Links to lessons

You can access the material that i used in my classes following the links (in white) that I will post below.

  • 2011-06-21, Earthquake excitation, inelastic response, ductility and inelastic design spectra.
  • 2011-06-14, Earthquake excitation, elastic response spectra and elastic design spectra.
  • 2011-06-07, Continuous systems with an infinite number of degrees of freedom.
  • 2011-05-31, Derived Ritz Vectors, Direct Numerical Integration, Multiple Support Excitation.
  • 2011-05-27, Modal Contributions and Truncation Errors analysis. Static Correction Procedure.
  • 2011-05-10:
    Superposition repeat, Matrix Iteration
    • a first approach to the issue with modal truncation errors
    • the idea of Matrix Iteration
    • convergence to 1st eigenvector/value
    • forcing convergence to higher mode, sweeping matrix
    • a first optimization, inverse iteration
    • the shifted eigenvalue problem
    • Rayleigh-Ritz procedure
  • 2011-05-03:
    Structural Matrices in multi DOF systems
    • further relationships of orthogonality
    • flexibility and stiffness
    • strain energy, symmetry
    • mass matrix, consistent or lumped>
    • damping matrix by linear combination
    • static condensation procedure
  • 2011-04-19:
    Multiple DOF systems
    • equations of dynamic equilibrium
    • vector equation of equilibrium, matrix formulation
    • homogeneous equation, eigenvalues and eigenvectors
    • vector orthogonality
    • modal expansion, uncoupled equations of motion
  • 2011-04-12:
    Generalized SDOF
    • articulated rigid bodies, generalised properties
    • separation of variables, deformable bodies
    • Rayleigh method
    • Rayleigh method's refinement
  • 2011-04-05:
    Step by step integration of the EOM
    • Intro, pros and cons
    • Piecewise exact integration
    • Explicit methods (central differences)
    • Implicit methods (Newmark Beta family)
    • Modified Newton-Raphson algorithm
  • 2011-03-29:
    Frequency domain, Fourier transform, FFT
    • Fourier transform pair,
    • Discrete Fourier Transform, aliasing
    • response in frequency domain, complex frequency response function, caveats
    • impulse response and frequency response form a Fourier Transform pair.
  • 2011-03-22:
    Response to Periodic and
    Non-Periodic Loading
      • periodic loadings as Fourier Series
      • frequency response function
      • superposition of effects
      • infinitesimal response to a force of infinitesimal duration (impulse)
      • response to a non-periodic load by superposition of infinitesimal responses: Duhamel Integral
  • 2011-03-15:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2011-03-08:
    Introduction - Free Vibrations
    • an introduction to dynamics of structures,
    • the SDOF linear oscillator,
    • free vibrations of undamped and damped SDOF oscillators.

The In the Summertime homework assignment announce

The In the Summertime homework, due 2011-09-08, is finally available; it consists of six problems and is equivalent to the first 2010-2011 homework, that was due in May 2011.
  1. You must do the new homework and submit it in due time if you have not submitted the first homework or you scored less than 60 points;
  2. you can do the new homework if you want to better your score; please beware that you cannot choose the best score between this homework and the first one, i.e., submitting a new homework you make me forget the first one.

2nd homework assignment announce

2011-06-23: ERRATA CORRIGE

In problem 3, it is not kn = k (200-5n-n2) but it is kn = k (210-5n-n2). Otherwise k12 < 0!

2011-06-20: the 2nd homework assignment is available for download.

You shall submit your solutions (printed or nicely handwritten) when you will do your oral exam, will it be in June, July, September or even February 2012.

1st assignment's degrees

I posted your degrees, note that for every wrong solution there is a comment: hover the nmouse over a cell to see the comment.

1st assignment's solutions

2011-05-18: if you want to compare your solutions of the problems of the 1st assignment with mine, you can.

My solutions are here. I'll post all the scripts that i used to prepare the solutions as soon as i have added some comments here and there.

2011-05-19: I have posted a figure that was not included in the solutions above: the virtual displacements for prob. 4.

2011-06-06: i have a more or less polished version of the scripts i used to produce the paper with the solutions.

Suspension on 2011-05-17 and hand in

The lessons and all the activities will be suspended on 2011-05-17 and 18 to give place to the University Conference on Teaching.

Your assignment being due for 2011-05-17 there is a little problem as I will not be in Lecco to receive your papers... in any case you can submit your papers by email, or if you prefer to hand in a printed/handwritten copy you can give it to Ms Elena Corbetta, in the office next to the reception.

Clarification on the assignment due on 2011-05-17

If you plan to do your exam in September or later, you're not strictly requested to submit the assigment due 2011-04-17. As an alternative, you will receive, near the end of August, a different assignment on the same arguments of this one, due a week before the September exam.

On the other hand, if you want to do the exam in July you must submit the current assigment in due terms

The second assignment, that I'll publish before the end of the course, is due on the day of your exam, in July or in September or later.

First homework assignment

The first homework assignment is due for 2010-05-17. The text of the problems is online.

Introduction

The course deals with the dynamical response of mechanical systems, linear and non-linear, under the assumption of small displacements.

Focus is given to analytical and numerical methods for the integration of the equations of motion, both in time and in frequency domain, to the numerical methods for the eigen-analysis of multiple degrees of freedom systems and to earthquake engineering applications.

Classes include tutorials and computational exercises.

During the semester, you'll be given two home assignments, the former approximately in mid may, the latter at the end of the lessons. Starting from this year, Dynamics of Structures is a 5 credits module in the 15 credits course Structural Analysis, so that, at the moment, I can only say that a part of your degree will be based on the home assignments. I'll be more specific as soon as possible

The slides i use in classes will be made available on this page in a short time after the classes, but older versions of the slides are already available, as i taught the same course, more or less the same course, in the previous academic years. Would you like to peek at the material i posted during the previous years, here it is...

Recommended books

  • Ray W,Clough, Joseph Penzien, Dynamics of Structures.
    Please note that the standard 2nd edition is out of press, but the link points to a software house that has bought the rights and issued a revised edition of the classic book by Clough and Penzien.
  • Anil K.Chopra, Dynamics of Structures (Theory and Applications to Earthquake Engineering), 3rd ed.
    A number of copies of this book are available from the campus library in Lecco.
    Should you find a hugely discounted used copy of the 2nd edition of prof. Chopra's book, don't worry and buy it, the 2nd ed. is good enough for our purposes.

90% of the material you'll see in the first part of the course is inspired from Clough and Penzien's book, 90% of the material of the second part of the course from Chopra's.

While it is very easy to follow the first part of the course using Chopra as your reference (the difference being in some notational convention and a more liberal use of examples in Chopra), following the second part of the course using the book from Clough and Penzien as a reference may be difficult.

In short, should you prefer to buy a single book, buy Chopra's.

Home assignments

The home assignments that you're supposed to do require no more than pencil and paper (a hand-held calculatur being however useful for eventuality of a sine or a cosine...).

Of course, you may feel more at ease and be more productive using a decent spreadsheet program, Excel or OpenOffice Calc (aka LibreOffice Calc) being the names that spring to mind...

Of course, anything more sophisticated than Excel (think of Mathematica and Matlab, or of Matlab's free clone Octave) can be helpful.

When i work preparing and solving the assignments, my setup comprises:

  • Calc, an advanced calculator that gives you symbolic manipulation of matrices, symbolic derivation and integration and much, much more.
    Not Mathematica, very different from Mathematica, however Calc is useful and free.
  • Scipy, a library of scientific tools for the programming language Python.
    I prefer using a set of libraries from inside a full blown programming language (Python) than using an ad hoc programming language (Matlab, Octave).
    Of course, Matlab has different assets... (toolkits, i.e.)
  • Python itself, an interpreted, interactive, object-oriented, extensible programming language.
  • Gnuplot, a command-line driven interactive data and function plotting utility, for 2- and 3-D graphs, Gnuplot does also parametric plots, model fitting...

Giacomo Boffi