Structural Dynamics: Structural Damping Flashcards
Dicuss the nature of dissipation forces.
All structures have a mechanism that leads to the dissipation of energy. The dissipation is responsible for the decay of harmonic oscillations when set in motion.
There are several sources of dissipation:
1) Dissipation associated with straining of the material, due to viscoelastic behaviour of the constitutive law (very limited in a metallic material, more relevant in composite structures due to matrix and micro-sliding of fibres)
2) Dissipation due to friction for sliding in inner joints, support connections etc.
What is the Kelvin-Voigt viscoelastic behavior?
We have a case where a spring and a damper are in parallel.
(ue=ud=u, F=Fe+Fd)
F = ku + cu. The solution for F const is u = -F/k(1 - e^-k/ct). The slope of the curve is equal to k/c.
What is the Maxwell viscoelastic behavior?
We have a case where a spring and a damper are in series.
(F=Fe=Fd, u=ue+ud)
u_dot = F_dot/k + F/c. Applying a constant force, we can see creeping behaviour (after the initial deformation given by the elastic part, there is a slow increase of the displacement). If u is const, then the behaviour is damped.
What is equivalent damping?
The behaviour of dissipation is complex, nonlinear, not deterministic and characterized by a dependance on many parameters and operative conditions, consequently, it is extremely difficult to be modelled starting from first principles.
For small aplitude oscillations, the mechanism can be represented through a linear contribution proportional to the time derivative of the free coordinates. Fd = Cu_dot.
Discuss about the viscous damping
Calculateing the reaction force on the ground R for a harmonic damped oscillator, we can calculate the energy dissipated which is: Wc = πcxo^2ωn, which indicates that the viscous damping is proportional to the frequency. Wk = 1/2kx0^2 is equal to the energy associated with the elastic forces.
The energy dissipation coefficient ed can be defined which is equal to the dissipative forces divided by 2pi times the elastic forces, which ultimately is equal to 2ξ. Therefore, the equivalent damping can be solved as ξ=Wc/4piWk.
In principle, it is reasonable to considure the possibility to evaluate the intrinsic material damping. We can perform oscillatory tests and identify some equivalent damping that could be sufficient to characterize our material.
What is modal damping?
Mq_dot_dot + Cq_dot + Kq = 0, q = Φz
ΦTMΦz_dot_dot + ΦΤCΦz_dot + ΦΤKΦz = 0. The normal damping is defined as the case where the modal damping matrix is diagonal ie ΦΤCΦ = [2ξiμiωi] , with ξi the modal damping ratio.
The hypothesis is that C is diagonal. The displacement due to a damping force of a single mode doesnt affect the displacement of the other modes. There is no reason to say that C is diagonal, however, we can see experimentally that the extra-diagonals terms are very small, so neglectable.
What is Rayleigh damping?
We suppose that the damping matrix is a linear combination of the mass and stiffness matrix:
C = αM + βK -> ξi = 1/2(α/ωi + βωi)
The constants α and β could be computed starting from the experimental evaluation of modal damping factors ξi. The two constants are sufficient to obtain a perfect modal diagonal damping only if two modes are consisdered, otherwise a “best fit” that minimized the error (eventually weighted to approximate the most important mode) should be used.
What is Coulomb friction?
Damping comes out from sources like viscoelastic behaviour, but of course it can come also from friction.
Fd = sign(x_dot)μN, with N the normal reaction force and μ the friction force. Wk = 1/2kx0^2 and Wc = 4μNx0. Consequently the equivalent damping ξ can be calculated.
The behaviour is highly nonlinear, as when the position x0 is changed, the amplitude of the force is changed and therefore the equivalent damping is different.
What is hysteretic damping?
A lot of researchers identified that many structured behave differently from either a proportional damping behavior or coulomb damping behavior, showing a behavior that is different from the two.
Indeed, most structures show a behaviour that is hysteretic, characterized by a damping factor that is proportional to the square of the amplitude (nonlinear and independent of the frequency). The hysteretic damping is often represented as a complex stiffness (in frequency domain) showing a dependence on amplitude x and not on frequency. η is the loss factor.
Therefore: Wc = πhx0^2, where h is a generic hysteric damping coefficient and Wc is proportional to the amplitude. So the equivalent damping can be found, the damping forces and the equation of motion.
