2.3 Damping
Models used to represent dissipation at the local material level and
at the global system level should typically be different. Simple
viscous behavior is very often not appropriate to describe material
damping while a viscous model is appropriate in the normal mode model
format (see details in Ref. [3]).
2.3.1 Viscous damping in the normal mode model form
In the normal mode form, viscous damping is represented by the modal
damping matrix G which is typically used to represent all the
dissipation effects at the system level.
Models with modal damping assume that a diagonal G is
sufficient to represent dissipation at a system level. The non-zero
terms of G are then usually expressed in terms of damping
ratios Gjj = 2 
j 
j. The damping ratio
j are accepted by most SDT functions instead of
a full G. The variable name damp is then used instead of ga in the documentation.
For a model with modal damping, the matrices in (6.6) are diagonal so
that the contributions of the different normal modes are uncoupled and
correspond exactly to the spectral decomposition of the model (see
cpx
for the definition of complex
modes). The rational fraction expression of the dynamic compliance matrix
(transfer from the inputs {u} to displacement outputs {y})
takes the form
| |
[ (s)] =
|
| N |
 |
| j=1 |
|
|
=
|
| N |
|
| j=1 |
|
|
(2.5) |
where the contribution of each mode is characterized by
the pole frequency j, damping ratio j, and the residue
matrix Tj (which is
equal to the product of the normal mode output shape matrix {c 
j} by the normal mode input shape matrix {jTb}).
Modal damping is used when lacking better information. One will
thus often set a uniform damping ratio (j=1% or damp =
0.01) or experimentally determined damping ratios that are different
for each pole (po=ii_pof(po,3); damp=po(:,2);).
Historically, modal damping was associated to the proportional damping
model introduced by Lord Rayleigh which assumes the usefulness of a
global viscously damped model with a dynamic stiffness of the form
[
Z(
s)] = [
Ms2+(
M +
K)
s+
K]
While this model indeed leads to a modally damped normal mode model, the and
coefficients can only be adjusted to represent physical damping
mechanisms over very narrow frequency bands.
Using a diagonal [G] can introduce significant errors when
normal mode coupling through the spatial distribution of damping mechanisms is
possible. The condition
proposed by Hasselman [4], gives a good indication of when
modal coupling will not occur. One will note that a structure with a
group of modes separated by a few percent in frequency and levels of
damping close to 1% does not verify this condition. The uncoupling assumption can however still be applied to blocks of modes [5].
A normal mode model with a full G matrix is said to be non-proportionally damped and is clearly more general/accurate than
the simple modal damping model. The SDT leaves
the choice between the non-proportional model using a matrix ga
and the proportional model using damping ratio for each of the pole
frequencies (in this case one has ga=2*diag(damp.*freq) or ga=2*damp*diag(freq)
if a scalar uniform damping ratio is defined).
For identification phases, standard approximations linked to the
assumption of modal damping are provided by (id_rc, id_rm and
res2nor), while id_nor provides an original algorithm of the determination
of a full G matrix. Theoretical aspects of this algorithm and
details on the approximation of modal damping are discussed in
[5]).
2.3.2 Damping in finite element models
Standard damped finite element models allow the incorporation of viscous and structural damping in the form of real C and complex K matrices respectively.
fe_mk could assemble a viscous damping matrix with user defined elements that would support matrix type 3 (viscous damping) using a call of the form
fe_mk(MODEL,'options',3) (see section 7.14 for new element creation). Viscous damping models are rarely appropriate at the finite element level [3], so that it is only supported by celas and cbush elements. Piece-wise Rayleigh damping where the viscous damping is a combination of element mass and stiffness on element subsets
|
C = |
| NS |
|
| j=1 |
|
[jS MjS + jS KjS]
(2.6) |
model=demosdt('demogartfe');
model=stack_set(model,'info','Rayleigh', ...
[10 0 .01 0.0; ... % Elements of group 10 (masses)
9 0 0.0 0.1; ... % Elements of group 9 (springs)
0 1 0 .1; ... % Elements with MatId 1
0 2 0 .01]); % Elements with MatId 2
c=feutilb('Rayleigh',model); figure(1);spy(c);
Such damping models are typically used in time integration applications.
Structural or hysteretic damping represents dissipation by giving a loss factor at the element level leading to a dynamic stiffness of the form
|
Z(s) = [Ms2+K+iB] = Ms2 + |
| NE |
|
| j=1 |
|
[Kje] |
( |
1+i je |
) |
(2.7) |
.dt by 2p the maximum potential energy Ep=1/2E.
Using complex valued constitutive parameters will not work for most element functions. Hysteretic damping models can thus be assembled using the Rayleigh command shown above (to assemble the imaginary part of K rather than C or using upcom (see section 6.3). The following example defines two loss factors for group 6 and other elements of the Garteur FEM model. Approximate damped poles are then estimated on the basis of real modes (better approximations are discussed in [6])
clear global Up; upcom('load GartUp'); upcom('plotelt'); cf=feplot;
upcom('ParStackreset');
upcom('ParStackadd k','Constrained Layer','group6');
upcom('ParStackadd k','Main structure','group~=6');
% type cur min max vtype
par = [ 1 1.0 0.1 3.0 1 ; ...
1 1.0 0.1 3.0 1 ];
upcom('ParCoef',par);
% assemble using different loss factors for each parameter
B=upcom('assemble k coef .05 .01');
[m,k]=upcom('assemble coef 1.0 1.0');
Case=fe_case(Up,'getcase');
% Estimate damped poles on real mode basis
def=fe_eig({m,k,Case.DOF},[6 20 1e3]);
mr=def.def'*m*def.def; % this is the identity
cr=zeros(size(mr));
kr=def.def'*k*def.def+i*(def.def'*B*def.def);
[psi,lambda]=fe_ceig(mr,cr,kr);
cf.def={def.def*psi,def.DOF,lambda/2/pi}
Note that in this model, the poles 
j are not complex conjugate since the hysteretic damping model is only valid for positive frequencies (for negative frequencies one should change the sign of the imaginary part of K).
Given a set of complex modes you can compute frequency responses with res2xf, or simply use the modal damping ratio found with fe_ceig. Continuing the example, above one uses
Up=fe_case(Up,'Dofload','Point loads',[4.03;55.03], ...
'SensDof','Sensors',[4 55 30]'+.03);
Sens=fe_case(Up,'sens'); Load=fe_load(Up);
np=size(mr,1);
RES=struct('res',[],'po',ii_pof(lambda(7:np)/2/pi,3), ...
'idopt',idopt('new'));
RES.idopt.residual=2;RES.idopt.fitting='complex';
for j1=7:np; ... % deal with flexible modes
Rj=(Sens.cta*def.def*psi(:,j1)) * ... % c psi
(psi(:,j1).'*def.def'*Load.def); ... % psi^T b
RES.res(j1-6,:)=Rj(:).'; ...
end
% Rigid body mode residual
RES.res(end+1,:)=0;
for j1=1:6; ...
Rj=(Sens.cta*def.def(:,j1))*(def.def(:,j1)'*Load.def); ...
RES.res(end,:)=RES.res(end,:)+Rj(:).'; ...
end
IIw=linspace(5,60,2048);
nor2xf(def,[zeros(6,1);RES.po(:,2)],Up,IIw, ...
'hz iiplot XF(''Normal'') -reset -po');
[ci,XF]=iiplot;
XF('Res2xf')=res2xf(RES,IIw);
damp=[zeros(6,1);RES.po(:,2)];
def.data=sqrt(real(def.data.^2)).*sqrt(1+i*damp*2);
nor2xf(def,[],Up,IIw,'hz iiplot XF(''Hysteretic'')');
iicom('submagpha');
Note that the presence of rigid body modes, which can only be represented as residual terms in the pole/residue format (see section 2.6), makes the example more complex. The plot illustrates differences in responses obtained with true complex modes, viscous modal damping or hysteretic modal damping (case where one uses the pole of the true complex mode with a normal mode shape) . Viscous and hysteretic modal damping are nearly identical. With true complex modes, only channels 2 and 4 show a visible difference, and then only near anti-resonances.
To incorporate static corrections, you may want to compute complex modes on bases generated by fe2ss, rather than simple modal bases obtained with fe_eig.
The use of a constant loss factor can be a crude approximation for materials exhibiting significant damping. Methods used to treat frequency dependent materials are described in Ref. [7].
©1991-2007 by SDTools
