Identifying phase-varying periodic behaviour in conservative nonlinear systems

Nonlinear normal modes (NNMs) are a widely used tool for studying nonlinear mechanical systems. The most commonly observed NNMs are synchronous (i.e. single-mode, in-phase and anti-phase NNMs). Additionally, asynchronous NNMs in the form of out-of-unison motion, where the underlying linear modes have a phase difference of 90°, have also been observed. This paper extends these concepts to consider general asynchronous NNMs, where the modes exhibit a phase difference that is not necessarily equal to 90°. A single-mass, 2 d.f. model is firstly used to demonstrate that the out-of-unison NNMs evolve to general asynchronous NNMs with the breaking of the geometrically orthogonal structure of the system. Analytical analysis further reveals that, along with the breaking of the orthogonality, the out-of-unison NNM branches evolve into branches which exhibit amplitude-dependent phase relationships. These NNM branches are introduced here and termed phase-varying backbone curves. To explore this further, a model of a cable, with a support near one end, is used to demonstrate the existence of phase-varying backbone curves (and corresponding general asynchronous NNMs) in a common engineering structure.


Introduction
Nonlinearities in mechanical structures can cause a wide variety of complex dynamic phenomena, such as modal interactions, localization, bifurcations and instability [1][2][3]. As such, identifying the existence of these phenomena, and addressing the difficulties they pose to the design, performance analysis and prediction of the behaviour of nonlinear systems can be challenging. For example, cables can exhibit complex nonlinear behaviours [4][5][6][7]; this renders cable-supported structures, e.g. cable-stayed bridges and floating offshore systems, susceptible to unwanted vibrations during operation [8,9].
To understand these nonlinear phenomena, linear theory is often insufficient, or even invalid, since the nonlinear behaviours can significantly differ from the linearized ones. In this context, extending linear theory to account for nonlinear behaviours is needed. To address this, the concept of a nonlinear normal mode (NNM) was defined by Rosenberg [10][11][12] as an in-unison, or synchronous, periodic resonance for a conservative nonlinear system. It requires that the displacements of components all reach their extreme values and pass through equilibrium points simultaneously during periodic resonances. As such, these NNMs may be represented in terms of their initial displacements, where the initial velocities are set to zero. Such synchronous NNMs include single-mode, in-phase and anti-phase responses, which have been observed in a variety of nonlinear systems, e.g. the two-mass oscillators [13][14][15], beam structures [16,17], rotor systems [18] and cable structures [19].
As well as synchronous (single-mode, in-phase and anti-phase) resonances, nonlinear systems can exhibit asynchronous resonances, where the displacements of components do not reach their extreme values and pass through equilibrium points simultaneously while remaining periodic. To account for such asynchronous resonances, an extension to Rosenberg's definition was proposed in [20,21], where an NNM is defined as a (non-necessarily synchronous) periodic response of the conservative system; this definition of an NNM is considered throughout this paper. One example of an asynchronous NNM is the whirling motion observed in cable structures [6,19] and rotor systems [18]. This whirling motion is an out-of-unison response in which one coordinate reaches an extremum while another passes through the equilibrium point. As such, for a twomode system, this motion may be represented by the initial displacement of one coordinate and the initial velocity of the other (while the respective initial velocity and displacement are simultaneously zero). Considering the phase of the underlying coordinates, in-phase and antiphase motions are characterized by a phase difference of 0 • or 180 • , while out-of-unison motion is characterized by a phase difference of ±90 • . Besides this special case of out-of-unison motion, to the best of our knowledge, a more general asynchronous NNM, characterized by a general phase relationship (i.e. non-necessarily 90 • out-of-phase), has not been identified in the literature. If such general asynchronous NNMs exist, they represent a large family of nonlinear responses that may affect the performance of the nonlinear systems, and may potentially be exploited. Such behaviours may be considered to be more complex than synchronous or out-of-unison motions, as they may include responses where no velocities or displacements are simultaneously zero. Their existence also indicates that the phase relationships between modal coordinates are crucial parameters to be determined when computing the NNMs.
Here, we hypothesize and then demonstrate the existence of general asynchronous NNMs using the simple motivating example of a two-mode 1 single-mass oscillator. In addition to this being a new solution type, this observation highlights the need to consider phase as a free variable (rather than constrained to 0, ±90 • or 180 • ) when searching for NNMs using analytical techniques, and using numerical approaches that rely on systematic investigation of the initial conditions. Section 2 first revisits the concept of synchronous and asynchronous NNMs, distinguished by the phase relationships between the modal coordinates of a two-mode system with 1 : 1 internal resonance. A numerical method is then used to demonstrate that the hypothisized general asynchronous NNMs can exist for the single-mass oscillator. As with the cable structure and in-line two-mass oscillator studied in [19], numerical results show that the single-mass model possesses out-of-unison NNMs when it has a geometrically orthogonal layout. With the breaking of the orthogonal configuration, the out-of-unison NNMs can evolve to more general asynchronous cases, where the phase difference is neither 0 • , 180 • nor ±90 • . This demonstrates that such motions may exist in a nonlinear mechanical structure.  on this finding, in §3, an analytical technique is used to further quantify the characteristics of asynchronous NNMs in the single-mass model. Analytical phase relationships of the backbone curves, i.e. the branches of NNMs, verify the results found in §2, and further reveal that, with the breaking of orthogonal configurations, the out-of-unison backbone curve evolves to one that consists of asynchronous NNMs whose phase relationships vary along the backbone curve. This class of backbone curve is defined here as a phase-varying backbone curve, and represents the loci of general asynchronous NNMs. Using insights obtained from the single-mass model, the existence of phase-varying backbone curves in a cable model (a common mechanical structure that exhibits out-of-unison backbone curves [19]) is then considered in §4. A reduced-order cable model is first derived and verified, using an existing analytical model [8]. The addition of a support near the cable root is then considered-this resembles the engineering practice of installing external devices to suppress vibrations [6,22,23]. This support breaks the orthogonal configuration of the cable and, as with the single-mass model, causes the out-of-unison (i.e. whirling) motions to evolve into general asynchronous motions on a phase-varying backbone curve. Finally, conclusions are presented in §5.
2. Nonlinear normal modes of a two-mode system with 1 : 1 internal resonance In this section, the NNMs of a two-mode system with 1 : 1 internal resonance are first revisited, where an NNM is defined as a periodic response of a conservative system [20,21]. NNMs can be divided into synchronous and asynchronous solutions. Examples of the synchronous responses are shown in figure 1a,b, where the lines represent the oscillations over time in linear modal space (q 1 and q 2 denote the first and second linear modal coordinates, respectively). The extrema of q 1 and q 2 are marked by circles and crosses, respectively. Figure 1a represents the simplest type of NNM solution-single-mode responses that contain contributions from only q 1 or q 2 . Figure 1b shows the synchronous mixed-mode NNMs that, in contrast to the singlemode NNMs in figure 1a, arise from modal interactions and consist of contributions from both linear modal coordinates, q 1 and q 2 . For synchronous mixed-mode NNMs, the modal coordinates reach extrema and equilibrium points simultaneously, as shown in figure 1b. Such NNMs can be characterized using the phase relationship, θ d , between the fundamental components of q 1 and q 2 , which can be either in-phase (θ d = 0) or anti-phase (θ d = π ); note that this phase relationship is undefined for the single-mode cases in figure 1a. They can also be represented by their initial   Figure 2. A schematic diagram of a single-mass two-mode system. A mass, with mass value m, has x and y denoting horizontal and vertical in-plane displacements, respectively. This mass is grounded by two horizontal springs with coefficients k 1 and k 3 , and unstretched lengths L 1 and L 3 . Another spring, with coefficient k 2 and unstretched length L 2 , grounds the mass, with angle δ representing the angle between k 1 and k 2 . The orthogonal case is shown here where δ = 90 • (Online version in colour.) modal conditions at extrema-non-zero displacements with zero velocities. These synchronous cases can be seen in a variety of nonlinear systems; see [13][14][15][16][17][18][19].
For asynchronous NNMs, an example is shown in figure 1c where one modal coordinate reaches an extremum when the other passes through the equilibrium point. In this case, the linear modal coordinates have ±π/2 out-of-phase relationships (the '+' and '−' signs here denote the clockwise and anticlockwise motions, respectively). Such motions can also be characterized by the initial conditions at extrema-a non-zero displacement for one coordinate with a nonzero velocity for the other, or vice versa. This class of NNM is termed an out-of-unison NNM in [19], and includes, for example, whirling motions of cables and rotor systems [18,19]. The commonly observed NNM motions can, therefore, be categorized as single-mode (figure 1a, where θ d is undefined), in-phase and anti-phase (figure 1b, θ d = 0, π , respectively) and out-ofunison (figure 1c, θ d = ±π/2). It seems logical, therefore, to pose the question: Can other phase relationships exist between the linear modes of NNM responses? Such an NNM is depicted in figure 1d, and corresponds to a general asynchronous response where the phase relationship between linear modal coordinates is θ d out-of-phase (to differentiate it from these previously discussed cases, θ d = 0, π , ±π/2). Unlike the synchronous and out-of-unison NNMs, this NNM represents responses where displacements and velocities cannot simultaneously be zero. To the best of our knowledge, this more general asynchronous NNM has not been identified in the literature.
To explore the existence of this general asynchronous NNM, and further characterize its features, a two-mode single-mass system, schematically shown in figure 2, is firstly considered. This example system consists of one mass, with mass value m, and has displacements x and y, denoting horizontal and vertical in-plane motions, respectively. This mass is grounded by three linear springs with coefficients k 1 , k 2 and k 3 , and with unstretched lengths L 1 , L 2 and L 3 , respectively. At equilibrium, all the springs are unstretched and springs k 1 and k 3 are lying in the x-direction, while the angle between k 1 and k 2 is denoted δ (when δ = 90 • , spring k 2 is orthogonal to springs k 1 and k 3 ). Such a system can exhibit nonlinear behaviours because of geometric nonlinearity. To investigate the nonlinear dynamic behaviours, the equations of motion can be obtained via the Euler-Lagrange equations For details of the derivation, readers can refer to appendix A. Using this two-mode model, we investigate one potential mechanism, i.e. the breaking of the orthogonality, 2 that may lead to the existence of the general asynchronous resonance. This is achieved by comparing the backbone curves, i.e. branches of NNMs, for the orthogonal and non-orthogonal cases. These backbone curves are computed using the numerical continuation software COCO [24], and hence no analytical approximation is required for the results shown in this section.
(a) NNMs of the system with an orthogonal configuration First, we consider the orthogonal case, where the two-mode single-mass system has m = 1, k 1 = k 3 = 0.5, k 2 = 1.005, L 1 = L 2 = L 3 = 1 and δ = 90 • . The backbone curves of this system are shown in figure 3 in the projection of the response frequency, Ω, against the absolute displacement amplitude of the mass, X 2 + Y 2 , where X and Y are the maximum amplitudes of displacements x and y, respectively. In this region, there are two single-mode backbone curves  Figure 4. Backbone curves obtained via numerical continuation for the two-mode system in figure 2 with a non-orthogonal configuration. The backbone curves are shown as solid curves in the projection of the response frequency Ω against absolute displacement of the mass, √ X 2 + Y 2 , for a system with m = 1, k 1 = k 3 = 0.5, k 2 = 1.005, L 1 = L 2 = L 3 = 1 and δ = 89.5 • ; and the branch point bifurcation is denoted as a solid dot, labelled 'BP2' . Four embedded plots, in the projection of modal coordinates, q 1 (t) against q 2 (t), represent the responses of NNMs on the corresponding backbone curves. The extreme displacement values of modal coordinates q 1 (t) and q 2 (t) are marked by circles and crosses, respectively, in these embedded plots. Arrows in the embedded plot linked to S ±v 1 denote clockwise and anticlockwise motions. For comparison, the backbone curves for the orthogonal case in figure 3 are shown as grey dash-dotted curves with branch point bifurcations denoted as open dots. (Online version in colour.) S 1 and S 2 ; 3 and two mixed-mode backbone curves, S + 1 and S − 1 , bifurcating from S 1 through the branch point bifurcation 'BP1' (the subscripts of S + 1 and S − 1 indicate the backbone curve from which they bifurcate, in this case from S 1 ). Note that, because of the symmetry of the configuration, S + 1 and S − 1 are superimposed in this projection. The NNMs on these single-mode and mixed-mode backbone curves exhibit synchronous resonances-see the time-parameterized responses of NNMs on these backbone curves in the embedded plots, which are analogous to those shown in figure 1a,b, respectively. Besides these synchronous backbone curves, asynchronous backbone curves, i.e. the out-of-unison backbone curves, S ±90 1 , bifurcate from S 1 through the branch point bifurcation 'BP2'. The embedded plot, near S ±90 1 in figure 3, describes the NNMs on S ±90 1 : q 1 reaches its extreme value when q 2 has a zero value and vice versa, and the arrows on the curve describe the clockwise motion (θ d = +π/2) and anticlockwise motion (θ d = −π/2), respectively. The out-of-unison NNMs in this plot are similar to those shown in figure 1c, and the out-of-unison motions reported in [19].

(b) Nonlinear normal modes of the system with a non-orthogonal configuration
With δ perturbed away from 90 • , the orthogonality of the system is broken. The effect of breaking the orthogonality on the backbone curves is shown in figure 4 for the case where δ = 89.5 • , while other parameters remain unchanged. For comparison, backbone curves for the orthogonal case are also presented using dash-dotted grey curves in this figure. It can be seen that the branch point bifurcation, 'BP1', splits to generate one primary in-phase backbone curve, S + 1 , and one isolated anti-phase backbone curve, S − 1 [26]. The contribution of the first linear modal coordinate, q 1 , to the single-mode backbone curve, S 2 , increases from 0 and leads to an in-phase backbone curve, S + 2 . These three mixed-mode backbone curves still consist of synchronous NNMs-see the embedded plots of time-parameterized responses. The other bifurcation point, 'BP2', remains, and connects the anti-phase backbone curve S − 1 to S ±v 1 . Backbone curves, S ±v 1 , can be seen as evolutions from the out-of-unison backbone curves, S ±90 1 , with the breaking of orthogonality. It is shown in the embedded plot linked to the S ±v 1 curves that, similar to the out-of-unison backbone curves, the NNMs on S ±v 1 also exhibit asynchronous responses, where the arrows denote the clockwise motion (+θ d ) and anticlockwise motion (−θ d ), respectively; however, the phase relationship between the two modal coordinates, θ d , are not ±π/2, but instead are similar to the asynchronous ones shown in figure 1d. This is highlighted by the dots and crosses, shown in the embedded plots, which illustrate that the extrema and equilibria are reached at different times. This demonstrates that NNMs with a general asynchronous motion can exist. The phase relationships, θ d , of NNMs on these backbone curves show amplitude-dependent characteristics; in other words, phase relationships of NNMs vary along the backbone curves. Hence they are termed phase-varying backbone curves, denoted with the superscript '±v'. This will be discussed in detail in the next section.
In this section, the periodic responses, i.e. the NNMs, of a two-mode system with 1 : 1 internal resonance were firstly reviewed, emphasizing the less studied asynchronous NNMs. A specific example of such asynchronous NNMs is the out-of-unison NNM, studied in [19], where the modal coordinates have ±π/2 phase difference. To explore the existence of a more general case, where the NNM has a phase difference θ d = 0, π , ±π/2 between linear modal coordinates, a simple two-mode system, shown in figure 2, has been considered. We found that the breaking of orthogonality can transform the out-of-unison NNMs to the more general asynchronous ones. In the next section, analytical studies are carried out to further study the dynamic characteristics of the asynchronous NNMs.

Analytical analysis of the asynchronous backbone curves
In this section, using the harmonic balance technique, the backbone curves of the single-mass system are found analytically and further used to characterize the asynchronous responses. To simplify this analytical study, the full model, described by equation (2.1a), is first expanded to a polynomial one using Maclaurin expansion, and then truncated by retaining nonlinear terms up to the cubic order. The obtained equations of motion in linear modal space arë where ω n1 and ω n2 denote the first and second linear natural frequencies, respectively, and where Ξ 1 , Ξ 2 , . . . , Ξ 4 and Ψ 1 , Ψ 2 , . . . , Ψ 5 are quadratic and cubic nonlinear coefficients, respectively. For details of this derivation, see appendix A.
To apply the harmonic balance method, 4 it is assumed that the modal displacements may be approximated by a single harmonic, i.e. frequencies of the two modes are equal, i.e. ω r1 = ω r2 = Ω, which accounts for 1 : 1 internal resonance. Following the procedure described in [16]-with the substitution of expressions (3.2) into the equations of motion (3.1) and the non-resonant terms removed-one can obtain the time-independent solutions from These equations can then be used to compute the backbone curves of the twomode system depicted in figure 2. Note that the quadratic terms, presented in equations (3.1), do not lead to 1 : 1 internally resonant components [27], hence they do not appear in equations (3.3).

(a) Backbone curves for systems with orthogonal configurations
For orthogonal configurations of the single-mass system, one can find that Ψ 2 = Ψ 4 = 0-see appendix A for details. This further reduces equations (3.3) to and There are six different sets of solutions to equations (3.4), representing the backbone curves of the system. The solution U 1 = U 2 = 0 is trivial, corresponding to the case where the system has no motion. Besides this trivial case, there are two single-mode solutions: one is the backbone curve, S 1 , with U 2 = 0 and U 1 = 0; the other is the backbone curve, S 2 , which has U 1 = 0 and U 2 = 0. Their amplitude-frequency relationships are and The NNMs on these single-mode branches, as discussed previously, are schematically shown in figure 1a.
Cases where U 1 = 0 and U 2 = 0 are related to mixed-mode backbone curves, hence the phase relationship (equation (3.4c)) between two linear modal coordinates must be determined before finding their amplitude-frequency relationships. Two sets of phase relationships can be found to satisfy equation (3.4c). One requires sin(θ d ) = 0, i.e. θ d = nπ (where n ∈ Z), which corresponds to the in-phase and anti-phase cases. With sin(θ d ) = 0 substituted into equations (3.4), the amplitudefrequency relationships for in-phase and anti-phase backbone curves are given by , Like the single-mode backbone curves, NNMs on these mixed-mode, in-phase and anti-phase backbone curves relate to synchronous responses, schematically shown in figure 1b. Note that S ± 2 does not exist for the system considered in figure 3. The other phase relationship, satisfying equation (3.4c), has cos(θ d ) = 0, i.e. θ d = (2n + 1)π/2. This solution branch represents the outof-unison backbone curves S ±90 (3.8) NNMs on these backbone curves exhibit ±π/2 out-of-phase asynchronous resonances, whose responses are schematically shown in figure 1c. Again, we note that S ±90 2 does not exist for the system considered in figure 3. (b) Backbone curves for systems with non-orthogonal configurations For non-orthogonal configurations of the single-mass system, the nonlinear coefficients, Ψ 2 and Ψ 4 , are not necessarily equal to 0. With non-zero values of Ψ 2 and Ψ 4 , the single-mode solutions (U 2 = 0 and U 1 = 0, or U 1 = 0 and U 2 = 0) can no longer be achieved in equations (3.3). Hence, in this case, only mixed-mode backbone curves can be found. Similar to the orthogonal case, the phase relationship needs to be determined first by considering equation (3.3c). This can be satisfied when sin(θ d ) = 0, i.e. θ d = nπ , which corresponds to the in-phase and anti-phase backbone curves S ± 1 and S ± 2 , similar to those discussed previously. The amplitude-frequency relationships are given by rearranging equations (3.3a) and (3.3b) as where p = cos(nπ ). For even n, p = +1 and this indicates the in-phase backbone curves S + 1 and S + 2 ; whereas for odd n, p = −1, representing anti-phase backbone curves S − 1 and S − 2 . Besides the in-phase and anti-phase cases, the other phase relationship relates to a zero value of the terms in the bracket of equation (3.3c), which can be rearranged as This expression indicates that the phase relationship (θ d ) is dependent upon the amplitude, suggesting that the phase difference between the modal coordinates varies along the backbone curve. Here, we term the asynchronous NNM branch with an amplitude-dependent phase relationship between modal coordinates as a phase-varying backbone curve. To find the expressions of this phase-varying backbone curve, the phase relationship (3.10) is substituted into equations (3.3a) and (3.3b); after some algebraic manipulation, one has , (3.11) As previously discussed, an orthogonal configuration can lead to Ψ 2 = 0 and Ψ 4 = 0. Substituting these into expressions (3.11), the amplitude-frequency relationships describing phase-varying backbone curves can be reduced to those describing out-of-unison backbone curves in equations (3.8). Furthermore, with Ψ 2 = 0 and Ψ 4 = 0, the phase relationship, described by the expression (3.10), can be reduced to cos(θ d ) = 0. This phase relationship is again identical to that for the out-of-unison backbone curves. This therefore indicates that the phase-varying backbone curve is an evolution from the out-of-unison backbone curve with the orthogonality breaking, through the phase-amplitude coupling, described by equation ( Figure 5b shows the phase relationships on these backbone curves in the projection of the response frequency, Ω, against the phase difference, θ d , between the two fundamental components of modal coordinates, q 1 and q 2 . For the orthogonal case, as expected, different NNMs on any one backbone curve share a fixed phase relationship between q 1 and q 2 , indicated by the dash-dotted straight lines in figure 5b. For the non-orthogonal case, the backbone curves, S + 1 , S − 1 and S + 2 , have fixed phase relationships; however, the phase relationships of backbone curves, S ±v 1 , vary with frequency. One branch of these phase-varying backbone curves, S +v 1 , has a phase relationship varying from θ d = π (on the branch point bifurcation on S − 1 ) to θ d = π/2, with the decrease in response frequency (along with the increase in displacement amplitude-see figure 5a). The embedded plots, labelled (i), (ii), . . . , (vi), present the time-parameterized responses of a selection of NNMs on S +v 1 . It can be seen that the NNMs evolve from an anti-phase NNM (θ d = π on the branch point bifurcation) towards a clockwise out-of-unison NNM (θ d = π/2). The other phase-varying backbone, S −v 1 , shows similar behaviours, except for having NNMs exhibiting anticlockwise motions.
In this section, the harmonic balance technique has been used to find the analytical expressions of backbone curves for the single-mass system with orthogonal and non-orthogonal configurations. Analytical analysis shows that the general asynchronous backbone curve, discussed in §2, has an amplitude-dependent phase relationship between the linear modal coordinates. This backbone curve is termed a phase-varying backbone curve, and it can be seen as an evolution from the out-of-unison backbone curve through the breaking of the orthogonality. The existence of such backbone curves indicates that phase relationships between modal coordinates are crucial parameters to be determined in finding NNMs, a key implication of which is when applying the harmonic balance method numerically to compute NNMs. In the next section, numerical analysis is carried out to investigate the existence of phase-varying backbone curves in a cable model.

Phase-varying backbone curves for a cable model
In this section, phase-varying behaviour is investigated using a horizontal cable, taut between two fixed end points. An additional elastic support connects the cable to the ground, near one of the fixed ends, as shown in figure 6. The dynamics of the cable system are modelled based on a lumped-mass approach, similar to the method in [28]. A brief description is given here for completeness.
The model is formulated by discretizing the cable into n identical elastic elements, connected in series between n + 1 nodes. The two end nodes are fixed, resulting in a total of 3(n − 1) degrees of freedom in three-dimensional space. The mass of the cable is equally distributed between the elements, and for each element half of its mass is lumped on either end. The elements are assumed to be undamped and linearly elastic. The cable has an unstretched length of L 0 , uniform density ρ, Young's modulus E, and a constant cross-section of diameter d. Axial stress is assumed to be uniformly distributed over the cross-sectional area, and a static axial pre-tension with a horizontal component T is applied at both cable ends. The forces acting on the cable are due to gravity and elasticity, while viscous and aerodynamic effects are neglected.
An additional undamped, linearly elastic element is attached to the cable at a position z s along its span. This element lies within the cross-sectional (x-y) plane, at an angle δ from the horizontal. It has a length l, stiffness k, and is unstretched when the system is at equilibrium.
A 2 d.f. nonlinear reduced-order model of the cable system, which captures its salient dynamic behaviour near the first two natural frequencies, is obtained using a force-based indirect reduction method [29]. This involves a projection of the equations of motion of the full model onto a 2 d.f. reduced basis. The reduction/projection basis consists of the first out-of-plane and the first inplane transverse mass-normalized linear modeshapes of the cable about its equilibrium position. As such, the equations of motion of the reduced-order model can be written as where f 1 and f 2 are the nonlinear restoring forces. For linear elastic finite-element models with geometric nonlinearities, the forcing functions typically take the form of quadratic and cubic polynomials [29][30][31], i.e.
and f 2 (q 1 , q 2 ) = Ξ 2 q 2 1 + 2Ξ 3 q 1 q 2 + 3Ξ 4 q 2 2 + Ψ 2 q 3 1 + 2Ψ 3 q 2 1 q 2 + 3Ψ 4 q 1 q 2 2 + 4Ψ 5 q 3 2 . (4.2b) Note that linear dependencies are imposed on the coefficients in equations (4.2), such that the energy in the system is conserved [31,32], similar to equations (3.1). The linear properties in equations (4.1) can be obtained directly through an eigenanalysis of the full system. However, the coefficients of the nonlinear terms in equations (4.2) are computed in a non-intrusive manner, using a set of static solutions of the lumped-mass model. 5 The static solutions are obtained by applying a set of prescribed static loads and computing the corresponding displacements. The selected loading cases consist of scaled linear combinations of the retained modes. For each load case, the computed static displacement of the full system is then projected onto the reduced modal space. Finally, the coefficients of the nonlinear terms in equations (4.2) are estimated through regression analysis in a least-squares manner, using the modal force-modal displacement dataset. A validation of the developed lumped-mass cable model, and of the corresponding reduced-order model obtained using the indirect reduction method described above, can be found in appendix B.
We now consider a 40-element, 117 d.f. cable model with the following physical parameters: L 0 = 1.5 m, d = 5 mm, ρ = 3000 kg m −3 , E = 200 GPa. The system is subjected to a static pre-tension with a horizontal component T = 100 N, and is additionally constrained by an elastic element with the following properties: l = 0.2 m, k = 10 5 N m −1 and z s = 0.15 m. Two additional support layouts are considered-one corresponds to the case when the spring is aligned with the y-axis, i.e. when δ = 90 • , and this is denoted as the orthogonal case; the other relates to the case when δ = 60 • , and this is denoted as the non-orthogonal case. The estimated parameters of either model can be found in table 1.    Table 1. Values of the estimated parameters of the reduced-order model for the orthogonal case and the non-orthogonal case.  curves for the single-mass system discussed in §3. A selection of NNMs on S −90 1 , i.e. NNMs marked by '+' signs in figure 7a,b, are presented in figure 7c,d, where the time-parameterized responses are shown in modal coordinates and physical coordinates, respectively. Because of the variation in the tension of a cable during oscillation, a non-resonant q 2 component arises from the nonlinear quadratic terms in expressions (4.2). This leads to a shift of the extrema q 1 (t) (marked by circles in figure 7c,d) along the backbone curve [25]. Nonetheless, an anticlockwise out-of-unison (θ d = −90 • ) phase relationship between q 1 and q 2 can still be seen, similar to the one in figure 1c. Likewise, similar behaviours can be expected for the other out-of-unison backbone curve, S +90 1 , except for having clockwise motions. Figure 8a shows the backbone curves for the non-orthogonal case. The corresponding phase relationships on the backbone curves are shown in figure 8b. With δ perturbed away from 90 • , it leads the single-mode backbone curves, S 1 and S 2 , to mixed-mode backbone curves, S + 1 and S − 2 , respectively. The out-of-unison backbone curves, S ±90 1 , like these for the single-mass system discussed in previous sections, evolve into phase-varying backbone curves, S ±v 1 , and bifurcate from the in-phase mixed-mode backbone curve, S + 1 . The phase-varying backbone curves show that phase relationships evolve from θ d = 0 (in-phase motions at the bifurcation) to θ d ≈ ±π/2 (nearly out-of-unison motions), as depicted in figure 8b. One example of this evolution of motions is presented in figure 8c, which describes the modal motions of NNMs marked with '+' signs

Conclusion
NNMs represent important tools in the analysis of nonlinear phenomena. This paper has investigated the less commonly studied asynchronous NNMs. It has been shown that out-ofunison NNMs (characterized by a ±90 • phase relationship between modal coordinates) are a special case of a more general, and previously unreported, NNM solution set where the phase relationship may assume any value. The existence of this general asynchronous NNM was first explored using a single-mass model via numerical analysis and it was demonstrated that the breaking of orthogonality causes the out-of-unison NNMs to evolve into general asynchronous NNMs (i.e. the phase difference between the modes evolves from ±90 • to a general phase difference). An analytical method was then used to find the expressions of backbone curves, i.e. branches of NNMs, of the single-mass model. The analytical phase relationship between modal coordinates revealed that, along with the breaking of orthogonality, the out-of-unison backbone curves evolve to ones on which the phase relationships exhibit amplitude-dependent characteristics. This newly identified NNM branch is defined as a phase-varying backbone curve. The existence of phase-varying backbone curves was then investigated in a cable model, through the attachment of a near-cable-end support. This support has the effect of breaking the orthogonal geometry of the cable and, as with the single-mass example, cause the out-of-unison (whirling) motions to evolve into general asynchronous motions.
The existence of phase-varying backbone curves, and the accompanying general asynchronous NNMs, represents a new set of nonlinear phenomena in mechanical systems. These NNMs may represent significant responses in nonlinear systems that may be critical in understanding their performance (such as the evolution from whirling in a cable with non-orthogonal geometry) or that may be exploited to improve the performance of such systems. The existence of such NNMs also indicates that phase relationships between modal coordinates are crucial parameters to be determined when computing nonlinear responses, and should be carefully considered when computing nonlinear responses. These motions may be considered more complex than the more commonly observed synchronous or out-of-unison motions, as their displacement and velocity coordinates are never simultaneously zero. A key implication is that, when using the harmonic balance technique to compute the NNMs, the phase relationships, being the critical parameters describing the nonlinear phenomena (similar to the role of the harmonic amplitudes), should be seen as unknowns to be determined. Acknowledgements. We gratefully acknowledge the financial support of the EPSRC and CSC.
Appendix A. Obtaining the truncated model of the single-mass system Using L i to denote the stretch or compression of springs k i , the Lagrangian of the one-mass two-mode system, schematically shown in figure 2, can be written as where m is the mass value; k 1 , k 2 and k 3 are the coefficients of linear springs with lengths L 1 , L 2 and L 3 , respectively; and δ denotes the angle between springs k 1 and k 2 . Applying the Euler-Lagrange equation, the equations of motion can be obtained as and This full model can then be expanded as polynomial equations using Maclaurin expansion, and further simplified by retaining nonlinear terms up to cubic orders. In this way, the equations of motion can be written as where M and K are the mass and linear stiffness matrices, respectively; N x is a vector of nonlinear terms; and x is a vector representing physical displacements. They can be written as M = m 0 0 m , K = k 1 + k 2 cos 2 (δ) + k 3 k 2 sin(δ) cos(δ) k 2 sin(δ) cos(δ) k 2 sin 2 (δ) (A 4) and N x = 3β 1 x 2 + 2β 2 xy + β 3 y 2 + 4γ 1 x 3 + 3γ 2 x 2 y + 2γ 3 xy 2 + γ 4 y 3 β 2 x 2 + 2β 3 xy + 3β 4 y 2 + γ 2 x 3 + 2γ 3 x 2 y + 3γ 4 xy 2 + 4γ 5 y 3 , x = where the coefficients of nonlinear terms, β 1 , β 2 , . . . , β 4 , γ 1 , γ 2 , . . . , γ 5 , are 2L 2 2 and γ 5 = cos 2 (δ)L 2 This underlying linear model can be used for linear modal analysis to find the modal parameters, allowing the truncated system (A 3) to be translated into linear modal space. This is achieved by using substitution x = Φq, where Φ is the modeshape matrix and q is a vector of linear modal coordinates, written as where the first and second columns of Φ denote the first and second linear modeshapes, respectively. After applying the linear modal transform, the equations of motion can be written and and where ω ni denotes the ith linear natural frequency, and nonlinear modal coefficients are 2 22 , 3 22 , Appendix B. Validation of lumped-mass cable model and reduced-order model The lumped-mass discretization approach as well as the subsequent reduction method described in §4 are validated by comparing the backbone curves of a reduced-order model with those obtained using an analytically derived dynamic model of a small-sag cable system [8]. The analytical model in [8] is applicable to highly stressed cables with a small weight-to-tension ratio, such that axial modal motions can be neglected. The cable considered for validation purposes has  Table 2. Comparison between the values of the estimated model parameters, using the analytical cable model in [8], and the 2 d.f. reduced-order model (ROM) via the reduction method described in §4.  the same properties as that described in §4, but the applied axial pre-tension is increased from T = 100 N to T = 200 N, in order to satisfy the aforementioned requirement.
A two-mode model based on [8] was previously studied in [19], where it was shown that the whirling motion of a cable corresponds to out-of-unison resonance between its first out-of-plane and first in-plane transverse modes. The results obtained using the corresponding reduced-order model are in close quantitative and qualitative agreement with these observations, as shown in figure 9. The corresponding parameters in the equations of motion (4.1), computed using the reduction method described in §4, as well as those obtained using the analytical model, are shown in table 2.