 1999 Computational Methods in Engineering'99
Eds.: P. M. Pimenta; R. M. L. R. F. Brasil; E. S. Almeida N.
FREQUENCY AND TIME DOMAIN DYNAMIC ANALYSIS
CONVERGENCE AND CAUSALITY
F. Venancio Filho*, F.S. Barbosa**, and A.M. Clarets
*
Department of Applied Mechanics and Structures
School of Engineering, Federal University of Rio de Janeiro
P.O. Box 68529, 21945-970, Rio de Janeiro RJ, Brazil
**
Department of Civil Engineering
COPPE/UFRJ, Graduate Institute of Federal University of Rio de Janeiro
P.O. Box 68506, 21945-970, Rio de Janeiro RJ, Brazil
s
Department of Civil Engineering
School of Mines UFOP, Federal University of Ouro Preto
35400-000, Ouro Preto MG, Brazil
This paper is dedicated to V. Souza Lima - master, colleague
and friend - in the occasion of his 70th and 50th anniversary
of outstanding academic and engineering work.
Key words: Dynamic Structural Analysis, Frequency Domain, Time Domain, Convergence,
Causality.
Abstract: The relation of frequency and time domain (FD and TD) methods of dynamic
structural analysis is presented in this paper. In the FD the analysis is performed through a
matrix formulation which leads to the Implicit Fourier Transform (ImFT) concept. By this
concept the direct and inverse Fourier transforms (FT) are implicitly executed through a
simple triple product which provides the final response in the time domain. From the ImFT
concept it is indicated that, when the number of sampling terms, N, in the FT is even, there is
a complex term in the response. A proof is given that, with increasing N, the modulus of that
term tends to zero and, moreover, the response obtained through the FD method tends to the
exact solution. The analysis in the TD is performed through the unit-impulse transfer matrix.
The relation between this and the ImFT matrix is established. In conclusion a proof of the
important causality property in FD and TD analysis is suggested.
F. Venancio Filho, F.S. Barbosa and A.M. Claret
1 INTRODUCTION
The dynamic analysis of structural systems can be performed by TD and by FD methods.
TD methods stem from the unit-impulse transfer function and the convolution integral while
FD ones stem from the complex frequency response function through Fourier Transforms. The
strict relation between both methods is a consequence that the unit-impulse transfer function
and the complex-frequency response function constitute a pair of Fourier Transforms.
Venancio-Filho and Claret1,2 developed a matrix formulation of the FD analysis of SDOF
systems through the concept of ImFT or complex-frequency response matrix. According to
this concept the direct and inverse discrete FT are implicitly performed by a compact and
elegant matrix expression. Moreover, the number of sampling intervals in the FT, N, can be
arbitrarily selected. Conversely, in the most common Fast Fourier Transform algorithms N
must be a power of 2. In this way, with the ImFT formulation, the analyst has more flexibility,
in the selection of the number of sampling intervals. It was verified that, when N is even, a
complex terms appears in the response2. In the present paper a proof of convergence is given
which indicates that, with N increasing to infinity, the modulus of that complex term tends to
zero and that the calculated response tends to the real solution. Numerical results support these
conclusions
The matrix formulation of the TD analysis of SDOF systems is herein presented through
the unit-impulse response matrix. The relation between this matrix and the complex-frequency
response matrix is established which indicates again the strict compability between both
methods. Finally a proof of the importante property of causality of the response by TD and FD
methods is presented.
The SDOF analysis is naturally applied for the analysis MDOF systems when the modal
equations are treated in the FD3. On the other hand systems with frequency-dependent
properties, like interaction systems (soil-structure, fluid-structure) and with hysteretic damping
can only be rigorously analyzed by FD methods. This is an important asset of these methods
which is not present in TD methods4.
2 MATRIX FORMULATION OF FREQUENCY-DOMAIN ANALYSIS
The response of a SDOF system is obtained through a FD analysis by the ImFT concept.
This concept is expressed by the following equation1:
v=
1 ∗
1
E H E p= e p
N
N
(1)
where e is the complex-frequency response matrix
In Equation 1
p = {p (t 0 ), p (t 1 ), p(t 2 ),K, p(t n ),K , p (t N −1 )}t
259.2
(2)
F. Venancio Filho, F.S. Barbosa and A.M. Claret
is the vector of the excitation at the discrete times t n = n ∆t =
Tp
N
(n = 0 ,1,2 ,K, N − 1) and
v = {v(t 0 ),v(t 1 ),v(t 2 ),K,v(t n ),K,v(t N −1 )}t
(3)
is the vector of the response at the same discrete times; E and E* are (NxN) matrices whose
−imn
2π
N
E *mn
imn
2π
N
and
=e
; H is a (NxN) diagonal
generic terms are, respectively, E mn = e
matrix composed with the complex frequency response functions at the discrete frequencies
2π
ω m = m ∆ω = m (m = 0 ,1,2 ,K , N − 1); and Tp is the extended period. The generic term of H is
Tp
{[
H (ω m ) = k 1 − β m2 + i (2ξβ m + sgn λ )
]}
−1
(4)
ωm
∆ω
=m
;ω is the natural frequency, ξ is the
ω
ω
damping ratio; and λ is the hysteretic damping factor. It is worthwhile to mention that the
complex frequency response function of Equation 4 can take into account viscous and
hysteretic damping. The discrete frequencies ω m must be considered in accordance with
Tables 1 and 2 where the symmetry of positive and negative frequencies is expressed.
where k is the system stiffness; β m =
Table 1. Discrete frequencies (N odd)
m
0
1
2
…
(N-1)/2
(N+1)/2
…
N-2
N-1
Table 2. Discrete frequencies (N even)
ωm
0
∆ω
2 ∆ω
…
[(N-1)/2] ∆ω
[-(N-1)/2] ∆ω
…
-2 ∆ω
- ∆ω
m
0
1
2
…
(N/2)-1
(N/2)
(N/2)+1
…
N-2
N-1
ωm
0
∆ω
2 ∆ω
…
[(N/2)-1] ∆ω
(N/2) ∆ω
-[(N/2)+1] ∆ω
…
-2 ∆ω
- ∆ω
3 MATRIX FORMULATION OF TD DYNAMIC ANALYSIS
The TD abalysis is performed through the unit-impulse transfer function h(t) which is the
inverse Fourier Transform of the complex-frequency transfer function H (ω ) . Thus, by
definition,
259.3
F. Venancio Filho, F.S. Barbosa and A.M. Claret
h(t )=
1
2π
t
∫0
H (ω )P(ω )e iω t dω
(5)
where P(ω ) is the FT of the loading p(t).
The discrete form of Equation 5 is
∆ω
h(t s − t r )= hsr =
2π
N −1
∑ H (ω m )e iω
(
m t s −t r
) (s ≥ r ).
(6)
m =0
Taking into account that ts=s∆t, tr= s∆t, and ω = m∆ω , and separating the positive and
negative exponents, Equation 6 transforms into
∆ω
h(t s − t r )= hsr =
2π
N −1 ism 2π
e N
∑
H (ω m ) e
−irm
2π
N
.
(7)
m =0
E *sm
ism
2π
N
−irm
2π
N
, respectively,
=e
and E mr = e
(NxN) matrices E* and E with generic terms
are now introduced. Considering these matrices and the (NxN) diagonal matrix H whose
generic term is given in Equation 4, the unit-impulse transfer matrix derives from Equation 7
as
h=
∆ω ∗
E HE .
2π
(8)
Substituting now E* H E from the foregoing equation into Equation 1 and considering that
∆ω ∆t 1
= , leads to
2π
N
v = ∆t h p
(9)
which expresses the TD response.
The response in the generic time ts is, from Equation 9,
h(t s )=∆t
N −1
∑ p(t r ) h(t s − t r ).
(10)
s =0
Equation 10 is the discrete form of the convolution equation v(t )=
t
∫0
p(τ )h(t −τ )dτ .
Comparing now Equations 1 and 9 and taking into account that ∆ω ∆t =
h=
1
e .
∆t
259.4
2π
one obtains
N
(11)
F. Venancio Filho, F.S. Barbosa and A.M. Claret
Equation 11 expresses the relation between the ImFT matrix (analysis in the FD domain) and
the unit-impulse transfer matrix (analysis in the TD domain).
4 CONVERGENCE ANALYSIS
The convergence analysis is presented along two lines. First, it is shown that, when N is
even, there is an imaginary term in the generic response v(tn) and then, proven that this term
tends to zero when N tends to infinity. In the sequel a proof is given that, with increasing N,
v(tn) converges to the real solution.
The generic response v(tn), in the generic time tn, is derived from Equation 1 as
1 N −1 i 2π
e
v(t n ) =
N m =0
∑
mn
N H
N −1
(ω m ) ∑ p (t n ) e
−i 2π
mn
N
.
(12)
n =0
 N −1
Taking into account the discrete frequencies of Table 1 (N odd), the first summation 

 m =0
Equation 12 is a summation of pairs of complex conjugates whose generic pair is
∑
C m ,m = e
± i 2π
mn
N H
N −1
(ω m ) ∑ p(t n )
e
m i 2π
mn
N

 in


(13)
n =0
where the subscripts m and m correspond, respectively, to positive and negative frequencies
(Tables 1 and 2).
When N is even (Table 2) a complex term, CN/2, associated with the Nyquist frequency,
N
ωN/2, is present in Equation 12. Substituting
for m in Equation 13 leads to
2
N −1
C N / 2 = cos nπ H (ω N / 2 )
∑ p(t n )cos nπ .
(14)
n =0
As N is even the summation in the RHS of Equation 7 can be developed as a summation of
pairs as follows:
N −1
∑ p(t n )cos nπ = [ p(t0 )− p(t1 )]+ [ p(t 2 )− p(t 3 )]+K+
n =0
[ p(t r )− p(t r +1 )]+K+ [ p(t N − 2 )− p(t N −1 )] ,
(15)
with r even. All the pairs in the previous equation can be expressed, like the generic one, as




[ p(t r )− p(t r +1 )]=  p(t r )− p t r +
259.5
T p 
 .
N  
(16)
F. Venancio Filho, F.S. Barbosa and A.M. Claret
In the limit, when N tends to infinity, the following result is obtained:
Tp


lim  p(t r )− p t r +
N →∞ 
N



 = p (t r )− p (t r )= 0 .


(17)
Therefore, when N tends to infinity, the central complex term given by Equation 14 tends to
zero.
Consider now the modulus of the terms of the generic pair of Equation 13
N −1
∑ p(t n )
C m = C m = H (ω m )
.
(18)
n =0
where, from Equation 4,
H (ω m ) =
1
k 
2
 1 − β m
(
1
)
2
2
+ (2ξβ m + sgn λ ) 

1/ 2
.
(19)
The maximum value of H (ω m ) occurs only when one of the discrete frequencies ω m
coincides, eventually, with the natural frequency ω, or when βm=1. Therefore from
Equation 19, the following situation is verified:
H (ω m ) ≤
1
1
k 2ξ + sgn λ
(ω m = 0,1,2,…, N - 1) .
(20)
N −1
Taking into that
∑ p(t n ) , Equation 18, tends to zero, according to Equations 15, 16 and 17
n =0
and that H (ω m ) , Equation 20, is limited, C m and C m , Equation 18, tend also to zero.
Therefore v(tn), Equation 12, converges to the real solution.
5 CAUSALITY OF THE RESPONSE
The causality of the response is the property that the response at any time t is not influenced
by the excitation at all the times greater than t. With the consideration of Equations (1) and (9)
the causality property corresponds to the lower triangularity of matrices e and h. In order that
h be lower triangular the load p(tr) (r>s) should not contribute to the response at time ts with is
equivalent to h(ts-tr)=0 for the r>s. To prove this consider Equation 7 with the exponents
gathered:
∆ω
h(t s − t r )= hsr =
2π
N −1
∑ H (ω m )e
m =0
259.6
−im (r − s )
2π
N
.
(21)
F. Venancio Filho, F.S. Barbosa and A.M. Claret
The maximum absolute value of H (ω m ) occurs when, eventually, ω m = ω . Then from
Equation 4:
H (ω m )
max
= H (ω ) =
1
.
k i 2ξ
(22)
Therefore the following inequality is verified:
H (ω m ) ≤ H (ω ) .
(23)
Introducing this inequality in Equation 21 the modulus hsr of hsr is expressed as
hsr
∆ω
≤
H (ω )
2π
N −1
∑Zm .
(24)
(r > s ) .
(25)
m =0
where
Zm = e
Now the series
∑Z m
−im( r − s )
2π
N
is a geometric one with ratio Z = e
−i ( r − s )
2π
N
and it is proven that it is
absolutely convergent 6, that is,
1− Z N
1+ Z + Z +L+ Z =
(Z ≠ 1) .
1− Z
2
N
On the other hand, from Equation 25, ZN=1. Then from Equation 26 the series
(26)
∑Zn
converges to zero. In consequence, from Equation 25, the terms such as hsr (r>s) converge to
zero as N tends to infinity. This conclusion proves finally the causality of the response in the
TD and FD dynamic analysis expressed, respectively, by Equations 9 and 1.
The causality of the response is closely related to the period extension. When the extended
period tends to infinity a causual response is obtained7.
6 NUMERICAL EXAMPLES OF CONVERGENCE AND CAUSALITY
The response of a SDOF system to a short duration impulsive load with time duration
td=0.275 sec whose peak is 15kN. The properties of the system are: natural frequency: ω=5.71
rd/sec; natural period: T=1.1sec; damping ratio ξ=0.10. The response is calculated from
4.605
Equation 1 with an extended period given by T p =
= 8.07 s . This expression for Tp
ξω
259.7
F. Venancio Filho, F.S. Barbosa and A.M. Claret
corresponds the time in which the value of an initial condition in free vibration reduced to
1%6. Results considering several values for N even and N odd are presented, respectively, in
Figures 1 and 2. The results for the greater values of N show a fairly good coincidence with
the exact solution5.
20,00
15,00
displacement (cm)
10,00
N = 512
N = 256
N = 128
N = 64
5,00
0,00
-5,00
-10,00
-15,00
0,00
1,00
2,00
3,00
4,00
time (s)
Figure 1. Response for N even (viscous damping):
20,00
15,00
displacement (cm)
10,00
N = 453
N = 229
5,00
N = 117
N = 59
0,00
-5,00
-10,00
-15,00
0,00
1,00
2,00
3,00
time (s)
Figure 2. Response for N odd (viscous damping):
259.8
4,00
F. Venancio Filho, F.S. Barbosa and A.M. Claret
Figure 3 compares the solutions with viscous and hysteretic damping (λ=2ξ) the
differences being quite small.
12,00
8,00
displacement (cm)
4,00
viscous
hysteretic
0,00
-4,00
-8,00
-12,00
0,00
1,00
2,00
3,00
4,00
time (s)
Figure 3. Comparison of the responses with viscous and hysteretical damping:
The fairly good convergence to zero, with arbitrarily increasing N, of the modulus of the
central complex term, Equation 14, is represented in Figure 4.
12,00
|C N/2 | x 10
-3
viscous
8,00
hysteretic
4,00
0,00
0,00
2,00
4,00
log2(N)
6,00
Figure 4. Convergence of modulus of the central complex term:
259.9
8,00
F. Venancio Filho, F.S. Barbosa and A.M. Claret
The response of a SDOF system with natural period T=0,20sec, damping ratio ξ=0,10
submitted to a shurt-duration half-since pulse with duration td=0,03sec is displayed in
Figures 5a (extended period Tp=1.46sec) and 5b(Tp=0.64sec). The response with an
insuficient extended period is non-causal.
Figure 5. a) causal response; b) noncausal response
7 CONCLUSIONS
The equivalence of formulations of FD and TD methods for dynamic structural analysis of
SDOF with viscous and hysteretic damping was presented. Proofs of the convergence of the
response analysed by FD methods and of the causality property were developed. Numerical
examples indicate the very fast convergence of the response when with increasing the number
of sampling intervals, in the FT. A response calculated with an insufficient extend period is
non-causal.
REFERENCES
[1] F.Venancio-Filho and A.M. Claret, “Matrix Formulation of the Dynamic Analysis of
SDOF in the Frequency Domain”, Computers and Structures, 42, 853-855 (1992).
[2] F. Venancio-Filho and A.M. Claret, “Frequency Domain Dynamic Analysis of MDOF
Systems”, Computers and Structures, 56, 189-191 (1995).
[3] W.G. Ferreira, F. Venancio-Filho, W.J. Mansur and A.M. Claret, “A Frequency Domain
Method for Nonlinear Dynamic Structural Analysis, Applied Mechanics in the Americas,
edited by Djenane Pamplona, Carlos Steele, Hans I. Weber, Paulo B. Gonçalves, Iwona
Jasiuka and Luiz Bevilacqua, 8, 1343-1346 (1999).
[4] F.Venancio-Filho, “Frequency Domain Dynamic Analysis of Systems with Viscous and
Hysteretic Damping”, Nonlinear Dynamics, Chaos, Control and Their Application to
259.10
F. Venancio Filho, F.S. Barbosa and A.M. Claret
Engineering Sciences, edited by José Manoel Balthazar, Paulo Batista Gonçalves and
Julio Clayssen, 2, 136-151 (1999).
[5] Ray W. Clough and Joseph Penzien, Dynamics of Structures, 2nd edition, McGraw Hill
(1993).
[6] I.S. Solkonikoff and R.M. Redheffer, Mathematics of Physics and Modern Engineering,
McGraw Hill (1996).
[7] F.Venancio-Filho and W. Graça Ferreira, ´Frequency-domain Analysis for Short Duration
Impulsive Loads”, Applied Mechanics in the Americas, edited by L.A. Godoy, M. Rysz
and L.E. Suarez, 4, 387-390 (1997).
259.11