PDF Index| SDT-visc
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Functions
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For an input [b]{u(s)} characterized by the frequency domain characteristics of u(s) and spatial content of b, component mode synthesis and substructuring methods provide approximations of the solution of problems (??), (??), or (??).
For damped problems, one should distinguish
All the problems that where introduced earlier can, in the frequency domain, be represented as frequency response computations of the form
| (2.44) |
which involve the inverse of the dynamic stiffness Z(s).
In the very general case where complex moduli are supposed to be analytic functions in the complex plane (locally regular), the dynamic stiffness Z(s) is also an analytic function. Observation and input matrices c and b being constant, the poles (i.e. singularities) of H(s) correspond to non zero solutions of
| (2.45) |
which defines the generalized non linear eigenvalue problem associated with a viscoelastic model.
Near a given pole, analytic functions have a unique Laurent's development
| (2.46) |
where γ is a arbitrary closed direct contour of the singularity λ.
As a result, near an isolated pole λj one has
| (2.47) |
where the normalization coefficient αj depends on the choice of a norm when solving (??) and is determined by
| (2.48) |
To simplify writing, it is desirable to use αj=1 which si the usual scaling for constant matrix eigenvalue problems described in the next section.
Having determined the set of poles in a given frequency band, having normalized the associated modes so that αj=1, one obtains a first order development in s
| (2.49) |
where the [c][Z(0)]−1 [b] terms correspond to the exact static response to loads associated with the input shape matrix b. This static correction term is well known in component mode synthesis applications and is analyzed in section ??.
It is often useful to consider a representation using residual flexibility
| (2.50) |
The solution of the non linear eigenvalue problem (??) is difficult (see section ??). Solution algorithms are thus greatly simplified by restating the problem as a classical first order eigenvalue problem with constant matrices.
For models with viscous and structural damping (??) or viscoelastic models of form (??), on thus generally solves the eigenvalue problem associated with (??), that is
| (2.51) |
Because of the block form of this problem, one can show that
| (2.52) |
and one thus gives the name complex mode both to θj and ψj.
The existence of 2N eigenvectors that diagonalize the matrices of (??) is equivalent to the verification of two orthonormality conditions
| (2.53) |
For a model represented in is state space form, such as (??), one solves the left and right eigenvalue problems
| (2.54) |
and uses standard orthonormality conditions
| (2.55) |
In the complex plane, one should distinguish complex poles associated with vibration modes and real poles which can be related with a damping ratio above 1 (supercritical damping that is not common) or to material relaxation (be linked to poles of the complex modulus). To understand this distinction one considers the viscoelastic oscillator (??). The viscoelastic behavior leads to one real pole β, whereas supercritical damping corresponds to ζ>1 and leads to two real poles
| (2.56) |
In a viscoelastic computation where the constitutive law contains real poles the root locus of the solution is similar to that shown in figure ??. One must thus distinguish the classical spectrum of vibration modes and the real poles associated with material relaxation
It is important to note that the number of real poles is directly associated with the number of DOFs in the internal states qv. The number of these poles will the increase with mesh refinement. In practice, one thus cannot expect to compute the real poles and associated modes in the area of relaxation poles. This selection of convergence area is the aspect that needs to be accounted for in the development of partial eigenvalue solvers for damped problems.
A second consequence of the increase in the number of real poles is the potential impossibility to compute modes with supercritical damping. The author's experience is that this limitation is mostly theoretical since in practice modes with supercritical damping are rare.
The complex modulus, being the Fourier transform of a real valued relaxation function, should be symmetric in frequency (E(−ω)=(E(ω)*). For a modulus representation that does not verify this hypothesis, as is the case of structural damping, only poles with positive imaginary parts have a meaning. For response synthesis, one will thus take the conjugates λj*,ψj* of modes computed with positive imaginary parts.
The computation of complex modes can be used to approach transfer functions using the developments (??) and (??). Orthogonality conditions given above correspond to the αj=1 normalization.
To simulate the dynamic response it is not useful and rarely possible from a numerical cost standpoint to use the full model (??) for direct time simulations (see section ??). Model reduction methods (modal analysis, substructuring, component mode synthesis, ...) seek an approximate solution within a restricted subspace. One thus assumes
| (2.57) |
and seek solution of (??) whose projection on the one the dual subspace TT is zero (this congruent transformation corresponds to a Ritz-Galerkin analysis). Transfer functions are the approximated by
| (2.58) |
One can note that for a non-singular transformation T (when {q} =[T] {qR} is bijective) the input u / output y relation is preserved. One says that the transfer functions are objective quantities (they are physical quantities that are uniquely defined) while DOFs q are generally not objective.
Classical bases used for model reduction combine modes and static responses to characteristic loads []. One distinguishes
| (2.59) |
For component mode synthesis (component model reduction to prior to a coupled system prediction), free modes have been used by [], MacNeal [], and many others.
| (2.60) |
For CMS, the use of static terms only is called Guyan condensation []. Adding fixed interface modes leads to the Craig Bampton method [].
| (2.61) |
Section ?? showed that the modal damping assumption was justified under the hypothesis of modal separation (??). When dealing with a real basis, it is often possible to compute an equivalent viscous damping model approximating the damped response with good accuracy.
To build this equivalence, one distinguishes in reduction bases (??), (??), or other, blocs Tm associated with modes whose resonance is within the frequency band of interest and Tr associated with residual flexibility terms. As shown in figure ??, one is interested in the modal contribution of the first while for the others only the asymptotic contribution counts.
For a real basis reduction, one is thus interested in an approximation of the form
| (2.62) |
The objective in distinguishing Tm and Tr is to guarantee that TrT Z(s) Tr does not present singularities within the band of interest (it represents residual flexibility not resonances).
To validate this hypotheses, one defines a reference elastic problem characterized by a stiffness K0 (one will use K0=Re(Z(ω0))+Mω02 avec ω0 with typically better results for a higher frequency value).
By computing modes in a subspace generated by Tr of the representative elastic problem
| (2.63) |
one can verify this uncoupling: if ω1r is within the band of interest the decoupling is not verified.
One can always choose bases Tm and Tr so as to diagonalize the reference problem. Furthermore, by writing Δ Z(s) = Z(s)−K−s2M, the transfer function is approximated by
| (2.64) |
The Modal Strain Energy method (MSE) with damping ratio given by (??), corresponds to the following approximation
| (2.65) |
One can easily generalize this approximation by building an equivalent viscous damping matrix by enforcing
| (2.66) |
for a characteristic frequency ωjr for diagonal terms (j=k) (ωjr+ωkr)/2 otherwise. For a viscous damping model, it is a simple projection (computation of TmT C Tm). For a structural damping model, there is a degree of approximation.
The validity of this approximation is discussed in [] where it is shown that building the equivalence in generalized coordinates and using term by term characteristic frequencies is efficient.
For coupling terms TmT Z(s) Tr, damping only has low influence (see the discussion on non proportional damping in section ??) and can thus be neglected.
For residual terms TrT Δ Z(s) Tr damping can be neglected for frequency analyses. For transient analyses, the presence of high frequency undamped modes (linked to the ωjr) induces non physical oscillations since these modes are introduced to approximate low frequency contributions and not high frequency resonances. It is thus good practice to introduce a significant modal damping for residual modes. For example one uses ζ=1/√(2), which leads to assume
| (2.67) |
The reduction is also applicable for viscoelastic models detailed in section ??. Indeed all matrices used in the formulation of the dynamic stiffness (??) can be projected. State-space or second order representations can be generated by replacing each matrix by its reduced version M by TTMT, etc. This reduction form will be used for eigenvalue solvers discussed in section ??.
The equivalent viscous damping model building strategy detailed in the previous section cannot be generalized since the real part of TmT Δ Z(s) Tm undergoes significant variations as the storage modulus changes with frequency. In other terms, the ωjm associated with K0 differ, possibly significantly, for frequencies of the non linear eigenvalue problem [Re(K(ωj))−ωj2M]{φj}=0. For modal synthesis or transient response computations, one will thus prefer representations associated with the spectral decomposition (??) as detailed in section section ??.
