The influence of phase-locking on internal resonance from a nonlinear normal mode perspective

When a nonlinear system is expressed in terms of the modes of the equivalent linear system, the nonlinearity often leads to modal coupling terms between the linear modes. In this paper it is shown that, for a system to exhibit an internal resonance between modes, a particular type of nonlinear coupling term is required. Such terms impose a effect of additional modes that are not coupled via phase-locking terms is then investigated by considering the backbone curves of the system. Using the example of a twomode model of a taut horizontal cable, the backbone curves are derived for both the case where phase-locked coupling terms exist, and where there are no phase-locked coupling terms. Following this, an analytical method for determining stability is used to show that phase-locking terms are required for internal resonance to occur. Finally, the effect of nonphase-locked modes is investigated and it is shown that they lead to a stiffening of the system. Using the cable example, a physical interpretation of this is provided. & 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Weakly nonlinear systems typically have an underlying linear structure and the underlying linear modes are often used to describe the fundamental components of the system. In this paper we investigate the coupling terms that are present in nonlinear systems when projected onto these linear modes. Specifically, we show how phase-locking conditions in these terms influence the internally resonant dynamic behaviour. Such phenomena are often observed in forced, lightly damped, and weakly nonlinear systems with multiple degrees-of-freedom, which represent a variety of physical applications, see for example [1][2][3][4]. Here, internal resonance is defined as the triggering of a dynamic response of a linear mode of the system that is not subjected to direct external excitation. Many previous authors have considered problems of this type, see for example [3,4] and references therein.
In this work we will use the normal form technique proposed by [5] for multi-degree-of-freedom forced, damped, weakly nonlinear systems. This approach leads naturally to the analysis of backbone curves, which define the dynamic behaviour of periodic motions in the unforced and undamped equivalent system, in the amplitude vs frequency plane. These are equivalent to the loci of the nonlinear normal modes (NNMs), represented in the amplitude vs frequency projection.
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Resonant equations of motion
Weakly nonlinear, multi-degree-of-freedom systems are often expressed in terms of the modal coordinates for the linearised version of the system, as the linear terms will then be decoupled. However, decoupling of the nonlinear terms is typically not achieved via a linear modal transform, and hence the modes will not, in general, match the NNMs of the nonlinear system. Note that here we use the term modes to refer to the modes of the linearised system. A multimodal nonlinear system may be written in modal coordinates, q, as € q þ Γ _ q þ ΛqþNðqÞ ¼ f: Assuming linear modal damping, the kth diagonal elements in diagonal matrices Γ and Λ are 2ζ k ω nk and ω nk 2 respectively and the vector N contains the nonlinear stiffness terms, and f represents the external excitation vector in modal coordinates.
Here, ζ k and ω nk are used to denote the linear damping ratio and linear natural frequency of the kth mode respectively.
To analyse weakly nonlinear systems it is helpful to transform the equations of motion into a new set of coordinates, u, which describe only the resonant components of the response. The dynamic equation in u is termed the resonant equation of motion. This can then be used to find steady-state solutions, in terms of modal amplitudes, via an exact harmonic balance using trial solutions of the form u k ¼ U k cos ω rk t À ϕ k À Á ¼ U k 2 e j ω rk t À ϕ k ð Þ þ U k 2 e À j ω rk t À ϕ k ð Þ ¼ u pk þ u mk ; where ω rk is the response frequency of the kth mode and subscripts p and m indicate positive and negative (minus) complex exponential terms respectively. The introduction of ω rk allows for the detuning of the kth mode from the linear natural frequency, ω nk . For a resonant response, this response frequency is typically selected such that it is close to the linear natural frequency of the mode in question, ω rk % ω nk . Note, however, that response frequencies that are not close to the linear natural frequencies may also be selected, as the assumption that this detuning is small is not a requirement of the technique. Normal form analysis allows us to find the periodic responses of a system, as is assumed when computing NNMs, or the steady-state response to a sinusoidal forcing. Taking the period to be T ¼ 2π=Ω, the response frequency of the kth mode, ω rk , is an integer multiple of Ω. Likewise, if the system is forced at a single frequency, the forcing frequency is an integer multiple of Ω.
If forcing is near-resonant, i.e. the frequency of the forcing acting on any mode is close to the natural frequency of that mode, or if there is no forcing, then the resonant equation of motion is found by applying a nonlinear near-identity transform to Eq. (1) to give where h is a vector of harmonic components. In the formulation shown in Eq. (3), it is assumed that the forcing and damping terms are resonant, and so are retained in the equation for u. For further discussion of how non-resonant terms are handled see [5,15]. Additionally, details of how the harmonics h and the transformed nonlinear terms N u are found are given in the Appendix.

Resonant equations of the example system
As an example we consider the dynamics of a horizontal taut cable, shown schematically in Fig. 1. This is described using the modal equations of motion, derived in [16] and discussed in detail in [4], which include gravitational sag and tension effects. The dynamics for the kth in-and out-of-plane modes, denoted z k and y k respectively (where in-plane relates to the plane in which gravitational sag occurs with the sag acting in the positive direction) may be expressed as where it is assumed that an infinite number of modes exist in both the in-plane and out-of-plane directions. We note that k denotes the mode number, but for this structure there are two modes corresponding to each mode number: the kth inplane, and the kth out-of-plane mode, as is reflected in the equations above. Here we use tilde,, to denote an out-of-plane parameter, and later it is also used to distinguish the out-of-plane modal variables. Linear modal damping is assumed, with damping ratios ζ k andζ k for the kth in-and out-of-plane modes respectively. It can also be seen that forcing takes the form of an out-of-plane support motion, v ℓ , at the x ¼ ℓ support, where ℓ is the distance between supports. The parameter m describes the modal mass, and the nonlinear parameters γ ik and ν ik arise from the gravitational sag and from the dynamic tension respectively. These nonlinear parameters are defined in [16] where it is shown that ν ik ¼ i 2 k 2 ν 11 and that, for even k, γ ik ¼ 0.
The linear natural frequencies of the kth in-and out-of-plane modes are written ω nk andω nk respectively. Note that, for all out-of-plane modes and the even in-plane modes, these natural frequencies are proportional to the mode number k, while the natural frequencies of the odd in-plane modes are slightly higher than the equivalent out-of-plane ones. These equations have been used, for example, to analyse the onset of internal resonance [17,18] and whirling motion [12].
Here we consider the interactions between the ath unforced in-plane mode and the bth forced out-of-plane mode of the cable. Using q ¼ z a y b À Á T ¼ q aqb À Á T , where the tilde notation is again used to denote the out-of-plane modes, the equations of motion for these two modes may be written in the matrix form of Eq. (1) with where it has been assumed that the support motion is sinusoidal, such that v ℓ ¼ V cos bΩt ð Þ, and hence R b ¼ Àm 2ð À 1Þ b π bV. Additionally, it is assumed that bΩ is close to the out-of-plane linear natural frequencyω nb and so may be treated as resonant.
As part of the normal form technique, we write N as a function of u, rather than of qsee [5] for further details. Therefore, using u k ¼ u pk þ u mk , gives ; Here we have written the response frequencies as ω ra ¼ aΩ andω rb ¼ bΩ, recalling that Ω ¼ 2π=T, where T is the period of the response. The elements in β Ã containing dashes are not needed as they correspond to elements in n Ã with a value of zero.
From this expression for β Ã it can be seen that some terms are zero regardless of the values of a and b, these are referred to as unconditionally resonant terms [11]. In addition, there are conditionally resonant terms, which are zero if a relationship between a and b is met. From inspection of β Ã it can be seen that conditionally resonant terms occur if a ¼b (elements ½1; 10; ½1; 14; ½2; 4 and ½2; 12) or if a ¼ 2b (elements ½1; 28; ½1; 30; ½2; 24 and ½2; 26). However for the case where a ¼ 2b, the corresponding values in n Ã are all a function ofγ ba which is zero if a is even, as it must be to satisfy a ¼ 2b. Hence a ¼ 2b does not result in non-zero conditionally resonant terms.
To proceed, therefore, we consider two cases; (i) the case where a ¼b which contains conditionally resonant terms and (ii) the more general case where a a b which only contains unconditionally resonant terms. Keeping the resonant terms in the equation of motion and removing the non-resonant ones using the q-u transform results in the resonant equation of motion € 3ν aa u pa u ma u a þ 2ν abũpbũmb u a þ δ a;b ν ab u paũ and where δ a;b is the Kronecker Delta function, where δ a;b ¼ 0 when a ab, and δ a;b ¼ 1 when a ¼ b.

Backbone curves
To calculate the backbone curves, the trial solution, Eq. (2), is substituted into the unforced, undamped equations of motion, € u þΛu þ N u ðuÞ ¼ 0. Making this substitution into the kth equation, € u k þω 2 nk u k þN uk ðuÞ ¼ 0, and balancing the e jω rk t terms gives where, making use of the fact that the transformed dynamic equation contains only resonant terms, we have written Note that balancing the e À jω rk t gives the complex conjugate of this expression and so offers no extra information when used to compute a solution. The imaginary parts of these equations (for 1 r k rn, where n is the number of modes) may be used to find any phase conditions between the modes. The real parts may then be solved to find expressions for the backbone curves.
In the case of the cable example, Eq. (9) may be written as For the general case, a a b, there is no phase relationship imposed by the equations of motion (as the Kronecker delta functions set the imaginary parts of Eq. (9) to zero) and we have the amplitude-response frequency expressions for a a b: Recall that, for a single frequency resonant response, Ω ¼ ω ra =a ¼ω rb =b. However for the special case where a ¼ b, to satisfy the imaginary part of Eq. (9), the phase condition sin 2φ a Àϕ a h i ¼ 0 must be met. This leads to either the phase relationshipφ a Àϕ a ¼ 0; π; … with e 2jφ a À ϕ a À Á ¼ 1 orφ a À ϕ a ¼ π=2; 3π=2; … with e 2jφ a À ϕ a À Á ¼ À1. Hence, we say that the modes are phase-locked when for a ¼ b: where p ¼1 if the modes are in-phase or in anti-phase,φ a Àϕ a ¼ 0; π, and p ¼ À1 if the modes are out-of-phase, ϕ a À ϕ a ¼ π=2; 3π=2.
We find that there are two single-mode backbone solutions, which are independent of the relationship between modes a and b. These backbone curves are also independent of p, which only appears in mixed-mode terms. By inspection of Eqs. (13) and (14) and of Eqs. (15) and (16), these solutions (in terms of the resonant response) are the in-plane-only mode S1: and the out-of-plane-only mode In addition to these, there are mixed-mode solutions. To find these, firstly the special case of a ¼b will be considered followed by the general case a a b.

The phase-locked case: a ¼b
Taking the case where a ¼ b, U a andŨ a denote the amplitudes of the in-and out-of-plane modes respectively. There are potentially four types of solution, as has been discussed for the case where a ¼ b ¼ 1 in [12]: two single-mode solutions, S1 and S2; and two mixed-mode solutions. These mixed-mode solutions may be found by setting the contents of the bracketed terms in Eqs. (15) and (16) to zero and eliminating the response frequency giving the amplitude relationship using the fact that ω na Zω na . The case where p ¼ 1 can only give valid solutions if ω na ¼ω na , which occurs when a is even. In this case there is no relationship between the amplitudes of the two modes. Using Eq. (15), for this case we can write for even a; (20) where, for S3 þ and S3 -, the phase difference isφ a À ϕ a ¼ 0 andφ a À ϕ a ¼ π respectively. Note that, for even values of a, solutions S1 and S2 are special cases of solution S3 7 . Physically, in a plane intersecting the chord line (see Fig. 1 Here, using Eqs. (15) and (16), the resonant frequency may be written as where for S4 þ and S4 À the phase isφ a À ϕ a ¼ π=2 andφ a À ϕ a ¼ Àπ=2 respectively, such that the cable is whirling clockwise or anti-clockwise. Note that U a ¼0 is a valid solution to Eq. (22), and in this case the solution lies on S2. This point represents a bifurcation between S2 and the whirling S4 7 solutions, see [12]. Note also for the case where a is even, such that ω na ¼ω na , S4 7 describes responses in which U a ¼Ũ a and the bifurcation occurs at zero-amplitude.
As an example, we now consider a cable with a length of 1.5 m, a diameter of 5 mm, a density of 3000 kg m À 3 , Young's Modulus of 2 Â 10 11 Pa and a static tension of 200 N, as originally described in [12]. Fig. 2 shows the backbone curves for the case where a ¼ b ¼ 1 for the first in-and out-of-plane modes. The non-zero S1 and S2 solutions are confined to U 1 , panel (a), andŨ 1 , panel (b), respectively. At approximately Ω ¼ 124:8 rad s À 1 , the mixed-mode backbone S4 7 bifurcates off the S2 solution branch with U 1 ¼ 0, with increasing U 1 andŨ 1 away from the bifurcation point.

The phase-unlocked case: a ab
Considering the general case where a a b, Eqs. (13) and (14), we again have the two single-mode solutions S1 and S2 (Eqs. (17) and (18)). In addition, as with a¼ b, it is possible to have solutions where both modes are present. Taking Eqs. (13) and (14) and where it has been recalled that ν ij ¼ i 2 j 2 ν 11 . Note that ω na Zða=bÞω nb so the bracketed term is positive or zero. Using this equation and Eq. (14) to calculate the resonant frequency, the mixed-mode solution may be summarised as Note that there is no phase condition imposed on the modal contributions, this is indicated by the Ã superscript. Here U a ¼0 is a valid solution and at this point the solution lies on S2, hence this solution branch bifurcates from S2.  panel (b) of Fig. 2, although it should be noted that on the S5 Ã branch the corresponding response in the first in-plane mode, with response amplitude U 1 , will be at one-third the frequency of theŨ 3 component. We note that, to get a full picture of the dynamic interactions in the cable, all combinations of modes must be considered; however here we show an example interaction between a pair of modes.

Internal resonance
Now we consider the potential for the system to exhibit internal resonance, i.e. to respond in a mode that is not directly excited. This will be achieved by considering the zero-amplitude response stability of an unforced mode. If the zeroamplitude response solution is unstable, due to the presence of another mode, then an internal resonance of the unforced mode will be observed. The findings will be applied to our example system by considering the response of the ath in-plane mode when the cable is excited out-of-plane close to the frequency of the bth out-of-plane mode. From the backbone analysis, it is revealed that the two-mode cable system is likely to exhibit internal resonance due to the presence of mixedmode backbone curves. This potential internal resonance is examined by considering the stability of the zero-amplitude response solution to the ath in-plane mode.
To assess the stability of the zero-solution of an unforced mode in a general system, we must consider the response of the mode when subjected to a perturbation away from zero. This is achieved by allowing the amplitude and phase of the response to vary slowly in time which, for the kth mode, may be expressed using where, from comparison with Eq. (2), U pk ¼ U k e À jϕ k and U mk ¼ U k e jϕ k . Here we have two time-scales: the oscillatory motion at frequency ω rk , and a slowly varying amplitude represented by the amplitude dependence on εt where ε is a bookkeeping aid [19] used to indicate a small parameter (and may be removed at an appropriate stage by setting it to unity). The use of this type of solution is discussed at length in [4,11]. Using a fast-and slow-time approach the derivatives of this solution can be written as Here the first derivatives are expressed to order ε 0 as they will only be used in small terms in the equations of motion (damping is assumed to be small), whereas the second derivatives are expressed to order ε 1 as they will be used in large terms, hence preserving the accuracy of the solution to order ε 1 .
To proceed, this solution is substituted into the resonant equation of motion for an unforced mode € u k þ 2ζ k ω nk _ u k þ ω 2 nk u k þ N uk ðuÞ ¼ 0, and then the e jω rk t and e À jω rk t terms in the resulting equation are balanced (as the equation is resonant there are no other complex exponential terms in time so this balance is exact) to give Fig. 3. Backbone curves for a cable, with parameters defined in [12], considering the case where resonance occurs in either or both of the first in-plane and third out-of-plane modes. These backbone curves are found using the parameters m ¼ 0:04418 kg, ω n1 ¼ 122:04 rad s À 1 ,ω n3 ¼ 366:1 rad s À 1 and ν 11 ¼ 1:417 Â 10 7 kg m À 2 s À 2 .
Expressing this in the form _ z ¼ f ðz; tÞ where z ¼ fU pk U mk g T , steady-state solutions, z s , may be found by solving f ðz s ; tÞ ¼ 0. The stability of these steady-state solutions may be assessed, to the first order of accuracy of a Taylor series expansion, by examining the eigenvalues of Df z ðz s ; tÞ, where Df z ðz; tÞ is the Jacobian of f ðz; tÞ; if the real part of the eigenvalues are positive then the solution is unstable, see for example [4]. Using Eq. (27), the Jacobian may be expressed as where we have used Eq. (10) to reflect that fact that N uk ðuÞ has e jω rk t and e À jω rk t terms. Since we are interested in the zeroamplitude response solution, we need to set z ¼ z s ¼ ½0 0 T . Additionally, it can be assumed that the perturbed solution is small, and hence any terms containing U i pk U j mk where iþ j 41 are considered to be negligible.

The form of the resonant nonlinearities
Before proceeding, it is worth considering further the form of N uk ðuÞ. Firstly, we know that all terms contain at least one U pk 1 or U mk 1 term, otherwise the zero-amplitude response solution would not exist. Secondly, we know that these terms are resonant, but, considering Eq. (A.1), we can see that there are two ways of satisfying this: either via a positive or a negative root. We also know that the higher order terms of ½U pk ; U mk do not need to be considered in the stability assessment as they disappear when evaluating the Jacobian for z ¼ z s ¼ ½0 0 T . Finally, we know that N uk ðuÞ is real. Considering a multimode system with the zero-amplitude response solution in mode k, which has resonant frequency ω rk ¼ r k Ω, there are four possible types of resonant term which are functions of either U pk 1 or U mk 1 , these are Possibilities p1 and p4 place no condition on the response frequency ω rk and hence are phase-unlocked terms with respect to the kth mode. They also result in a positive and negative complex exponential in time for the u pk and u mk terms respectively, and so appear in the leading-diagonal terms of the Jacobian. Furthermore, as these terms may be written N u uk u k , where N uk u is time-independent, it may be seen that N uk u must be real. In contrast, p2 and p3 require a frequency relationship between the kth mode and the other modes in the system. Using Eq. (2), this results in a phase of the nonlinear term, relative to the linear term, of À 2ϕ k þ P i;i a k ½ϕ i ðs pi À s mi Þ ¼ nπ for both solutions. As p2 is a negative complex exponential in time and p3 a positive one, these phase-locked terms will appear in the off-diagonals of the Jacobian, and these terms form a complex conjugate pair, ensuring that N uk ðuÞ is real.
Using the response solution given by Eq. (25), this allows us to write N uk ðuÞ ¼ N u uk U pk e jω rk t þ N u uk U mk e À jω rk t þ N l uk U pk e À jω rk t þ N l uk U mk e jω rk t þ h:o:t ., where superscripts u and l indicate phase-unlocked and locked terms respectively, and the overbar indicates the complex conjugate, and we note that N u uk is real. Additionally, note that these terms are in the order p1, p4, p2 and p3. Using this, the form of the resonant nonlinearities allows the Jacobian matrix, Eq. (28), to be rewritten as

Zero-amplitude response stability
Using the form of the resonant nonlinearities given in Eq. (33) the eigenvalues of the Jacobian matrix may be written as ð2ω rk λÞ 2 þ 4ζ k ω nk ω rk ð2ω rk λÞþ4ζ 2 k ω 2 nk ω 2 rk þðω 2 nk À ω 2 rk þ 2N u uk Þ 2 À 4N l uk N l uk ¼ 0: It follows that for the real part of an eigenvalue, λ, to be positive we require This implies that a necessary, but not sufficient, requirement for an unstable zero-amplitude response solution for the kth mode is that N l uk must be non-zero. Hence, for the triggering of a response in an unforced mode, phase-locked terms are required.

Application to the cable example
Now, considering the cable example when the bth out-of-plane mode is excited directly, we determine whether the ath in-plane mode may respond via internal resonance. Using the first line in Eq.
Considering Eq. (36) it can be seen that an unstable zero-amplitude response solution for the bth mode can only exist when a ¼b. Therefore, for a a b, while we have calculated mixed-mode backbone curves (solutions S5 Ã a a b ), no internally resonant forced response along these backbone curves will be observed. For the case where a ¼ b, using Eq. (36), the point of instability of zero-amplitude response solution of the ath in-plane mode, i.e. the point at which a response of the in-plane mode is triggered, occurs when the forced response solution of the ath out-of-plane mode crosses the resonant tongue defined by where we have noted thatŨ paŨ ma ¼Ũ 2 a by comparing Eqs. (2) and (25). Note that in the cable analysis we could have considered the forcing of many out-of-plane modes. This would have shown that the ith in-plane modes can only be triggered, but will not necessarily be triggered, by a response in the ith outof-plane mode as it is just this mode which provides a phase-locked coupling. It is also interesting to note that there are no phase-locked terms between different in-plane or between different out-of-plane modes (the equations take the same form as between an in-and an out-of-plane mode). A special case is when the cable is inclined, when additional parametric forcing terms are introduced due to the inclination, allowing a 1:2 resonance, see for example [17,18]. Fig. 4(a) shows the forced response of the cable due to out-of-plane support excitation when the first in-plane and the first out-of-plane modes are considered. For this, and subsequent figures, the forced response was calculated using numerical continuation, whereas the backbone curves were generated from the analytical solutions derived earlier. Note that forced responses may also be found analytically using the second-order normal form techniquesee, for example, [5,14]. Here, however, numerical continuation is employed to validate the assumptions used to find the backbone curves.
The transitions between the white and the green background to Fig. 4(a) mark the resonant tongue, given by Eq. (38). The points at which the primary forced response curve, containing only the out-of-plane mode, meets this boundary, coincide with bifurcation points. An additional branch, containing both the in-and out-of-plane modes, emerges from these bifurcation points and envelops the S4 7 backbone. In the case of the upper bifurcation point, it can be seen that the solution following S2 becomes unstable at the bifurcation, indicating the loss of stability of the zero-amplitude response solution of the in-plane mode. In contrast, for the case where the first in-plane and the third out-of-plane modes are considered, Fig. 4 (b), there is no resonant tongue, and the response envelops S2 throughout.

Stiffening effects
We have seen that, to observe internal resonance between a forced and an unforced mode, phase-locking resonant terms are necessary but not sufficient. In the case of the cable example such terms only exist between the kth in-and kth out-ofplane modes. The resulting motion envelops a mixed-mode backbone curve (or NNM branch), namely the S4 7 solution branch for the cable example. We now consider the effect of other modes on the onset of internal resonance between a forced and an unforced mode that have phase-locking resonant terms. For this discussion we consider the cable example and, specifically, we investigate the influence of an additional out-of-plane mode that is subjected to near-resonant forcing.
The cth out-of-plane mode may experience near-resonant excitation when the support motion is where R ij ¼ Àm 2ð À 1Þ i iπ j 2 V j , and where, from Eq. (4), q a ¼z a ,q b ¼ y b andq c ¼ y c . To fully apply the normal form technique to the forced system requires a forcing transform, q ¼ v þ e, prior to the nonlinear transform, u ¼ v þ h, [20,15]. This transform removes non-resonant forcing terms from each modal equation, by introducing parametric excitation terms into the nonlinear vector. However, a first approximation is to assume that the out-of-plane response terms are large compared to the forcing terms and so e can be neglected when compared to v. This allows us to find, albeit approximate, solutions using the analysis outlined in the previous sections.
Using this approximation, we now consider how the presence of the cth out-of-plane mode influences the modal interactions between the ath in-plane and bth out-of-plane modes (where a¼b). The analysis used to identify the NNMs relating to the ath in-plane and bth out-of-plane modes is modified to include terms due to the presence of the directly excited cth out-of-plane mode, due to the ν ac q aq 2 c and ν bcqbq 2 c terms in the equations of motion. Redefining the resonant response vector as u ¼ ½u aũbũc > ¼ ½u aũaũc > , the vector containing the resonant nonlinear terms for the three modes, formerly Eq. (8), is given by Here we have made use of the fact that we are interested in the case where a¼b (and so a ac). Considering the unforced system € u þ ΛuþN u ðuÞ ¼ 0, the S2 (i.e. ath out-of-plane-only response) and S4 7 (i.e. ath inplane and ath out-of-plane phase-locked response) now have the additional condition thatŨ c ¼ 0. However, if the cth mode exhibits a response, the backbone curves are affected due to the action of the last term in each of the first two rows of N u . The phase-locked, free response equations, Eqs. (15) and (16), are now Hence the response solution is modified away from the backbone expression for S2 due to the presence of theŨ c a 0 Backbone curves are shown in dark grey, stable and unstable forced responses are shown in blue and red respectively and the loss of stability of the zero-amplitude response solution for the in-plane mode is shown as a transition between the white and the green backgrounds. These backbone curves correspond to those shown in Figs. 2 and 3, and are computed using the parameters m ¼ 0:04418 kg, ω n1 ¼ 122:04 rad s À 1 , ω n1 ¼ 123:9 rad s À 1 ,ω n3 ¼ 366:1 rad s À 1 and ν 11 ¼ 1:417 Â 10 7 kg m À 2 s À 2 . Additionally, to compute the forced responses, the nonlinear parameter values γ 11 ¼ 1:188 Â 10 4 kg m À 1 s À 2 and γ 31 ¼ 1:069 Â 10 5 kg m À 1 s À 2 are used. In both forcing cases, the modal damping is ζ 1 ¼ζ 1 ¼ζ 3 ¼ 0:002 and the amplitude of displacement of the support is V ¼ 0:02 mm. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) response resulting in It can be seen that the effect of theŨ c response is to shift the solutions in the frequency plane. Note that these solutions are not necessarily backbones capturing a NNM response of the system, as NNMs require a periodic motion, whereas here the frequency of the cth out-of-plane mode is unconstrained, i.e. it is not necessary thatω rc ¼ cΩ. However they do demonstrate that the presence of other modes makes the system appear stiffer. Physically, this can be interpreted as a change in the tension of the cable, due to the presence of other modesthe dynamic tension of a cable contains zero-frequency response terms of the form θ 2 k U k , where θ k is the modeshape of the kth mode [4]. Related to this, the final example in [14] shows that, when excited out-of-plane, the zero-frequency cable response in all the in-plane modes is equivalent to a correction to the static sag. It can be shown that the forced response of the ath out-of-plane mode is similarly affected, namely the cable appears stiffer. We now consider the effect of the cth out-of-plane mode on the resonant tongue, recalling that this marks the onset of a response in the ath in-plane mode, triggered via the corresponding ath out-of-plane mode, due to phase-locking terms between these modes. Due to the additional nonlinear terms for the ath in-plane mode, due to the cth out-of-plane mode, see Eq. (40), the N u ua term, see Eq. (33), is modified to Using Eq. (36), the stability condition, for the case where there is a constantŨ c response, may be written as As with the solution curves, the effect of a non-zeroŨ c can be viewed as a shift in the frequency domain, equivalent to having an adjusted natural frequency,ω na , of the form This adjustment to the effective linear natural frequency is identical to the adjustment for the solution curves. Considering the cable system used in the previous figures, we can view the effect of the cth out-of-plane mode. Firstly we explore the somewhat artificial example where the forcing is such that the resonant response amplitude of the cth out-ofplane mode,Ũ c , is constant and the harmonics are assumed to be small. Fig. 5 considers the case where a ¼ b ¼ 1 and c ¼ 3, and shows the response of the cable in terms of the resonant amplitude of the ath, i.e. first out-of-plane mode. The singlemode response to the forcing is shown, along with the bifurcation points at which the mixed-mode response is triggered (black dots) and the subsequent mixed-mode response. To aid comparison with the case where the resonant response amplitude of the third out-of-plane mode,Ũ 3 , is zero, the backbone curves and resonant tongue for theŨ 3 ¼ 0 case, Fig. 4 Fig. 5. Forced response of the cable using a three-mode model consisting of the first in-plane mode, the first and third out-of-plane modes, for the artificial case where the resonant response amplitude of the third out-of-plane mode,Ũ 3 , is constant, atŨ 3 ¼ 0:5 mm. These responses have been computed using the following parameter values: m ¼ 0:04418 kg, ω n1 ¼ 122:04 rad s À 1 ,ω n1 ¼ 123:9 rad s À 1 , ν 11 ¼ 1:417 Â 10 7 kg m À 2 s À 2 , γ 11 ¼ 1:188 Â 10 4 kg m À 1 s À 2 , γ 31 ¼ 1:069 Â 10 5 kg m À 1 s À 2 , ζ 1 ¼ζ 1 ¼ 0:002 and V ¼ 0:015 mm.
(a), are shown in light grey and green respectively. It can be seen that the resonant tongue and unforced-undamped responses are both shifted in frequency due to the presence of the third out-of-plane mode. Likewise, the forced response is shifted, but here we note that the peak amplitude of the response is also affected due to the increased frequency at which the peak response occurs (this can be seen when compared with Fig. 4(a)).
These responses can be represented on a 3D plot, see Fig. 6. Here the backbone curve (whereŨ 3 ¼ 0) has been projected onto the unforced, undamped system response surface (shaded in blue) and the resonant tongue onto a resonant tongue surface (shaded in green). The two forced responses (corresponding to Figs. 4(a) and 5) are also shown for the case where there is no in-plane response, and where black dots indicate the points at which the in-plane response is triggered. Now, considering the more realistic case in which the third out-of-plane mode is excited at a constant amplitude of support motion, due to the interactions between the two out-of-plane modes the stiffening effect varies with excitation frequency. Fig. 7 shows the response for the case where the in-plane resonant motion is zero, along with the internally resonant branches. It can be seen that the resonant tongue, which shifts asŨ 3 increases, due to the stiffening effect, captures these trigger points well.
This discussion indicates that, when multiple out-of-plane modes are present, while the in-plane mode is only triggered by the corresponding out-of-plane mode, i.e. when a ¼ b, the presence of other out-of-plane modes will alter the behaviour Fig. 6. Forced response of the cable using a three-mode model consisting of the first in-plane mode, the first and third out-of-plane modes, showing the plane of unforced undamped responses and the resonant tongue. This represents the artificial case where the resonant response amplitude of the third outof-plane mode,Ũ 3 , is constant. These responses have been computed using the same parameters as those used in Fig. 5; i.e. m ¼ 0:04418 kg, ω n1 ¼ 122:04 rad s À 1 ,ω n1 ¼ 123:9 rad s À 1 , ν 11 ¼ 1:417 Â 10 7 kg m À 2 s À 2 , γ 11 ¼ 1:188 Â 10 4 kg m À 1 s À 2 , γ 31 ¼ 1:069 Â 10 5 kg m À 1 s À 2 , ζ 1 ¼ζ 1 ¼ 0:002 and V ¼ 0:015 mm. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) Fig. 7. Forced response of the cable using a three-mode model consisting of the first in-plane mode, and the first and third out-of-plane modes, for the case where the excitation amplitude of the forcing near the third out-of-plane mode, V 3 , is constant. Blue curves denote stable solutions and red curves, unstable solutions. Black dots indicate the intersection with the resonant tongue, and the bifurcations onto the internally resonant responses. The parameters used here are identical to those used in Figs. 5 and 6; i.e. m ¼ 0:04418 kg, ω n1 ¼ 122:04 rad s À 1 ,ω n1 ¼ 123:9 rad s À 1 , ν 11 ¼ 1:417 Â 10 7 kg m À 2 s À 2 , γ 11 ¼ 1:188 Â 10 4 kg m À 1 s À 2 , γ 31 ¼ 1:069 Â 10 5 kg m À 1 s À 2 and ζ 1 ¼ζ 1 ¼ 0:002. Additionally, the excitation amplitudes V 1 ¼ 0:015 mm and V 3 ¼ 0:01 mm are usedas described in Eqs. (39). See Fig. 8 for projections from this plot onto theŨ 1 vs Ω, andŨ 3 vs Ω planes. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) of this interaction. Therefore, these non-resonant out-of-plane modes must be considered in order to accurately predict the dynamic behaviour, and the points at which an internal resonance is triggered.

Conclusions
In this paper, the phenomenon of phase-locking between modes, and its influence on internal resonance has been considered in detail. In particular, using the example of a taut cable, expressions for both phase-locked and phase-unlocked backbone curves were developed. This analysis was initially based on two modesone in-plane and one out-of-plane. From this it was shown that phase-locking may only occur between corresponding modes, i.e. the ath in-plane and ath out-ofplane, while two non-corresponding modes do not exhibit phase-locking. Both the phase-locked and the phase-unlocked cases lead to mixed-mode backbone curves that bifurcate from single-mode backbone curves. However, these mixed-mode backbone curves are distinct as, notably, no specific phase-condition between the linear modes is present for the phaseunlocked case.
Taking a general, two-mode, weakly nonlinear system, a stability analysis was used to show that phase-locking is necessary for an internal resonance to occur. Considering the out-of-plane forced responses of the cable, it was shown that the mixed-mode, phase-locked backbone curves correspond to internally resonant interactions, whereas the phaseunlocked backbone curves do not. This stability analysis was also applied to the cable system to demonstrate that the resonant tongue, described using the analysis, could accurately predict the onset of internal resonance.
As phase-unlocked interactions do not lead to internal resonance, it might, at first, appear that they are of little interest in the analysis of nonlinear behaviours. However, in the final section of this paper, it was demonstrated that the presence of phase-unlocked modes may lead to stiffening effects in the system. These were shown to influence the internally resonant responses observed between other, phase-locked, modes. This was demonstrated by extending the two-mode model of the cable, where corresponding in-and out-of-plane modes exhibit phase-locked responses, to include a third, phase-unlocked, out-of-plane mode. The stiffening effect of this third, phase-unlocked mode was shown to cause an apparent shift in the response frequency of the interactions between the other modes.  Fig. 7, the parameters used here are: m ¼ 0:04418 kg, ω n1 ¼ 122:04 rad s À 1 , ω n1 ¼ 123:9 rad s À 1 , ν 11 ¼ 1:417 Â 10 7 kg m À 2 s À 2 , γ 11 ¼ 1:188 Â 10 4 kg m À 1 s À 2 , γ 31 ¼ 1:069 Â 10 5 kg m À 1 s À 2 , ζ 1 ¼ζ 1 ¼ 0:002, V 1 ¼ 0:015 mm and V 3 ¼ 0:01 mm. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) For the somewhat artificial case where the fundamental response of the third mode was held at a constant amplitude, this frequency shift was demonstrated to be constant. The more realistic case, where the response of the third mode varies under forcing, was then shown to exhibit more complex behaviours. However, these behaviours were accurately predicted by the analytical descriptions of the unforced, undamped equivalent responses, and the onset of internal resonance was predicted using the resonant tongues. This demonstrated that, while the phase-unlocked modes do not lead to internal resonance, they can alter the nature of the internal resonance between phase-locked modes.