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2 SDOF MODULE “JESDOF”
The most fundamental concept in structural dynam-
ics is the single degree-of-freedom (SDOF) oscilla-
tor, whose understanding is crucial for students. This
was the rationale behind the development of software
module “jesdof”. More specifically, four subcate-
gories of data must be defined: (a) elastic proper-
ties (stiffness), mass and energy-loss (damping); (b)
initial conditions for the displacement and velocity;
(c) definition of the external source of excitation (dy-
namic loading), either as an ASCII file containing dis-
crete values at equally spaced time intervals, or by a
mathematical expression; and (d) parameters associ-
ated with the time integration algorithm used to nu-
merically solve the governing differential equation of
the system.
The governing differential equation for dynamic
equilibrium is
m¨x(t) + c˙x(t) + kx(t) = F
0
f(t) (1)
where m, c, k are the mass, damper, and stiffness (in
tons, kN-sec/m and kN/m, respectively) of the struc-
tural system, while x(t) (in m) is its response to a
dynamic load of magnitude F
0
(in kN) and dimen-
sionless time variation f(t). Note that overdots de-
note derivatives with respect to time t. Furthermore,
the natural frequency and fundamental period of the
SDOF system are ω
2
0
= k/m, T = 2/ω
0
(in rad/sec and
sec, resp.). It is standard practice to define a dimen-
sionless damping ratio ζ = c/2mω
0
in lieu of damper
values. Finally, the closed form solution of the SDOF
is given by Duhamel’s integral as
x(t) = x
0
cos(ω
0
t) + ( ˙x
0
/ω
0
)sin(ω
0
t) +
x
ST
ω
0
Z
t
0
f(t)sin
ω
0
(t − τ)
dτ (2)
where x
0
, ˙x
0
are initial displacement and velocity of
the system, while x
ST
= F
0
/k is its equivalent static
displacement. For ground accelerations ¨x
g
(t), the
forcing function becomes F(t) = −m¨x
g
(t) and the
SDOF kinematics are understood to be relative to the
ground motion. Solution is accomplished through nu-
merical integration of Eq. 2 using the Newmark beta
algorithm, with the time interval of interest subdi-
vided into N increments of size equal to or less than
T/10. The structural response is given in a simple
text area. Finally, the user may visualize the motion
of the system in the graph area of the GUI by select-
ing as abscissa and ordinate variables in combination:
time, external force, displacement, velocity and accel-
eration.
3 RESPONSE SPECTRA “ESPEC”
This application produces elastic response spectra
from ground motion recordings, defined as the plot
of maximum response (for either kinematic or force
variables) versus frequency (or period) of all possi-
ble SDOF systems under a specific type of external
force (usually a seismically-induced ground motion).
More specifically, response spectra are parametric in
terms of damping (e.g., ζ = 0− 10%) and are com-
puted from Duhamel’s integral of Eq. 2 by varying
the natural period of an SDOF system so as to sweep
an acceptable range that corresponds to real struc-
tures, i.e., 0 ≤ T ≤ 3.0 (in sec). Following computa-
tion, the results may be presented as absolute pseudo-
acceleration, relative pseudo-velocity or relative dis-
placement related as
S
A
= ωS
V
= ω
2
S
D
(3)
The above an exact relation for a damping ratio of
zero and approximate otherwise. Module “espec” uti-
lizes the Newmark-beta numerical integration algo-
rithm for integrating the equation of motion in time.
4 3-STORY FRAME “JFRAME”
Application module “jframe” does modal analysis of
a simple three story plane frame that translates into a
three DOF system, whose properties are user defined
as the elastic and inertial characteristics plus the ge-
ometrical configuration. Upon entering the input pa-
rameters, this program module computes and presents
in simple text area the following fundamental data:
stiffness matrix, mass matrix, period and frequency
associated with each eigenmode, the components for
each eigenvector, plus the participation factor, gener-
alized mass and effective mass corresponding to each
eigenmode. The underlying computation procedures
are based on the finite element method, but this does
not require any detailed knowledge from the part of
the students. Finally, there is an option to specify
an acceleration response spectrum, in which case the
program proceeds to calculate the spectral pseudo-
acceleration, relative pseudo-velocity and the relative
displacement for each eigenmode. More specifically,
the governing equation of motion for a multiple DOF
system is Eq. 1 again, except that the quantities of
interest are now matrices and vectors (of size N, the
number of independent DOF) in lieu of scalars. Of
course, the number of eigenvalues (natural frequen-
cies) and eigenvectors (corresponding modal shapes)
recovered is again N.
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