288
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
Adding the three equations of Eq. (7.59), we get
3
j=1
m j p
2
r φ
(r )
j = k 1 φ
(r )
1
(7.60)
Multiplying both sides of Eq. (7.60) by B r S
(r )
d , we get
3
j=1
m j p
2
r φ
(r )
j B r S
(r )
d = k 1 φ
(r )
1 B r S
(r )
d
(7.61a)
Combining Eqs. (7.56) and (7.61a), we get
3
j=1
m j φ
(r )
j B r S
(r )
a = k 1 φ
(r )
1 B r S
(r )
d = Base shear = V
(r )
3
(7.61b)
Similarly, adding last two equations of Eq. (7.59), multiplying both sides by B r S
(r )
d
and substituting the relation given by Eq. (7.56), we get
3
j=2
m j φ
(r )
j B r S
(r )
a = k 2
φ
(r )
2 − φ
(r )
1
B r S
(r )
d = V
(r )
2
(7.62)
Equations (7.61a) and (7.62) conclusively prove that Eqs. (7.54a–7.54b) and
(7.58) are identical.
7.8 Use of Response Spectra for Designing MDF Systems
The various steps for designing MDF systems by using response spectra of the type
shown in Fig. 7.3 are as follows:
(1) Compute the lowest few modes and natural periods of the structural system.
For multi-storeyed buildings, first four modes are usually sufficient. First two
modes are predominant in the case of masonry dams. First five or six modes
may be necessary for the analysis of earth dams.
Let the time periods be T 1 , T 2 , T 3, …., T N and the modes be denoted by
φ
(1)
i , φ
(2)
i , φ
(3)
i , . . . . φ
(N )
i
for the ith mass for first N modes,.
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
Adding the three equations of Eq. (7.59), we get
3
j=1
m j p
2
r φ
(r )
j = k 1 φ
(r )
1
(7.60)
Multiplying both sides of Eq. (7.60) by B r S
(r )
d , we get
3
j=1
m j p
2
r φ
(r )
j B r S
(r )
d = k 1 φ
(r )
1 B r S
(r )
d
(7.61a)
Combining Eqs. (7.56) and (7.61a), we get
3
j=1
m j φ
(r )
j B r S
(r )
a = k 1 φ
(r )
1 B r S
(r )
d = Base shear = V
(r )
3
(7.61b)
Similarly, adding last two equations of Eq. (7.59), multiplying both sides by B r S
(r )
d
and substituting the relation given by Eq. (7.56), we get
3
j=2
m j φ
(r )
j B r S
(r )
a = k 2
φ
(r )
2 − φ
(r )
1
B r S
(r )
d = V
(r )
2
(7.62)
Equations (7.61a) and (7.62) conclusively prove that Eqs. (7.54a–7.54b) and
(7.58) are identical.
7.8 Use of Response Spectra for Designing MDF Systems
The various steps for designing MDF systems by using response spectra of the type
shown in Fig. 7.3 are as follows:
(1) Compute the lowest few modes and natural periods of the structural system.
For multi-storeyed buildings, first four modes are usually sufficient. First two
modes are predominant in the case of masonry dams. First five or six modes
may be necessary for the analysis of earth dams.
Let the time periods be T 1 , T 2 , T 3, …., T N and the modes be denoted by
φ
(1)
i , φ
(2)
i , φ
(3)
i , . . . . φ
(N )
i
for the ith mass for first N modes,.