Links to lessons

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

  • 2018-02-27:
    Introduction - Free Vibrations
    • An introduction to dynamics of structures,
    • the single degree of freedom (SDOF) linear oscillator,
    • the free vibrations of SDOF oscillators.
  • 2018-03-01:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2017-03-06:
    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.
  • 2018-03-08:
    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.
  • 2018-03-13:
    Generalized SDOF
    • Assemblages of Rigid Bodies, Principle of Virtual Displacements
    • Deformable Bodies, Separation of Variables
    • Rayleigh Quotient Method
    • Refinements of Rayleigh Quotient Estimates
  • 2018-03-15:
    Tutorial and homework assignment #1
  • 2018-03-20:
    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
  • 2018-03-22:
    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
  • 2018-03-27
    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
  • 2018-04-05:
    Tutorial and homework assignment #2
  • 2018-04-10
    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
      •
  • 2018-04-12
    Continuous Systems
    • Continuous Systems
    • Beams
    • EoM
    • Separation of Variables
    • Free Vibrations, Eigenvalue Problem, Orthogonality
    • Forced Response
      •
  • 2018-04-17
    • 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

Written test, January 29 2019

You can find the text of the written test and the solutions

Written test, September 6 2018

Please find the text and the solution of the first problem.
PDFs of all the solutions are available too: Should you like it, you can see the original document/programs (and download the source code from the page top menu)

Written test, July 16 2018

The text of the written test and its solution.

Written test, July 02 2018

The text of the written test and its solution.

Homeworks

Homework #3. I have posted the solutions; were you interested in the source code, the notebooks used to generate the solutions, a couple of links are present inside the PDF document.

Homework #3, Errata Corrige:
the displacement for $\tau>2$ is not equal to $0$ but to $1$; I have also clarified the dimensionality of $u_\mathcal{A}$. The PDF file linked below has been updated to reflect the corrections.

2018-04-19: The text of the problems.

Homework #2, 2018-04-05: the text, the solution of problem n.1 and the solution of problem n.2.

Homework #1, 2018-03-15: the text and the solutions.

I have also separately posted the EP solution of the third problem , 'cause it's longish altough simple.

Exam

The exam consists in a classroom written test and an oral test.

The written test, mostly an admission test for the oral, consists in a set of problems that require no software for their solution.
The problems will be similar to the homeworks' ones (see below) but simpler. Last year's written tests, as well as reference solutions, are available on site.

The oral test consists in a Q&A session, you are expected to be able to discuss all the different topics of the course.

Organization of the course

This year we'll have 14 classes, starting February 27th and ending April 19th, three of which will be tutorial sessions (Thu Mar 15th, Thu Apr 5th and Thu Apr 19th)

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 the written test.

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

[1] Other classroom materials means, mostly, small computer programs.

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 & Software

Quite a few homework problems 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