Civil Engineering Reference
In-Depth Information
There are 10 equations, the half-bandwidth is 3
and the skyline storage is 31
The eigenvalues are:
0.3823E-02 0.1278E+00 0.8511E+00 0.2765E+01 0.6323E+01 0.1147E+02
0.1748E+02 0.2324E+02 0.2756E+02 0.3225E+02
Eigenvector number 1 is:
0.6303E-01 0.1499E+00 0.2275E+00 0.2539E+00 0.4579E+00 0.3154E+00
0.7229E+00 0.3423E+00 0.1000E+01 0.3485E+00
Eigenvector number 2 is:
0.2307E+00 0.4535E+00 0.5431E+00 0.2347E+00 0.5094E+00 -0.3411E+00
0.2270E-01 -0.8308E+00 -0.7287E+00 -0.1000E+01
Eigenvector number 3 is:
0.2888E+00 0.4155E+00 0.3179E+00 -0.3976E+00 -0.1579E+00 -0.5696E+00
-0.2684E+00 0.3779E+00 0.3425E+00 0.1000E+01
Figure 10.3 Results from Program 10.1 example
2
) are then calculated using Jacobi's method, which
employs library subroutines
bandred
and
bisect
.
The first
nmodes
eigenvectors are then extracted by finding the relevant non-trivial
solutions to the homogeneous equations and transforming them back to the correct vector
space. In order to do this the global matrix is converted to skyline vector storage form (
kv
)
and the eigenvectors solved for using subroutines
sparin_gauss
and
spabac_gauss
.
In this example, the first three eigenvectors are computed and normalised to a vector length
(Euclidean norm) of unity. The results are all shown in Figure 10.3.
As a check, the computed results indicate a fundamental frequency
eigenvalues of this band matrix (
ω
ω
=
√
0
.
0038
=
0
062 which should be compared with the analytical result (e.g. Chopra, 1995) for a slender
beam of,
.
EI
ρA
=
3
.
516
L
ω
=
0
.
063
(10.4)
2
Program 10.2 Eigenvalue analysis of an elastic solid in plane strain using 4- or 8-node
rectangular quadrilaterals. Lumped mass. Mesh numbered in
x
-or
y
-direction.
PROGRAM p102
!-------------------------------------------------------------------------
! Program 10.2 Eigenvalue analysis of an elastic solid in plane strain
! using 4- or 8-node rectangular quadrilaterals. Lumped mass.
! Mesh numbered in x- or y-direction.
!-------------------------------------------------------------------------
USE main; USE geom; IMPLICIT NONE
INTEGER,PARAMETER::iwp=SELECTED_REAL_KIND(15)
INTEGER::i,idiag,iel,ifail,j,k,nband,ndim=2,ndof,nels,neq,nmodes,nn,nod, &
nodof=2,nprops=3,np_types,nr,nxe,nye
REAL(iwp)::area,etol=1.e-30_iwp,one=1.0_iwp,penalty=1.e20_iwp,zero=0.0_iwp
CHARACTER(LEN=15)::element='quadrilateral'
!-----------------------dynamic arrays------------------------------------
INTEGER,ALLOCATABLE::etype(:),g(:),g_g (:,:),g_num(:,:),kdiag(:),nf(:,:),&
num(:)
REAL(iwp),ALLOCATABLE::coord(:,:),diag(:),g_coord(:,:),kh(:),km(:,:),
&
ku(:,:),kv(:),mm(:,:),prop(:,:),rrmass(:),udiag(:),x_coords(:),
&
y_coords(:)
!-----------------------input and initialisation--------------------------
