Links to lessons

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

  • 2020-03-10:
    Introduction - Free Vibrations
    • An introduction to dynamics of structures,
    • the single degree of freedom (SDOF) linear oscillator,
    • the free vibrations of SDOF oscillators.
  • 2020-03-12:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2020-03-17:
    Response to Periodic and Non-Periodic Loading
    • Analysis in the frequency domain, Fourier Series and Fourier Transform, Discrete Fourier Transform and Aliasing, fast algorithm for DFT (FFT).
    • Analysis in the time domain, Duhamel integral.
  • 2020-03-19:
    Impulsive loads, Step by Step methods
    • Impulsive Loads
    • definition, usual assumptions,
    • analytical solutions,
    • approximate analysis.
    • Step by Step Methods
    • general discussion, pros & cons,
      • the piecewise linear method,
      • the central differences method,
      • the constant acceleration method,
      • the linear acceleration method.
  • 2020-03-24:
    Generalized SDOF
    • Assemblages of Rigid Bodies, Principle of Virtual Displacements
    • Deformable Bodies, Separation of Variables
    • Rayleigh Quotient Method
    • Refinements of Rayleigh Quotient Estimates
  • 2020-03-26:
    Tutorial and homework assignment #1
  • 2020-03-27:
    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
  • 2020-03-31:
    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
  • 2020-04-02:
    Modal Contribution Factors, Numerical Integration, Multiple Support Excitation.
    • Modal Partecipation Factors $\Gamma_i$
    • Modal Contributions to Load Vector
    • Modal Partecipation Factors
    • Static Correction
    • Wilson's $\theta$ method
    • Multiple Support Excitation
    • Influence Matrix
  • 2020-04-07
    Truncation, Matrix Iteration
    • modal responses as coefficients in eigenvector expansion of response
    • truncated eigenvector expansion
    • matrix iteration
      • is about refining equilibrium
      • converges to the first eigenvector
      • can be forced to converge to each eigenvector in sequence
    • Ritz Coordinates - reduction of dimensionality
    • Rayleigh Quotient as a function of Ritz coordinates - reduced eigenvalue problem
    • Choice of base - Subspace Iteration procedure
  • 2020-04-09:
    Tutorial and homework assignment #2
  • 2020-04-16
    Continuous Systems
    • Continuous Systems
    • Beams
    • EoM
    • Separation of Variables
    • Free Vibrations, Eigenvalue Problem, Orthogonality
    • Forced Response
  • 2020-04-17
    • Earthquakes
    • Response Spectra
    • Tripartite Plots
    • Idealized Response Spectra
    • Elastic Design Spectra
    • Inelastic Design
    • Inelastic Earthquake Response
    • Inelastic Response Spectra
    • Inelastic Design Spectra
  • 2018-04-19:
    Tutorial and homework assignment #3
  • -->

Organization of the course

This year we'll have 12 or 13 classes (depending on the end of the semester) starting March 10th and two of these classes will be tutorial sessions (the dates of the tutorials will be announced later).

You will receive your final mark after a colloquium with me (oral exam).
To be admitted to the oral exam you must pass a written test or submit a well received report on your final project(see below).

The written test provides that you have 4 hours to work on two or three problems designed to be solved using paper, pen and a hand-held calculator. An interesting fact is that you can use all the written and printed materials you can bring to the examination room…

Following each tutorial you will receive homework problems to solve (possibly working with your colleagues) and hand in. Even if the homeworks have no direct influence on the admission to the oral or on your final mark, they are useful to help you focus your preparation and get ready for either the common Computational Mechanics + Dynamics of Structures project (see below) or the written test.

At the end of the course (mid April) you will be assigned an individual project, designed to check your understanding of both Dynamics of Structures and Computational Mechanics. You will produce a report on your project and this report will be evaluated, possibly leading to your admission to the final exam.
If we'll be back to normality then, it is possible that the work on the project will be started in a common workshop with the teacher of Computational Mechanics and me.

The slides I use will be posted here after each class, as well as solved problems and other classroom materials [1], if you want to have a look at the slides before a class, last year's slides are a click away[2].

Please understand that the slides , etc, are not a substitute for a real textbook.

[1] Other classroom materials means, mostly, small computer programs.
[2] You can go back for quite a few years of materials modifying the URL above.


The 2 homeworks are strictly optional and have no direct influence on your final marks. On the other hand, working on them lets you test your understanding of the subject and prepare for either the project or the written test. You can find the homeworks and the solutions here.

Recommended textbooks

  • Anil K.Chopra, Dynamics of Structures (Theory and Applications to Earthquake Engineering), 4th ed.
  • Ray W. Clough, Joseph Penzien, Dynamics of Structures.
    This classic text is solely sold by the software house linked above.
    You can find an used copy with a bit of luck.

The course is mostly inspired by Clough and Penzien's book, but for many topics it follows Chopra's approach and examples. Should you prefer to buy a single book, my advice is "Buy Chopra's".

Note that

  • former editions of Chopra's book are perfectly OK for my course and can be found at lower prices, both used or new,
  • a fair number of copies of Chopra's book (3rd ed.) are available from the Campus Library in Lecco.

Homeworks, Final Project & Software

Most homework problems as well as the final project will require you to plot a time series or to perform some matrix algebra, but most spreadsheets (say Excel or Calc) can do these things.

However it may be simpler to write some computer programs that solve the problems, using Mathematica or MATLAB (or Matlab's free clone Octave) or Python or ... to perform the computations and generate the plots.

Politecnico's students can freely download the proprietary software I mentioned (Excel, Mathematica, MATLAB ) as well as many other proprietary applications and development tools, from a specific Politecnico site. OTOH Calc, Octave, Python etc are Open Source software.

The software examples that I'll discuss in class are in Python (really easy to read even for the inexperienced). To have an idea of the language here it is the solution of a recently assigned problem presented as a notebook, while the software used in tutorials is MATLAB .

Giacomo Boffi