48 5. HARMONIC FREQUENCY-AMPLITUDE CHARACTERISTICS
plitude of A
1=4
is presented for the non-travelable period-4 motions. ree paired bifurcation
trees for symmetric period-4 motions to chaos exists. Two paired asymmetric period-4 motions
independently exist, which are not from the symmetric period-4 motions. e corresponding
quantity levels are A
1=4
9:0. e harmonic amplitude A
1=2
is presented in Fig. 5.6(iii). e
quantity levels of such harmonic amplitudes are A
1=2
1:8. In Fig. 5.6(iv), the harmonic ampli-
tude of A
3=4
is presented for the non-travelable period-4 motions, which is similar to harmonic
amplitude A
1=4
. However, the corresponding quantity levels for A
1=4
and A
3=4
are different.
e primary harmonic amplitude A
1
is presented in Fig. 5.6(v). e corresponding quantity
level is A
1
0:6 for 2 .2:0; 4:0/ and A
1
0:01 for 2 .4:0; 8:0/. Similarly, the harmonic
amplitudes of A
5=4
; A
3=2
; A
7=4
are presented in Figs. 5.6(vi)–(viii). e corresponding quantity
levels are A
5=4
0:5; A
3=2
0:15; A
7=4
0:3 for 2 .2:0; 4:0/. e harmonic amplitudes of
A
k
(k D 2; 3; 4; 9 ) are presented in Figs. 5.6(ix)–(xii), respectively. e quantity levels for non-
travelable period-4 motions are A
2
0:12; A
3
0:01; A
4
3 10
3
; and A
9
5 10
6
for
2 .2:0; 4:0/ and A
2
10
2
; A
3
10
4
, A
4
5 10
5
, and A
9
10
10
for 2 .4:0; 6:0/.
For a travelable period-4 motion to chaos, the constant term a
.m/
0
D (m D 4) of veloc-
ity is presented in Fig. 5.7(i) for the positive branch. For the negative branch of the asymmetric
period-4 motion to chaos, a
.m/
0
D is not presented herein. In Fig. 5.7(ii), the harmonic
amplitude of A
1=4
is presented for period-4 motions with the quantity level of A
1=4
3:0. In
Fig. 5.7(iii), harmonic amplitude A
1=2
is presented for period-4 motions with the quantity level
of A
1=2
1:5. In Fig. 5.7(iv), the A
5=4
0:7; A
3=2
0:4; A
7=4
0:6 harmonic amplitude of
A
3=4
is presented for period-4 motions, and the quantity level is A
3=4
1:5. In Fig. 5.7(vii),
the primary harmonic amplitude A
1
is presented for period-4 motions with the quantity level
of A
1
0:6. 2 .4:0; 8:0/. Similarly, the harmonic amplitudes of A
5=4
; A
3=2
; A
7=4
are pre-
sented in Figs. 5.7(vi)–(viii). e corresponding quantity levels are A
5=4
0:7, A
3=2
0:4,
and A
7=4
0:6. e harmonic amplitudes of A
k
(k D 2; 3; 4; 9 ) are presented in Figs. 5.7(ix)–
(xii), respectively. e quantity levels for travelable period-4 motions are A
2
0:36; A
3
0:18;
A
4
0:03; A
9
5 10
5
for 2 .2:0; 4:0/ and A
2
0:2; A
3
0:02; A
4
4 10
3
; and
A
9
3 10
8
for 2 .4:0; 6:0/.
e harmonic frequency-amplitudes for independent period-5, period-6, period-8,
period-10, and period-12 motions can be analyzed similarly.
5.6. PERIOD-4 MOTIONS 49
(i) (ii)
Ω
Ω
π
π
π
π
(iii) (iv)
Ω
Ω
(v) (vi)
ΩΩ
Figure 5.7: Harmonic frequency-amplitude characteristics for bifurcation trees of travelable
period-4 motion to chaos based on velocity: (i) a
.m/
0
(m D 3; 6). (ii)–(vi) A
k=m
(m D 6; k D
1; 2 ; : : : ; 8; 12; 16; 36); Parameters: (˛ D 4:0; ı D 0:1; Q
0
D 5:0; 2 .2:0; 6:0/). (Continues.)
50 5. HARMONIC FREQUENCY-AMPLITUDE CHARACTERISTICS
(vii) (viii)
ΩΩ
(ix) (x)
ΩΩ
(xi) (xii)
Ω
Ω
Figure 5.7: (Continued.) Harmonic frequency-amplitude characteristics for bifurcation trees
of travelable period-4 motion to chaos based on velocity: (vii)–(xii) A
k=m
(m D 6; k D
1; 2 ; : : : ; 8; 12; 16; 36); Parameters: (˛ D 4:0; ı D 0:1; Q
0
D 5:0; 2 .2:0; 6:0/).
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