4.4 Time Domain Analysis: Proportional Damping
85
which gives
T m ¨
q(t) +
T k q(t) =
T f(t),
by left multiplication with T . The orthogonality condition of Eq. 4.10 implies
diag{ ˜
m i } ¨
q(t) + diag{ ˜
k i } q(t) =
T f(t) or
˜
m i ¨
q i (t) + ˜
k i q i (t) = f i (t), i = 1, . . . , n,
(4.47)
where f i (t) =
T f(t)
i
denotes the ith component of the n-dimensional vector
T f(t). The latter equation becomes
¨
q i (t) + ω
2
i q i (t) = f i (t)/ ˜
m i , i = 1, . . . , n,
(4.48)
by using Eq. 4.9 following division with ˜
m i .
These equations describe the forced vibrations of n undamped SDOF systems with natural frequencies {ω i } subjected to the forcing functions {f i (t) =
T f(t)
i
}. Their solutions have the form (see results on SDOF systems)
q i (t) = q i,0 cos(ω i t) +
˙
q i,0
ω i
sin(ω i t) + q p,i (t), i = 1, . . . , n,
(4.49)
where (q i,0 , ˙
q i,0 ) are the initial conditions for these oscillators, which can be
obtained from, e.g., Eq. 4.31, and q p,i (t) are particular solutions that depend on
the types of forcing functions.
The forced vibration solution of the MDOF system has the expression
x(t) =
n
i=1
i q i (t) =
n
i=1
i
q i,0 cos(ω i t) +
˙
q i,0
ω i
sin(ω i t) + q p,i (t)
.
(4.50)
Example 4.4 Consider the 2-DOF structure in Fig. 4.7 in Example 4.5. The system
is at rest at the initial time and is subjected to the force sin(ν t), ν = ω i , i = 1, 2,
applied at the second degree of freedom, as illustrated in the figure. The motion of
this system is described by Eq. 4.24, n = 2, with the forcing function
f(t) =
0
1
sin(ν t).
The calculation of the displacement vector x(t) involves the following three
steps.
– Step 1: Modal analysis: The modal shapes and frequencies are calculated in
Example 4.5 of the following subsection and are used here.
85
which gives
T m ¨
q(t) +
T k q(t) =
T f(t),
by left multiplication with T . The orthogonality condition of Eq. 4.10 implies
diag{ ˜
m i } ¨
q(t) + diag{ ˜
k i } q(t) =
T f(t) or
˜
m i ¨
q i (t) + ˜
k i q i (t) = f i (t), i = 1, . . . , n,
(4.47)
where f i (t) =
T f(t)
i
denotes the ith component of the n-dimensional vector
T f(t). The latter equation becomes
¨
q i (t) + ω
2
i q i (t) = f i (t)/ ˜
m i , i = 1, . . . , n,
(4.48)
by using Eq. 4.9 following division with ˜
m i .
These equations describe the forced vibrations of n undamped SDOF systems with natural frequencies {ω i } subjected to the forcing functions {f i (t) =
T f(t)
i
}. Their solutions have the form (see results on SDOF systems)
q i (t) = q i,0 cos(ω i t) +
˙
q i,0
ω i
sin(ω i t) + q p,i (t), i = 1, . . . , n,
(4.49)
where (q i,0 , ˙
q i,0 ) are the initial conditions for these oscillators, which can be
obtained from, e.g., Eq. 4.31, and q p,i (t) are particular solutions that depend on
the types of forcing functions.
The forced vibration solution of the MDOF system has the expression
x(t) =
n
i=1
i q i (t) =
n
i=1
i
q i,0 cos(ω i t) +
˙
q i,0
ω i
sin(ω i t) + q p,i (t)
.
(4.50)
Example 4.4 Consider the 2-DOF structure in Fig. 4.7 in Example 4.5. The system
is at rest at the initial time and is subjected to the force sin(ν t), ν = ω i , i = 1, 2,
applied at the second degree of freedom, as illustrated in the figure. The motion of
this system is described by Eq. 4.24, n = 2, with the forcing function
f(t) =
0
1
sin(ν t).
The calculation of the displacement vector x(t) involves the following three
steps.
– Step 1: Modal analysis: The modal shapes and frequencies are calculated in
Example 4.5 of the following subsection and are used here.