Vibration stabilization by a nonresonant secondary limit cycle oscillator

Abstract

Many vibrating systems, over some ranges of parameter values, exhibit a single unstable mode. Adding a small resonant secondary system to the unstable system is a well-known stabilization strategy. Here we show that even a nonresonant secondary system, if equipped with a limit cycle of its own, can stabilize the unstable mode of the primary system. The primary system is modeled here as a linear spring block system with negative damping. The secondary system is a van der Pol oscillator. Smallness of the latter’s parameters allows use of the method of multiple scales. The resulting slow amplitude equations decouple from the phases and a two-dimensional system is obtained. The secondary system’s amplitude evolves faster than that of the primary system, which simplifies analysis. A parameter-dependent transformation casts the system in a canonical form with a single free parameter $c_1>0$ in addition to the small perturbation parameter. The canonical phase portrait involves two key straight lines. When $c_1 < 4$ these lines intersect and a separatrix passes through that intersection. Solutions on one side of the separatrix show quenching of the primary instability with limit cycle oscillation of the secondary system. Solutions on the other side of the separatrix show significant oscillations of the primary system at its natural frequency, with the secondary limit cycle being quenched. When $c_1 > 4$, stabilization fails for all initial conditions. In summary, for the case of a negatively damped oscillator interacting with a small nonresonant secondary limit cycle oscillator, we show stabilization, provide a pair of canonical equations with one free parameter, and present a complete qualitative characterization of the dynamics.

Appendices

Broader literature review

We have conducted a broader literature search and found 40 papers, of which 39 of them were published between 1980 and 2022. These papers each have the van der Pol oscillator coupled to either the van der Pol oscillator or some other oscillator. These papers are included in this article as references [20, 24, 25, 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64]. These references can be grouped in various ways depending on which aspect of their contribution we wish to focus on. Each such grouping serves to emphasize how our work is different.

Many of these papers involve two coupled oscillators that are resonant, i.e., the natural frequencies are close. From our 40 references, it turns out that references [20, 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46] are in this first category. Since our two oscillators have frequencies that are not close, we do not fall in this category of resonant coupled oscillators.

Another way to classify the papers is to identify those (27 out of 40) where the two oscillators’ inertia terms are of equal or nearly equal magnitude, and their coupled dynamics, which may be weakly or strongly nonlinear, are then examined. References [24, 25, 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 46,47,48,49,50,51,52,53,54,55] fall in this category. We do not fall in this category because our motivation is an untuned vibration stabilizer where the added system is necessarily small compared to the primary system, and it still affects the dynamics.

Another way to group some papers is to note those wherein two oscillators, by mathematical design, are capable of having perfectly synchronized solutions. In abstract first-order form, if we have

$$\begin{aligned} \dot{x} = f(x) + g(x,y) \quad \text{ and } \quad \dot{y} = f(y) + g(y,x), \end{aligned}$$

then $x=y$ is a possible solution satisfying

$$\begin{aligned} \dot{x} = f(x) + g(x,x). \end{aligned}$$

Whether that phase-locked solution is stable or not then becomes important. If it is stable, the basin of attraction is important, and so on. Out of the 40, 9 papers fall in this category [29, 31,32,33, 36, 38, 39, 42, 46]. Since our two oscillators do not have this form, our paper does not fall into this category.

Then there are coupled van der Pol and Duffing systems, for which some papers study the dynamics near 1:1 resonance [41, 44, 49, 56, 57], phase-locked motions [49, 54], or assume comparable inertia [25, 41, 42, 49, 52,53,54]. As explained above, our paper is very different from these papers.

There are also studies of coupled van der Pol and Duffing systems that involve 1:2 or 1:3 resonances: see [58,59,60,61]. We focus on nonresonant interactions and so we do not fall in this category.

We note that nonresonant interactions were studied in [51,52,53], but there the inertias of the two oscillators were the same (as noted above), and they reported decoupled amplitude dynamics of the coupled oscillators, while our results with disparate inertias show a potentially useful coupled dynamics. Therefore we differ from those papers.

Finally, for parameter values that are not small and where analytical progress was limited, numerical or experimental approaches have been used [47, 50, 54, 62,63,64] to observe nonlinear responses like multistability and chaotic solutions. Our paper is not in this category.

In this way, there is at least one key difference, and usually more than one key difference, between our paper and these 40 papers.

Expressions from multiple scale analysis

The solution for $Z_{1}$ is

$$\begin{aligned}{} & {} Z_{1}(T_{0}, T_{1}, T_{2}) = -\frac{1}{4 (1-{\omega _{p}}^{2})^{4}}\nonumber \\{} & {} \quad \left( (2 \gamma {\omega _{p}}^{4} - 4 \gamma {\omega _{p}}^{2} + 2 \gamma )R Q^{2} - (4 \gamma {\omega _{p}}^{4} -8 \gamma {\omega _{p}}^{2} \right. \nonumber \\{} & {} \quad \left. + 4 \gamma - \gamma R^{2}) R \right) \cos (T_{0} + \phi ) + \frac{m {\omega _{p}}^{4} R}{(1-{\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \sin (T_{0} + \phi )+ \frac{\gamma R^{3}}{4 (9 - {\omega _{p}}^{2})(1-{\omega _{p}}^{2})^{3} } \cos (3 T_{0} + 3 \phi )\nonumber \\{} & {} \quad - \frac{\gamma Q^{3}}{32 \omega _{p}} \cos (3 \omega _{p} T_{0} + 3 \psi )\nonumber \\{} & {} \quad + \frac{(1-2 \omega _{p}) \gamma Q^{2} R}{4 (1 - 3 \omega _{p}) (1 + \omega _{p}) (1-\omega _{p})^{2}} \nonumber \\{} & {} \quad \cos (T_{0} - 2 \omega _{p} T_{0} + \phi - 2 \psi )\nonumber \\{} & {} \quad + \frac{(1+2 \omega _{p}) \gamma Q^{2} R}{4 (1 + 3 \omega _{p}) (1 - \omega _{p}) (1+\omega _{p})^{2}} \nonumber \\{} & {} \quad \cos (T_{0} + 2 \omega _{p} T_{0} + \phi + 2 \psi )\nonumber \\{} & {} \quad + \frac{\gamma (2 -\omega _{p}) Q R^{2}}{16 (1-\omega _{p}) (1-{\omega _{p}}^{2})^{2}} \nonumber \\{} & {} \quad \cos (2 T_{0} - \omega _{p} T_{0} + 2 \phi -\psi )\nonumber \\{} & {} \quad -\frac{\gamma (2 +\omega _{p}) Q R^{2}}{16 (1+\omega _{p}) (1-{\omega _{p}}^{2})^{2}} \nonumber \\{} & {} \quad \cos (2 T_{0} + \omega _{p} T_{0} + 2 \phi +\psi ). \end{aligned}$$

(B.1)

Recall Eq. (27), the expressions are

$$\begin{aligned}{} & {} L_{1}=\frac{\partial ^{2} X_{2}}{\partial {T_{0}}^{2}} + X_{2} -\left( \frac{m^{2} {\omega _{p}}^{4} (1+ 3{\omega _{p}}^{2}) R}{4 (1-{\omega _{p}}^{2})^{3}}\right. \nonumber \\{} & {} \left. \quad +2 R \frac{\partial \phi }{\partial T_{2}}\right) \sin (T_{0} + \phi )\nonumber \\{} & {} \quad + \biggl (c R + \frac{m \gamma \left( 2(Q^{2}-2)(1-{\omega _{p}}^{2})^{2} + R^{2}\right) R}{4 (1- {\omega _{p}}^{2})^{4}} \nonumber \\{} & {} \quad + 2 \frac{\partial R}{\partial T_{2}} \biggr ) \cos (T_{0} +\phi )\nonumber \\{} & {} \quad -\frac{m \gamma \omega _{p} \left( (Q^{2}-4) (1-{\omega _{p}}^{2})^{2} + 2 R^{2}\right) Q}{4 (1-{\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \cos (\omega _{p} T_{0} + \psi )\nonumber \\{} & {} \quad + \frac{m^{2} {\omega _{p}}^{6} Q}{(1-{\omega _{p}}^{2})^{2}} \sin (\omega _{p} T_{0} + \psi )+ \frac{9 m \gamma \omega _{p} Q^{3}}{32} \nonumber \\{} & {} \quad \cos (3 \omega _{p} T_{0} + 3 \psi )\nonumber \\{} & {} \quad {- \frac{9 m \gamma R^{3}}{4 (9-{\omega _{p}}^{2})(1-{\omega _{p}}^{2})^{3}} \cos (3 T_{0} + 3 \phi )}\nonumber \\{} & {} \quad - \frac{m \gamma (1+2 \omega _{p})^{3} Q^{2} R}{4 (1 + 3 \omega _{p}) (1 - \omega _{p}) (1+\omega _{p})^{2}} \nonumber \\{} & {} \quad \cos (T_{0} + 2 \omega _{p} T_{0} + \phi + 2 \psi )\nonumber \\{} & {} \quad - \frac{m \gamma (1-2 \omega _{p})^{3} Q^{2} R}{4 (1 - 3 \omega _{p}) (1 + \omega _{p}) (1-\omega _{p})^{2}} \nonumber \\{} & {} \quad \cos (T_{0} - 2 \omega _{p} T_{0} + \phi - 2 \psi )\nonumber \\{} & {} \quad + \frac{m \gamma (2+\omega _{p})^{3} Q R^{2}}{16 (1+\omega _{p}) (1-{\omega _{p}}^{2})^{2}}\nonumber \\{} & {} \quad \cos (2 T_{0} + \omega _{p} T_{0} + 2 \phi +\psi )\nonumber \\{} & {} \quad - \frac{m \gamma (2-\omega _{p})^{3} Q R^{2}}{16 (1-\omega _{p}) (1-{\omega _{p}}^{2})^{2}} \nonumber \\{} & {} \quad \cos (2 T_{0} - \omega _{p} T_{0} + 2 \phi -\psi ), \end{aligned}$$

(B.2)

and

$$\begin{aligned}{} & {} L_{2}=\frac{\partial ^{2} Z_{2}}{\partial {T_{0}}^{2}} + {\omega _{p}}^{2} Z_{2} + \frac{\partial ^{2} X_{2}}{\partial {T_{0}}^{2}} \nonumber \\{} & {} \quad + \frac{{\omega _{p}}^{2}}{8 (1-{\omega _{p}}^{2})^{5}}\biggl (m \gamma (6 {\omega _{p}}^{6} - 10 {\omega _{p}}^{4} + 2 {\omega _{p}}^{2} \nonumber \\{} & {} \quad + 2 ) Q^{2} R + m \gamma (-12 {\omega _{p}}^{6} + 20 {\omega _{p}}^{4} - 4 {\omega _{p}}^{2} \nonumber \\{} & {} \quad -4 + 7 {\omega _{p}}^{2} R^{2} + R^{2}) R +(-16 {\omega _{p}}^{8} + 64 {\omega _{p}}^{6}\nonumber \\{} & {} \quad -96 {\omega _{p}}^{4} + 64 {\omega _{p}}^{2} -16) \frac{\partial R}{\partial T_{2}}\biggr ) \cos (T_{0} + \phi )\nonumber \\{} & {} \quad +\frac{1}{16 (9-{\omega _{p}}^{2}) (1-9 {\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{6}}\nonumber \\{} & {} \quad \biggl ((-45 {\omega _{p}}^{10} + 518 {\omega _{p}}^{8} -1110 {\omega _{p}}^{6} + 864 {\omega _{p}}^{4} \nonumber \\{} & {} \quad -245 {\omega _{p}}^{2} + 18) \gamma ^{2} R^{2} Q^{2} + (-18 {\omega _{p}}^{14} \nonumber \\{} & {} \quad + 264 {\omega _{p}}^{12} -1150 {\omega _{p}}^{10} + 2356 {\omega _{p}}^{8} \nonumber \\{} & {} \quad -2574 {\omega _{p}}^{6} + 1504 {\omega _{p}}^{4} -418 {\omega _{p}}^{2} + 36 ) \gamma ^{2} Q^{4} \nonumber \\{} & {} \quad + (18 {\omega _{p}}^{4} -92 {\omega _{p}}^{2} +10) \gamma ^{2} R^{4} \nonumber \\{} & {} \quad + (72 {\omega _{p}}^{14} -1016 {\omega _{p}}^{12} +4072 {\omega _{p}}^{10}\nonumber \\{} & {} \quad -7640 {\omega _{p}}^{8} +7640 {\omega _{p}}^{6} \nonumber \\{} & {} \quad -4072 {\omega _{p}}^{4} +1016 {\omega _{p}}^{2} -72) \gamma ^{2} Q^{2} \nonumber \\{} & {} \quad + (-72 {\omega _{p}}^{8} + 800 {\omega _{p}}^{6} -1456 {\omega _{p}}^{4} \nonumber \\{} & {} \quad + 800 {\omega _{p}}^{2}-72)\gamma ^{2} R^{2}\nonumber \\{} & {} \quad + m^{2} (-36 {\omega _{p}}^{16} + 292 {\omega _{p}}^{14} +472 {\omega _{p}}^{12} \nonumber \\{} & {} \quad -1784 {\omega _{p}}^{10} + 1164 {\omega _{p}}^{8} -108 {\omega _{p}}^{6} ) + \gamma ^{2} (144 {\omega _{p}}^{12}\nonumber \\{} & {} \quad -1888 {\omega _{p}}^{10} +6256 {\omega _{p}}^{8} -9024 {\omega _{p}}^{6} \nonumber \\{} & {} \quad + 6256 {\omega _{p}}^{4} -1888 {\omega _{p}}^{2} + 144) +(-288 {\omega _{p}}^{16}\nonumber \\{} & {} \quad + 4064 {\omega _{p}}^{14}-16288 {\omega _{p}}^{12}\nonumber \\{} & {} \quad + 30560 {\omega _{p}}^{10}-30560 {\omega _{p}}^{8}+16288 {\omega _{p}}^{6} -4064 {\omega _{p}}^{4} \nonumber \\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad + 288 {\omega _{p}}^{2} )\frac{\partial \phi }{\partial T_{2}}\biggr ) R \sin (T_{0} + \phi ) +\frac{\omega _{p}}{4 (1-{\omega _{p}}^{2})^{4}}\nonumber \\{} & {} \quad \biggl (({\omega _{p}}^{8} -3 {\omega _{p}}^{6} + 3 {\omega _{p}}^{4} -{\omega _{p}}^{2} ) m \gamma Q^{3} \nonumber \\{} & {} \quad + (-2 {\omega _{p}}^{4} -2 {\omega _{p}}^{2}) m \gamma R^{2} Q + (-4 {\omega _{p}}^{8} + 12 {\omega _{p}}^{6} \nonumber \\{} & {} \quad -12 {\omega _{p}}^{4} + 4 {\omega _{p}}^{2} ) m \gamma Q + (8 {\omega _{p}}^{8}-32 {\omega _{p}}^{6} \nonumber \\{} & {} \quad +48 {\omega _{p}}^{4}-32 {\omega _{p}}^{2} +8 ) \frac{\partial Q}{\partial T_{2}}\biggr )\cos (\omega _{p} T_{0} + \psi ) \nonumber \\{} & {} \quad -\frac{1}{128 (1-9 {\omega _{p}}^{2}) (1- {\omega _{p}}^{2})^{5}} \biggl ((336 {\omega _{p}}^{8} \nonumber \\{} & {} \quad -896 {\omega _{p}}^{6}+ 800 {\omega _{p}}^{4} -256 {\omega _{p}}^{2} + 16) \gamma ^{2} R^{2} Q^{2} \nonumber \\{} & {} \quad + (63 {\omega _{p}}^{12} -322 {\omega _{p}}^{10} +665 {\omega _{p}}^{8} -700 {\omega _{p}}^{6} \nonumber \\{} & {} \quad +385 {\omega _{p}}^{4} -98 {\omega _{p}}^{2} +7) \gamma ^{2} Q^{4}\nonumber \\{} & {} \quad + (108 {\omega _{p}}^{4} -84 {\omega _{p}}^{2}+8) \gamma ^{2} R^{4} \nonumber \\{} & {} \quad + (-288 {\omega _{p}}^{12} + 1472 {\omega _{p}}^{10} -3040 {\omega _{p}}^{8} + 3200 {\omega _{p}}^{6} \nonumber \\{} & {} \quad -1760 {\omega _{p}}^{4}+ 448 {\omega _{p}}^{2}-32) \gamma ^{2} Q^{2}\nonumber \\{} & {} \quad + (-288 {\omega _{p}}^{8} + 896 {\omega _{p}}^{6} -960 {\omega _{p}}^{4}\nonumber \\{} & {} \quad +384 {\omega _{p}}^{2} -32) \gamma ^{2} R^{2} \nonumber \\{} & {} \quad + m^{2} (-864 {\omega _{p}}^{14} + 2688 {\omega _{p}}^{12} -2880 {\omega _{p}}^{10} \nonumber \\{} & {} \quad + 1152 {\omega _{p}}^{8} -96 {\omega _{p}}^{6} )\nonumber \\{} & {} \quad + \gamma ^{2} (288 {\omega _{p}}^{12} -1472 {\omega _{p}}^{10} + 3040 {\omega _{p}}^{8} \nonumber \\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad -3200 {\omega _{p}}^{6}+ 1760 {\omega _{p}}^{4} -448 {\omega _{p}}^{2} + 32)\nonumber \\{} & {} \quad +(2304 {\omega _{p}}^{13} -11776 {\omega _{p}}^{11} + 24320 {\omega _{p}}^{9} \nonumber \\{} & {} \quad -25600 {\omega _{p}}^{7} +14080 {\omega _{p}}^{5}-3584 {\omega _{p}}^{3} \nonumber \\{} & {} \quad + 256 \omega _{p})\frac{\partial \psi }{\partial T_{2}}\biggr ) Q \sin (\omega _{p} T_{0} + \psi ) \nonumber \\{} & {} \quad + \frac{m \gamma {\omega _{p}}^{2} (7 {\omega _{p}}^{4} -46 {\omega _{p}}^{2} -9) R^{3}}{8 (9-{\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{5}} \cos (3 T_{0} + 3 \phi )\nonumber \\{} & {} \quad - \frac{3 \gamma ^{2} R^{3}}{16 (9- {\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{6}} \left( ({\omega _{p}}^{8} -17 {\omega _{p}}^{6} +69 {\omega _{p}}^{4} \right. \nonumber \\{} & {} \quad -91 {\omega _{p}}^{2}+ 38) Q^{2} +(-3 {\omega _{p}}^{2} +11) R^{2} + 8 {\omega _{p}}^{6} \nonumber \\{} & {} \quad \left. -56 {\omega _{p}}^{4}+ 88 {\omega _{p}}^{2} -40\right) \sin (3 T_{0} + 3 \phi ) - \frac{5 m \gamma {\omega _{p}}^{3} Q^{3}}{32 (1-{\omega _{p}}^{2})} \nonumber \\{} & {} \quad \cos (3 \omega _{p} T_{0} + 3 \psi )+ \frac{\gamma ^{2} Q^{3}}{128 (1-9 {\omega _{p}}^{2}) (1- {\omega _{p}}^{2})^{3}}\nonumber \\{} & {} \quad \left( (9 {\omega _{p}}^{8} -28 {\omega _{p}}^{6} + 30 {\omega _{p}}^{4} -12 {\omega _{p}}^{2} +1 ) Q^{2}\right. \nonumber \\{} & {} \quad + (540 {\omega _{p}}^{4} -152 {\omega _{p}}^{2} - 4) R^{2} + 72 {\omega _{p}}^{8} \nonumber \\{} & {} \left. \quad -224 {\omega _{p}}^{6} + 240 {\omega _{p}}^{4} -96 {\omega _{p}}^{2} + 8\right) \sin (3 \omega _{p} T_{0} + 3 \psi ) \nonumber \\{} & {} \quad + \frac{5 \gamma ^{2} R^{5}}{16 (9-{\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{5}} \sin (5 T_{0} + 5 \phi )\nonumber \\{} & {} \quad -\frac{5 \gamma ^{2} Q^{5}}{128} \sin (5 \omega _{p} T_{0} + 5 \psi ) \nonumber \\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad - \frac{m \gamma {\omega _{p}}^{2} (22 {\omega _{p}}^{4} + 3 {\omega _{p}}^{3} -3 {\omega _{p}}^{2} + \omega _{p} + 1) R Q^{2}}{8 (1 + 3 \omega _{p}) (1-{\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \cos (T_{0} + 2 \omega _{p} T_{0} + \phi + 2 \psi )\nonumber \\{} & {} \quad - \frac{m \gamma {\omega _{p}}^{2} (22 {\omega _{p}}^{4} - 3 {\omega _{p}}^{3} -3 {\omega _{p}}^{2} - \omega _{p} + 1) R Q^{2}}{8 (1 - 3 \omega _{p}) (1-{\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \cos (T_{0} - 2 \omega _{p} T_{0} + \phi - 2 \psi )\nonumber \\{} & {} \quad + \frac{m \gamma {\omega _{p}}^{2} ({\omega _{p}}^{4} + 7 {\omega _{p}}^{3} + 14 {\omega _{p}}^{2} -2 \omega _{p} + 4) Q R^{2}}{16 (1-{\omega _{p}}^{2})^{4}} \nonumber \\{} & {} \quad \cos (2 T_{0} + \omega _{p} T_{0} + 2 \phi + \psi ) \nonumber \\{} & {} \quad - \frac{m \gamma {\omega _{p}}^{2} ({\omega _{p}}^{4} - 7 {\omega _{p}}^{3} + 14 {\omega _{p}}^{2} +2 \omega _{p} + 4) Q R^{2}}{16 (1-{\omega _{p}}^{2})^{4}} \nonumber \\{} & {} \quad \cos (2 T_{0} - \omega _{p} T_{0} + 2 \phi - \psi )\nonumber \\{} & {} \quad + \frac{\gamma ^{2} (4 + \omega _{p}) ({\omega _{p}}^{3} + 2 {\omega _{p}}^{2} -17 \omega _{p} -26) Q R^{4} }{64 (9-{\omega _{p}}^{2}) (1+\omega _{p}) (1-{\omega _{p}}^{2})^{4}} \nonumber \\{} & {} \quad \sin (4 T_{0} + \omega _{p} T_{0} + 4 \phi + \psi )\nonumber \\{} & {} \quad + \frac{\gamma ^{2} (4 - \omega _{p}) ({\omega _{p}}^{3} - 2 {\omega _{p}}^{2} -17 \omega _{p} +26) Q R^{4} }{64 (9-{\omega _{p}}^{2}) (1-\omega _{p}) (1-{\omega _{p}}^{2})^{4}} \nonumber \\{} & {} \quad \sin (4 T_{0} - \omega _{p} T_{0} + 4 \phi - \psi )\nonumber \\{} & {} \quad + \frac{\gamma ^{2} (1 + 4 \omega _{p}) (11 {\omega _{p}}^{2} + 8 \omega _{p} + 1) Q^{4} R}{64 \omega _{p} (1 + 3 \omega _{p}) (1+\omega _{p}) (1-{\omega _{p}}^{2})} \nonumber \\{} & {} \quad \sin (T_{0} + 4 \omega _{p} T_{0} + \phi + 4 \psi )\nonumber \\{} & {} \quad - \frac{\gamma ^{2} (1 - 4 \omega _{p}) (11 {\omega _{p}}^{2} - 8 \omega _{p} + 1) Q^{4} R}{64 \omega _{p} (1 - 3 \omega _{p}) (1-\omega _{p}) (1-{\omega _{p}}^{2})} \nonumber \\{} & {} \quad \sin (T_{0} - 4 \omega _{p} T_{0} + \phi - 4 \psi )\nonumber \\{} & {} \quad + \frac{\gamma ^{2} R Q^{2}}{64 \omega _{p} (1-9 {\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \left( (5 {\omega _{p}}^{2}+54 {\omega _{p}}^{7} + 135 {\omega _{p}}^{6} -186 {\omega _{p}}^{5} -141 {\omega _{p}}^{4} \right. \nonumber \\{} & {} \quad + 146 {\omega _{p}}^{3} -14 \omega _{p} +1) Q^{2} + (-36 {\omega _{p}}^{4} + 78 {\omega _{p}}^{3} \nonumber \\{} & {} \quad -2 {\omega _{p}}^{2} -4 \omega _{p} ) R^{2} + 48 {\omega _{p}}^{7} -32 {\omega _{p}}^{2} \nonumber \\{} & {} \quad -224 {\omega _{p}}^{6} + 112 {\omega _{p}}^{5} + 256 {\omega _{p}}^{4} -176 {\omega _{p}}^{3} \left. + 16 \omega _{p} \right) \nonumber \\{} & {} \quad \sin (T_{0} - 2 \omega _{p} T_{0} + \phi - 2 \psi )\nonumber \\{} & {} \quad + \frac{\gamma ^{2} R Q^{2}}{64 \omega _{p} (1-9 {\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \left( (- 5 {\omega _{p}}^{2}+54 {\omega _{p}}^{7} - 135 {\omega _{p}}^{6} -186 {\omega _{p}}^{5} +141 {\omega _{p}}^{4} \right. \nonumber \\ \end{aligned}$$

$$\begin{aligned}{} & {} \quad + 146 {\omega _{p}}^{3} -14 \omega _{p} -1) Q^{2} + (36 {\omega _{p}}^{4} + 78 {\omega _{p}}^{3} \nonumber \\{} & {} \quad +2 {\omega _{p}}^{2} -4 \omega _{p} ) R^{2} + 48 {\omega _{p}}^{7} +224 {\omega _{p}}^{6}\nonumber \\{} & {} \quad \left. + 112 {\omega _{p}}^{5} - 256 {\omega _{p}}^{4} -176 {\omega _{p}}^{3} +32 {\omega _{p}}^{2} + 16 \omega _{p} \right) \nonumber \\{} & {} \quad \sin (T_{0} + 2 \omega _{p} T_{0} + \phi + 2 \psi )\nonumber \\{} & {} \quad - \frac{\gamma ^{2} Q R^{2}}{32 (1-3 \omega _{p}) (9- {\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{5}} \nonumber \\{} & {} \quad \left( (3 {\omega _{p}}^{10}-18 {\omega _{p}}^{9}-48 {\omega _{p}}^{8}+283 {\omega _{p}}^{7} \right. \nonumber \\{} & {} \quad + 193 {\omega _{p}}^{6}-1277 {\omega _{p}}^{5}+ 7 {\omega _{p}}^{4} + 1777 {\omega _{p}}^{3} \nonumber \\{} & {} \quad -416 {\omega _{p}}^{2}-765 \omega _{p} +261) Q^{2} \nonumber \\{} & {} \quad + (-6 {\omega _{p}}^{5}-22 {\omega _{p}}^{4}+116 {\omega _{p}}^{3}+84 {\omega _{p}}^{2}\nonumber \\{} & {} \quad -334 \omega _{p} +98) R^{2} + 12 {\omega _{p}}^{9}+ 44 {\omega _{p}}^{8}-256 {\omega _{p}}^{7}\nonumber \\{} & {} \quad -448 {\omega _{p}}^{6} +1592 {\omega _{p}}^{5} +440 {\omega _{p}}^{4} -2464 {\omega _{p}}^{3} \nonumber \\{} & {} \quad \left. + 288 {\omega _{p}}^{2}+1116 \omega _{p}-324\right) \nonumber \\{} & {} \quad \sin (2 T_{0} - \omega _{p} T_{0} + 2 \phi - \psi )\nonumber \\{} & {} \quad + \frac{\gamma ^{2} Q R^{2}}{32 (1+3 \omega _{p}) (9- {\omega _{p}}^{2}) (1-{\omega _{p}}^{2})^{5}} \nonumber \\{} & {} \quad \left( (3 {\omega _{p}}^{10}+18 {\omega _{p}}^{9}-48 {\omega _{p}}^{8}-283 {\omega _{p}}^{7} + 193 {\omega _{p}}^{6}\right. \nonumber \\{} & {} \quad +1277 {\omega _{p}}^{5} + 7 {\omega _{p}}^{4} - 1777 {\omega _{p}}^{3}-416 {\omega _{p}}^{2}+765 \omega _{p} \nonumber \\{} & {} \quad +261) Q^{2} + (6 {\omega _{p}}^{5}-22 {\omega _{p}}^{4}-116 {\omega _{p}}^{3}+84 {\omega _{p}}^{2}\nonumber \\{} & {} \quad +334 \omega _{p}+98) R^{2} - 12 {\omega _{p}}^{9}+ 44 {\omega _{p}}^{8}+256 {\omega _{p}}^{7}\nonumber \\{} & {} \quad -448 {\omega _{p}}^{6}-1592 {\omega _{p}}^{5} +440 {\omega _{p}}^{4} \nonumber \\{} & {} \quad \left. +2464 {\omega _{p}}^{3}+ 288 {\omega _{p}}^{2}-1116 \omega _{p}-324\right) \nonumber \\{} & {} \quad \sin (2 T_{0} + \omega _{p} T_{0} + 2 \phi +\psi )\nonumber \\{} & {} \quad -\frac{\gamma ^{2} (2 + 3 \omega _{p}) (6 {\omega _{p}}^{3} + 49 {\omega _{p}}^{2} + 24 \omega _{p} +1) R^{2} Q^{3}}{128 \omega _{p} (1 + 3 \omega _{p}) (1 + \omega _{p}) (1-{\omega _{p}}^{2})^{2}} \nonumber \\{} & {} \quad \sin (2 T_{0} + 3 \omega _{p} T_{0} + 2 \phi + 3 \psi )\nonumber \\{} & {} \quad +\frac{\gamma ^{2} (2 - 3 \omega _{p}) (6 {\omega _{p}}^{3} - 49 {\omega _{p}}^{2} + 24 \omega _{p} -1) R^{2} Q^{3}}{128 \omega _{p} (1 - 3 \omega _{p}) (1 - \omega _{p}) (1-{\omega _{p}}^{2})^{2}} \nonumber \\{} & {} \quad \sin (2 T_{0} - 3 \omega _{p} T_{0} + 2 \phi - 3 \psi )\nonumber \\{} & {} \quad - \frac{\gamma ^{2} (3 +2 \omega _{p}) (3 {\omega _{p}}^{4} + 11 {\omega _{p}}^{3} -29 {\omega _{p}}^{2} -107 \omega _{p} -38) R^{3} Q^{2}}{32 (1 + 3 \omega _{p}) (9-{\omega _{p}}^{2}) (1+\omega _{p}) (1- {\omega _{p}}^{2})^{3}} \nonumber \\{} & {} \quad \sin (3 T_{0} + 2 \omega _{p} T_{0} \nonumber \\{} & {} \quad + 3 \phi + 2 \psi )\nonumber \\{} & {} \quad - \frac{\gamma ^{2} (3 -2 \omega _{p}) (3 {\omega _{p}}^{4}- 11 {\omega _{p}}^{3} -29 {\omega _{p}}^{2} +107 \omega _{p} -38) }{32 (1 - 3 \omega _{p}) (9-{\omega _{p}}^{2}) (1-\omega _{p}) (1- {\omega _{p}}^{2})^{3}} \nonumber \\{} & {} R^{3} Q^{2} \sin (3 T_{0} - 2 \omega _{p} T_{0} + 3 \phi -2 \psi ). \end{aligned}$$

(B.3)

The solution for $X_{2}$ is

$$\begin{aligned}{} & {} X_{2}(T_{0},T_{1},T_{2})\nonumber \\{} & {} \quad ={\frac{ ((Q^{2}-4) (1-{\omega _{p}}^{2})^{2}+2 R^{2}) m \gamma Q \omega _{p}}{4 (1-{\omega _{p}}^{2})^{4}}} \nonumber \\{} & {} \quad \cos (\omega _{p} T_{0} + \psi )\nonumber \\{} & {} \quad {-\frac{Q {m}^{2}{\omega _{p}}^{6}}{(1-{\omega _{p}}^{2}) ^{3}}} \sin (\omega _{p} T_{0} + \psi ) \nonumber \\{} & {} \quad -{\frac{9 m \gamma \omega _{p} Q^{3}}{32 (1-9{\omega _{p}}^{2})}} \cos (3 \omega _{p} T_{0} + 3 \psi )\nonumber \\{} & {} \quad -{\frac{9 m \gamma R^{3}}{32 (9-{\omega _{p}}^{2})(1-{\omega _{p}}^{2})^{3}}} \cos (3 T_{0} + 3 \phi )\nonumber \\{} & {} \quad -{\frac{ (1+2 \omega _{p})^{3} m \gamma R Q^{2}}{ 16 \omega _{p} (1+3 \omega _{p}) (1-\omega _{p}) (1+\omega _{p})^{3}}}\nonumber \\{} & {} \quad \cos (T_{0}+\phi +2\omega _{p} T_{0} + 2\psi )\nonumber \\{} & {} \quad +{\frac{ (1-2 \omega _{p})^{3} m \gamma R Q^{2}}{ 16 \omega _{p} (1-3 \omega _{p}) (1+\omega _{p}) (1-\omega _{p})^{3}}}\nonumber \\{} & {} \quad \cos (T_{0}+\phi -2\omega _{p} T_{0} - 2\psi )\nonumber \\{} & {} \quad +{\frac{ (2+\omega _{p})^{3} m \gamma R^{2} Q}{ 16 (3+\omega _{p}) (1-\omega _{p})^{2}(1+\omega _{p})^{4}}} \nonumber \\{} & {} \quad \cos (2 T_{0}+2 \phi +\omega _{p} T_{0} + \psi )\nonumber \\{} & {} \quad -{\frac{ (2-\omega _{p})^{3} m \gamma R^{2} Q}{ 16 (3-\omega _{p}) (1+\omega _{p})^{2}(1-\omega _{p})^{4}}} \nonumber \\{} & {} \quad \cos (2 T_{0}+2 \phi -\omega _{p} T_{0} - \psi ) \end{aligned}$$

(B.4)