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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html>
<head>
<title>Dynamics of Structures</title>
<meta http-equiv="content-type" content="text/html;charset=latin-1"></>
<link rel="stylesheet" href="css/2013.css" type="text/css"></link>
</head>
<body>
<div id="container">
<div id="banner">
<a href="http://www.polimi.it">
<img style="border: 0px solid ; width: 158px; height: 65px;"
src="logopolired.png" alt="Politecnico di Milano" ></img>
</a>
<h1>Dynamics of Structures<br /></h1>
</div>
<!-- banner -->
<div id="outer">
<div id="left">
<h2>Links to lessons</h2>
<p>You can access the material that i used
in my classes following the links <font color="white">(in
white)</font> that I will post below.</p>
<ul>
<li><a href="dati_2013/_12.html">2013-06-11</a>
Applications to Earthquake Engineering: Response
Spectra, Design Spectra, Ductility Spectra.
<li><a href="dati_2013/_11.html">2013-06-04</a>
Continuous systems with an infinite number of degrees of
freedom. </li>
<li><a href="dati_2013/_10.html">2013-05-28</a>
Derived Ritz Vectors, Direct Numerical Integration,
Multiple Support Excitation.</li>
<li><a href="dati_2013/_09.html">2013-05-21:</a>
<br>Matrix Iteration Repeat, Modal Contributions and
Static Correction,</br>
<ul><small>
<li>Recall onRitz-Rayleigh and Subspace
Iteration</li>
<li>Modal Partecipation Factor</li>
<li>Modal contributions to the load vector</li>
<li>Modal contributions to pseudo-static
response</li>
<li>Pseudo-displacements</li>
<li>Peak factors</li>
<li>Static correction</li>
</small></ul>
</li>
<li><a href="dati_2013/_08.html">2013-05-14:</a>
<br>Superposition repeat, Matrix Iteration</br>
<ul><small>
<li>a first approach to the issue with modal
truncation errors</li>
<li>the idea of Matrix Iteration</li>
<li>convergence to 1st eigenvector/value</li>
<li>forcing convergence to higher mode, sweeping
matrix</li>
<li>a first optimization, inverse iteration</li>
<li>the shifted eigenvalue problem</li>
<li> Rayleigh-Ritz procedure</li>
</small></ul></li>
<li><a href="dati_2013/_07.html">2012-05-07:</a>
<br>Structural Matrices in multi DOF systems</br>
<ul><small>
<li>further relationships of orthogonality</li>
<li>flexibility and stiffness</li>
<li>strain energy, symmetry</li>
<li>mass matrix, consistent or lumped></li>
<li>damping matrix by linear combination</li>
<li>static condensation procedure</li>
</small></ul>
</li>
<li><a href="dati_2013/_06.html">2013-04-30:</a>
<br>Multiple DOF systems</br>
<ul><small>
<li>equations of dynamic equilibrium</li>
<li>vector equation of equilibrium, matrix
formulation</li>
<li>homogeneous equation, eigenvalues and
eigenvectors</li>
<li>vector orthogonality</li>
<li>modal expansion, uncoupled equations of motion</li>
</small></ul>
</li>
<li><a href="dati_2013/_05.html">2013-04-16:</a>
<br>Generalized SDOF</br>
<ul><small>
<li>articulated rigid bodies, generalised properties</li>
<li>separation of variables, deformable bodies</li>
<li>Rayleigh method</li>
<li>Rayleigh method's refinement</li>
</small></ul>
</li>
<li><a href="dati_2013/_04.html">2013-04-09:</a>
<br>Step by step integration of the EOM</br>
<ul><small>
<li>Piecewise exact integration</li>
<li>Explicit methods (central differences)</li>
<li>Implicit methods (Newmark Beta family)</li>
<li>Modified Newton-Raphson algorithm</li>
</small></ul>
</li>
<li><a href="dati_2013/_03.html">2013-03-26:</a>
<br>Impulsive loads, intro to Step by Step methods</br>
<ul>
<li>Impulsive Loads</li>
<small>
<li>definition, usual assumptions,</li>
<li>analytical solutions,</li>
<li>approximate analysis.</li>
</small>
<li>Step by Step Methods</li>
<small>
<li>general discussion, pro"cons,</li>
<li>e.g., the piecewise linear method</li>
</small>
</ul>
<li><a href="dati_2013/_02.html">2013-03-19:</a>
<br>Response to Periodic and Non-Periodic Loading</br>
<small>
<ul>
<li>Analysis in the frequency domain, Fourier Series
and Fourier Transform, Discrete Fourier Transform
and Aliasing, fast algorithm for DFT (FFT).</li>
<li>Analysis in the time domain, Duhamel integral,
relationship between the two solutions.</li>
</ul>
</small>
</li>
<li><a href="dati_2013/_01.html">2013-03-12:</a>
<br>Response to Harmonic Loading</br>
<ul><small>
<li>undamped response, dynamic amplification factor,
resonant response</li>
<li>damped response</li>
<li>accelerometer</li>
<li>vibration isolation</li>
<li>measuring equivalent damping ratio</li>
</small></ul>
</li>
<li><a href="dati_2013/_00.html">2013-03-05:</a>
<br>Introduction - Free Vibrations</br>
<ul><small>
<li>An introduction to dynamics of structures,</li>
<li>the single degree of freedom (SDOF) linear
oscillator,</li> <li>the free vibrations of SDOF
oscillators.</li>
</small></ul>
</li>
</div>
<!-- HERE START THE RIGHT COLUMN -->
<div class="onwhite" id="content">
<h2>The Third Homework</h2> <a
href="dati_2013/ha01/ha03.pdf">The text of the third
homework</a> is now available. This homework is due if
you haven't submitted the first homework in due terms.
<h2>The Second Homework</h2>
<a href="dati_2013/ha01/ha02.13.pdf">The text of the second
homework</a> is now available. You must hand in your
solutions, printed or nicely handwritten, the day of your
oral exam.
<h2><big>The First Homework</big></h2>
<dl>
<dt>2013-05-07</dt>
<dd><a href="dati_2013/ha01/ha01.pdf">The text of the
first homework</a> is now available.</dd>
<dt>2013-06-04</dt> <dd>
The <a href="dati_2013/ha01/hasol01.pdf">solutions</a> of
the first homework exercises are now available.</dd>
</dl>
<h2>Homeworks explained</h2>
<p>
You have to present TWO homeworks, on different subjects,
out of THREE homework assignments:
<ul>
<li>the FIRST or the THIRD, one of them, and </li>
<li>the SECOND one.</li>
</ul> The FIRST assigment is about SDOF systems and a
touch of MDOF, and is due on May 28. The SECOND will be
released at the end of the classes, will be about MDOF
systems and is due for the day of your oral examination.
</p>
<div>The THIRD assignment will be about SDOF systems and a
touch of MDOF's (that is, exactly the same subjects of the
1st one), will be released at the end of July and will be
due at the end of August.</div>
<font color="black">If you aren't ready for the first
homework, if you plan to do the exam in September or in
February, if you have other things to do in May, then you
can do the THIRD assignment instead of the FIRST one.</font>
<h2>Objectives</h2>
<p> The course deals with the dynamical response of mechanical
systems, linear and non-linear, under the assumption of small
displacements.</p>
<p>Focus is given to
<ul>
<li> analytical and numerical methods for the
integration of the equations of motion, both in time and
in frequency domain,</li>
<li> the numerical methods for the eigen-analysis of
multiple degrees of freedom systems and</li>
<li> earthquake engineering applications.</li>
</ul>
</p>
<h2>Organization</h2>
<p>This year we'll have 14 or 15 weekly classes, inclusive
of tutorials and computational exercises.</p>
<p> You will be evaluated on two homeworks and a final oral
test. </p>
<p> All the slides used in classes will be (almost)
immediately made available on this page after each
class.</p>
<p> If you want to have a look at the slides <em>before</em>
a class, you can: <a href="sd_2012.html">last year's
slides</a> are still available. Remember, however, that
I'm still slowly changing the content and the organization
of my lessons and that the reference material is only what
I post <a href="sd_2013.html">here</a>.
</p>
<h2>Recommended books</h2>
<ul>
<li>Ray W. Clough, Joseph Penzien,
<a href="http://orders.csiberkeley.com/ProductDetails.asp?ProductCode=DOC%2DDYNOFSTR">
Dynamics of Structures</a>.<br> This classic text is
sold solely by the software house linked above.</br></li>
<li> Anil K.Chopra,
<a href="http://www.amazon.com/Structures-Prentice-Hall-International-Engineering-Mechanics/dp/0132858037/">
Dynamics of Structures </a> (Theory and Applications
to Earthquake Engineering), 4th ed.
</li>
</ul>
<p>The first part of the course follows Clough and Penzien's
book, the second part follows Chopra's. Should you prefer to
buy a single book, my advice is buy Chopra's.</p>
<p><small> Note that <strong>a fair number of copies of
Chopra's book (3rd ed.) are available from the campus
library in Lecco</strong>, note also that Chopra's 2nd
and 3rd editions are widely available at lower prices
and are perfectly appropriate for my course.</small></p>
<h2>Homeworks</h2>
<p> Each homework comprises a set of exercises. While many
of these exercises won't require more than paper, pencil
and a hand-held calculator, the remaining ones will
require plotting a time series or a bit of really
elementary matrix algebra, i.e., things that every
half-decent spreadsheet can do.
</br>Of course specialized programs, let's say <a
href="http://www.wolfram.com/mathematica/">Mathematica</a>,
<a href=
"http://www.mathworks.com/products/matlab/">Matlab</a>
or Matlab's free clone <a href="http://www.octave.org">
Octave</a> can be helpful.</p>
<p> A pleasant alternative to proprietary stuff is the <a
href="http://ipython.org/notebook.html">IPython
notebook</a>, an example of which is <a title="Exercise 4:
Assemblage of Rigid Bodies" href=
"http://nbviewer.ipython.org/url/boffi.github.io/nb/04_Rigid_Bodies.ipynb">
the solution</a> of an exercise I've assigned in 2012.
</p>
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<h1><a href="http://www.dica.polimi.it">Dipartimento di
Ingegneria Civile ed Ambientale</a></h1>
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<address><a href="mailto:giacomo.boffi@polimi.it">Giacomo Boffi</a></address>
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