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 [1–6]. 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 [7–13]. Among these applications, the nonlinear tuned mass damper (NLTMD) has been extensively studied in the literature with its advantageous performance demonstrated [14–17].
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 [16,21–23]. 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 [32–34]. 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-mode^{2} conservative system, with cubic nonlinearities, is firstly considered, before considering a specific two-mode system. The Lagrangian of this general system may be written
From equation (2.2b), when the coefficient of q^{3}_{1} equals 0 (i.e. when Ψ_{1} = 0), q_{2} = 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, q_{1}, by solving
(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 m_{1} and m_{2}, and displacements x_{1} and x_{2}, respectively. These masses are grounded via two linear springs, with coefficients k_{1} and k_{3}, respectively, and are connected by another linear spring with coefficient k_{2}. This system also contains three nonlinear cubic springs with coefficients α_{1}, α_{2} and α_{3}, 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, Ψ_{i}, and the physical parameters of the model, as derived in appendix A.
Here, the symmetry of the system is divided into two parts: linear symmetry (LS), where both m_{1} = m_{2} and k_{1} = k_{3}; and nonlinear symmetry (NS), where α_{1} = α_{3}. Note that the LS–NS case leads to Ψ_{1} = 0 and Ψ_{2} = 0, as shown in appendix A—see equation (A 9) with α_{1} = α_{3}. 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 S_{1} and S_{2} consist of only the first and second modal coordinates, respectively; while S^{+}_{2} and S^{−}_{2} represent mixed-mode backbone curves containing both linear modal coordinates. The subscripts of S^{+}_{2} and S^{−}_{2} indicate the backbone curve from which they bifurcate (i.e. from S_{2} 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].
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, S_{1} and S_{2}. | ||||
(ii) | two mixed-mode backbone curves; either S^{±}_{2}, emerging from a perfect bifurcation on S_{2}, or S^{±}_{1}, emerging from a perfect bifurcation on S_{1}. |
(b) Breaking either the nonlinear or the linear parameter symmetry
Figure 2a represents the nonlinear parameter space of α_{1} against α_{3}—i.e. the parameters of the two nonlinear grounding springs—for the case where the system has LS. The nonlinear parameters leading to Ψ_{1} = 0 and Ψ_{2} = 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 Ψ_{1} = Ψ_{2} = 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. α_{1}≠α_{3}, both Ψ_{1}≠0 and Ψ_{2}≠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 Ψ_{1}≠0 and Ψ_{2}≠0.
Similarly, breaking the LS, while retaining the NS, can also break the dynamic symmetry. With the breaking of the LS (m_{2} = 0.8m_{1} and k_{3} = 0.7k_{1}), the orientations of Ψ_{1} = 0 and Ψ_{2} = 0 are changed, and are no longer overlapping, as shown in figure 3a. If the NS is retained, i.e. α_{1} = α_{3} (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 Ψ_{1} = 0 and Ψ_{2} = 0, where one can find two single-mode backbone curves, is not on the line representing α_{1} = α_{3} (i.e. the point at which the green and purple lines in figure 3a cross does not correspond to the grey line).
(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 α_{3} is reduced from the NS-case to the point where Ψ_{2} = 0, marked as a purple dot labelled (b) in figure 4a. As expected, this leads to a single-mode backbone curve S_{2}; however, as Ψ_{1}≠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 S^{−}_{2} which bifurcate off S_{2}. Further reducing α_{3} leads to the point where Ψ_{1} = 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 S_{1} (as predicted by the Ψ_{1} = 0 condition) but with a primary and an isolated backbone curve, S^{+}_{2} and S^{−}_{2}. As shown in figure 4a, Ψ_{1} = Ψ_{2} = 0 may still be satisfied for this case if α_{1} and α_{3} are on the intersection of Ψ_{1} = 0 and Ψ_{2} = 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.
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 Ψ_{1} = 0 and Ψ_{2} = 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.
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, S_{1} and S_{2}, which requires that both Ψ_{1} = 0 and Ψ_{2} = 0 in the equations of motion (2b). The expressions of Ψ_{1} and Ψ_{2} for the NLTMD system are given in equations (A 5), and may be written in matrix form as
Figure 6a shows the nonlinear parameter space α_{1} against α_{2}, for the case where k_{1}/k_{2} = 21, and where the mass values satisfy equation (3.6). The overlapping green and purple lines represent the parameter relationships that lead to Ψ_{1} = 0 and Ψ_{2} = 0, respectively (in this instance the case α_{1}/α_{2}=441, satisfies equation (3.7) and hence Ψ_{1} = Ψ_{2} = 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 Ψ_{1}, …, Ψ_{5} (A 5) can be simplified as
The in-phase solutions, corresponding to θ_{d} = 0 are denoted S^{+}_{1} and S^{+}_{2}, while the anti-phase solutions, corresponding to θ_{d} = π are denoted S^{−}_{1} and S^{−}_{2} (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
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 S_{2}, which are denoted S^{±}_{2}, or on S_{1}, which are denoted S^{±}_{1}. For the perfect bifurcation point on S_{2}, the amplitude of the first modal coordinate U_{1} = 0; likewise, the second modal amplitude U_{2} = 0 for the perfect bifurcation point on S_{1}. Using these conditions, the amplitude and frequency of these two bifurcation points can be obtained, from equations (3.13), as
(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
Systems with nonlinear parameters that do not lie on Ψ_{1} = 0 or Ψ_{2} = 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(θ_{d}) = 0, leading to the phase relationship θ_{d} = θ_{1} − θ_{2} = nπ, where even and odd n values denote in-phase and anti-phase modal relationships, respectively. Further defining the phase parameter p as
Figure 8 presents the mixed-mode backbone curves of systems with linear parameters m_{1} = 1, m_{2} = 0.05, k_{1} = 1, k_{2} = k^{opt}_{2} ≈ 0.0454 in the nonlinear parameter space α_{1} against α_{2}. Backbone curves on Ψ_{1} = 0 and Ψ_{2} = 0, labelled in this figure, are the same as those in figures 7c,b, respectively. This space is divided by the parameter axes α_{1} = 0 and α_{2} = 0 into the following four classes of system:
(i) | a hardening system (the first quadrant) Perturbing the nonlinear parameters clockwise from Ψ_{2} = 0 breaks the perfect bifurcation on S_{2}, shown in figure 7b. The breaking of this bifurcation generates one isolated backbone curve, S^{+}_{2}, between two primary backbone curves, S^{−}_{2} and S^{+}_{1}, ^{4} seen in figure 8a. Likewise, if nonlinear parameters are perturbed anticlockwise from Ψ_{2} = 0, the perfect bifurcation on S_{2} breaks in a different direction, resulting in one isolated backbone curve, S^{−}_{2}, below two primary backbone curves, depicted in figure 8b. Further varying the nonlinear parameters in the anticlockwise direction towards Ψ_{1} = 0, the contribution of the second modal coordinate, U_{2}, to the mixed-mode, in-phase backbone curve, S^{+}_{1}, gradually decreases to zero. This results in a single-mode backbone curve S_{1}, seen from the evolution of backbone curves from figure 8b to 7c. Finally, perturbing anticlockwise from Ψ_{1} = 0, the U_{2} component of S_{1} increases, leading to a mixed-mode, anti-phase backbone curve S^{−}_{1}, as shown in figure 8c. | ||||
(ii) | a hardening structure with a softening attachment (the second quadrant) Further decreasing α_{2} until α_{2} < 0, causes S^{−}_{1} 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 S^{−}_{1}. Continuing anticlockwise, from above Ψ_{2} = 0 to below it, leads to a similar behaviour to the hardening system (the first quadrant) as it crosses Ψ_{1} = 0. The contribution from U_{1} to the mixed-mode, in-phase backbone curve S^{+}_{2} may gradually decrease, reaching zero at Ψ_{2} = 0, leading to a single-mode backbone curve S_{2}. The contribution then increases, resulting in a mixed-mode, anti-phase backbone curve S^{−}_{2}. 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 S^{−}_{1}. Further varying the nonlinear parameters towards Ψ_{1} = 0, the isolated and primary backbone curves, S^{+}_{1} and S^{−}_{1}, merge into a single-mode backbone curve, S_{1}, with a perfect bifurcation. Anticlockwise of Ψ_{1} = 0, the perfect bifurcation breaks and generates an isolated backbone curve, S^{−}_{1}, below two primary backbone curves, S^{+}_{1} and S^{−}_{2}, 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, S^{−}_{1}, at zero frequency, as shown in figure 8i,j. Additionally, the mixed-mode backbone curve, S^{−}_{2}, 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 Ψ_{1} = 0 and Ψ_{2} = 0; | ||||
(i1) | the emergence and breaking of a perfect bifurcation on S_{2} for a hardening system when crossing Ψ_{2} = 0, and on S_{1} for a softening system when crossing Ψ_{1} = 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 Ψ_{2} = 0 breaks the perfect bifurcation on S_{2}, 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 9b–d. The isolated backbone curve first increases in frequency and amplitude at a limited rate, seen from figure 9b,c as α_{2} changes from 7.5 × 10^{−4} to 7.0 × 10^{−4}. It then shifts to infinite frequency and amplitude as α_{2} approaches a critical value of approximately 6.76 × 10^{−4}, shown in figure 9d. This corresponds to the vanishing (or emergence, if α_{2} 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.
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 U_{1}, and it can be written as
As the isolated backbone curve reaches the vanishing point, it has infinite amplitude; thus, letting ${U}_{1}\to \mathrm{\infty}$, gives
Equation (4.5) represents conditions between α_{1} and α_{2} 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 Ψ_{1} = 0 describe:
(i) | the shaded area anticlockwise of Ψ_{1} = 0: two primary backbone curves, S^{−}_{1} and S^{+}_{2}, with one isolated backbone curve, S^{−}_{2}, below those two—shown in figure 9b,c; | ||||
(ii) | the unshaded area anticlockwise of Ψ_{1} = 0: two primary backbone curves, S^{−}_{1} and S^{+}_{2}, without an isolated backbone curve—depicted in figure 9d. |
The new regions clockwise of Ψ_{2} = 0 describe:
(i) | the shaded area clockwise of Ψ_{2} = 0: two primary backbone curves, S^{+}_{1} and S^{−}_{2}, with one isolated backbone curve, S^{+}_{2}, between those two—shown in figure 9e; | ||||
(ii) | the unshaded area clockwise of Ψ_{2} = 0: two primary backbone curves, S^{+}_{1} and S^{−}_{2}, 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
To interpret the modeshape elements ϕ_{ij} by physical parameters, the linear modal analysis is then carried out by finding the eigenvalues and eigenvectors of M^{−1}K, leading to
For the system shown in figure 5 with Ψ_{1} = 0 and Ψ_{2} = 0, it satisfies parameter conditions described in equations (3.6) and (3.7). The modeshapes in expressions (A 7) can be further simplified as
For the specific system, shown schematically in figure 1, with symmetric linear parameters, i.e. m_{1} = m_{2} and k_{1} = k_{3}, the modeshape elements satisfy ϕ_{11} = ϕ_{12} = ϕ_{21} = − ϕ_{22}, obtained from expressions (A 7). Thus, Ψ_{1} and Ψ_{2}, expressed in equations (A 5), can be reduced to
Appendix B. List of coefficients
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.
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