4.4 Time Domain Analysis: Proportional Damping
81
and c = 0.0505 m + 0.0270 k, which is at rest at the initial time, i.e., x 0 = 0 and
˙
x 0 = 0. The modal frequencies and modal shapes of the system are ω 1 = 0.6027
and ω 2 = 3.1043 and
=
0.3111 0.8366
0.9504 −0.5478
.
The system is subjected to the harmonic force f (t) = sin(ν t) applied at the second
degree of freedom so that
f(t) =
0
1
sin(ν t).
The modal coordinates are solutions of Eq. 4.26 with zero initial conditions since
x 0 = 0 and ˙
x 0 = 0 (see Eqs. 4.31 or 4.32). The forcing functions f i (t) =
T f(t)
i
are f i (t) = i,2 sin(ν t), i = 1, 2, where i,r denotes the rth component of i .
The modal coordinate can be obtained from Eq. 4.27 or results for SDOF systems
given in Sect. 2.4.5.
The left panel of Fig. 4.5 shows with solid and dashed lines the first and second
modal responses 1 q 1 (t) and 2 q 2 (t) for ν = 2. The right panel shows the system
displacements x 1 (t) and x 2 (t). Similar plots are in Fig. 4.6 for ν = 3. The system
displacements {x i (t)} for ν = 2 are dominated by the first mode response as it can
be seen by comparing modal and system responses in the two panels of the figure.
This observation is also supported by the fact that the displacements x 1 (t) and x 2 (t)
have the same sign at all times. For ν = 3, the second mode response becomes
dominant in time since the forcing frequency ν and modal frequency ω 2 are nearly
equal. Note also that the signs of the displacements x 1 (t) and x 2 (t) differ at times,
which indicates that the second mode contributes significantly to the overall system
displacement.
0
5
10
15
20
25
30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
Φ
i q
i (t)
0
5
10
15
20
25
30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
x
i (t)
Fig. 4.5 Modal responses 1 q 1 (t) and 2 q 2 (t) (solid and dashed lines) for ν = 2 and the
corresponding system displacements {x i (t)} (left and right panels)
81
and c = 0.0505 m + 0.0270 k, which is at rest at the initial time, i.e., x 0 = 0 and
˙
x 0 = 0. The modal frequencies and modal shapes of the system are ω 1 = 0.6027
and ω 2 = 3.1043 and
=
0.3111 0.8366
0.9504 −0.5478
.
The system is subjected to the harmonic force f (t) = sin(ν t) applied at the second
degree of freedom so that
f(t) =
0
1
sin(ν t).
The modal coordinates are solutions of Eq. 4.26 with zero initial conditions since
x 0 = 0 and ˙
x 0 = 0 (see Eqs. 4.31 or 4.32). The forcing functions f i (t) =
T f(t)
i
are f i (t) = i,2 sin(ν t), i = 1, 2, where i,r denotes the rth component of i .
The modal coordinate can be obtained from Eq. 4.27 or results for SDOF systems
given in Sect. 2.4.5.
The left panel of Fig. 4.5 shows with solid and dashed lines the first and second
modal responses 1 q 1 (t) and 2 q 2 (t) for ν = 2. The right panel shows the system
displacements x 1 (t) and x 2 (t). Similar plots are in Fig. 4.6 for ν = 3. The system
displacements {x i (t)} for ν = 2 are dominated by the first mode response as it can
be seen by comparing modal and system responses in the two panels of the figure.
This observation is also supported by the fact that the displacements x 1 (t) and x 2 (t)
have the same sign at all times. For ν = 3, the second mode response becomes
dominant in time since the forcing frequency ν and modal frequency ω 2 are nearly
equal. Note also that the signs of the displacements x 1 (t) and x 2 (t) differ at times,
which indicates that the second mode contributes significantly to the overall system
displacement.
0
5
10
15
20
25
30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
Φ
i q
i (t)
0
5
10
15
20
25
30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
x
i (t)
Fig. 4.5 Modal responses 1 q 1 (t) and 2 q 2 (t) (solid and dashed lines) for ν = 2 and the
corresponding system displacements {x i (t)} (left and right panels)