\(N1\) modal interactions of a threedegreeoffreedom system with cubic elastic nonlinearities
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Abstract
In this paper the \(N1\) nonlinear modal interactions that occur in a nonlinear threedegreeoffreedom lumped mass system, where \(N=3\), are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. \(\omega _{n1}: \omega _{n2}:\omega _{n3} \approx 1:1:1\)) is considered. However, due to the symmetries of the system under consideration, only \(N1\) modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in the frequency domain. The forced responses are computed using the continuation software AUTO07p. A comparison of the results gives insights into the multimodal interactions and shows how the frequency response of the system is related to those branches of the backbone curves that represent such interactions.
Keywords
3DoF nonlinear oscillator Backbone curve Nonlinear modal interaction Secondorder normal form method1 Introduction
The motivation for this study is the possibility for modes, in multidegreeoffreedom nonlinear systems, to interact with each other [17]. These types of modal interaction have been previously studied because they are often related to unwanted vibration effects in engineering structures [16]. The majority of the existing literature is for undamped, unforced systems, and includes structural elements such as beams, cables, membranes, plates and shells—see for example [1, 15, 27]. Several different analytical and numerical approaches have been used to approach this type of problem, such as perturbation methods [18], nonlinear normal modes [13, 22, 23, 25] or normal form analysis [2, 11, 21]. The majority of work in the literature on modal interaction is based on 2DoF nonlinear systems where two modes interact, see for example [3, 5, 8, 9, 10, 13, 29], although for continuous systems higher numbers of modes can typically be retained in the approximation, see for example [24]. Some work has been carried out on 3DoF lumped mass systems in the context of nonlinear vibration suppression [12].
In a system with N modes, it is possible for response solutions to exist in which some ( i.e. \(Ni \ge 2\)) and/or all N of the modes interact. In this paper, we consider the case where this type of \(Ni\) modal interaction can occur. More specifically, the case where \(N = 3\) is analysed, i.e. only two ( i.e. \(N1\)) modes can interact. To do this we have chosen a specific configuration of an inline 3DoF nonlinear oscillator with small forcing and light damping. Due to the structural symmetry, one of the modes of this system is linear and the other two modes are nonlinear. So even though all the modal natural frequencies are close, the linear mode is not coupled with the other two. Hence the study of the 3DoF system is reduced to the modal interaction of two coupled modes. Our analysis follows the approach developed by [3, 9] to consider nonlinear modal interactions of a 2DoF nonlinear oscillator. However, the effects of two modes interacting in this 3DoF system has subtle differences from those described in the previous literature. This will be explained in Sect. 4.
The paper is structured as follows. In Sect. 2 we describe the configuration of this inline oscillator. Then, we apply a normal form transformation method to the 3DoF system to obtain its potential backbone curves expressions. In Sect. 3 the backbone curves of the system with the hardening nonlinearity are computed. These curves are then used to infer the dynamic behaviour of the system, which in turn can be used to interpret the forced, damped behaviour. In the Sect. 4, the backbone curves of the system with softening nonlinearity are presented. Lastly, the stability of the backbone branches are analysed and their relation with the forced response are shown. Conclusions are drawn in Sect. 5.
2 System description and analytical method application
3 Hardening case
3.1 Backbone curves
When the nonlinear stiffness is positive, \(\mu > 0\), the solutions for Eqs. 16a and 18a are complex. Therefore, for the hardening case, the backbone branches \(S4^\pm \) and \(S5^\pm \) have no physical meaning and only S1, S2 and S3 exist. This means that there is no nonlinear modal interaction between the three backbone curves.
3.2 Forced response
To show how the backbone curves can help facilitate the interpretation of the modal interaction of the nonlinear system, the forced response amplitude of three masses, \(X_1\), \(X_2\) and \(X_3\), in the frequency domain for the hardening nonlinear system is shown in Fig. 3. To illustrate the relationship, the corresponding backbone curves in Fig. 2 are also shown. Here, a damping ratio \(\zeta \simeq 0.001\) is chosen for all modes and the external force amplitude is \([P_1, P_2, P_3]^{T} = [3, 1, 1]^{T} \times 10^{3}\), which corresponds to the situation where all three modes are excited at the same amplitude, i.e. \([P_{m1}, P_{m2}, P_{m3}]^{T} = [1, 1, 1]^{T} \times 10^{3}\). The forced response has been computed from an initial steady state solution, found with numerical integration in MATLAB, which is then continued in forcing frequency using the software AUTO07p [4].

For the first resonance, the response curves of three masses are similar to that of the linear oscillator and are centred around S1.

For the second resonance, the familiar shape of the responses of a typical Duffing oscillator are following S2 and the jump phenomenon can be also observed on the right part of the curve for mass 1 and 3. Note that due to the special values of the linear modeshape of mode 2, i.e. \(\{1, 0, 1 \}^T\), there is no peak for the response of mass 2 in Fig. 3b within this bandwidth.

For the third resonance, in Fig. 3a, the curve around S3 contains a loop where the upper trajectory is unstable which occurs from the addition and subtraction of the modal contributions. In Fig. 3b, c, the response of the typical Duffing oscillator can be observed enveloping S3.
4 Softening case
4.1 Backbone curves
For the softening case, \(\mu <0\), Eqs. 16a and 18a have real solutions. Therefore, the inunison, \(S4^\pm \), and outofunison, \(S5^\pm \), resonant backbone curves are physical. In Fig. 4, the backbone curves for the softening case where \(\omega _{n1} = 1\), \(\omega _{n2} = 1.005\), \(\omega _{n3} = 1.015\) and \(\kappa = 0.05\) are shown. The first and second columns show the backbone curves of the modal states, \(u_1\), \(u_2\) and \(u_3\), and the physical displacements, \(x_1\), \(x_2\) and \(x_3\), respectively. As with the hardening case, the singlemode backbone curves, S1, S2 and S3, for mass 1 and 3 are the same. However, the symmetry is broken as the \(S4^{\pm }\) comes with the position of \(S4^+\) and \(S4^\) swapped. Here, the mixedmode backbone curves \(S4^\pm \) and \(S5^\pm \), where both mode 2 and 3 are activated, are of primary interest.
4.2 Stability of the backbone curve
We now consider the stability of the backbone curves. Here, only the stability analysis of the backbone branch S2 is given in detail due to the fact that both \(S4^\pm \) and \(S5^\pm \) intersect with it, as shown in Fig. 4. Note that on backbone curve S2 which is the solution of Eq. 12, \(u_3\) is equal to zero. So the stability of S2 can be determined by considering the dynamics of \(u_3\) about its zero solution (Note that \(u_1\) is not considered here due to its independence of \(u_2\) and \(u_3\)). When the zero solution of \(u_3\) is unstable, the S2 solution is also unstable.
Using the same approach, it can be also shown that the parts of S2 below bifurcation point \(\hbox {BP}_1\) and above \(\hbox {BP}_2\) are stable and the part between the two bifurcation points is unstable.
4.3 Analysis of the forced response
4.3.1 Forced response

For the small force amplitude situation, Fig. 5a, there is only one response curve which is centred around S2. This curve is the response of a typical softening Duffing oscillator and only composed the response of mode 2, \(u_2\). For this case, the force is insufficient to trigger the modal interaction or jump.

For the medium force amplitude situation, Fig. 5b, there are three response curves (1 blue and 2 green). The two green curves following \(S4^{\pm }\) bifurcate from the singlemode response curve (blue one) at two secondary bifurcation points, respectively, and they are composed of the response of both mode 2, \(u_2\), and mode 3, \(u_3\).

For the large force amplitude situation, Fig. 5c, there are two additional response curves (the black ones) surrounding \(S5^{\pm }\) which also bifurcates from the singlemode response curve of mode 2. On these two curves, both mode 2 and 3 are present as well.
4.3.2 Discussion of the force amplitude for triggering the modal interaction
Now, we consider at what forcing amplitude the modal interaction is triggered for the singlemode excitation situation. In Fig. 5b, c, it can be seen that the mixedmode response curves emanate from the singlemode response curve at the secondary bifurcation points. We infer from this that the modal interaction response curves will appear from the singlemode response curve when only one linear mode is forced [2, 9]. So, the existence of modal interaction is studied by looking for bifurcation points on the singlemode response curve. The theory used for detecting the secondary bifurcation points is the same used for the stability of backbone curves in Sect. 4.2. When the zero solution for the response of the unforced mode is stable (unstable), the singlemode response curve of the other forced mode is also stable (unstable). As a result the point of neutral stability is the secondary bifurcation point. The case where only mode 2 is forced has been considered as an example here. Along the response curve composed of only \(u_2\) (blue curves in Fig. 5), the modal coordinate \(u_3\) is zero, so we are going to study the stability of the zero solution of \(u_3\) [6].
 (1)
if there is zero or one intersection point, there will be no modal interaction. Fig. 6a.
 (2)
if there are two or three intersection points, the modal interaction response following the inunison backbone curves will exist. Fig. 6b.
 (3)
if there are four intersection points, both the modal interaction response following the inunison and outofunison backbone curves will occur. Figure 6c.
5 Conclusion
In this paper, we have considered the \(N1\) modal interactions that occur in a threedegreefreedom lumped mass system. In particular we considered the potential modal interactions of the system by analysing the backbone curves of the undamped, unforced system. This is an important topic because in lightly damped structures the dynamic behaviour is largely determined by the properties of the underlying undamped dynamic system.
First the undamped, unforced case was considered. In particular the modal interaction case that occurs when all the underlying linear modal frequencies are close was examined (i.e. \(\omega _{n1}: \omega _{n2}:\omega _{n3} \approx 1:1:1\)). In this case the first mode is linear because of the symmetry of the system and the other two modes will potentially interact with each other when the special parameters are chosen. We showed how this system can be analysed using a normal form transformation to obtain the nonlinear backbone curves of the undamped, unforced response. Following this, the response in the frequency domain of the corresponding lightly damped and harmonically forced system was obtained using the continuation software AUTO07p. This results were compared with the backbone curves to show its validity for predicting the nonlinear resonant behaviour of the system during \(N1\) modal interactions.
Notes
Acknowledgments
The authors would like to acknowledges the support of the Engineering and Physical Sciences Research Council. S.A.N is supported by EPSRC Fellowship EP/K005375/1. D.J.W is supported by EPSRC grant EP/K003836/2.
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