Normal mode models

Contents Functions

5.2 Normal mode models

The spectral decomposition is a key notion for the resolution of linear differential equations and the characterization of system dynamics. Predictions of the vibrations of structures are typically done for linear elastic structures or, for non-linear cases, refer to an underlying tangent elastic model.

Spectral decomposition applied to elastic structures leads to modal analysis. The main objective is to correctly represent low frequency dynamics by a low order model whose size is typically orders of magnitude smaller than that of the finite element model of an industrial structure.

The use of normal modes defined by the spectral decomposition of the elastic model and corrections (to account for the restricted frequency range of the model) is fundamental in modal analysis.

Associated models are used in the normal mode model format

[[I] s² + [Γ] s + [Ω²]]{p(s)} = [φ^T b]{u(s)}

{y(s)} = [cφ] {p(s)}

(5.4)

where the modal masses (see details below) are assumed to be unity.

The nor2res, nor2ss, and nor2xf functions are mostly based on this model form (see nor2ss theory section). They thus support a low level entry format with four arguments

om	modal stiffness matrix Ω². In place of a full modal stiffness matrix om, a vector of modal frequencies freq is generally used (in rad/s if Hz is not specified in the type string). It is then assumed that om=diag(freq.`^`2). om can be complex for models with structural damping (see the section on damping ).
ga	modal damping matrix Γ (viscous). damping ratios damp corresponding to the modal frequencies freq are often used instead of the modal damping matrix ga (damp cannot be used with a full om matrix). If damp is a vector of the same size as freq, it is then assumed that ga=diag(2freq.damp). If damp is a scalar, it is assumed that ga=2dampdiag(freq). The application of these models is discussed in the section on damping ).
pb	modal input matrix {φ_j}^T[b] (input shape matrix associated to the use of modal coordinates).
cp	modal output matrix [c]{φ_j} (output shape matrix associated to the use of modal coordinates).

Higher level calls, use a data structure with the following fields

.freq	frequencies (units given by .fsc field, 2*pi for Hz). This field may be empty if a non diagonal nor.om is defined.
.om	alternate definition for a non diagonal reduced stiffness. Nominally om contains diag(freq.^2).
.damp	modal damping ratio. Can be a scalar or a vector giving the damping ratio for each frequency in nor.freq.
.ga	alternate definition for a non diagonal reduced viscous damping.
.pb	input shape matrix associated with the generalized coordinates in which nor.om and nor.ga are defined.
.cp	output shape matrix associated with the generalized coordinates in which nor.om and nor.ga are defined.
.dof_in	A six column matrix where each row describes a load by [SensID NodeID nx ny nz Type] giving a sensor identifier (integer or real), a node identifier (positive integer), the projection of the measurement direction on the global axes (if relevant), a Type.
.lab_in	A cell array of string labels associated with each input.
.dof_out	A six column matrix describing outputs following the .dof_in format.
.lab_out	A cell array of string labels associated with each output.

General load and sensor definitions are then supported using cases (see section 4.5.3).

Transformations to other model formats are provided using nor2ss (state-space model), nor2xf (FRFs associated to the model in the xf format), and nor2res (complex residue model in the res format). The use of these functions is demonstrated in demo_fe.

Transformations from other model formats are provided by fe2ss, fe_eig, fe_norm, … (from full order finite element model), id_nor and res2nor (from experimentally identified pole/residue model).