For many problems this method combines the best features of the real eigensolution methods and the Complex Hessenberg method. The real eigensolutions from the structural stiffness and mass matrices are found first. The Lanczos and SINV real eigenvalue methods use a Sturm sequence technique that guarantees that all roots in a range have been found or identified as being missing. The tridiagonal real eigensolution methods also find all roots of the problem, so that none can be missing. Unfortunately, there is no similar technique for roots in the complex plane. Using a modal approach provides the following benefits:
n4
r⁄10
( )nb2 b r
1. The user can be assured that all the undamped roots of the system in a region specified by the user have been found with any of the real methods.
2. The complex eigenvalues are usually of the same order of magnitude as the real eigenvalues.
3. The real eigenvectors are then used to reduce the damping matrix and any other special dynamic effects that may be present to h-set size matrices.
4. The h-set mass matrix is guaranteed to be well conditioned for the inverse operation unless very strange effects are input with the M2PP matrix or the counterpoint terms of transfer functions, so there is no need to omit massless physical DOF, a task which can be laborious on a complicated model. (It is necessary, however, to ensure that any extra points that are used have nonnull mass terms.)
5. All eigensolutions are then found in this reduced basis. The user is assured that there are no solutions which have been skipped over.
6. The economic trends are also favorable because the real eigensolutions do take advantage of matrix sparsity and banding (the Real Lanczos Method is the best in this respect today), and they reduce the problem to a much smaller size before the Hessenberg solution is attempted.
However, if strong nonstructural effects are present, such as those caused by direct input matrices or transfer functions, a large number of real modes may be necessary to account for these forces. An example would be a servo control that acts as a large mass on the structure. This mass could reduce the frequency of some local high frequency modes and move them into the important range.
The Complex Lanczos and Inverse Power methods may also be used in the modal method, but the economic rewards are less, and they may also miss roots. They should be used as backup methods in this context, used to check the solution produced by the Hessenberg method when there is some question about its correctness.
User Interface
Various form of the eigenvalues are used as inputs and are provided as outputs. The real eigensolution outputs the eigenvalues in three forms. The complex eigenvalue, , is formed from the real quantities and where . If , the radian frequency,
of complex eigenvalue analysis is the same as that of real eigensolutions. The real part, , is a measure of the decay rate of a damped structure, or if negative, the rate of
divergence of an unstable system. The imaginary part, , is the modified frequency in radians/unit time. However, roots with negative values of should be treated as special terms. The output labeled f is the circular frequency in cycles per unit time. It is equal to
.
The CMETHOD = [SID] Case Control command selects the EIGC Bulk Data entry, which has the form
p α ω p = α+iω α = 0. ω α ω ω ω⁄( )2π
NORM, G, and C have to do with the specification of the method of normalization. The eigenvectors may be normalized either to a unit value at grid point G for coordinate C, or for the largest term to be of unit magnitude. E is used to specify the convergence criterion of the solution. Each method has a different default value for this criterion, and each is adequate for most problems. ND1 is the number of eigenvectors to be computed by the Hessenberg method. Data in this field is only allowed when there are no continuation entries, while the Hessenberg method ignores the data on the continuation entries. The shift points for the Lanczos method are defined on the continuation entries. The recommended practice is to specify one point and at the lower bound of the expected range of eigenvalues, but not at 0.0 A second shift may be input at and
at the upper bound of the expected range. All ALPHABj and OMEGABj must be blank. The number of Ritz vectors computed for the J-th shift is three times the number entered in NDJ. While each Ritz vector may be capable of defining an eigenvector, it is more likely that some of the Ritz vectors will be discarded because they are not orthogonal to
lower-numbered Ritz vectors during the Lanczos reduction process. After the poor vectors are discarded, the remaining vectors are used to compute the eigensolutions.
The number of modes computed from all shifts up to the j-th shift may be more or less than NDJ. If there are less, the processing continues with the next shift; otherwise, the process ends and all eigensolutions are output. The ratio of discarded vectors to retained vectors is problem dependent. However, the recommended practice is to ask for all of the desired eigenvectors for ND1 on the first shift, but put in additional shifts. If it is possible to compute all solutions with one shift, this is the most economical option. If more shifts are needed, they will be used.
The subregions for searching for roots with the Inverse Power method are also specified on the continuation entries. (ALPHAAJ, OMEGAAJ) define one point in the complex plane, and (ALPHABJ, OMEGABJ) a second point. A line is drawn between these points, and a box of width LJ is placed around this line. NEJ is larger than the number of roots expected in the subregion, and NDJ the number desired. This defines the first subregion for searching for eigenvalues. More continuation entries may be used to define more subregions. Again, if more eigenvalues are computed than are requested, all are output. The eigenvalue output for a sample problem is shown below:
C O M P L E X E I G E N V A L U E S U M M A R Y
ROOT EXTRACTION EIGENVALUE FREQUENCY DAMPING NO. ORDER (REAL) (IMAG) (CYCLES) COEFFICIENT 1 9 -5.806441E+01 5.750383E+03 9.152019E+02 2.019497E-02 2 8 -6.294888E+01 6.293917E+03 1.001708E+03 2.000309E-02 3 7 -6.910709E+01 6.844852E+03 1.089392E+03 2.019243E-02 4 1 -3.300980E+02 1.667092E+04 2.653260E+03 3.960164E-02 5 5 -3.565692E+02 1.823559E+04 2.902285E+03 3.910694E-02 . . .
1 2 3 4 5 6 7 8 9 10
EIGC SID METHOD NORM G C E ND1
ALPHAAJ OMEGAAJ ALPHABJ OMEGABJ LJ NEJ NDJ
α1 = 0.0 ω1
α2 = 0.0
The column labeled (REAL) contains , and the column labeled (IMAG) contains . The column labeled (FREQUENCY) contains the circular frequency. The last column is the damping coefficient computed from the equation
which is approximately twice the value of the conventional modal damping ratio. This form was more popular with the aeroelastic flutter specialists who were the primary users of this capability.
Note that if the magnitude of this term is computed to be less than , it is reset to zero.
For small values, the damping coefficient is twice the fraction of critical damping for the mode. The eigenvalues are sorted on , with the negative values sorted first (there are none in this example), sorted on increasing magnitude, followed by the eigenvalues with positive , again sorted on magnitude. Roots with equal values are sorted next on .