doc LIDA
Documentation of the LIDA function.
helpFun('LIDA')
Linear Implicit Dynamic Analysis
[U,UT,UTT,EI] = LIDA(DT,XGTT,OMEGA,KSI,U0,UT0,ALGID,RINF)
Description
Linear implicit direct time integration of second order differential
equation of motion of dynamic response of linear elastic SDOF systems
The General Single Step Single Solve (GSSSS) family of algorithms
published by X.Zhou & K.K.Tamma (2004) is employed for direct time
integration of the general linear or nonlinear structural Single
Degree of Freedom (SDOF) dynamic problem. The optimal numerical
dissipation and dispersion zero order displacement zero order
velocity algorithm designed according to the above journal article,
is used in this routine. This algorithm encompasses the scope of
Linear Multi-Step (LMS) methods and is limited by the Dahlquist
barrier theorem (Dahlquist,1963). The force - displacement - velocity
relation of the SDOF structure is linear. This function is part of
the OpenSeismoMatlab software. It can be used as standalone, however
attention is needed for the correctness of the input arguments, since
no checks are performed in this function. See the example
example_LIDA.m for more details about how this function can be
implemented.
Input parameters
DT [double(1 x 1)] is the time step
XGTT [double(1:nstep x 1)] is the column vector of the acceleration
history of the excitation imposed at the base. nstep is the
number of time steps of the dynamic response.
OMEGA [double(1 x 1)] is the eigenfrequency of the structure in
rad/sec.
KSI [double(1 x 1)] is the ratio of critical damping of the SDOF
system.
U0 [double(1 x 1)] is the initial displacement of the SDOF system.
UT0 [double(1 x 1)] is the initial velocity of the SDOF system.
ALGID [char(1 x :inf)] is the algorithm to be used for the time
integration. It can be one of the following strings for superior
optimally designed algorithms:
'generalized a-method': The generalized a-method (Chung &
Hulbert, 1993)
'HHT a-method': The Hilber-Hughes-Taylor method (Hilber,
Hughes & Taylor, 1977)
'WBZ': The Wood–Bossak–Zienkiewicz method (Wood, Bossak &
Zienkiewicz, 1980)
'U0-V0-Opt': Optimal numerical dissipation and dispersion
zero order displacement zero order velocity algorithm
'U0-V0-CA': Continuous acceleration (zero spurious root at
the low frequency limit) zero order displacement zero order
velocity algorithm
'U0-V0-DA': Discontinuous acceleration (zero spurious root at
the high frequency limit) zero order displacement zero order
velocity algorithm
'U0-V1-Opt': Optimal numerical dissipation and dispersion
zero order displacement first order velocity algorithm
'U0-V1-CA': Continuous acceleration (zero spurious root at
the low frequency limit) zero order displacement first order
velocity algorithm
'U0-V1-DA': Discontinuous acceleration (zero spurious root at
the high frequency limit) zero order displacement first order
velocity algorithm
'U1-V0-Opt': Optimal numerical dissipation and dispersion
first order displacement zero order velocity algorithm
'U1-V0-CA': Continuous acceleration (zero spurious root at
the low frequency limit) first order displacement zero order
velocity algorithm
'U1-V0-DA': Discontinuous acceleration (zero spurious root at
the high frequency limit) first order displacement zero order
velocity algorithm
'Newmark ACA': Newmark Average Constant Acceleration method
'Newmark LA': Newmark Linear Acceleration method
'Newmark BA': Newmark Backward Acceleration method
'Fox-Goodwin': Fox-Goodwin formula
RINF [double(1 x 1)] is the minimum absolute value of the eigenvalues
of the amplification matrix. For the amplification matrix see
eq.(61) in Zhou & Tamma (2004).
Output parameters
U [double(1:nstep x 1)] is the time-history of displacement
UT [double(1:nstep x 1)] is the time-history of velocity
UTT [double(1:nstep x 1)] is the time-history of acceleration
EI [double(1:nstep x 1)] is the time-history of the seismic input
energy per unit mass. See: {Uang, C. M., & Bertero, V. V. (1990).
Evaluation of seismic energy in structures. Earthquake
engineering & structural dynamics, 19(1), 77-90} for more
details.
Example (Figure 6.6.1 in Chopra, Tn=1sec)
dt=0.02;
fid=fopen('elcentro.dat','r');
text=textscan(fid,'%f %f');
fclose(fid);
xgtt=text{1,2};
Tn=1;
omega=2*pi/Tn;
ksi=0.02;
u0=0;
ut0=0;
AlgID='U0-V0-Opt';
rinf=1;
[u,ut,utt] = LIDA(dt,xgtt,omega,ksi,u0,ut0,AlgID,rinf);
D=max(abs(u))/0.0254
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Copyright (c) 2018-2023
George Papazafeiropoulos
Major, Infrastructure Engineer, Hellenic Air Force
Civil Engineer, M.Sc., Ph.D.
Email: gpapazafeiropoulos@yahoo.gr
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