Links to lessons

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

  • 2016-04-26 Applications to Earthquake Engineering: Response Spectra, Design Spectra, Ductility Spectra.
  • 2016-04-21 Continuous systems with an infinite number of degrees of freedom.
  • 2015-05-26 Derived Ritz Vectors, Direct Numerical Integration, Multiple Support Excitation.
  • 2016-04-14:
    Matrix Iteration again, Modal Contributions and Static Correction,
    • An application of the Ritz-Rayleigh procedure
    • the Subspace Iteration procedure
    • Modal Partecipation Factor
    • Modal contributions to the load vector
    • Modal contributions to pseudo-static response
    • Pseudo-displacements
    • Peak factors
    • Static correction
  • 2016-04-12:
    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
    • the Rayleigh-Ritz procedure
  • 2016-04-08:
    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
  • 2016-04-07:
    Step by Step methods, non linear

    Rayleigh Quotient method

    MDOF Systems
    • EoM for a 2 DOF System
    • 2DOF s-s response to harmonic excitation
    • Equation of Free Vibrations
    • Eigenvalues and Eigenvectors
    • Ortogonality of Eigenvectors
    • Modal Coordinates, Uncoupled EoM
  • 2016-03-31:
    Step by Step methods

    Generalized SDOF
    • Assemblages of rigid bodies, Principle of Virtual Displacements
    • Separation of variables, deformable bodies
  • 2016-03-18:
    Impulsive loads, intro to Step by Step methods
    • Impulsive Loads
    • definition, usual assumptions,
    • analytical solutions,
    • approximate analysis.
    • Step by Step Methods
    • general discussion, pro"cons,
    • e.g., the piecewise linear method
  • 2016-03-17:
    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.
  • 2016-03-11:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2016-03-10:
    Introduction - Free Vibrations
    • An introduction to dynamics of structures,
    • the single degree of freedom (SDOF) linear oscillator,
    • the free vibrations of SDOF oscillators.

Homework Assignments

January Homework Assignments

The submission term for the January Homework Assigment has expired

You can look at the solutions, 2 DOF System, Support Motion and Rayleigh Quotient.

I'd like to show you also an alternative procedure for the solution of the first problem.

The January Homework Assigment has been published.

To be admitted to February exams you must submit your solutions not later than January 31st and score more than 60 points over a maximum of 100.

Detailed instructions are enclosed in the text of the assignment.

August Homework Assignment

The August Homework Assigment's term has expired.

The solutions of the exercises are now partially available.

Rayleigh Quotient

Support Motion

The solution of the Isolation problem will folllow soon.

The August Homework Assigment has been published.

To be admitted to September exams you must submit your solutions not later than August 31st and score more than 60 points over a maximum of 110.

Detailed instructions are enclosed in the text of the assignment.

June Homework Assignment

The June Homework Assigment's term has expired.

The solutions of the exercises are now partially available.

Rayleigh Quotient

Support Motion

The solution of the Impact problem will folllow soon.

Objectives

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,
  • the numerical methods for the eigen-analysis of multiple degrees of freedom systems and
  • earthquake engineering applications.

Organization

This year we'll have 13 or 14 weekly classes, inclusive of tutorials and computational exercises.

All the slides used in classes will be (almost) immediately made available on this page after each class.

If you want to have a look at the slides before a class, last year's slides 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 here.

Exam

The exam is based on an oral test and a preliminary written test.
The oral test consists in a discussion of a few of the topics that we have touched during our classes.
The preliminar written test consists in a homework, comprising a variable number of problems, assigned ten days or so in advance of the exam call.
The final mark will be based mainly on the oral test.

Recommended books

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

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.

Note that a fair number of copies of Chopra's book (3rd ed.) are available from the campus library in Lecco, note also that Chopra's 2nd and 3rd editions are widely available at lower prices and are perfectly appropriate for my course.

Homeworks

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.
Of course specialized programs, let's say Mathematica, Matlab or Matlab's free clone Octave can be helpful, to say the least...

A pleasant alternative to proprietary stuff is the IPython notebook, an example of which is the solution of an exercise I've assigned in 2012.



















































Giacomo Boffi