Mode-superposition method is a method of using the natural frequencies and mode shapes from the modal analysis (ANTYPE,MODAL) to characterize the dynamic response of a structure to transient (ANTYPE,TRANS with TRNOPT,MSUP, Transient Analysis), or steady harmonic (ANTYPE,HARM with HROPT,MSUP, Harmonic Analysis) excitations.
The following topics are available:
The equations of motion may be expressed as in Equation 15–5:
 | (14–122) |
is
the time-varying load vector, given by
 | (14–123) |
where:
 = time varying nodal forces |
 = load vector scale factor (input
on LVSCALE command) |
 = load
vector from the modal analysis (see below) |
The load vector 
is computed
when doing a modal analysis and its generation is the same as for
a substructure load vector, described in Substructuring Analysis.
The following development is similar to that given by Bathe ([2]):
Define a set of modal coordinates 
such that
 | (14–124) |
where:
 = the ith mode shape |
 = the number of modes to be used (input as MAXMODE on TRNOPT or HROPT commands) |
Note that Equation 14–124 hinders
the use of nonzero displacement input, since defining 
in terms of 
is not straight forward. The inverse relationship
does exist (Equation 14–124) for the
case where all the displacements are known, but not when only some
are known. Substituting Equation 14–124 into Equation 14–122,
 | (14–125) |
Premultiply by a typical mode shape 
:
 | (14–126) |
The orthogonal condition of the natural modes states that
 | (14–127) |
 | (14–128) |
In the mode-superposition method using the Lanczos and other extraction methods, only Rayleigh or constant damping is allowed so that:
 | (14–129) |
Applying these conditions to Equation 14–126, only the i = j terms remain:
 | (14–130) |
The coefficients of
,
, and 
, are derived as follows:
Coefficient of
: By the normality condition (Equation 15–54),
(14–131)
Coefficient of
: The damping term is based on treating the modal coordinate as a single DOF system (shown in Equation 14–122) for which:
(14–132)
and
(14–133)
Equation 14–133 can give a definition of
: 
(14–134)
From (Tse [69]),
(14–135)
where:
= fraction of critical damping
for mode j and,
(14–136)
where:
= natural circular frequency of
mode j Combining Equation 14–134 thru Equation 14–131 with Equation 14–132,
(14–137)
Coefficient of yj:
From Equation 15–51,
(14–138)
Premultiply by
,
(14–139)
Substituting Equation 14–131 for the mass term,
(14–140)
For convenient notation, let
(14–141)
represent the right-hand side of Equation 14–130. Substituting Equation 14–131, Equation 14–137, Equation 14–140 and Equation 14–141 into Equation 14–130, the equation of motion of the modal coordinates is obtained:
(14–142)
Since j represents any mode, Equation 14–142 represents n uncoupled equations in the n unknowns
. The advantage of the uncoupled system
(ANTYPE,TRAN with TRNOPT,MSUP)
is that all the computationally expensive matrix algebra has been
done in the eigensolver, and long transients may be analyzed inexpensively
in modal coordinates with Equation 14–124. The
are converted back into geometric displacements 
(the system response to the loading) by using Equation 14–124. That is, the individual modal
responses 
are superimposed to obtain the actual response,
and hence the name "mode-superposition".
For the QR damped mode extraction method, the general equations apply except the differential equations of motion in modal coordinate are deduced from Equation 14–230 with the right hand side force vector of Equation 14–126. They are written as:
 | (14–143) |
where:
 = real eigenvector matrix normalized with
respect to mass coming from the LANCZOS run of QRDAMP (see QR Damped Method for more details). |
 = diagonal matrix containing the eigenvalues  . |
 = unsymmetric part of the stiffness matrix. |
It can be seen that if 
is
arbitrary and/or 
is
unsymmetric, the modal matrices are full so that the modal equations
are coupled.
When using the unsymmetric eigensolver, the matrices are unsymmetric. Both left and right normalized eigenmodes are used to decouple the modal equations as follows.
Equation 14–130 becomes:
 | (14–144) |
where:
= the jth left mode shape
= the jth right mode shape (also called
the jth mode shape). For more information,
see Unsymmetric Method.
This equations leads to the modal coordinate equation (Equation 14–142) when Rayleigh damping is considered.
The modal damping, 
, is the combination of several Mechanical APDL damping inputs, as described in Equation 14–41.
In modal superposition analysis, the dynamic response will be approximate when the applied loading excites the higher frequency modes of a structure. To improve the accuracy of the dynamic response, the residual vector method employs additional modal transformation vectors (designated as residual vectors) in addition to the eigenvectors in the modal transformation (Equation 14–124).
The residual vector method (RESVEC,ON) uses extra residual vectors computed in the modal analysis (ANTYPE,MODAL) to characterize the high frequency response of a structure to dynamic loading. It applies to the following mode-superposition analyses:
Because of the improved convergence properties of this method, fewer eigenmodes are required from the eigensolution.
The dynamic response of the structure can be divided into two terms:
 | (14–145) |
where:
 = lower mode contributions (Equation 14–124) |
 = higher mode contributions, which can
be expressed as the combination of residual vectors. |
First, the flexibility matrix can be expressed as:
 | (14–146) |
where:
 = generalized inverse matrix of stiffness matrix [K] (see Geradin and Rixen
[365]) |
 = elastic normal modes |
 = total degree of freedom of the
system |
|
The residual flexibility matrix is given by:
 | (14–147) |
Define residual vectors as:
 | (14–148) |
where:
 =
matrix of force vectors |
Orthogonalize the residual vectors with respect to the retained
elastic normal modes gives orthogonalized residual vectors 
, that is, pseudo-modes with associated frequencies
that are orthogonal to the mass and stiffness matrices (see Dickens
et al. [420]).
Then the basis vectors for modal subspace are formed by:
 | (14–149) |
This method also applies to the component mode synthesis (CMS) generation pass (ANTYPE,SUBSTR with CMSOPT) when a load is applied to internal degrees of freedom (all DOFs that are not master DOFs) of the substructure. The basis of normal modes used in the three available CMS methods is updated following Equation 14–149 to include the resulting residual vector.
If rigid-body modes exist, pseudo-constraints are required for the calculation of Equation 14–148. Issue the D,,,SUPPORT command to specify only the minimum number of pseudo-constraints necessary to prevent rigid-body motion. In Equation 14–148, the matrix of force vectors [F] is replaced by
 | (14–150) |
where:
: rigid-body modes normalized to the mass matrix 
.
The residual vector method should not be used when the force vectors [F] are only due to inertia loads (such as ACEL). In this case, the matrix of force vectors in Equation 14–150 is near zero, leading to unstable residual vectors.
In mode-superposition analysis, the dynamic response is approximate when the applied loading excites the higher frequency modes of a structure. To improve the accuracy of the dynamic response, the residual response method can be used. It is similar to the residual vector method except that it directly uses the residual static responses calculated with residual flexibility.
The residual response method (RESVEC,,,,,ON) applies to the following mode-superposition analyses:
When the equations are symmetric, the residual responses are defined by Equation 14–148.
When the equations are unsymmetric (see Equations for Unsymmetric Eigensolver Based Analysis), Equation 14–146 becomes:
 | (14–151) |
The residual flexibility matrix is given by:
 | (14–152) |
The residual responses are calculated with Equation 14–148 with the residual flexibility defined by Equation 14–152.
If rigid-body modes exist, see comments at the end of Residual Vector Method.