In this section, the complexification-averaging method is used to obtain the first-order equation and analyze the distribution of the system response in frequency bands. To this end, the following transformations are introduced:
Lets\({u_1}={x_1}\), \({u_k}={x_{k - 1}} - {x_k}\), \(k=2\,,\;3\,,\;4\), and introduce the following transformations:
\({\dot {u}_k}+i\Omega {u_k}={\partial _k}{e^{i\Omega t}}\), \(k=1\,,\; \cdots \,,\;4\) (6)
Subsequently, these expressions can be obtained:
\({u_k}=\frac{{{\partial _k}{e^{i\Omega t}} - {{\bar {\partial }}_k}{e^{ - i\Omega t}}}}{{2i\Omega }}\), \({\dot {u}_k}=\frac{{{\partial _k}{e^{i\Omega t}}+{{\bar {\partial }}_k}{e^{ - i\Omega t}}}}{2}\),
$${\ddot {u}_k}=\frac{{{{\dot {\partial }}_k}{e^{i\Omega t}}+i\Omega {e^{i\Omega t}}{\partial _k}+{{\dot {\bar {\partial }}}_k}{e^{ - i\Omega t}} - i\Omega {e^{i\Omega t}}{{\bar {\partial }}_k}}}{2}$$
7
Substituting expression (7) into Eq. (5) and retaining the slow flow parts results in the following expressions:
$${\dot {\partial }_1}+i\Omega {\partial _1}+\mu {\partial _1}+\frac{{{\partial _1}}}{{i\Omega }}+{\lambda _1}{\partial _2} - \frac{{3{k_1}{\partial _2}^{2}{{\bar {\partial }}_2}}}{{4{i^3}{\Omega ^3}}}=f$$
8a
,
$${\varepsilon _1}\left( {{{\dot {\partial }}_1}+i\Omega {\partial _1} - {{\dot {\partial }}_2} - i\Omega {\partial _2}} \right) - {\lambda _1}{\partial _2}+\frac{{3{k_1}{\partial _2}^{2}{{\bar {\partial }}_2}}}{{4{i^3}{\Omega ^3}}}+{\lambda _2}{\partial _3} - \frac{{3{k_2}{\partial _3}^{2}{{\bar {\partial }}_3}}}{{4{i^3}{\Omega ^3}}}=0$$
8b
,
$${\varepsilon _2}\left( {{{\dot {\partial }}_1}+i\Omega {\partial _1} - {{\dot {\partial }}_2} - i\Omega {\partial _2} - {{\dot {\partial }}_3} - i\Omega {\partial _3}} \right) - {\lambda _2}{\partial _3}+\frac{{3{k_2}{\partial _3}^{2}{{\bar {\partial }}_3}}}{{4{i^3}{\Omega ^3}}}+{\lambda _3}{\partial _4} - \frac{{3{k_3}{\partial _4}^{2}{{\bar {\partial }}_4}}}{{4{i^3}{\Omega ^3}}}=0$$
8c
,
$${\varepsilon _3}\left( {{{\dot {\partial }}_1}+i\Omega {\partial _1} - {{\dot {\partial }}_2} - i\Omega {\partial _2} - {{\dot {\partial }}_3} - i\Omega {\partial _3} - {{\dot {\partial }}_4} - i\Omega {\partial _4}} \right) - {\lambda _3}{\partial _4}+\frac{{3{k_3}{\partial _4}^{2}{{\bar {\partial }}_4}}}{{4{i^3}{\Omega ^3}}}=0$$
8d
Introducing \({\partial _k}={a_k}+i{b_k}\), \(k=1\,,\; \cdots \,,\;4\) into Eq. (8) yields:
\({\dot {a}_1}+i{\dot {b}_1}+i\Omega \left( {{a_1}+i{b_1}} \right)+\mu \left( {{a_1}+i{b_1}} \right)+\frac{{{a_1}+i{b_1}}}{{i\Omega }}+{\lambda _1}\left( {{a_2}+i{b_2}} \right)\)
$$- \frac{{3i{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right)\left( {{a_2}+i{b_2}} \right)}}{{4{\Omega ^3}}}=f$$
9a
,
\({\varepsilon _1}\left[ {{{\dot {a}}_1}+i{{\dot {b}}_1}+i\Omega \left( {{a_1}+i{b_1}} \right) - {{\dot {a}}_2} - i{{\dot {b}}_2} - i\Omega \left( {{a_2}+i{b_2}} \right)} \right] - {\lambda _1}\left( {{a_2}+i{b_2}} \right)+\frac{{3i{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right)\left( {{a_2}+i{b_2}} \right)}}{{4{\Omega ^3}}}\) \(+{\lambda _2}\left( {{a_3}+i{b_3}} \right) - \frac{{3i{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right)\left( {{a_3}+i{b_3}} \right)}}{{4{\Omega ^3}}}=0\), (9b)
\({\varepsilon _2}\left[ {{{\dot {a}}_1}+i{{\dot {b}}_1}+i\Omega \left( {{a_1}+i{b_1}} \right) - {{\dot {a}}_2} - i{{\dot {b}}_2} - i\Omega \left( {{a_2}+i{b_2}} \right) - {{\dot {a}}_3} - i{{\dot {b}}_3} - i\Omega \left( {{a_3}+i{b_3}} \right)} \right]\)
$$- {\lambda _2}\left( {{a_3}+i{b_3}} \right)+\frac{{3i{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right)\left( {{a_3}+i{b_3}} \right)}}{{4{\Omega ^3}}}+{\lambda _3}\left( {{a_4}+i{b_4}} \right) - \frac{{3i{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right)\left( {{a_4}+i{b_4}} \right)}}{{4{\Omega ^3}}}$$
9c
,
\({\varepsilon _3}\left[ {{{\dot {a}}_1}+i{{\dot {b}}_1}+i\Omega \left( {{a_1}+i{b_1}} \right) - {{\dot {a}}_2} - i{{\dot {b}}_2} - i\Omega \left( {{a_2}+i{b_2}} \right) - {{\dot {a}}_3} - i{{\dot {b}}_3} - i\Omega \left( {{a_3}+i{b_3}} \right)} \right.\)
$$\left. { - {{\dot {a}}_4} - i{{\dot {b}}_4} - i\Omega \left( {{a_4}+i{b_4}} \right)} \right] - {\lambda _3}\left( {{a_4}+i{b_4}} \right)+\frac{{3i{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right)\left( {{a_4}+i{b_4}} \right)}}{{4{\Omega ^3}}}=0$$
9d
Then the real and imaginary parts of Eq. (9) can be separated in the form below:
$${\dot {a}_1} - \Omega {b_1}+\mu {a_1}+\frac{{{b_1}}}{\Omega }+{\lambda _1}{a_2}+\frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){b_2}}}{{4{\Omega ^3}}}=f$$
10a
,
$${\dot {b}_1}+\Omega {a_1}+\mu {b_1} - \frac{{{a_1}}}{\Omega }+{\lambda _1}{b_2} - \frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){a_2}}}{{4{\Omega ^3}}}=0$$
10b
,
$${\varepsilon _1}\left[ {{{\dot {a}}_1} - \Omega {b_1} - {{\dot {a}}_2}+\Omega {b_2}} \right] - {\lambda _1}{a_2} - \frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){b_2}}}{{4{\Omega ^3}}}+{\lambda _2}{a_3}+\frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){b_3}}}{{4{\Omega ^3}}}=0$$
10c
,
$${\varepsilon _1}\left[ {{{\dot {b}}_1}+\Omega {a_1} - {{\dot {b}}_2} - \Omega {a_2}} \right] - {\lambda _1}{b_2}+\frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){a_2}}}{{4{\Omega ^3}}}+{\lambda _2}{b_3} - \frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){a_3}}}{{4{\Omega ^3}}}=0$$
10d
,
\({\varepsilon _2}\left[ {{{\dot {a}}_1} - \Omega {b_1} - {{\dot {a}}_2}+\Omega {b_2} - {{\dot {a}}_3}+\Omega {b_3}} \right] - {\lambda _2}{a_3} - \frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){b_3}}}{{4{\Omega ^3}}}\)
$$+{\lambda _3}{a_4}+\frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){b_4}}}{{4{\Omega ^3}}}=0$$
10e
,
\({\varepsilon _2}\left[ {{{\dot {b}}_1}+\Omega {a_1} - {{\dot {b}}_2} - \Omega {a_2} - {{\dot {b}}_3} - \Omega {a_3}} \right] - {\lambda _2}{b_3}+\frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){a_3}}}{{4{\Omega ^3}}}\)
$$+{\lambda _3}{b_4} - \frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){a_4}}}{{4{\Omega ^3}}}=0$$
10f
,
$${\varepsilon _3}\left[ {{{\dot {a}}_1} - \Omega {b_1} - {{\dot {a}}_2}+\Omega {b_2} - {{\dot {a}}_3}+\Omega {b_3} - {{\dot {a}}_4}+\Omega {b_4}} \right] - {\lambda _3}{a_4} - \frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){b_4}}}{{4{\Omega ^3}}}=0$$
10g
,
$${\varepsilon _3}\left[ {{{\dot {b}}_1}+\Omega {a_1} - {{\dot {b}}_2} - \Omega {a_2} - {{\dot {b}}_3} - \Omega {a_3} - {{\dot {b}}_4} - \Omega {a_4}} \right] - {\lambda _3}{b_4}+\frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){a_4}}}{{4{\Omega ^3}}}=0$$
10h
.
Then these equations can be derived based on Eq. (10):
$${\dot {a}_1}=\Omega {b_1} - \mu {a_1} - \frac{{{b_1}}}{\Omega } - {\lambda _1}{a_2} - \frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){b_2}}}{{4{\Omega ^3}}}+f$$
11a
,
$${\dot {b}_1}= - \Omega {a_1} - \mu {b_1}{\text{+}}\frac{{{a_1}}}{\Omega } - {\lambda _1}{b_2}+\frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){a_2}}}{{4{\Omega ^3}}}$$
11b
,
\({\dot {a}_2}= - \mu {a_1} - \frac{{{b_1}}}{\Omega } - {\lambda _1}{a_2} - \frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){b_2}}}{{4{\Omega ^3}}}+f+\Omega {b_2} - \frac{{{\lambda _1}{a_2}}}{{{\varepsilon _1}}} - \frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){b_2}}}{{4{\Omega ^3}{\varepsilon _1}}}\)
$$+\frac{{{\lambda _2}{a_3}}}{{{\varepsilon _1}}}+\frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){b_3}}}{{4{\Omega ^3}{\varepsilon _1}}}$$
11c
,
\({\dot {b}_2}= - \mu {b_1}+\frac{{{a_1}}}{\Omega } - {\lambda _1}{b_2}+\frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){a_2}}}{{4{\Omega ^3}}} - \Omega {a_2} - \frac{{{\lambda _1}{b_2}}}{{{\varepsilon _1}}}+\frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){a_2}}}{{4{\Omega ^3}{\varepsilon _1}}}\)
$$+\frac{{{\lambda _2}{b_3}}}{{{\varepsilon _1}}} - \frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){a_3}}}{{4{\Omega ^3}{\varepsilon _1}}}$$
11d
,
\({\dot {a}_3}=\frac{{{\lambda _1}{a_2}}}{{{\varepsilon _1}}}+\frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){b_2}}}{{4{\Omega ^3}{\varepsilon _1}}} - \frac{{{\lambda _2}{a_3}}}{{{\varepsilon _1}}} - \frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){b_3}}}{{4{\Omega ^3}{\varepsilon _1}}}+\Omega {b_3} - \frac{{{\lambda _2}{a_3}}}{{{\varepsilon _2}}}+\frac{{{\lambda _3}{a_4}}}{{{\varepsilon _2}}}\)
$$- \frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){b_3}}}{{4{\Omega ^3}{\varepsilon _2}}}+\frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){b_4}}}{{4{\Omega ^3}{\varepsilon _2}}}$$
11e
,
\({\dot {b}_3}=\frac{{{\lambda _1}{b_2}}}{{{\varepsilon _1}}} - \frac{{3{k_1}\left( {{a_2}^{2}+{b_2}^{2}} \right){a_2}}}{{4{\Omega ^3}{\varepsilon _1}}} - \frac{{{\lambda _2}{b_3}}}{{{\varepsilon _1}}}+\frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){a_3}}}{{4{\Omega ^3}{\varepsilon _1}}} - \Omega {a_3} - \frac{{{\lambda _2}{b_3}}}{{{\varepsilon _2}}}+\frac{{{\lambda _3}{b_4}}}{{{\varepsilon _2}}}\)
$$+\frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){a_3}}}{{4{\Omega ^3}{\varepsilon _2}}} - \frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){a_4}}}{{4{\Omega ^3}{\varepsilon _2}}}$$
11f
,
\({\dot {a}_4}=\frac{{{\lambda _2}{a_3}}}{{{\varepsilon _2}}} - \frac{{{\lambda _3}{a_4}}}{{{\varepsilon _2}}}+\frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){b_3}}}{{4{\Omega ^3}{\varepsilon _2}}} - \frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){b_4}}}{{4{\Omega ^3}{\varepsilon _2}}}+\Omega {b_4}\)
$$- \frac{{{\lambda _3}{a_4}}}{{{\varepsilon _3}}} - \frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){b_4}}}{{4{\Omega ^3}{\varepsilon _3}}}$$
11g
,
\({\dot {b}_4}=\frac{{{\lambda _2}{b_3}}}{{{\varepsilon _2}}} - \frac{{{\lambda _3}{b_4}}}{{{\varepsilon _2}}} - \frac{{3{k_2}\left( {{a_3}^{2}+{b_3}^{2}} \right){a_3}}}{{4{\Omega ^3}{\varepsilon _2}}}+\frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){a_4}}}{{4{\Omega ^3}{\varepsilon _2}}} - \Omega {a_4}\)
$$- \frac{{{\lambda _3}{b_4}}}{{{\varepsilon _3}}}+\frac{{3{k_3}\left( {{a_4}^{2}+{b_4}^{2}} \right){a_4}}}{{4{\Omega ^3}{\varepsilon _3}}}$$
11h
.
In order to obtain the steady-state response of the system, we set \({\dot {a}_n}=0\), \({\dot {b}_n}=0\), \(n=1,\;2,\;3,\;4.\) Furthermore, the amplitude of an oscillator can be calculated by \(A=\frac{{\sqrt {a_{n}^{2}+b_{n}^{2}} }}{\Omega }\).
To verify the accuracy of the calculation, the results of the complexification-averaging method are compared with that of the Runge-Kutta method. To this end, the steady-state response equations of the system (11) are solved by using a code based on least square method and Eq. (5) is solved directly by using Runge-Kutta method, respectively. It should be indicated that these equations are both solved in the Matlab software environment. In Fig. 4, the black and red points denote the stable and unstable solutions from complexification-averaging method and least square method, and blue circles denotes the solutions from Runge-Kutta method. Figure 4 reveals that the results in the stable region of Fig. 4(a) and in the lower stable branch of Fig. 4(b) obtained by the two methods are consistent. It is worth noting that the values of the blue circles are the maximal values of the displacement in the selected time window. Therefore, there are some deviations in unstable regions and the stable higher branch. More specifically, when the excitation amplitude is large enough (e.g., \(f=0.014\)), the stability analysis between the semi-analytical and numerical methods is unsatisfactory.
The reduced model is expected to generate complex dynamic behaviors because it still has three strong nonlinearities. However, when only the frequency responses of the reduced model are compared with those of a system consisting of one linear oscillator and one cubic oscillator [32, 33], multiple unstable branches arise around the resonance frequency and two unstable branches are extended to the frequency band beyond the resonance frequency for a large excitation amplitude. Figure 5 shows the frequency responses of the cubic oscillators for \(f=0.006\). Similar to the linear oscillator, it is observed that unstable branches of the first cubic oscillator are connected. However, unstable branches of the second and third oscillators are detached. In the present study, the semi-analytical method is used to find the branches of the studied system and further to guide the investigations on the differences in the energy transfer between these branches.
Figure 6 shows the system displacements for \(\Omega {\text{=}}1\) and \(f=0.006\). It is found that the system always has chaotic dynamics and its maximal response amplitude only changes slightly independent of the initial conditions. Accordingly, only a waveform with zero initial conditions is presented. Although the system has three unstable semi-analytical solutions whose values are quite different for this excitation, there is no significant difference between the responses with different initial conditions obtained by Runge-Kutta method. Figure 7 reveals that the responses of the system with \(\Omega {\text{=0}}{\text{.98}}\) are periodic. The vibrations are unstable at the beginning of the response (also called the transient response) and energy transfers to small scales. However, the vibrations become stable rapidly and energy mainly localizes in the linear oscillator.
To further prove the chaotic state of the system, Lyapunov exponents are obtained where the responses in the last 1000 of 3000 (non-dimensional time) are used. The system (2) is transferred into state-space form involving eight variables, where four variables are related to the displacement and the other four variables reflect the velocity. Consequently, eight Lyapunov exponents are obtained. The results reveal that when \(\Omega {\text{=}}1\), five Lyapunov exponents are positive while all Lyapunov exponents are negative for \(\Omega {\text{=0}}{\text{.98}}\). Figure 9 illustrates the distribution of the average energy and the normalized energy flux which can be applied to evaluate the energy transfer between the oscillators. The input energy flux (\({E_i}_{n}\)) and normalized energy flux (\({\hat {E}_n}\)) between neighboring oscillators can be expressed as follows:
$${E_i}_{n}=\int_{{{t_0}}}^{{{t_0}+T}} {f\cos (\Omega t)\,{{\dot {x}}_1}\,dt}$$
12a
\({\hat {E}_n}=\frac{1}{{{E_i}_{n}{\varepsilon _n}}}{\text{ }}\int_{{{t_0}}}^{{{t_0}+T}} {{k_n}{{({x_n} - {x_{n+1}})}^3}{{\dot {x}}_{n+1}}\,dt}\), \(n=1,\;2,...,N\) (12b)
Figure 9(a) shows that as chaotic responses occur, energy distributes in the entire chain, while it is concentrated on the linear oscillator when periodic responses generate. It is found that the average energy of the smaller oscillator is lower than that of the larger oscillator even if the chain produces chaotic responses. This phenomenon is attributed to the hierarchy of scales (i.e., the smaller oscillator has smaller mass and stiffness). However, the black straight line with a positive slope in Fig. 9(b) demonstrates that the energy transfer between the smaller oscillators is more intense than that between larger oscillators in a chaotic state. As expected, the energy transfer between smaller oscillators is weak in a periodic state.
Figure 10 shows the frequency responses of the cubic oscillators for ƒ = 0.014. It is observed that the maximal unstable solution in the resonance band is close to solutions of the stable higher branch, which is quite different from the frequency response of the linear oscillator in Figure 4(b). Figures 11-13 show three typical responses. The responses in Figure 11 are obtained in the resonance band. The responses in Figure 12 and 13 are obtained for the same excitation frequency which is out of the resonance band, and different initial conditions. Considering the Lyapunov exponents in Figure 14, it is concluded that the three responses are chaotic. The responses obtained from the Runge-Kutta method in the chain with random initial conditions are chaotic, while the semi-analytical solutions for this excitation frequency are stable. It is noted that the results of stability analysis are consistent obtained by numerical and semi-analytical methods for relatively small excitation amplitudes. However, there exists some deviations between them for large excitation amplitudes. Even so, the semi-analytical results still can be used to assist the search of the higher branches in the numerical study. Interestingly, the two significantly different responses obtained by Runge-Kutta method under the same excitation frequency are both chaotic.
In this section, the characteristics of the cross-scale energy transfer in the higher branch are compared with those in the lower branch. Figure 15(a) reveals that the energy distributions in the chain for \(\Omega {\text{=}}1\) and \(\Omega {\text{=0}}{\text{.95}}\) with zero initial conditions are slightly different and the distribution rates in the four oscillators are consistent (black and red lines has the similar shape). Moreover, it is observed that the average energy in the chain for the higher branch of \(\Omega {\text{=0}}{\text{.95}}\) is much larger than those in the other cases. The same result can be achieved from the waveforms. Furthermore, the energy fluxes in the two cases with zero initial conditions are almost identical, while the energy flux in the case with non-zero initial condition is much lower than that in the former two cases, shown in Fig. 15(b). Accordingly, it is concluded that high levels of energy do not necessarily lead to a high energy transfer.
Then it is necessary to investigate the influence of the excitation amplitude on the cross-scale energy transfer of the oscillator chain with chaotic responses. In this regard, the results of two case studies are shown in Figs. 16 and 17. It is observed that as the excitation increases, the distributed energy in the three cubic oscillators monotonically increases in the resonance band (e.g., \(\Omega =1\)) except a special case (e.g., \(f=0.016\)). Moreover, the energy level of four oscillators increases slightly as the system generates a higher branch. This phenomenon may be attributed to difficulties in increasing the amplitude for a higher branch. The initial conditions for obtaining high branches in Fig. 17 are listed in Table 1. It is worth noting that the normalized energy fluxes almost remain constant in the two cases, indicating that the variations of the energy transfer between oscillators have a linear correlation with the increase of the input energy.
Table 1
Initial conditions for obtaining high branches in Fig. 17
|
Initial conditions
|
0.014
|
0.569
|
-0.522
|
-0.765
|
-0.211
|
-0.105
|
0.489
|
-0.091
|
-0.255
|
0.016
|
-0.693
|
-0.636
|
-0.203
|
-0.147
|
0.181
|
-0.533
|
-0.106
|
0.762
|
0.018
|
0.513
|
-0.268
|
0.118
|
-0.321
|
-0.038
|
0.311
|
-0.785
|
0.182
|
0.02
|
0.099
|
-0.165
|
0.492
|
0.209
|
0.283
|
-0.582
|
-0.308
|
0.423
|
Figures 18 and 19 show the influences of the scale (mass ratio) between cubic oscillators on the cross-scale energy transfer and the initial conditions are listed in Table 2. The oscillator chains produce chaotic responses. It is found that the energy level distributed in the cubic oscillators decreases with the increase of the mass ratio in both excitation conditions. Moreover, the distribution of normalized energy fluxes have the same slope and almost overlap with each other in Figs. 18(b) and 19(b). Recalling Fig. 15(b), the normalized energy flux of the oscillator chain has a significant difference as it produces two different vibration states. The responses in the lower and higher branches are deemed as two vibration states. However, as the chain vibrates in one state, the normalized energy flux is almost independent of the variations of the input energy and the scale between cubic oscillators.
Table 2
Initial conditions for obtaining high branches in Fig. 19
Mass ratio
|
Initial conditions
|
7
|
0.705
|
-0.177
|
0.024
|
-0.356
|
0.208
|
-0.591
|
0.164
|
0.209
|
8
|
0.569
|
-0.522
|
-0.765
|
-0.211
|
-0.105
|
0.489
|
-0.091
|
-0.255
|
9
|
0.062
|
0.381
|
0.527
|
-0.441
|
0.334
|
0.355
|
0.461
|
-0.261
|
10
|
-0.436
|
0.423
|
0.572
|
-0.102
|
0.263
|
-0.065
|
-0.028
|
0.089
|