92
4 Multi-Degree of Freedom (MDOF) Systems
Note that (1) ABS-based responses always exceed structural responses since
maxima of modal responses do not occur simultaneously, (2) ABS- and SRSS-based
responses differ significantly if modal response maxima are similar, e.g., they are
n R 1 and
√
n R 1 if R i = R 1 , i ≥ 2, and (3) ABS- and SRSS-based responses are
similar if one of the modal responses is dominant, e.g., they are approximately equal
to R 1 if R 1 R i , i ≥ 2.
Example 4.6 Suppose the 2-DOF system with mass and stiffness matrices given in
Example 4.3, and a damping matrix c = α m + β k, where α = 0.03 and β = 0.05,
is subjected to the El Centro ground motion. The modal periods and damping ratios
are T 1 = 10.425 s, T 2 = 2.0240 s, ζ 1 = 0.04, and ζ 2 = 0.0824. The response
spectral values are S d (ω 1 , ζ 1 ) = 16 in and S d (ω 2 , ζ 2 ) = 8 in. They can be obtained
from the response spectrum of this seismic event shown in Fig. 2.17. The modal
participation factors are 1 = 1.4338 and 2 = 0.6621 so that the maxima of
modal displacements are R 1 = 22.94 in and R 2 = 5.30 in. The structural responses
{ i R i } in the two modes are
1 R 1 =
7.14
21.86
in and 2 R 2 =
4.43
−2.90
in.
The ABS- and SRSS-estimates of the drift are
ABS:|21.80| + | − 2.90| = 24.70in
SRSS:
21.80 2 + (−2.90) 2 1/2 = 21.99in.
The ABS- and SRSS-estimates of the inter-story displacement are
ABS: |21, 80 − 7.14| + |4, 43 − (−2.90)| = 21.99in,
SRSS:
(21, 80 − 7.14) 2 + (4, 43 − (−2.90) 2 )
1/2 = 16.39in.
Note that drift estimates by the ABS and SRSS rules are similar. In contrast, the
inter-story estimates by these rules differ significantly. Since the modal periods T 1
and T 2 are not closely spaced (a qualitative statement!), it is likely that the SRRS
rule would be used for design. As expected, the ABS-based estimates are larger than
the SRSS-based estimates.
4.4.6 Experimental Determination of Modal Frequencies
Consider a MDOF system with n degrees of freedom, mass matrix m, and stiffness
matrix k, which is placed on a shake table oscillating with the acceleration sin(ν t).
The frequency ν of the forcing function is increased slowly, and the resulting
structural responses are monitored. We show that this sequence of experiments can
be used to estimate the modal frequencies {ω i }. This application relates to Sect. 4.4.3
on the undamped systems in forced vibration.
The equation of motion of the system is (see Eq. 4.56)
4 Multi-Degree of Freedom (MDOF) Systems
Note that (1) ABS-based responses always exceed structural responses since
maxima of modal responses do not occur simultaneously, (2) ABS- and SRSS-based
responses differ significantly if modal response maxima are similar, e.g., they are
n R 1 and
√
n R 1 if R i = R 1 , i ≥ 2, and (3) ABS- and SRSS-based responses are
similar if one of the modal responses is dominant, e.g., they are approximately equal
to R 1 if R 1 R i , i ≥ 2.
Example 4.6 Suppose the 2-DOF system with mass and stiffness matrices given in
Example 4.3, and a damping matrix c = α m + β k, where α = 0.03 and β = 0.05,
is subjected to the El Centro ground motion. The modal periods and damping ratios
are T 1 = 10.425 s, T 2 = 2.0240 s, ζ 1 = 0.04, and ζ 2 = 0.0824. The response
spectral values are S d (ω 1 , ζ 1 ) = 16 in and S d (ω 2 , ζ 2 ) = 8 in. They can be obtained
from the response spectrum of this seismic event shown in Fig. 2.17. The modal
participation factors are 1 = 1.4338 and 2 = 0.6621 so that the maxima of
modal displacements are R 1 = 22.94 in and R 2 = 5.30 in. The structural responses
{ i R i } in the two modes are
1 R 1 =
7.14
21.86
in and 2 R 2 =
4.43
−2.90
in.
The ABS- and SRSS-estimates of the drift are
ABS:|21.80| + | − 2.90| = 24.70in
SRSS:
21.80 2 + (−2.90) 2 1/2 = 21.99in.
The ABS- and SRSS-estimates of the inter-story displacement are
ABS: |21, 80 − 7.14| + |4, 43 − (−2.90)| = 21.99in,
SRSS:
(21, 80 − 7.14) 2 + (4, 43 − (−2.90) 2 )
1/2 = 16.39in.
Note that drift estimates by the ABS and SRSS rules are similar. In contrast, the
inter-story estimates by these rules differ significantly. Since the modal periods T 1
and T 2 are not closely spaced (a qualitative statement!), it is likely that the SRRS
rule would be used for design. As expected, the ABS-based estimates are larger than
the SRSS-based estimates.
4.4.6 Experimental Determination of Modal Frequencies
Consider a MDOF system with n degrees of freedom, mass matrix m, and stiffness
matrix k, which is placed on a shake table oscillating with the acceleration sin(ν t).
The frequency ν of the forcing function is increased slowly, and the resulting
structural responses are monitored. We show that this sequence of experiments can
be used to estimate the modal frequencies {ω i }. This application relates to Sect. 4.4.3
on the undamped systems in forced vibration.
The equation of motion of the system is (see Eq. 4.56)