Civil Engineering Reference
In-Depth Information
Figure 11.4 is subjected to a harmonic vertical force of cos
ωt
at node 18. The damping
ratio
ζ
(see equation (3.120)), called
dr
in the program is 0
.
05 or 5% applied to all modes
of the system.
The forcing frequency
3, which is deli-
berately chosen to be close to the second natural frequency of the undamped system
(Note:
ω
(called
omega
in the program) is set at 0
.
ω
2
≈
.
0
3 from Figure 10.7), so at this frequency the influence of damping should
be significant.
The structure of the program is essentially the same as Program 10.2 up to the end
of the section headed “extract the mass normalised eigenvectors”, however the current
program uses a different eigenvector normalisation strategy that reduces the modal mass
matrix to a unit matrix. It should be remembered that the eigenvectors first computed as
udiag
are those of the transformed problem and the true eigenvectors must be recovered
prior to normalisation and storage in the eigenvector matrix
bigk
.
When the time stepping loop is entered, the cosine loading of a single degree of freedom
system is introduced, and the modal contributions superposed at each degree of freedom.
A structure chart for the modal superposition algorithm is given in Figure 11.5.
The example uses the first six eigenmodes, read as
nmodes=6
to synthesise the
time response. Users are invited to examine the sensitivity of the response to different
values of
nmodes
up to a maximum of 30 in this case. The output from the analy-
sis is shown in Figure 11.6 and the response in the
y
-direction at node 18 is plotted in
Figure 11.7.
There are 30 equations and the half-bandwidth is 15
Result at node 18
time load x-disp y-disp
0.0000E+00 0.1000E+01 0.0000E+00 0.0000E+00
0.1000E+01 0.9553E+00 0.2188E+02 0.2612E+02
0.2000E+01 0.8253E+00 0.2825E+02 0.3727E+02
0.3000E+01 0.6216E+00 0.3210E+02 0.4508E+02
0.4000E+01 0.3624E+00 0.3308E+02 0.4887E+02
0.5000E+01 0.7074E-01 0.3110E+02 0.4829E+02
0.6000E+01 -0.2272E+00 0.2635E+02 0.4340E+02
0.7000E+01 -0.5048E+00 0.1924E+02 0.3463E+02
0.8000E+01 -0.7374E+00 0.1042E+02 0.2277E+02
0.9000E+01 -0.9041E+00 0.6594E+00 0.8875E+01
0.1000E+02 -0.9900E+00 -0.9155E+01 -0.5813E+01
0.1100E+02 -0.9875E+00 -0.1815E+02 -0.1998E+02
0.1200E+02 -0.8968E+00 -0.2553E+02 -0.3237E+02
0.1300E+02 -0.7259E+00 -0.3062E+02 -0.4186E+02
0.1400E+02 -0.4903E+00 -0.3298E+02 -0.4761E+02
0.1500E+02 -0.2108E+00 -0.3240E+02 -0.4911E+02
0.1600E+02 0.8750E-01 -0.2892E+02 -0.4623E+02
0.1700E+02 0.3780E+00 -0.2285E+02 -0.3921E+02
0.1800E+02 0.6347E+00 -0.1475E+02 -0.2869E+02
0.1900E+02 0.8347E+00 -0.5325E+01 -0.1561E+02
0.2000E+02 0.9602E+00 0.4572E+01 -0.1135E+01
Figure 11.6
Results from Program 11.2 example
