4.4 Time Domain Analysis: Proportional Damping
91
differs slightly from that in Eq. 2.30 for SDOF systems with displacement functions
x(t) defined by the differential equation ¨
x + 2 ζ ω ˙
x + ω 2 x = f (t)/m since the
analogue of f (t) in this equation is −
i a(t)
˜
m i for the differential equation of
q i (t) so that its forced vibration component is (see Eq. 2.30)
q p,i (t) =
t
0
1
˜
m i ω d,i
e
−ζ i ω i (t−s) sin
ω d,i (t − s)
− i a(s) ˜
m i
ds
= − i
t
0
1
ω d,i
e
−ζ i ω i (t−s) sin
ω d,i (t − s)
a(s) ds,
(4.64)
which gives the unit impulse response function in Eq. 4.63.
4.4.5.2 Design Response Spectrum
Recall that response spectra are maxima of responses of SDOF systems, e.g., the
displacement response spectrum S d (ω, ζ ) is the maxima of the absolute values of
displacements of a family of SDOF systems indexed by their natural frequency
ω and damping ratio ζ that are subjected to the same ground acceleration a(t).
Accordingly, the maxima of the modal responses given by Eq. 4.60 or other forms
of the solution of Eq. 4.58 are R i = max t {|q i (t)|} = i S d (ω i , ζ i ), i = 1, . . . , n,
since we deal with linear differential equations.
The modal spectral displacements {R i } are well defined. However, system
response maxima cannot be obtained exactly from {R i } since maxima of different
modal coordinates occur at different times. To overcome this impasse, it is common
to use modal combination rules, i.e., formulas that map modal response maxima
into system response maxima. The rules are based on the qualitative arguments,
e.g., it is assumed that modal responses with closely spaced frequencies are in phase
so that their extremes occur simultaneously. The meaning of closely spaced is not
defined. Moreover, modal frequency closeness is affected by modal damping ratio
and the frequency content of a(t). The estimation of structural response maxima,
e.g., the displacement maxima, by modal combination rules involves two steps.
• Step 1: Find S d (ω i , ζ i ) from plots of response spectra for the ground acceleration
a(t) of interest and calculate modal response maxima from R i = max t {|q i (t)|} =
i S d (ω i , ζ i ).
• Step 2: Estimate structural response maxima from modal response maxima
{R i } by modal combination rules. Here are two of the most popular modal
combination rules.
– SRSS (Square root of the sum of the square of modal response maxima):
R =
n
i=1 R 2
i
1/2 if modal frequencies are not closely spaced.
– ABS (Sum of modal response maxima):
R =
n
i=1 R i , if modal frequencies are closely spaced.
91
differs slightly from that in Eq. 2.30 for SDOF systems with displacement functions
x(t) defined by the differential equation ¨
x + 2 ζ ω ˙
x + ω 2 x = f (t)/m since the
analogue of f (t) in this equation is −
i a(t)
˜
m i for the differential equation of
q i (t) so that its forced vibration component is (see Eq. 2.30)
q p,i (t) =
t
0
1
˜
m i ω d,i
e
−ζ i ω i (t−s) sin
ω d,i (t − s)
− i a(s) ˜
m i
ds
= − i
t
0
1
ω d,i
e
−ζ i ω i (t−s) sin
ω d,i (t − s)
a(s) ds,
(4.64)
which gives the unit impulse response function in Eq. 4.63.
4.4.5.2 Design Response Spectrum
Recall that response spectra are maxima of responses of SDOF systems, e.g., the
displacement response spectrum S d (ω, ζ ) is the maxima of the absolute values of
displacements of a family of SDOF systems indexed by their natural frequency
ω and damping ratio ζ that are subjected to the same ground acceleration a(t).
Accordingly, the maxima of the modal responses given by Eq. 4.60 or other forms
of the solution of Eq. 4.58 are R i = max t {|q i (t)|} = i S d (ω i , ζ i ), i = 1, . . . , n,
since we deal with linear differential equations.
The modal spectral displacements {R i } are well defined. However, system
response maxima cannot be obtained exactly from {R i } since maxima of different
modal coordinates occur at different times. To overcome this impasse, it is common
to use modal combination rules, i.e., formulas that map modal response maxima
into system response maxima. The rules are based on the qualitative arguments,
e.g., it is assumed that modal responses with closely spaced frequencies are in phase
so that their extremes occur simultaneously. The meaning of closely spaced is not
defined. Moreover, modal frequency closeness is affected by modal damping ratio
and the frequency content of a(t). The estimation of structural response maxima,
e.g., the displacement maxima, by modal combination rules involves two steps.
• Step 1: Find S d (ω i , ζ i ) from plots of response spectra for the ground acceleration
a(t) of interest and calculate modal response maxima from R i = max t {|q i (t)|} =
i S d (ω i , ζ i ).
• Step 2: Estimate structural response maxima from modal response maxima
{R i } by modal combination rules. Here are two of the most popular modal
combination rules.
– SRSS (Square root of the sum of the square of modal response maxima):
R =
n
i=1 R 2
i
1/2 if modal frequencies are not closely spaced.
– ABS (Sum of modal response maxima):
R =
n
i=1 R i , if modal frequencies are closely spaced.