6.2. ROTATIONAL PERIODIC MOTIONS 63
10
4
, and A
4
4:1120 10
5
. e main harmonic amplitudes of A
.2l1/=6
are A
1=6
8:5755, A
1=2
0:0145, A
5=6
0:0924, A
7=6
0:2005, A
3=2
0:0168, A
11=6
3:1535
10
3
, A
13=6
0:0257, A
5=2
0:0123, A
17=6
1:3600 10
3
, A
19=6
9:0329 10
4
, A
7=2
1:0250 10
3
, A
23=6
3:3377 10
4
, A
25=6
1:2023 10
4
, A
9=2
4:1120 10
4
, and
A
29=6
6:9155 10
5
. e corresponding harmonic phases of the two paired asymmetric mo-
tions pair are presented in Fig. 6.9d with
b
k=6
D mod.
r
k=6
C ; 2/ ( k D 1; 2; 3; : : :).
(a) (b)
π
ππππ
π
π
π
π
π
(c) (d)
A
1/8
A
3/8
A
5/8
A
127/8
π
π
φ
19/8
φ
3/8
φ
1/8
φ
0
φ
127/8
φ
k/8
φ
9/8
π
π
Figure 6.10: Non-travelable, rotational, symmetric period-8 motion ( D 3:33): (a) displace-
ment, (b) trajectory, (c) harmonic amplitudes, and (d) harmonic phases (˛ D 4:0, ı D 0:1,
Q
0
D 5:0). (t
0
D 0:0 x
0
1:2518, Px
0
4:4671).
6.2.2 SYMMETRIC PERIOD-8 MOTIONS
In Fig. 6.10, a non-travelable symmetric rotating period-8 motion is presented for D 3:33.
e initial condition of such motion is x
0
1:2518 and Px
0
4:4671. e displacement
64 6. NON-TRAVELABLE PERIODIC MOTIONS
response of the symmetric period-8 motion is presented in Fig. 6.10a. For eight excita-
tion periods, the symmetric periodic motion returns back the initial condition. e trajec-
tory of such symmetric period-8 motion possesses a libration with three full clockwise rota-
tions and then three full counter-clockwise rotations to complete a closed orbit, as shown in
Fig. 6.10b. Such trajectory is symmetric to the point .; 0/. e corresponding harmonic am-
plitudes of such a symmetric period-8 motion is presented in Fig. 6.10c. e constant har-
monic term possesses an amplitude of a
.8/
0
2 and mod.a
.8/
0
; 2/ 0. A
2l=8
D A
l=4
D 0
but A
.2l1/=8
¤ 0 (l D 1; 2; : : :). e main harmonic amplitudes of A
.2l1/=8
(l D 1; 2; : : :)
are A
1=8
10:5247, A
3=8
1:5374, A
5=8
0:3626, A
7=8
0:0871, A
9=8
0:2308, A
11=8
0:1182, A
13=8
0:0415, A
15=8
0:0226, A
17=8
0:0288, A
19=8
0:0145, A
21=8
0:0148,
A
23=8
1:7874 10
3
, A
25=8
6:7234 10
4
. For 127th harmonic terms, A
127=8
10
13
.
e harmonic phase distribution of
.2l1/=8
is presented in Fig. 6.10d.
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