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The standard spectral decomposition discussed for state-space models in the previous section can be applied directly to second order models of structural dynamics. The associated modes are called complex modes by opposition to normal modes which are associated to elastic models of structures and are always real valued.

Left and right eigenvectors, which are equal for reciprocal structural models, can be defined by the second order eigenvalue problem,

[Mλ_{j}^{2}+Cλ_{j}+K] {ψ_{j}} = {0}
(5.16) |

In practice however, mathematical libraries only provide first order eigenvalue solvers to that a transformation to the first order form is needed. Rather than the trivial state-space form (5.10), the following generalized state-space form is preferred

| (5.17) |

The matrices M,C and K being symmetric (assumption of reciprocity), the generalized state-space model (5.17) is symmetric. The associate left and right eigenvectors are thus equal and found by solving

⎛ ⎜ ⎝ | [ |
| ] λ_{j} +
[ |
| ] | ⎞ ⎟ ⎠ | {θ_{j}} = {0}
(5.18) |

Because of the specific block from of the problem, it can be shown that

{θ_{j}} = { |
| } (5.19) |

where it should be noted that the name complex modeshape is given to
both θ_{j} (for applications in system dynamics) and ψ_{j} (for
applications in structural dynamics).

The initial model being real, complex eigenvalues λ_{j} come
in conjugate pairs associated to conjugate pairs of
modeshapes {ψ_{j}}. With the exception of systems with real poles,
there are 2N
complex eigenvalues for the considered symmetric systems (ψ_{[N+1 … 2N]} = ψ_{[1 … N]} and λ_{[N+1 … 2N]} = λ_{[1 … N]}).

The existence of a set of 2N eigenvectors is equivalent to the verification of two orthogonality conditions

| (5.20) |

where in (5.20) the arbitrary diagonal matrix was chosen to be the identity because it leads to a normalization of complex modes that is equivalent to the collocation constraint used to scale experimentally determined modeshapes ([12] and section 2.8.2).

Note that with hysteretic damping (complex valued stiffness, see section 5.3.2) the modes are not complex conjugate but opposite. To use a complex mode basis one thus needs to replace complex modes whose poles have negative imaginary parts with the conjugate of the corresponding mode whose pole has a positive imaginary part.

For a particular dynamic system, one will only be interested in
predicting or measuring how complex modes are excited (modal input
shape matrix {θ_{j}^{T}B}={ψ_{j}^{T}b}) or observed (modal output
shape matrix {Cθ_{j}}={cψ_{j}}).

In the structural dynamics community, the modal input shape matrix
is often called modal participation factor (and noted L_{j}) and the
modal output shape matrix simply modeshape. A different
terminology is preferred here to convey the fact that both notions are dual
and that {ψ_{j}^{T}b_{l}}={c_{l}ψ_{j}} for a reciprocal structure
and a collocated pair of inputs and outputs (such that uẏ is
the power input to the structure).

For predictions, complex modes can be computed from finite element models using fe_ceig. Computing complex modes of full order models is typically not necessary so that approximations on the basis of real modes or real modes with static correction are provided. Given complex modes, you can obtain state-space models with res2ss. For further discussions, see Ref. [31] and low level examples in section 5.3.3.

For identification phases, complex modes are used in the form of residue matrices product [R_{j}]={cψ_{j}}{ψ_{j}^{T}b} (see
the next section). Modal residues are obtained by id_rc and separation of the modal input and output parts is obtained using id_rm.

For lightly damped structures, imposing the modal damping assumption, which forces the use of real modeshapes, may give correct result and simplify your identification work very much. Refer to section 2.8.3 for more details.

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