Informations générales
ECTSECTS
2.5
Goal(s)
Formulation of problems of structural dynamics using
Strength of Materials (columns and beams) and Mechanics of Continua,
with restriction to linear mechanical behaviors.
Derivation of a discretised version of the problems with the Finite Element Method.
Worked-out examples that can be solved by hand.
Content(s)
Chap. I. Principle of Hamilton and Lagrange equations for discrete systems
I.1 Principle of virtual work
I.2 Generalized coordinates
I.3 Classification of generalized forces
I.4 Principle of Hamilton.
Exercices:
I.1 Double pendulum
I.2 System with 3 degrees of freedom
I.3 System rotating at constant speed
I.4 Small motions of a system of 3 rigid bars under gravity
I.5 Single pendulum: energy conservation
Chap. II Undamped vibrations
II.1 Small oscillations about an equilibrium. Stability of an equilibrium.
II.2 Normal modes of vibrations
II.3 Free vibrations with non homogeneous initial conditions
II.4 Forced harmonic vibrations. Dynamical influence matrix.
Exercices:
II.1 Coupled pendula: stability of equilibrium and flutter
II.2 Rotating mass
II.3 Buckling of a system composed of 3 rigid bars
Chap III Damped vibrations
III.1 Use of the eigenmodes of the underlying undamped system.
Small damping. Diagonal damping.
III.2 Responses in the set of state variables. Arbitrary and harmonic homogeneous cases.
Matrix of dynamical influence. Structural identification.
Exercices:
III.1 Caughey and Rayleigh dampings
III.2 Forced harmonic response in presence of small damping.
III.3 Criterion of phase quadrature.
III.4 Method of characteristic phase lags
III.5 Method of stationarity of the reactive power
III.6 System with one degree of freedom
Chap. IV Analysis via the finite element method of systems composed of bars.
IV.1 Vibrations of a bar as a continuum
IV.2 Vibrations of a bar as discretised by the finite element method.
IV.3 Discretisation of a system of bars in the space.
Exercices:
IV.1 A single bar, fixed at one end, loaded at the other end.
Analytical response, wave propagation and wave reflection.
IV.2 Discretisation and spectral analysis of a truss
Chap. V Analysis via the finite element method of systems made of beams
(not accounting for shear force).
V.1 Vibrations of a beam as a continuum.
V.2 Vibrations of a beam as discretised by the finite element method.
V.3 Discretisation of a planar system of beams.
Exercices:
V.1 Convergence of eigenvalues with a refinement of the mesh: beam clamped at both ends.
V.2 Spectral analysis of a beam under bending and compression.
V.3 Spectral analysis: influence of an isolated elastic support
V.4 Spectral analysis: influence of a continuous elastic support
V.5 Spectral analysis: influence of pre-stress
V.6 Spectral analysis of a clamped portal frame
Prerequisites :
Mechanics of Continua
Elasticity
Strength of Materials
Spectral analysis
Finite Element Method
Test
Written exam 2h
Calendar
S2
Additional Information
Lectures: 7X2h=14h Tutorials:4X2h=8h Laboratory testing: 10h
Bibliography
Dynamics of structures, R.W. Clough and J. Penzien,
Mac Graw Hill Int. Student Edition, 1982
Vibration Theory and Applications, W.T. Thomson, Prentice Hall, 1965
French State controlled diploma conferring a Master's degree
