fe_ceig

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fe_ceig

Purpose

Computation and normalization of complex modes associated to a second order viscously damped model.

Syntax

[psi,lambda] = fe_ceig( ... )
lambda       = fe_ceig(m,c,k)
def          = fe_ceig( ... )
        ...  = fe_ceig(m,c,k)
        ...  = fe_ceig({m,c,k,mdof},eigopt)
        ...  = fe_ceig({m,c,k,T,mdof},eigopt)
        ...  = fe_ceig(model,eigopt)

Description

Complex modes are solution of the second order eigenvalue problem (see section 5.5 for details)

[M] _N× N {ψ_j} _N× 1 λ_j² + [C] {ψ_j} λ_j + [K] {ψ_j} = 0

where modeshapes psi=ψ and poles Λ=[^\ λ_j _\ ] are also solution of the first order eigenvalue problem (used in fe_ceig)

[

C	M
M	0

]_2N × 2N [

ψΛ

]_2N × 2N [Λ]_2N × 2N + [

K	0
0	−M

] [

ψΛ

]= [0]_2N × 2N

and verify the two orthogonality conditions

ψ^TCψ + Λψ^TMψ + ψ^TMψ Λ = I and ψ^TKψ − Λψ^TMψΛ = −Λ

[psi,lambda] = fe_ceig(m,c,k) is the old low level call to compute all complex modes. For partial solution you should use def = fe_ceig(model,ceigopt) where model can be replaced by a cell array with {m,c,k,mdof} or {m,c,k,T,mdof} (see the example below). Using the projection matrix T generated with fe_case('gett') is the proper method to handle boundary conditions.

Options give [CeigMethod EigOpt] where CeigMethod can be 0 (full matrices), 1 (real modes then complex ones on the same basis) 2 and 3 are refined solvers available with the VISCO extension. EigOpt are standard fe_eig options.

Here is a simple example of fe_ceig calls.

femesh('reset');
model=femesh('testubeam');model.DOF=[];
model=fe_case(model,'fixdof','Base','z==0');
[m,k,model.DOF]=fe_mk(model,'options',[0 2]);
Case=fe_case(model,'gett');

kc=k*(1+i*.002); % with hysteretic damping
def1=fe_ceig({m,[],kc,model.DOF},[1 6 10 1e3]);       % free modes
def2=fe_ceig({m,[],kc,Case.T,Case.DOF},[1 6 10 1e3]); % fixed modes

See also

fe_eig, fe_mk, nor2ss, nor2xf, section 5.3