Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification

Abstract

A system composed of two cubic nonlinear oscillators with close natural frequencies, and thus displaying a 1:1 internal resonance, is studied both theoretically and experimentally, with a special emphasis on the free oscillations and the backbone curves. The instability regions of uncoupled solutions are derived and the bifurcation scenario as a function of the parameters of the problem is established, showing in an exhaustive manner all possible solutions. The backbone curves are then experimentally measured on a circular plate, where the asymmetric modes are known to display companion configurations with close eigenfrequencies. A control system based on a Phase-Locked Loop (PLL) is used to measure the backbone curves and also the frequency response function in the forced and damped case, including unstable branches. The model is used for a complete identification of the unknown parameters and an excellent comparison is drawn out between theoretical prediction and measurements.

Appendices

Appendix 1: derivation of first-order equations

This appendix gives the full detail of the derivation of the first-order slow-scale equations for the system of cubic oscillators featuring 1:1 internal resonance using the multiple scales method. The derivation is written for the forced and damped problem and is then finally reduced to free vibration by cancelling the forcing and damping terms. This allows us to give a unified presentation for the two cases, following closely the derivation shown in [45]. It is also mandatory for our presentation since the derivation of the instability region for the free vibration is derived from the forced and damped case, as explained below.

The starting point is thus the following equations of motion:

$$\begin{aligned} {\ddot{X}}_1 + \omega _1^2 X_1 + \varepsilon \left[ 2 \mu _1 {\dot{X}}_1 + \varGamma _1 X_1^3 + C_1 X_1 X_2^2\right]&=\varepsilon F_1 \cos \varOmega t , \end{aligned}$$

(27a)

$$\begin{aligned} {\ddot{X}}_2 + \omega _2^2 X_2 +\varepsilon \left[ 2 \mu _2 {\dot{X}}_2 +\varGamma _2 X_2^3 + C_2 X_2 X_1^2\right]&=\varepsilon F_2 \cos \varOmega t. \end{aligned}$$

(27b)

These two equations generalizes the case of free vibration considered in (1), by adding two different damping factors for each oscillator, $\mu _1$ and $\mu _2$, and two forcing terms with amplitudes $F_1$ and $F_2$, scaled at order $\varepsilon$ since the primary resonance is investigated. These equations are close to those used in [45], except that two distinct damping terms are considered instead of a single one $\mu = \mu _1 = \mu _2$ selected in [45]. Note also that in [45], the nonlinear stiffness terms were at the right-hand side of the equations of motions, so that the comparison can be drawn by simply changing the signs of $\varGamma _1$, $\varGamma _2$, $C_1$ and $C_2$.

The two detunings are introduced as

$$\begin{aligned} \omega _2&= \omega _1 + \varepsilon \sigma _1, \end{aligned}$$

(28a)

$$\begin{aligned} \varOmega&= \omega _1 + \varepsilon \sigma _2. \end{aligned}$$

(28b)

The first detuning $\sigma _1$ quantifies the 1:1 internal resonance, while $\sigma _2$ expresses the fact that a primary resonance is investigated so that $\varOmega \simeq \omega _1$. The multiple scales method is introduced, with $T_0=t$ a fast time scale and $T_1=\varepsilon t$ a slow time scale. The unknown are expanded as $X_i = X_{i1}(T_0,T_1) + \varepsilon X_{i2}(T_0,T_1)$. The first-order solution is easy to find and reads:

$$\begin{aligned} X_{11}&= A(T_1) \exp (i \omega _1 T_0) + c.c., \end{aligned}$$

(29a)

$$\begin{aligned} X_{21}&= B(T_1) \exp (i \omega _2 T_0) + c.c., \end{aligned}$$

(29b)

where c.c. stands for complex conjugate. The solvability conditions write, for the two unknown complex amplitudes $A(T_1)$ and $B(T_1)$ :

$$\begin{aligned} -2 i \omega _1&(A^\prime + \mu _1 A) - 3 \varGamma _1 A^2 {\bar{A}} - C_1 ({\bar{A}}B^2 {{\,\mathrm{e}\,}}^{2i\sigma _1 T_1} + 2 A B {\bar{B}}) + \frac{F_1}{2} {{\,\mathrm{e}\,}}^{i\sigma _2 T_1} = 0, \end{aligned}$$

(30a)

$$\begin{aligned} -2 i \omega _2&(B^\prime + \mu _2 B) - 3 \varGamma _2 B^2 {\bar{B}} - C_2 ({\bar{B}}A^2 {{\,\mathrm{e}\,}}^{-2i\sigma _1 T_1} + 2 B A {\bar{A}}) + \frac{F_2}{2} {{\,\mathrm{e}\,}}^{i(\sigma _2 - \sigma _1) T_1} = 0, \end{aligned}$$

(30b)

where $(\,)^\prime$ denotes the derivative with respect to the slow time scale $T_1$. These two equations can be rewritten by considering the polar form for the two unknowns, such that $A = a(T_1) \exp (i \alpha (T_1))$ and $B = b(T_1) \exp (i \beta (T_1))$. The non-autonomous system for the amplitude and phases finally writes:

$$\begin{aligned}&a^\prime = - \mu _1 a - \frac{C_1 }{2\omega _1}ab^2 \sin (2\beta - 2\alpha + 2 \sigma _1 T_1) + \frac{F_1}{4 \omega _1}\sin (\sigma _2 T_1 - \alpha ) , \end{aligned}$$

(31a)

$$\begin{aligned}&\alpha ^\prime = \frac{3 \varGamma _1}{2 \omega _1} a^2 + \frac{C_1}{2 \omega _1}b^2 \left( 2 + \cos (2\beta - 2\alpha + 2 \sigma _1 T_1) \right) - \frac{F_1}{4 \omega _1 a } \cos (\sigma _2 T_1 - \alpha ) ,\end{aligned}$$

(31b)

$$\begin{aligned}&b^\prime = - \mu _2 b - \frac{C_2}{2 \omega _2} ba^2 \sin (-2\beta + 2\alpha - 2 \sigma _1 T_1) + \frac{F_2}{4 \omega _2} \sin \left( (\sigma _2 - \sigma _1) T_1 - \beta \right) ,\end{aligned}$$

(31c)

$$\begin{aligned}&\beta ^\prime = \frac{3 \varGamma _2}{2 \omega _2} b^2 + \frac{C_2}{2 \omega _2}a^2 \left( 2 + \cos (-2\beta + 2\alpha - 2 \sigma _1 T_1) \right) - \frac{F_2}{4 \omega _2 b } \cos \left( (\sigma _2 - \sigma _1) T_1 - \beta \right) . \end{aligned}$$

(31d)

Note that in order to make the system (31) autonomous, one needs to introduce the following two additional variables

$$\begin{aligned} \gamma _1&= \sigma _2 T_1 - \alpha , \end{aligned}$$

(32a)

$$\begin{aligned} \gamma _2&= (\sigma _2 - \sigma _1)T_1 - \beta . \end{aligned}$$

(32b)

When forcing and damping terms are removed, Eq. (31) depends on only one angular variable, so that numerous different choices can be selected in order to make the system autonomous. In order to stay close to the notations used for the forced and damped system, the following change of coordinate is selected as:

$$\begin{aligned} \gamma _1&= - \alpha , \end{aligned}$$

(33a)

$$\begin{aligned} \gamma _2&= - \sigma _1 T_1 - \beta . \end{aligned}$$

(33b)

This choice leads to the autonomous system given in Sect. 2, Eq. (6).

Appendix 2: instability region for the uncoupled solutions

In this section we derive the instability region of the uncoupled solutions for the free vibration case, from the analysis of the damped/forced system. The starting point is the instability regions derived in [39] for the forced/damped case, i.e. for the system (31), made autonomous using change of variable from Eq. (32). The analysis led in [45] shows that uncoupled solution where only the first mode is excited is unstable when this relationship is fulfilled :

$$\begin{aligned} \sigma _2 = \sigma _1 + \frac{C_2 a^2}{\omega _2}\sqrt{\frac{C_2^2 a^4}{3\omega _2^2} - \mu ^2} \end{aligned}$$

(34)

Cancelling the damping by letting $\mu =0$ in this equation leads to

$$\begin{aligned} \sigma _2 = \sigma _1 + (2+s)\frac{C_2 a^2}{\omega _2}\quad \hbox {with} \ s=\pm \, 1. \end{aligned}$$

(35)

The last step is to replace $\sigma _2$, which is defined by (28b). However cancelling the forcing would result in $\varOmega$ being undefined. Instead, one needs to map $\varOmega$ to the nonlinear oscillation frequency in the free regime. $\sigma _2$ is then the detuning allowing one to express this nonlinear frequency as a function of the linear eigenfrequency with $\omega _{NL} = \omega _1 + \varepsilon \sigma _2$. Replacing $\sigma _2$ with its expression as given by Eq. (35), one can finally obtain:

$$\begin{aligned} \omega _{NL} = \omega _2 + \varepsilon (2+s)\frac{C_2 a^2}{\omega _2}\quad \hbox {with} \ s=\pm \, 1. \end{aligned}$$

(36)

This equation shows that as soon as the nonlinear frequency of A-mode enters the region delimited by the two curves obtained with $s=\pm \,1$, then the uncoupled solution becomes unstable. In order to derive the instability region for the B-mode, the same reasoning is applied using symmetric relationships, leading to:

$$\begin{aligned} \omega _{NL} = \omega _1 + \varepsilon (2+s)\frac{C_1 b^2}{\omega _1}\quad \hbox {with} \ s=\pm \, 1. \end{aligned}$$

(37)

Appendix 3: stability of the coupled solutions

The stability of the coupled solution is derived classically from the jacobian matric of (6). The general jacobian ${{\mathcal {J}}}$ reads, with $S_{\gamma }=\sin 2(\gamma _1 - \gamma _2)$ and $C_{\gamma }=\cos 2(\gamma _1 - \gamma _2)$ in order to ease notations:

$$\begin{aligned} {\small {{\mathcal {J}}} = \left( \begin{array}{cccc} -\frac{C_1}{2\omega _1} b^2 S_{\gamma } &{} \quad -\frac{C_1}{\omega _1} ab^2 C_{\gamma } &{} \quad -\frac{C_1}{\omega _1} ab S_{\gamma } &{} \quad \frac{C_1}{\omega _1} ab^2 C_{\gamma } \\ -\frac{3\varGamma _1}{\omega _1} a &{} \quad \frac{C_1}{\omega _1} b^2 S_{\gamma } &{} \quad -\frac{C_1}{\omega _1} b (2 + C_{\gamma } ) &{} \quad -\frac{C_1}{\omega _1} b^2 S_{\gamma } \\ \frac{C_2}{\omega _2} ab S_{\gamma } &{} \quad \frac{C_2}{\omega _2} b a^2 C_{\gamma } &{} \quad \frac{C_2}{2\omega _2} a^2 S_{\gamma } &{} \quad -\frac{C_2}{\omega _2} b a^2 C_{\gamma } \\ -\frac{C_2}{\omega _2} a (2 + C_{\gamma } ) &{} \quad \frac{C_2}{\omega _2} a^2 S_{\gamma } &{} \quad -\frac{3\varGamma _2}{\omega _2} b &{} \quad -\frac{C_2}{\omega _2} a^2 S_{\gamma } \end{array} \right) . } \end{aligned}$$

(38)

The coupled solutions are characterized by specific relationships on the angles leading to simplification of ${{\mathcal {J}}}$. Indeed one has $\sin 2(\gamma _1 - \gamma _2) = 0$ and $\cos 2(\gamma _1 - \gamma _2) = s = \pm \, 1$. With these simplifications the $4\times 4$ determinant of the jacobian matrix ${{\mathcal {D}}} = \hbox {det} ({{\mathcal {J}}} - \lambda {{\mathcal {I}}})$ with ${{\mathcal {I}}}$ the identity matrix can be analytically derived as:

$$\begin{aligned} {{\mathcal {D}}} = \lambda ^2\left[ \lambda ^2 - 3 a^2b^2s \left( \frac{\varGamma _2 C_2}{\omega _2^2} + \frac{\varGamma _1 C_1}{\omega _1^2} \right) + 2 \frac{C_1 C_2}{\omega _1 \omega _2} b^2 a^2 s (2+s) \right] . \end{aligned}$$

(39)

Two eigenvalues are found to be zero which is logical for coupled solutions in four-dimensional phase space. The two other eigenvalues are solutions of

$$\begin{aligned} \lambda ^2 = 3 a^2b^2s \left[ \frac{\varGamma _2 C_2}{\omega _2^2} + \frac{\varGamma _1 C_1}{\omega _1^2} - \frac{2}{3}\frac{C_1 C_2}{\omega _1 \omega _2} (2+s)\right] , \end{aligned}$$

(40)

with $s=+1$ for normal mode and $s=-1$ for elliptic mode. Each mode (normal or alliptic) is stable as long as $\lambda ^2 < 0$, which leads to the conclusion that stablity is governed only by the value of the scalar $S_c = \frac{\varGamma _1 \omega _2}{C_2 \omega _1} + \frac{\varGamma _2 \omega _1}{C_1 \omega _2}$, the normal mode being stable as long as $S_c < 2$, and the elliptic mode as long as $S_c > 2/3$.

Appendix 4: parametric study: bifurcation scenario in the particular case without detuning

In this appendix, the particular case of perfectly equal eigenfrequencies $\omega _1 = \omega _2$ with a vanishing detuning $\sigma _1= 0$, is considered. In this case, the amplitude values for which the branch points $I_{Ea}$, $I_{Na}$, $I_{Eb}$ and $I_{Nb}$ (as defined in Eqs. (22) and (23)) are equal to zero: this means that the coupled solutions could exist from a vanishing amplitude. This is the direct consequence of the fact that, as $\sigma _1= 0$, the gaps between the starting point of the A-mode and the B-mode backbone curves and their instability regions, does not exist anymore. The second consequence is also that uncoupled solutions are either always stable or always unstable, whatever the amplitude. Considering the coupled solutions, cancelling the values of all branch points does not mean that NM and EM always exist. Indeed, Eq (14), which defines the amplitude relationships for coupled solutions, rewrites with $\sigma _1 = 0$

$$\begin{aligned} \left( 3\varGamma _1 - (2+s)C_2 \right) a^2 = \left( 3\varGamma _2 - (2+s)C_1 \right) b^2, \end{aligned}$$

(41)

with $s=\pm \, 1$ for NM and EM. Consequently coupled solutions can exist if and only if the respective coefficients in front of the square amplitude have the same sign.

The stability chart that gives all possible solutions as function of the nonlinear coefficients is thus modified and shown in Fig. 14a. The main difference with the detuned case where $\sigma _1 > 0$ is that the coupled solutions of finite extent can not exist anymore since all branch points have the same vanishing amplitude. This leads to modification of the lower right part of the stability chart to make it symmetric. The possible cases are discussed as function of the stability of the uncoupled mode, reported in Fig. 14a on the vertical and horizontal axes. Four cases exist:

• Case 1 A-mode and B-mode are stable. This means that the backbone curve of each uncoupled solution is outside its instability region. It corresponds to the four edges of the stablity chart, in upper left, upper right, lower left and lower right regions. Two cases are then possible:

  • Case 1.1: if $3\varGamma _1 < C_2$ and $\varGamma _2 > C_1$, or if $3\varGamma _2 < C_1$ and $\varGamma _1 > C_2$. This case means that the backbone curve of the A-mode stays on the left of the instablity region while backbone curve of the B-mode is on the right (or vice-versa). Then in this case the coefficients of Eq.(41) have opposite signs, thus no coupled solutions exist.

  • Case 1.2: if $3\varGamma _1 < C_2$ and $3\varGamma _2 < C_1$ (case 1.2.1), or if $\varGamma _1 > C_2$ and $\varGamma _2 > C_1$ (case 1.2.2), the backbone curves of the A-mode and the B-mode are respectively on the same side of their instability regions. Then in this case both coupled solutions exist, and inspection of the values of $S_c$ indicates that in case 1.2.1 NM is stable while EM is unstable, and case 1.2.2 leads to the contrary with NM unstable and EM stable.

• Case 2 The A-mode and the B-mode are unstable. This means that each backbone curve is totally inside the instability region, so that $C_2/3< \varGamma _1 < C_2$ and $C_1/3< \varGamma _2 < C_1$. Then in this case both coupled solutions exist and are stable.

• Case 3 The A-mode is unstable and the B-mode is stable. The instability of the A-mode is obtained thanks to the condition $C_2/3< \varGamma _1 < C_2$. Two subcases are then possible:

  • If $3\varGamma _2 < C1$, then EM does not exist, only NM is possible and is stable (case 3.1).

  • If $\varGamma _2 > C_1$, then NM does not exist, only EM is possible and is stable (case 3.2).

• Case 4 The B-mode is stable and the A-mode is unstable. This case can be simply deduced from the previous one by symmetry (changing the indices $1 \rightleftarrows 2$).

Figure 14b illustrates a case in the upper right region of the stability chart, in which both the A-mode and the B-mode are stable and the two coupled solutions exist: the NM is stable while the EM is unstable. Finally Fig. 14c, d shows the projection on the $(\omega ,a)$ and $(\omega ,b)$ for the case in the central region where both uncoupled solutions are unstable. In this case, the only stable solutions are the coupled branches, and both EM and NM are stable.

Fig. 14

Stability analysis and bifurcation scenario in the case $\sigma _1 = 0$, with fixed parameters as $\omega _1=\omega _2=\pi$, $\varGamma _1 = \varGamma _2=1$. a stability chart showing the possible solutions when varying the values of the coupling coefficients $C_1$ and $C_2$. b Three-dimensional representation of the solution branches in space $(\omega ,a,b)$, for $C_1=C_2 = 5$. c–d Two-dimensional projections in the planes $(\omega ,a)$ and $(\omega ,b)$ for the case $C_1=C_2 =25$. (Color figure online)