Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Conditions for the existence of isolated backbone curves

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Thomas L. Hill

Thomas L. Hill

Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK

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Simon A. Neild

Simon A. Neild

Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK

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Published:https://doi.org/10.1098/rspa.2019.0374

    Abstract

    Isolated backbone curves represent significant dynamic responses of nonlinear systems; however, as they are disconnected from the primary responses, they are challenging to predict and compute. To explore the conditions for the existence of isolated backbone curves, a generalized two-mode system, which is representative of two extensively studied examples, is used. A symmetric two-mass oscillator is initially studied and, as has been previously observed, this exhibits a perfect bifurcation between its backbone curves. As this symmetry is broken, the bifurcation splits to form an isolated backbone curve. Here, it is demonstrated that this perfect bifurcation, indicative of a symmetric structure, may be preserved when the symmetry is broken under certain conditions; these are derived analytically. With the symmetry broken, the second example—a single-mode nonlinear structure with a nonlinear tuned mass damper—is considered. The evolution of the system's backbone curves is investigated in nonlinear parameter space. It is found that this space can be divided into several regions, within which the backbone curves share similar topological features, defining the conditions for the existence of isolated backbone curves. This allows these features to be more easily accounted for, or eliminated, when designing nonlinear systems.

    1. Introduction

    Over recent decades, demand on the performance of engineering structures has been continually growing. Meeting this demand often requires extending the performance envelope of structures to regions where nonlinearity must be accounted for. This results in complex nonlinear dynamic phenomena, such as bifurcations, internal resonances and multiple solutions [16]. Despite the challenges these phenomena have posed to analysis and design, recent studies seek to exploit, rather than avoid, the nonlinear behaviours. Application areas include vibration absorbers and energy harvesting [713]. Among these applications, the nonlinear tuned mass damper (NLTMD) has been extensively studied in the literature with its advantageous performance demonstrated [1417].

    Den Hartog [18] proposed a widely adopted method to optimize the parameters of a linear tuned mass damper (LTMD). Later, a nonlinear generalization of Den Hartog's equal-peak method for an NLTMD was established in [16], where the performance between the NLTMD and the LTMD was compared. Gatti [17] highlights that the main advantage of introducing nonlinearity is the improvement of the bandwidth of the device. An NLTMD has, for example, been used to control a supercritical Hopf-bifurcation of aerofoil flutter in [19], and to control limit cycle oscillations of mechanical systems in [20].

    Besides these favourable properties, the use of an NLTMD may bring some undesirable dynamic phenomena, such as isolas, i.e. forced responses that are isolated from the primary response branches. Due to this feature, the existence of isolas can be difficult to determine; furthermore, these isolated solutions may represent significant, high-amplitude responses [16,2123]. An early study of isolas in engineering systems was carried out by Abramson [24] in 1955. Extensive work since this has focused on the mechanism for their creation, such as discontinuity [25,26], internal resonances [27] and symmetry breaking [28,29].

    Numerous approaches have been used to detect and trace isolas. One numerical method is continuation which uses special points, such as fold bifurcations and extremum points, to trace the evolution of isolas by varying specific parameters [30,31]. In combination with continuation methods, singularity theory can be used to provide complementary information in the prediction and identification of isolas [3234]. Methods based on continuation can efficiently find isolas; however, they require a good understanding of the system, and its responses, to select the appropriate continuation parameters. Another numerical method is global analysis, which may detect an isola by finding initial conditions which are within the basin of attraction of that isola [35]. This approach requires a large number of simulations of initial conditions, making it computationally expensive. In addition to these methods, experimental techniques are also used to detect isolas; for example, using control-based continuation, an isola is detected for a nonlinear beam structure in [36].

    An alternative approach to considering forced responses is to analyse the underlying backbone curves of the unforced, undamped system. Backbone curves, which are also known as nonlinear normal modes (NNMs), are also widely used in nonlinear modal analysis, reduced-order modelling and localization analysis (e.g. [5,6,37]). Backbone curves can be related to the forced responses of a system (including those that lie on isolas) using energy balance analysis [29,30,38]. This involves finding the points in the forced responses that lie on the backbone curves, such that the forcing energy in matches the damping energy loss. This approach reduces the isola-finding problem to an analytical and computationally simpler one; however, it requires that the backbone curves are known.

    To complicate matters, while less studied than the forced counterpart, backbone curves themselves can also be isolated.1 Isolated backbone curves have been demonstrated for a simple, near-symmetric two-mass oscillator [38], and it was shown that this isolated curve emerged due to the breaking of symmetry. Recently, an isolated backbone curve has also been measured experimentally for a cross-beam system using a nonlinear force appropriation technique [39], and again shown to evolve from the symmetry breaking in the system. Without a priori knowledge of such isolated backbone curves, any isolated forced responses that are associated with them may go undetected. While the aforementioned systems exhibit isolated backbone curves during symmetry-breaking, in practice, some systems cannot be symmetric, e.g. a grounded structure with an ungrounded NLTMD attached, and the existence of isolas can have a significant impact on their performance, as discussed above. A general methodology, describing the relationship between the symmetry of the system and the evolution of isolated backbone curves, has not yet been fully explored. To understand this relationship, a more general case needs to be considered. Such insights into this relationship would ensure that isolated backbone curves can be reliably predicted when designing nonlinear devices and structures.

    This paper presents a technique to determine the existence of isolated backbone curves for a two-mode system with cubic nonlinearities and a 1 : 1 internal resonance. This is motivated by the fact that much of the current literature on modal interactions consider systems where just two modes interact. The general two-mode model is related to two specific example systems: an in-line two-mass oscillator is used to explore the relationship between symmetry breaking and isolated backbone curves; the second motivating example is an NLTMD attached to a single-mode nonlinear structure. Exploiting the method developed for the general two-mode case, the evolution of isolated backbone curves in nonlinear parameter space is identified for the NLTMD system. To this end, the rest of this paper is organized as follows.

    Section 2 firstly revisits the backbone curves of a two-mode symmetric system, which has been considered extensively (e.g. [5,29,37,38,40,41]). This system exhibits two single-mode backbone curves with one perfect bifurcation leading to mixed-mode backbone curves. By breaking the symmetry of this system, the perfect bifurcation splits to form an isolated backbone curve. It is then shown that, similarly to a symmetric system, an asymmetric system may also exhibit single-mode backbone curves with a perfect bifurcation. The mechanisms by which symmetry breaking affects the modal equations are then explored. Section 3 considers a nonlinear structure with an NLTMD and, using the insights from §2, derives the analytical parameter relationships to obtain two single-mode backbone curves with one perfect bifurcation. Perturbing the linear and nonlinear parameters from these relationships, the evolution of backbone curves in nonlinear parameter space is then discussed, and the emergence and evolution of isolated backbone curve addressed. Focusing on the system with hardening springs, discriminant analysis is used in §4 to find the analytical conditions under which the isolated backbone curve may be removed, i.e. shifted to infinite frequency and amplitude. Analytical expressions found in §§3 and 4 serve as boundaries, distinguishing topological features of backbone curves in nonlinear parameter space, and defining conditions for the existence of isolated backbone curves. Lastly, this paper is closed with conclusion in §5.

    2. Breaking the symmetry of a nonlinear 2-d.f. oscillator

    In this section, a general two-mode2 conservative system, with cubic nonlinearities, is firstly considered, before considering a specific two-mode system. The Lagrangian of this general system may be written

    L=12q˙12+12q˙2212ωn12q1212ωn22q2214Ψ4q14Ψ1q13q212Ψ3q12q22Ψ2q1q2314Ψ5q24,2.1
    where qi, q˙i and ωni are the ith linear modal displacement, velocity and natural frequency respectively, and Ψ1, …, Ψ5 are the nonlinear coefficients. Note that the nonlinear coefficients are defined in this order for simplicity in later sections. Applying the Euler–Lagrange equation then leads to the following equations of motion:
    q¨1+ωn12q1+Ψ4q13+3Ψ1q12q2+Ψ3q1q22+Ψ2q23=02.2a
    and
    q¨2+ωn22q2+Ψ1q13+Ψ3q12q2+3Ψ2q1q22+Ψ5q23=0.2.2b
    Note that the use of the Lagrangian in deriving these expressions restricts the number of nonlinear parameters to five, while ensuring that the equations of motion remain conservative [42,43].

    From equation (2.2b), when the coefficient of q31 equals 0 (i.e. when Ψ1 = 0), q2 = 0 is a solution (although coupling between the two modes still exists via other terms). Substituting this into equation (2.2a) gives the single-mode solution, representing a single-mode backbone curve, or NNM branch, which consists of only the first linear modal coordinate, q1, by solving

    q¨1+ωn12q1+Ψ4q13=0.2.3
    Note that Ψ1 = 0 is the special case which results in this single-mode backbone, a solution which cannot exist when Ψ1≠0. In addition to this single-mode solution, backbone curves containing contributions from both linear modes, i.e. mixed-mode backbone curves, can also be found when Ψ1 = 0. Likewise, when the coefficient of q32 in equation (2.2a), Ψ2, equals 0, one can find the single-mode solution that consists of only the second linear modal coordinate, q2, from
    q¨2+ωn22q2+Ψ5q23=0.2.4
    Otherwise, when both Ψ1≠0 and Ψ2≠0, this system only has mixed-mode backbone curves. To find the backbone curves of the general two-mode system, the harmonic balance technique3 is used, firstly by assuming that the modal displacements may be written as
    qiui=Uicos(ωritθi),2.5
    where ui represents the fundamental response of qi, and where Ui, ωri and θi are amplitude, response frequency and phase of ui, respectively. Note that this trigonometric solution is equivalent to the exponential form used in [38,41]. It is further assumed that the fundamental frequencies of the two modes are equal, i.e. ωr1 = ωr2 = Ω, hence, the response frequency ratio is 1 : 1. With the substitution of expressions (2.5) into the equations of motion (2b), and the non-resonant terms removed, one can obtain the time-independent solutions from
    4(ωn12Ω2)U1+3Ψ4U13+Ψ3U1U22[1+2cos2(θd)]+3(Ψ2U23+3Ψ1U12U2)cos(θd)=0,2.6a
    4(ωn22Ω2)U2+3Ψ5U23+Ψ3U12U2[1+2cos2(θd)]+3(Ψ1U13+3Ψ2U1U22)cos(θd)=02.6b
    and[2Ψ3U1U2cos(θd)+3Ψ1U12+3Ψ2U22]sin(θd)=0,2.6c
    where θd = θ1 − θ2. These equations can then be used to compute the backbone curves of the general two-mode system.

    (a) The backbone curves of a symmetric two-mass oscillator

    To demonstrate the effect of symmetry breaking, a specific two-mode system, the two-mass oscillator shown schematically in figure 1, is now used. The system consists of two masses with mass values m1 and m2, and displacements x1 and x2, respectively. These masses are grounded via two linear springs, with coefficients k1 and k3, respectively, and are connected by another linear spring with coefficient k2. This system also contains three nonlinear cubic springs with coefficients α1, α2 and α3, as shown in figure 1. The backbone curves of this system can be computed from equations (2.6), using the relationship between the nonlinear modal coefficients, Ψi, and the physical parameters of the model, as derived in appendix A.

    Figure 1.

    Figure 1. A schematic diagram of a two-mode system in the form of a two-mass oscillator. Two masses, with mass values m1 and m2, have displacements x1 and x2, respectively, while linear and nonlinear cubic springs have coefficients ki and αi, respectively, where i = 1, 2, 3. (Online version in colour.)

    Here, the symmetry of the system is divided into two parts: linear symmetry (LS), where both m1 = m2 and k1 = k3; and nonlinear symmetry (NS), where α1 = α3. Note that the LS–NS case leads to Ψ1 = 0 and Ψ2 = 0, as shown in appendix A—see equation (A 9) with α1 = α3. As previously discussed, this leads to single-mode solutions. The backbone curves of the system with both LS and NS have been investigated in detail in [29,38,41], and an example is illustrated in figure 2c. The single-mode backbone curves S1 and S2 consist of only the first and second modal coordinates, respectively; while S+2 and S2 represent mixed-mode backbone curves containing both linear modal coordinates. The subscripts of S+2 and S2 indicate the backbone curve from which they bifurcate (i.e. from S2 in this case), and the superscripts+ and denote in-phase and anti-phase responses between the fundamental components of the linear modal coordinates, respectively. For details of how these backbone curves have been computed, using equations (2.6), see [29,38,41].

    Figure 2.

    Figure 2. The effect of breaking the nonlinear symmetry (NS), i.e. breaking the condition α1 = α3, for a system with linear symmetry (LS), i.e. m1 = m2 and k1 = k3. (a) The nonlinear parameter space, α1 against α3, for the system with LS when m1 = m2 = 1, k1 = k3 = 1, k2 = 0.3 and α2 = 0.05. The α1 and α3 values that lead to Ψ1 = 0 and Ψ2 = 0 are shown as a green dotted line and a purple dashed line, respectively. (b) Backbone curves for a system with linear symmetry and nonlinear asymmetry (NA) when α1 = 1, α3 = 0.5 (represented by a black dot labelled (b) in panel (a)). (c) Backbone curves for an LS–NS system with α1 = 1, α3 = 1, where the solid point represents the perfect bifurcation (denoted by a solid dot labelled (c) in panel (a)). (d) Backbone curves for an LS-NA system with α1 = 1, α3 = 1.5 (represented by a black dot labelled (d) in panel (a)). Panels (b)–(d) are shown in the projection of the response frequency, Ω, against the amplitude of the first mass, X1. (Online version in colour.)

    Here, we introduce the concept of dynamic symmetry to describe backbone curves with the features of an LS–NS system, specifically characterized by the following features:

    (i)

    two single-mode backbone curves, S1 and S2.

    (ii)

    two mixed-mode backbone curves; either S±2, emerging from a perfect bifurcation on S2, or S±1, emerging from a perfect bifurcation on S1.

    (b) Breaking either the nonlinear or the linear parameter symmetry

    Figure 2a represents the nonlinear parameter space of α1 against α3—i.e. the parameters of the two nonlinear grounding springs—for the case where the system has LS. The nonlinear parameters leading to Ψ1 = 0 and Ψ2 = 0 are denoted by green dotted and purple dashed lines, respectively. In this case, i.e. for LS, these lines correspond to NS. In other words, for this simple system, both LS and NS will always lead to Ψ1 = Ψ2 = 0, hence resulting in single-mode backbone curves which is one of the conditions for dynamic symmetry. For nonlinear parameter combinations with nonlinear asymmetry (NA), i.e. α1α3, both Ψ1≠0 and Ψ2≠0, as indicated by the dots labelled (b) and (d) in figure 2a. This symmetry breaking turns the single-mode backbone curves into mixed-mode ones, breaks the perfect bifurcation, and generates an isolated backbone curve. As shown in figure 2b,d (corresponding to parameters labelled (b) and (d) in figure 2a), an isolated backbone curve emerges from two primary mixed-mode backbone curves, as observed in [29]. As proven in appendix A, a system with LS and NA cannot exhibit dynamic symmetry as in this case Ψ1≠0 and Ψ2≠0.

    Similarly, breaking the LS, while retaining the NS, can also break the dynamic symmetry. With the breaking of the LS (m2 = 0.8m1 and k3 = 0.7k1), the orientations of Ψ1 = 0 and Ψ2 = 0 are changed, and are no longer overlapping, as shown in figure 3a. If the NS is retained, i.e. α1 = α3 (depicted by the grey line in figure 3a), the backbone curves, depicted in figure 3b, are similar in form to the ones for the LS–NA system in figure 2d, i.e. one isolated backbone curve between two primary mixed-mode backbone curves. The LA–NS system considered here cannot have dynamic symmetry since the intersection of Ψ1 = 0 and Ψ2 = 0, where one can find two single-mode backbone curves, is not on the line representing α1 = α3 (i.e. the point at which the green and purple lines in figure 3a cross does not correspond to the grey line).

    Figure 3.

    Figure 3. The effect of breaking the linear symmetry (LS) for a system with nonlinear symmetry (NS). (a) The nonlinear parameter space for the system with linear asymmetry (LA) when m1 = 1, m2 = 0.8, k1 = 1, k3 = 0.7, k2 = 0.3 and α2 = 0.05. The α1 and α3 values that lead to Ψ1 = 0 and Ψ2 = 0 are shown as a green dotted line and a purple dashed line, respectively, and parameters leading to NS are shown as a dash-dotted grey line. (b) Backbone curves for an LA–NS system with α1 = 1, α3 = 1 (represented by a grey dot labelled (b) in panel (a)) in the projection of the response frequency, Ω, against the amplitude of the first mass, X1. (Online version in colour.)

    (c) Breaking both the linear and nonlinear parameter symmetry

    Following from the LA–NS system considered in §2b, the NS is also broken to investigate the backbone curves of an LA–NA system. Figure 4b shows the backbone curves for the case where α3 is reduced from the NS-case to the point where Ψ2 = 0, marked as a purple dot labelled (b) in figure 4a. As expected, this leads to a single-mode backbone curve S2; however, as Ψ1≠0, the first primary backbone curve, S+1, contains a component of the second mode with in-phase modal coordinates. As such, this is not a dynamically symmetric case, despite sharing some characteristics, such as the backbone curves S+2 and S2 which bifurcate off S2. Further reducing α3 leads to the point where Ψ1 = 0, shown as a green dot labelled (c) in figure 4a, whose backbone curves are shown in figure 4c. These exhibit a single-mode backbone curve S1 (as predicted by the Ψ1 = 0 condition) but with a primary and an isolated backbone curve, S+2 and S2. As shown in figure 4a, Ψ1 = Ψ2 = 0 may still be satisfied for this case if α1 and α3 are on the intersection of Ψ1 = 0 and Ψ2 = 0, i.e. the point labelled (d) in figure 4a. Dynamic symmetry can therefore be obtained for such an LA–NA system, as can be seen from the backbone curves in figure 4d.

    Figure 4.

    Figure 4. Obtaining dynamic symmetry for an LA–NA system. (a) The nonlinear parameter space for a system with LA (linear parameters and α2 are equal to those considered in figure 3). The α1 and α3 values that lead to Ψ1 = 0 and Ψ2 = 0 are shown as a green-dotted and a purple-dashed lines, respectively, and parameters leading to NS are shown as a dash-dotted grey line. (b) Backbone curves for an LA–NA system with α1 = 1, α3 ≈ 0.6785 (represented by a purple dot labelled (b) in panel (a)). (c) Backbone curves for an LA–NA system with α1 = 1, α3 ≈ 0.5510 (denoted by a green dot labelled (c) in panel (a)). (d) Backbone curves for an LA–NA system with α1 ≈ 0.3333, α3 ≈ 0.1833 (represented by a solid dot labelled (d) in panel (a)). (Online version in colour.)

    As previously discussed, the concept of dynamic symmetry is defined as having similar characteristics to an LS–NS system; nonetheless, such behaviour can be observed in an LA–NA system, if parameters are appropriately selected. This means that an LA–NA system can exhibit the same dynamic characteristics as a fully symmetric system. Furthermore, defining conditions for the existence of single-mode solutions, expressions Ψ1 = 0 and Ψ2 = 0 also serve as boundaries, which divide the nonlinear parameter space into several regions, within which the backbone curves share similar topological features. These regions, in the nonlinear parameter space, allow the changes in the fundamental dynamic behaviours to be identified and predicted.

    3. Backbone curves for an NLTMD-inspired two-mode system

    In this section, a two-mode asymmetric system, depicted in figure 5, is considered. This system is equivalent to that shown in figure 1, but with the springs grounding the second mass removed. This system is representative of a nonlinear structure (first mass) with an NLTMD (second mass) attached.

    Figure 5.

    Figure 5. A schematic diagram of a two-mode asymmetric system, representative of a nonlinear structure with an NLTMD. (Online version in colour.)

    Isolated backbone curves represent particularly undesirable features in an NLTMD device [16,21,23], due to the difficulty of predicting them, and their potential to represent high-amplitude dynamic responses. To further understand their features, in this section the parameter conditions required for dynamic symmetry are first found. However, in practice, the optimized linear parameters of an NLTMD cannot usually satisfy such conditions. The evolution of backbone curves in nonlinear parameter space, for an optimized set of linear parameters, is then investigated and used to determine the conditions for the existence of isolated backbone curves.

    (a) Parameter conditions required for dynamic symmetry

    As discussed in §2, an LA–NA system can exhibit dynamic symmetry if the parameters are selected appropriately. One feature of dynamic symmetry is having two single-mode solutions, S1 and S2, which requires that both Ψ1 = 0 and Ψ2 = 0 in the equations of motion (2b). The expressions of Ψ1 and Ψ2 for the NLTMD system are given in equations (A 5), and may be written in matrix form as

    (Ψ1Ψ2)=[ϕ113ϕ12(ϕ11ϕ21)3(ϕ12ϕ22)ϕ11ϕ123(ϕ11ϕ21)(ϕ12ϕ22)3](α1α2),3.1
    where α3 = 0 has been substituted (i.e. no nonlinear spring grounding the second mass) and where ϕij are elements of the linear modeshape matrix Φ, defined as
    Φ=[ϕ11ϕ12ϕ21ϕ22].3.2
    Note that the first column of this matrix, i.e. ϕ11 and ϕ21, represents the modeshape of the first linear mode, while the second linear modeshape is captured by ϕ12 and ϕ22, in the second column of Φ. In order for Ψ1 = 0 and Ψ2 = 0 to be satisfied, equation (3.1) shows that either α1 = 0 and α2 = 0 (which would represent the trivial case where the system is linear), or that the determinant of the matrix in equation (3.1) must be zero, i.e.
    ϕ113ϕ12(ϕ11ϕ21)(ϕ12ϕ22)3ϕ11ϕ123(ϕ11ϕ21)3(ϕ12ϕ22)=0.3.3
    Note that for a system with an asymmetric configuration, ϕij are non-zero, and ϕ11ϕ21 and ϕ12ϕ22. Thus, equation (3.3) can be rearranged to
    ϕ112(ϕ12ϕ22)2ϕ122(ϕ11ϕ21)2=1,3.4
    which can be satisfied using the following conditions:
    ϕ11(ϕ12ϕ22)ϕ12(ϕ11ϕ21)=1:ϕ11ϕ21ϕ12ϕ22=03.5a
    and
    ϕ11(ϕ12ϕ22)ϕ12(ϕ11ϕ21)=1:ϕ21ϕ11+ϕ22ϕ12=2.3.5b
    Condition (3.5a) cannot be satisfied as it requires that the first and second modeshapes are equal; therefore, dynamic symmetry, i.e. when Ψ1 = Ψ2 = 0, may only be achieved when condition (3.5b) is satisfied. The relationship between the modeshape coefficients, ϕij, and the linear physical parameters is derived in appendix A. Substituting equations (A 7) into condition (3.5b) (also using equation (A 6)) leads to
    k1k2=m1+m2m2.3.6
    This demonstrates that, in order for the NLTMD-inspired system to exhibit dynamic symmetry, the linear stiffness coefficients must obey the ratio described by equation (3.6). As well as this condition for the linear parameters, the relationship between the nonlinear parameters may be found by substituting expressions (3.6) and (A 7) back into equation (3.1), leading to
    α1α2=(ϕ12ϕ22)4ϕ124=(ϕ11ϕ21)4ϕ114=(m1+m2m2)2.3.7

    Figure 6a shows the nonlinear parameter space α1 against α2, for the case where k1/k2 = 21, and where the mass values satisfy equation (3.6). The overlapping green and purple lines represent the parameter relationships that lead to Ψ1 = 0 and Ψ2 = 0, respectively (in this instance the case α1/α2=441, satisfies equation (3.7) and hence Ψ1 = Ψ2 = 0). Despite being a linearly asymmetric system, this has strong similarities to figure 2a, which represents an LS system, and indicates that multiple nonlinear parameter combinations will lead to dynamic symmetry. Note that when the parameter relationships (3.6) and (3.7) are satisfied, expressions for Ψ1, …, Ψ5 (A 5) can be simplified as

    Ψ1=0,Ψ2=0,Ψ3=6ϕ112ϕ122α1,Ψ4=2ϕ114α1,Ψ5=2ϕ124α1.3.8
    With Ψ1 = 0 and Ψ2 = 0, expressions for backbone curves (2.6) can be reduced to
    {4(ωn12Ω2)+3Ψ4U12+Ψ3U22[1+2cos2(θd)]}U1=0,3.9a
    {4(ωn22Ω2)+3Ψ5U22+Ψ3U12[1+2cos2(θd)]}U2=03.9b
    and2Ψ3U1U2cos(θd)sin(θd)=0.3.9c
    The case where U1 = 0 and U2 = 0 represent the trivial case where the system is stationary. Two sets of single-mode solutions, denoted S1 and S2, can be found with frequency–amplitude relationships described as
    S1:U2=0,Ω2=ωn12+34Ψ4U123.10
    and
    S2:U1=0,Ω2=ωn22+34Ψ5U22.3.11
    This system can also exhibit mixed-mode backbone curves. To compute these, the phase relationship, θd, between the fundamental components of the two modal coordinates, u1 and u2, needs to be determined. From equation (3.9c), this may be satisfied when θd = nπ/2, with nZ. The case where n is odd, which satisfies cos(θd) = 0, represents solutions exhibiting out-of-unison resonance [46], i.e. the two modes are ±90° out-of-phase. The case where n is even, satisfying sin(θd) = 0, represents in-phase and anti-phase solutions. Considering the out-of-unison solutions, the substitution of cos(θd) = 0 into equations (3.9a) and (3.9b), leads to the frequency–amplitude relationship
    S1,2±90:U12=4(ωn22ωn12)+(3Ψ5Ψ3)U223Ψ4Ψ33.12a
    and
    Ω2=4(3Ψ4ωn22Ψ3ωn12)+(9Ψ4Ψ5Ψ32)U224(3Ψ4Ψ3).3.12b
    With substitution of expressions (3.6), (3.8), (A 6) and (A 8) into equation (3.12b), one can find that the response frequency, Ω, of out-of-unison resonance, in this case, is always zero. This means that out-of-unison solutions cannot exist in the dynamically symmetric case for this system.
    Figure 6.

    Figure 6. Dynamic symmetry for an NLTMD-inspired system. (a) The nonlinear parameter space, α1 against α2, for a system with LA, i.e. m1 = 1, m2 = 0.05, k1 = 1 and k2 = 1/21, where Ψ1 = 0 and Ψ2 = 0 are two overlapping lines. (b) Backbone curves with dynamic symmetry for a hardening LA–NA system when α1 = 1 and α2 = 1/441 (represented by a solid dot labelled (b) in panel (a)). (c) Backbone curves with dynamic symmetry for a softening LA–NA system when α1 = − 1 and α2 = − 1/441 (represented by a solid dot labelled (c) in panel (a)). (Online version in colour.)

    The in-phase solutions, corresponding to θd = 0 are denoted S+1 and S+2, while the anti-phase solutions, corresponding to θd = π are denoted S1 and S2 (S±1,2 is used to denote all of them). For the dynamically symmetric case, these backbone curves all share the frequency-amplitude relationship, given by

    S1,2±:U12=4(ωn22ωn12)+3(Ψ5Ψ3)U223(Ψ4Ψ3)3.13a
    and
    Ω2=4(Ψ4ωn22Ψ3ωn12)+3(Ψ4Ψ5Ψ32)U224(Ψ4Ψ3).3.13b

    As well as two single-mode backbone curves, described by equations (3.10) and (3.11), dynamic symmetry also requires two mixed-mode backbone curves, described by equations (3.13), with a perfect bifurcation on either S2, which are denoted S±2, or on S1, which are denoted S±1. For the perfect bifurcation point on S2, the amplitude of the first modal coordinate U1 = 0; likewise, the second modal amplitude U2 = 0 for the perfect bifurcation point on S1. Using these conditions, the amplitude and frequency of these two bifurcation points can be obtained, from equations (3.13), as

    bifurcation point onS1:U12=4(ωn22ωn12)3(Ψ4Ψ3),Ω2=Ψ4ωn22Ψ3ωn12Ψ4Ψ33.14a
    and
    bifurcation point onS2:U22=4(ωn22ωn12)3(Ψ3Ψ5),Ω2=Ψ3ωn22Ψ5ωn12Ψ3Ψ5.3.14b
    As ωn2 > ωn1, to obtain positive solutions, i.e. positive amplitude and frequency, requires
    bifurcation point onS1:Ψ4Ψ3>0,Ψ4ωn22Ψ3ωn12>03.15a
    and
    bifurcation point onS2:Ψ3Ψ5>0,Ψ3ωn22Ψ5ωn12>0.3.15b
    Note that conditions (3.15) are valid for any system with cubic nonlinearities and a 1 : 1 resonance between two modes. To relate these to the NLTMD system, the expressions for modeshape elements (A 8) and the nonlinear parameter relationship (3.7) are substituted into these inequalities. This reveals that a perfect bifurcation from S1 onto S±1 exists when both nonlinear parameters are softening, i.e. α1 < 0 and α2 < 0; while a perfect bifurcation from S2 onto S±2 may be seen for hardening nonlinear parameters, i.e. α1 > 0 and α2 > 0 when m2 < m1/3. Figure 6b,c shows the backbone curves with dynamic symmetry, i.e. satisfying parameter conditions (3.6), (3.7) and (3.15) for systems with hardening and softening parameters, respectively (labelled with (b) and (c), respectively, in figure 6a).

    (b) Evolution of backbone curves in the nonlinear parameter space

    In the previous discussion, we did not consider the tuning of the NLTMD, but rather concentrated on whether a solution exists that exhibits dynamic symmetry and derived the linear and nonlinear parameter relationships, (3.6), (3.7) and (3.15) for this to occur. Now, to investigate the evolution of backbone curves in nonlinear parameter space, it is assumed that the linear parameters of the NLTMD are tuned to achieve optimal performance.

    The classical approach for optimizing the linear parameters of a TMD is known as the fixed-points method [18]. Instead of imposing two fixed points, using H optimization, a closed-form exact solution to obtain equal peaks in receptance curves of the underlying linear system is discussed in [47], where the linear stiffness of the NLTMD can be optimized using

    k2opt=8μk1[16+23μ+9μ2+2(2+μ)4+3μ]3(1+μ)2(64+80μ+27μ2),3.16
    where μ = m2/m1 is the mass ratio and kopt2 is the optimized linear spring coefficient of the NLTMD. This cannot satisfy relationship (3.6), and hence dynamic symmetry cannot be achieved. However, if nonlinear parameters are correspondingly selected on either Ψ1 = 0 or Ψ2 = 0, single-mode backbone curves, S1 and S2, respectively, can still be solved via amplitude–frequency relationships (3.10) and (3.11). To find the nonlinear parameter conditions that lead to either Ψ1 = 0 or Ψ2 = 0, the modeshape expressions (A 7) are substituted into the expressions for Ψi (A 5) (with α3 = 0). Letting Ψ1 = 0 and Ψ2 = 0, respectively, one has
    Ψ1=0:α1α2=(ωn12m1+k1)3(ωn22m1+k1)k243.17a
    and
    Ψ2=0:α1α2=(ωn12m1+k1)(ωn22m1+k1)3k24.3.17b
    As depicted in figure 7a, the curves Ψ1 = 0 and Ψ2 = 0 do not overlap, instead, these curves now are intersecting at the origin in the nonlinear parameter space (where the system is reduced to a linear one). Backbone curves for the system with Ψ1 = 0 and Ψ2≠0 (the point labelled (c) in figure 7a) are shown in figure 7c; these are similar to those shown in figure 4c, where a single-mode backbone curve S1 is also present. Likewise, backbone curves for the system with Ψ2 = 0 and Ψ1≠0 (the point labelled (b) in figure 7a) are shown in figure 7b, which have similarity to figure 4b.
    Figure 7.

    Figure 7. Optimizing the linear parameters of the NLTMD leads to the breaking of dynamic symmetry for an NLTMD-inspired system. (a) Nonlinear parameter space, α1 against α2, for a system with LA, i.e. m1 = 1, m2 = 0.05, k1 = 1 and k2 = kopt2 ≈ 0.0454. The α1 and α3 values that lead to Ψ1 = 0 and Ψ2 = 0 are shown as a green dotted line and a purple dashed line, respectively, and curves Ψ1 = 0 and Ψ2 = 0 are intersecting at the origin. (b) Backbone curves with the single-mode solution S2 for the LA–NA system when α1 = 1 and α2 ≈ 0.00256 (represented by a purple dot labelled (b) in panel (a)). (c) Backbone curves with the single-mode backbone curve S1 for the LA–NA system when α1 = 1 and α2 ≈ 0.00166 (represented by a green dot labelled (c) in panel (a)). (Online version in colour.)

    Systems with nonlinear parameters that do not lie on Ψ1 = 0 or Ψ2 = 0 exhibit mixed-mode backbone curves. As previously, the phase relationship between the two modal coordinates needs to be determined using equation (2.6c). This expression is satisfied with real and positive solutions by sin(θd) = 0, leading to the phase relationship θd = θ1 − θ2 = nπ, where even and odd n values denote in-phase and anti-phase modal relationships, respectively. Further defining the phase parameter p as

    p=cos(θd)=cos(nπ)={+1for even n1for odd n,3.18
    allows the expressions governing the modal amplitudes and frequencies, given in (2.6a,b), to be written
    4(ωn12Ω2)U1+3[Ψ4U13+Ψ3U1U22+p(Ψ2U23+3Ψ1U12U2)]=03.19a
    and
    4(ωn22Ω2)U2+3[Ψ5U23+Ψ3U12U2+p(Ψ1U13+3Ψ2U1U22)]=0.3.19b
    Rearranging these two equations gives the frequency-amplitude relationships
    Ω2=ωn12+34[Ψ4U13+Ψ3U22U1+p(Ψ2U23+3Ψ1U12U2)]U113.20a
    and
    0=(3pΨ2U11)U24+3(Ψ5Ψ3)U23+[9p(Ψ2Ψ1)U1]U22+[4ωn224ωn12+3(Ψ3Ψ4)U12]U2+3pΨ1U13,3.20b
    which may be solved to find the mixed-mode backbone curves.

    Figure 8 presents the mixed-mode backbone curves of systems with linear parameters m1 = 1, m2 = 0.05, k1 = 1, k2 = kopt2 ≈ 0.0454 in the nonlinear parameter space α1 against α2. Backbone curves on Ψ1 = 0 and Ψ2 = 0, labelled in this figure, are the same as those in figures 7c,b, respectively. This space is divided by the parameter axes α1 = 0 and α2 = 0 into the following four classes of system:

    (i)

    a hardening system (the first quadrant)

    Perturbing the nonlinear parameters clockwise from Ψ2 = 0 breaks the perfect bifurcation on S2, shown in figure 7b. The breaking of this bifurcation generates one isolated backbone curve, S+2, between two primary backbone curves, S2 and S+1, 4 seen in figure 8a. Likewise, if nonlinear parameters are perturbed anticlockwise from Ψ2 = 0, the perfect bifurcation on S2 breaks in a different direction, resulting in one isolated backbone curve, S2, below two primary backbone curves, depicted in figure 8b.

    Figure 8.

    Figure 8. The evolution of backbone curves in the nonlinear parameter space, α1 against α2, for a system with LA, i.e. m1 = 1, m2 = 0.05, k1 = 1 and k2 = kopt2 ≈ 0.0454. The α1 and α2 values that lead to Ψ1 = 0 and Ψ2 = 0 are shown as a green dotted line and a purple dashed line, respectively. The panels around the main figure show backbone curves topologies for the nonlinear regions in terms of response frequency and displacement amplitude of the first mass. Panels (a,b,c,e,g,h,i,k) show these topologies in regions indicated by the solid-grey arrows. Panels (d,f ,j,l) show the topologies corresponding to α1 and α2 axes, as indicated by the dash-dotted grey arrows. (Online version in colour.)

    Further varying the nonlinear parameters in the anticlockwise direction towards Ψ1 = 0, the contribution of the second modal coordinate, U2, to the mixed-mode, in-phase backbone curve, S+1, gradually decreases to zero. This results in a single-mode backbone curve S1, seen from the evolution of backbone curves from figure 8b to 7c. Finally, perturbing anticlockwise from Ψ1 = 0, the U2 component of S1 increases, leading to a mixed-mode, anti-phase backbone curve S1, as shown in figure 8c.

    (ii)

    a hardening structure with a softening attachment (the second quadrant)

    Further decreasing α2 until α2 < 0, causes S1 to bend leftward, as depicted in figure 8e. Note that no isolated backbone curve is predicted for systems in the second quadrant.

    (iii)

    a softening system (the third quadrant)

    Crossing from the second quadrant into the third causes S+2 to bend leftward, along with S1. Continuing anticlockwise, from above Ψ2 = 0 to below it, leads to a similar behaviour to the hardening system (the first quadrant) as it crosses Ψ1 = 0. The contribution from U1 to the mixed-mode, in-phase backbone curve S+2 may gradually decrease, reaching zero at Ψ2 = 0, leading to a single-mode backbone curve S2. The contribution then increases, resulting in a mixed-mode, anti-phase backbone curve S2. Such behaviour is shown in figure 8g,h. Simultaneously, the isolated backbone curve, S+1, emerges from zero frequency and draws closer to the primary backbone curve S1.

    Further varying the nonlinear parameters towards Ψ1 = 0, the isolated and primary backbone curves, S+1 and S1, merge into a single-mode backbone curve, S1, with a perfect bifurcation. Anticlockwise of Ψ1 = 0, the perfect bifurcation breaks and generates an isolated backbone curve, S1, below two primary backbone curves, S+1 and S2, shown in figure 8i.

    (iv)

    a softening structure with a hardening attachment (the fourth quadrant)

    Crossing from the third to the fourth quadrant leads to the disappearance of the isolated backbone curve, S1, at zero frequency, as shown in figure 8i,j. Additionally, the mixed-mode backbone curve, S2, bends rightward, shown in figure 8j,k.

    For further demonstration of the evolution of backbone curves in nonlinear parameter space, see the video, Evolution of backbone curves.avi, provided as electronic supplementary material.

    From figure 8, it can be seen that the hardening systems (the first quadrant) and softening systems (the third quadrant) share the following features:

    (i)

    a change of contribution of, and phase relationship between, two modal coordinates, from being in-phase, to single-mode, and then to anti-phase, or vice versa, when crossing Ψ1 = 0 and Ψ2 = 0;

    (i1)

    the emergence and breaking of a perfect bifurcation on S2 for a hardening system when crossing Ψ2 = 0, and on S1 for a softening system when crossing Ψ1 = 0.

    The perfect bifurcations denote critical conditions, perturbing from which leads to the onset of isolated backbone curves. These conditions are defined by relationships (3.17a,b) and thus represent the boundaries for the existence of isolated backbone curves. The following section explores additional boundaries that may exist.

    4. Additional topological boundaries

    As described in §3b, a perturbation from Ψ2 = 0 breaks the perfect bifurcation on S2, for a hardening system, and results in an isolated backbone curve, shown in figures 7b and 8a,b. Further deviation may cause the isolated backbone curve to move toward higher frequency and larger amplitude, as depicted in figure 9b,c, which are corresponding to systems on points labelled (b) and (c), respectively in figure 9(a); eventually, the isolated backbone curve may undergo swift change from finite frequency and amplitude to infinite values. One example of this change is depicted in figure 9bd. The isolated backbone curve first increases in frequency and amplitude at a limited rate, seen from figure 9b,c as α2 changes from 7.5 × 10−4 to 7.0 × 10−4. It then shifts to infinite frequency and amplitude as α2 approaches a critical value of approximately 6.76 × 10−4, shown in figure 9d. This corresponds to the vanishing (or emergence, if α2 is increased) of an isolated backbone curve, and this critical value defines another topological difference in backbone curves, i.e. systems with and without an isolated backbone curve.

    Figure 9.

    Figure 9. Two additional topological boundaries for the existence of isolated backbone curves. (a) The first quadrant of figure 8, along with two additional boundaries, shown as dash-dotted black lines, differentiating between regions with and without isolated backbone curves. Panels (b), …, (f ) are backbone curves of systems represented by black dots in Panel (a) for a system with α1 = 1 and varied α2. (Online version in colour.)

    To find the conditions defining such boundaries in nonlinear parameter space, one can trace the isolated backbone curve to seek conditions for its existence. It is observed that the isolated backbone curve vanishes when the amplitude of the minimum frequency solution becomes infinite; hence, the conditions that lead to this case are investigated here.5 Since the minimum frequency solution is related to a multiple root of amplitude for the frequency–amplitude equations (3.20), one can refer to the zero discriminant of the amplitude equation (3.20b) to trace the multiple root. The zero discriminant of the quartic equation (3.20b) is a sixth-order polynomial equation with respect to U1, and it can be written as

    DiscU2=0:f6U16+f5U15+f4U14+f3U13+f2U12+f1U1+f0=0,4.1
    where f6 is a function of nonlinear parameters, written as
    f6=g1α16+g2α15α2+g3α14α22+g4α13α23+g5α12α24+g6α1α25+g7α26,4.2
    and where coefficients g1, …, g7 are determined by the underlying linear system, some of which are given in appendix B. Note that f0, …, f5, g1, g2, g6 and g7 are not provided as they are not required for the following derivations.

    As the isolated backbone curve reaches the vanishing point, it has infinite amplitude; thus, letting U1, gives

    DiscU2f6=0.4.3
    After some algebraic manipulation, one can find coefficients g1 and g7 have factors (p − 1)2(p + 1)2, while coefficients g2 and g6 have factors (p − 1)(p + 1). Recalling that p = ± 1, defined in expression (3.18), it follows that g1 = g7 = g2 = g6 = 0, and equation (4.3) can be further simplified to give
    f6=α12α22(g3α12+g4α1α2+g5α22)=0.4.4
    Two non-zero solutions for α2 are
    α2=g4±g424g5g32g5α1.4.5
    Likewise, if frequency–amplitude relationships (3.20) are rearranged to give a quartic amplitude equation with respect to U1 rather than in U2, as currently, one can find same expression as (4.5) by following the procedure demonstrated above. This means U1 and U2 will shift to infinity simultaneously on the critical conditions described by expression (4.5). As Ω is explicitly determined by equation (3.20a), it will also shift to infinity when U1 and U2.

    Equation (4.5) represents conditions between α1 and α2 when the isolated backbone curve has infinite frequency and amplitude. This allows the first quadrant in figure 8, i.e. the hardening system, to be further divided into additional regions, as shown in figure 9a. The new regions anticlockwise of Ψ1 = 0 describe:

    (i)

    the shaded area anticlockwise of Ψ1 = 0: two primary backbone curves, S1 and S+2, with one isolated backbone curve, S2, below those two—shown in figure 9b,c;

    (ii)

    the unshaded area anticlockwise of Ψ1 = 0: two primary backbone curves, S1 and S+2, without an isolated backbone curve—depicted in figure 9d.

    The new regions clockwise of Ψ2 = 0 describe:

    (i)

    the shaded area clockwise of Ψ2 = 0: two primary backbone curves, S+1 and S2, with one isolated backbone curve, S+2, between those two—shown in figure 9e;

    (ii)

    the unshaded area clockwise of Ψ2 = 0: two primary backbone curves, S+1 and S2, without an isolated backbone curve—depicted in figure 9f .

    In summary, expressions (3.17a,b), combined with conditions (3.15), are boundaries for the existence of a perfect bifurcation for a hardening and a softening system, respectively, perturbing from which the bifurcation breaks and an isolated backbone curve emerges. Expressions (4.5) describe the other boundaries at which isolated backbone curves may vanish or emerge from infinite frequency and amplitude. The shaded area in figure 9a highlights the region in which an isolated backbone curve can exist.

    5. Conclusion

    Isolated backbone curves can be related to isolated forced responses, which can have a significant negative impact on the performance of nonlinear engineering systems. This paper has investigated the conditions for the existence of isolated backbone curves of a two-mode system with cubic nonlinearities and a 1 : 1 resonance. The concept of dynamic symmetry has been defined as the case where a system exhibits two single-mode backbone curves with one perfect bifurcation. By breaking the symmetry of a simple example system, we have found that dynamic symmetry is still obtainable when the system is asymmetric. This highlights that an asymmetric system may exhibit dynamic behaviour that is equivalent to that of a symmetric system. A specific two-mode asymmetric system, composed of a primary structure and an NLTMD, was then considered, and an analytical approach was used to demonstrate that dynamic symmetry may only be achieved when the linear parameters obey specific relationships. After optimizing the linear parameters for vibration suppression performance, we have demonstrated analytical methods that allow the nonlinear parameter space to be divided into several regions, within which backbone curves present similar topological features. The boundaries of these regions define conditions for the existence of the isolated backbone curves. We have then demonstrated how these regions can be further refined by considering whether the isolated backbone curves can exist for finite amplitudes and frequencies.

    The methodology used in this paper is based on a general two-mode model with cubic nonlinearities and a 1 : 1 internal resonance. While specific example systems have been considered, the approach used may be generalized to similar systems. This allows the existence of isolated backbone curves to be determined more rigorously when designing nonlinear systems.

    Data accessibility

    This article has no additional data.

    Authors' contributions

    D.H. led the development of the work, with supervisory support from T.L.H. and S.A.N. All authors contributed to the preparation of the manuscript.

    Competing interests

    We declare we have no competing interests.

    Funding

    S.A.N. is supported by the EPSRC (grant no. EP/R006768/1) and D.H is supported by a scholarship from the CSC.

    Acknowledgements

    We gratefully acknowledge the financial support of the EPSRC and CSC.

    Appendix A. Modal analysis for the two-mass oscillator in §2

    The equation of motion of the system shown in figure 1 is written as

    Mx¨+Kx+Nx=0,A 1
    where M and K are mass and linear stiffness matrices, respectively, Nx is a vector of nonlinear stiffness terms, and x is a vector representing physical displacements. They are written
    M=[m100m2],K=[k1+k2k2k2k2+k3]andNx=(α1x13+α2(x1x2)3α2(x2x1)3+α3x23),x=(x1x2).}A 2
    The system is translated into linear modal space by substituting x = Φq into equations (A 1), where Φ is the linear modeshape matrix, which is written
    Φ=[ϕ11ϕ12ϕ21ϕ22].A 3
    Further multiplying both sides by ΦT, the equations of motion can be obtained as
    ΦTMΦq¨+ΦTKΦq+Nq=0A 4
    where Nq = ΦTNx(Φq) and ΦTMΦ = I. As can be seen, after the linear modal transform, one can obtain the equations of motion in the same form as equations (2b) with
    Ψ1=ϕ113ϕ12α1+(ϕ11ϕ21)3(ϕ12ϕ22)α2+ϕ213ϕ22α3,Ψ2=ϕ11ϕ123α1+(ϕ11ϕ21)(ϕ12ϕ22)3α2+ϕ21ϕ223α3,Ψ3=3[ϕ112ϕ122α1+(ϕ11ϕ21)2(ϕ12ϕ22)2α2+ϕ212ϕ222α3],Ψ4=ϕ114α1+(ϕ11ϕ21)4α2+ϕ214α3andΨ5=ϕ124α1+(ϕ12ϕ22)4α2+ϕ224α3.}A 5

    To interpret the modeshape elements ϕij by physical parameters, the linear modal analysis is then carried out by finding the eigenvalues and eigenvectors of M−1K, leading to

    ωni2=[(k2+k3)/m2+(k1+k2)/m1]±[(k2+k3)/m2(k1+k2)/m1]2+4(k2/m1)(k2/m2)2,A 6
    and
    ϕ112=k22(k1+k2m1ωn12)2m2+m1k22,ϕ122=k22(k1+k2m1ωn22)2m2+m1k22A 7a
    and
    ϕ212=(k1+k2m1ωn12)2(k1+k2m1ωn12)2m2+m1k22,ϕ222=(k1+k2m1ωn22)2(k1+k2m1ωn22)2m2+m1k22.A 7b

    For the system shown in figure 5 with Ψ1 = 0 and Ψ2 = 0, it satisfies parameter conditions described in equations (3.6) and (3.7). The modeshapes in expressions (A 7) can be further simplified as

    ϕ112=12m1(1+μ+(1+μ)/μμ),ϕ122=12m1(1+μ(1+μ)/μμ)A 8a
    and
    ϕ212=((1+μ)/μ+1)22m1(1+μ+(1+μ)/μμ),ϕ222=((1+μ)/μ1)22m1(1+μ(1+μ)/μμ).A 8b
    where μ = m1/m2, which is the mass ratio.

    For the specific system, shown schematically in figure 1, with symmetric linear parameters, i.e. m1 = m2 and k1 = k3, the modeshape elements satisfy ϕ11 = ϕ12 = ϕ21 = − ϕ22, obtained from expressions (A 7). Thus, Ψ1 and Ψ2, expressed in equations (A 5), can be reduced to

    Ψ1=Ψ2=ϕ114(α1α3).A 9
    As discussed in §2, obtaining dynamic symmetry requires two single-mode solutions, i.e. satisfying Ψ1 = 0 and Ψ2 = 0. As seen from equations (A 9), Ψ1 = 0 and Ψ2 = 0 can only be satisfied when α1 = α3, which means having symmetric nonlinear parameters. In summary, to obtain dynamic symmetry for a system with linear symmetry requires nonlinear symmetry.

    Appendix B. List of coefficients

    g3=19683(ϕ112+ϕ122)4(ϕ11δ11+ϕ12δ12)2(ϕ11δ12ϕ12δ11)6,B 1
    g5=19683(δ112+δ122)4(ϕ11δ11+ϕ12δ12)2(ϕ11δ12ϕ12δ11)6B 2
    andg4=19683P(ϕ11δ12ϕ12δ11)6,B 3
    where
    P=(2ϕ1168ϕ114ϕ12223ϕ112ϕ124427ϕ126)δ116+(2ϕ1268ϕ112ϕ12423ϕ114ϕ122427ϕ116)δ126+(28ϕ115ϕ12883ϕ113ϕ12349ϕ11ϕ125)δ115δ12+(28ϕ11ϕ125883ϕ113ϕ12349ϕ115ϕ12)δ11δ125+(8ϕ116+90ϕ114ϕ1224049ϕ112ϕ12423ϕ126)δ114δ122+(8ϕ126+90ϕ112ϕ1244049ϕ114ϕ12223ϕ116)δ112δ124+(883ϕ115ϕ12+353627ϕ113ϕ123883ϕ125ϕ11)δ113δ123,B 4
    and where δ11 = ϕ11 − ϕ21 and δ12 = ϕ12 − ϕ22.

    Footnotes

    1 Note the distinction between isolated backbone curves and isolas, which exist in the forced responses.

    2 Note that the term mode is used here to refer to a mode of the underlying linear model of the system, whereas a Nonlinear Normal Mode (NNM) denotes a periodic response of the conservative nonlinear system [5].

    3 Other analytical methods, such as the second-order normal form technique [44] or the multiple-scales method [45], could alternatively be used.

    4 Note that in other projections, such as Ω against X2, the relative amplitudes of these backbone curves may differ.

    5 There may be other conditions that allow isolated backbone curves to vanish/emerge; however, this particular case is investigated here as an example of such behaviour, rather than as an exhaustive study.

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.4742093.

    Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

    References