- 1981: 2853--2862.
-
- \noindent{}Mandlebrot, Benoit B. The Fractal Geometry of Nature. New
- York, NY: W.H. FREEMAN AND COMPANY, NY, 1983.
-
- \noindent{}Pierce, John. Almost all about waves. Cambridge, Mass.: MIT
- Press, Massachusets, 1974.
-
- \noindent{}Wang, Y.P, et al. ``Development of Design Spectra for
- Actively Controlled Wall-Frame Buildings.'' Journal of Engineering
- Mechanics June, 1992: 1201--1220
-
- \noindent{}Yang, J.N, et al. ``Aseismic Hybrid Control of Nonlinear
- and Hysteretic Structures I.'' Journal of Engineering Mechanics
- July, 1992: 1423--1440.
-
- \noindent{}Yang, J.N, et al. ``Aseismic Hybrid Control of Nonlinear
- and Hysteric Structures II.'' Journal of Engineering Mechanics July,
- 1992: 1441--1456.
-
- \noindent{}Yang, J.N, et al. ``Stable Controllers for Instantaneous
- Optimal Control.'' Journal of Engineering Mechanics August, 1992: 1612--1630.
-
- \bye
-
-
-
-