Understanding targeted energy transfer from a symmetry breaking perspective
Abstract
Targeted energy transfer (TET) represents the phenomenon where energy in a primary system is irreversibly transferred to a nonlinear energy sink (NES). This only occurs when the initial energy in the primary system is above a critical level. There is a natural asymmetry in the system due to the desire for the NES to be much smaller than the primary structure it is protecting. This asymmetry is also essential from an energy transfer perspective. To explore how the essential asymmetry is related to TET, this work interprets the realization of TET from a symmetry breaking perspective. This is achieved by introducing a symmetrized model with respect to the generically asymmetric original system. Firstly a classic example, which consists of a linear primary system and a nonlinearizable NES, is studied. The backbone curve topology that is necessary to realize TET is explored and it is demonstrated how this topology evolves from the symmetric case. This example is then extended to a more general case, accounting for nonlinearity in the primary system and linear stiffness in the NES. Exploring the symmetry-breaking effect on the backbone curve topologies, enables the regions in the NES parameter space that lead to TET to be identified.
1. Introduction
Nonlinearity can be found in the majority of mechanical systems. It can lead to complex dynamic behaviours, such as quasi-periodic and chaotic responses, and pose challenges in understanding, modelling and simulating nonlinear systems [1–3]; however, nonlinearity can also be exploited. One field, where the nonlinearity is extensively used, is vibration suppression. For example, the nonlinear tuned mass damper (NLTMD) has been extensively considered for vibration mitigation in mechanical systems, i.e. to reduce the vibration of the system being protected (the primary system) through energy dissipation in the NLTMD [4–6]. To work efficiently, the natural frequency of an NLTMD is usually tuned with respect to a particular mode of the primary system based on resonant responses [5]. This indicates that such a device can provide optimal performance only for a narrowband frequency in the vicinity of the target mode. One way of overcoming the band-limited nature of the vibration suppression device is to employ multiple dampers targeting different modes [7].
The effective bandwidth of vibration suppression can be further extended by exploiting the nonlinearity in the form of nonlinear energy sinks (NESs). As with an NLTMD, an NES is composed of a small mass^{1} but is now attached to the primary system by a nonlinearizable spring (i.e. one lacking a linear stiffness component) [8]. Without a resonant frequency, the NES is able to interact resonantly with numerous modes of the primary system, and hence exhibits a broad frequency bandwidth performance [9–11]. In addition, one promising feature seen in applications of the NES is the targeted energy transfer (TET) phenomenon [9,11,12]—with an initial energy (above a critical level) in the primary system, the energy can be transferred in an irreversible manner from the primary system to the NES (or energy receiver), where the energy is dissipated. Owing to these advantages, a variety of NES schemes have been proposed to suppress vibrations of mechanical systems, e.g. a device with cubic nonlinearity [13,14], a nonlinear rotator [15,16], a vibro-impact oscillator [17,18], a tuned bistable NES [19] and a lever-type NES [20].
To understand the mechanism underpinning the phenomenon of TET, numerous studies have been undertaken, where the concept of nonlinear normal modes (NNMs), i.e. undamped and unforced periodic responses [21], has frequently been used to interpret the forced and damped responses [8,22,23]. These NNM-based frameworks reveal that the resonant capture, where the damped responses are captured in the neighbourhood of NNM manifolds, governs the strong energy transfer from the primary system to the NES [9,11]. In [22], the bifurcations and topologies of the NNM branches, i.e. backbone curves, were used in combination with numerical wavelet transforms to consider the time evolution of harmonic components and evaluate the energy transfer between modes. In [23], three mechanisms were identified to realize TET: fundamental, or 1 :1, resonant capture; subharmonic resonant capture; and resonant capture triggered by nonlinear beating. Later, an experimental investigation on the resonant capture was performed in [24]. Focusing on the fundamental resonant capture, the conditions of the system and external forcing that are required to exhibit efficient or optimal TET are quantified in [25,26]. To understand TET, the dynamics can also be approximated as a partition of slow and fast dynamics; as such, the phenomena of TET is captured by the evolutions of the slow-flow dynamics [23,26].
In the geometric sense, essential asymmetry, i.e. mass and potential asymmetry between the primary system and the NES, leads to localized NNMs (at the primary system and the NES, respectively), and hence brings about TET [10,27,28]. However, how the realization of TET is related to symmetry breaking from a symmetric case, to the best knowledge of the authors, has not been examined. This paper is devoted to understanding TET from a symmetry-breaking perspective. This is achieved by introducing a symmetrized model, i.e. a degenerated model to the original model that exhibits similar dynamics to a symmetric system, to study the evolutions of backbone curves due to symmetry breaking. Using this technique, the underlying mechanism to realize TET is explored, allowing the nonlinear systems that exhibit TET to be identified within the NES-parameter space. To this end, the rest of the paper is organized as follows.
In §2., an overview is given of the TET phenomenon in a classic example system that consists of a linear primary system and a nonlinearizable NES. Using backbone curves, it is demonstrated that their topology in energy localization captures the key features of TET – the critical energy level and irreversible energy transfer. In §3., along with the introduction of a symmetrized model, the backbone curve topology, underpinning the existence of TET, is studied by considering how it may be viewed as an evolution from a symmetric case. In §4., by extending the model to a more general case, accounting for the nonlinearity in the primary system and the linear stiffness in the NES, the bifurcation scenarios, determining the topologies of the backbone curves, are explored for the general case. Analytical boundaries, based on the symmetrized model, are derived to approximately distinguish the backbone curve topologies for the original model in the NES-parameter space. These divisions identify the systems with a special backbone curve topology that enables TET. Lastly, this paper is closed with conclusions in §5.
2. Targeted energy transfer
Realizing TET usually requires an NES to be attached to the primary system and for the energy to flow from the primary system to the NES, from which the energy is dissipated. In line with much of the literature, the primary system is modelled here as a mass spring oscillator that captures the targeted mode of the full system. Schematically shown in figure 1, this primary system, with mass value ${m}_{1}$, has displacement ${x}_{1}$, and is grounded by a linear spring, a linear damper and a cubic nonlinear spring with coefficients ${k}_{1}$, ${c}_{1}$ and ${\alpha}_{1}$, respectively (shown in blue in figure 1). The NES is represented as the second mass with mass value, ${m}_{2}$, where ${m}_{2}\ll {m}_{1}$, and displacement ${x}_{2}$. This device is attached to the primary system via a linear spring, a linear damper and a cubic nonlinear spring with coefficients ${k}_{2}$, ${c}_{2}$ and ${\alpha}_{2}$, respectively (shown in red in figure 1). The dynamics of the system can be described by the equations of motion
${m}_{1}$ | ${m}_{2}$ | ${k}_{1}$ | ${k}_{2}$ | ${\alpha}_{1}$ | ${\alpha}_{2}$ | ${c}_{1}$ | ${c}_{2}$ |
1 | 0.05 | 1 | 0 | 0 | 1 | 0.005 | 0.005 |
To study the phenomenon of TET, the primary system is given a non-zero initial velocity (while the system is in equilibrium and the velocity of the NES is zero), representing an energy impulse to the primary system. Figure 2a shows the time-domain responses of the two masses when the initial velocity of the primary system is ${\dot{x}}_{1}(0)=0.1$. The top panel demonstrates that the response of the primary system is much larger than that of the NES. This is also demonstrated by the instantaneous energy ratio in the NES,
— | nonlinear beating $(0\sim 50\hspace{0.17em}\text{s})$: for the first stage, the energy is transferring back and forth between the primary system and the NES; | ||||
— | resonant capture, or resonant decay $(50\sim 120\hspace{0.17em}\text{s})$: following the nonlinear beating, the energy is irreversibly transferred to the NES, until almost all energy is localized in the NES; | ||||
— | escape $(\text{after about 120, s})$: in the final stage, the majority of the remaining energy is returned back to the primary system, when the total energy has already decayed to a low level. |
For this high-initial-energy case, the energy, imparted to the primary system, is transferred irreversibly to the NES. Whether the majority of the initial energy remains in the primary system, shown in figure 2a, or is transferred to the NES, figure 2b, is determined by a critical energy level [9,23,25]. This energy level may be clearly seen in figure 3, which shows the energy ratio dissipated by the NES,
As we have seen, the realization of TET is determined by the initial energy. This phenomenon can also be observed by considering the underlying conservative dynamics, or NNMs, i.e. the undamped and unforced periodic solutions [22,23]. In figure 4a, the backbone curves, i.e. branches of NNMs, of the example system in initial velocity space, $({\dot{x}}_{2}(0),{\dot{x}}_{1}(0))$, are shown as solid curves, where one in-phase backbone curve, ${S}_{1,2}^{+}$, and one anti-phase backbone curve, ${S}_{1,2}^{-}$, can be seen. Note that, besides the fundamental branches where the NNMs exhibit 1:1 responses, tongue branches where the NNMs show $m:n$ subharmonic resonance ($m,n\in {\mathbb{Z}}^{+}$) can also be seen—the loops shown in the embedded plot of figure 4a. As will be shown in the following, the resonant capture, observed in the example system (figure 2b), is related to fundamental, or $1:1$, resonant decay following the fundamental branches. This represents the case considered in this paper, for other cases where TET is realized via subharmonic resonant capture, readers may refer to [23]. In figure 4a, these backbone curves are shown using a colour scale representing the energy in the NES, scaled by the total energy in the system for the NNM solutions passing through the equilibrium, i.e.
— | ${S}_{1,2}^{-}$: at low energy levels, the energy is localized in the primary system (i.e. blue on the colour scale), while at high energy levels, the energy is localized in the NES (red on the colour scale). | ||||
— | ${S}_{1,2}^{+}$: at low energy levels, the energy is localized in the NES (red), while at high energy levels, the energy is localized in the primary system (blue). |
In figure 4b, the corresponding energy ratio, dissipated by the NES, is compared to the backbone curves in the initial velocity space (a). It can be seen that the critical energy level is related to the fold on ${S}_{1,2}^{-}$. Besides the example system, whose backbone curves are shown as solid lines in figure 4, another case where ${\alpha}_{2}=0.5$ is also shown in figure 4 using dotted lines, where again, the critical energy level is related to the fold. It will be demonstrated in §3. that the generation of this fold may be seen as the result of bifurcation splitting due to symmetry breaking.
By projecting the time-domain responses in figure 2 to the subspace where ${x}_{1}(t)=0$ (when ${\dot{x}}_{1}>0$), the responses with an initial energy below the critical energy level (figure 2a) and above the critical energy level (figure 2b) are shown as red dots in figure 5a,b, respectively. These dots are connected via red solid lines to indicate the path of the decaying response. Combining the backbone curves with the response decays, the phenomena shown in figure 2 can be understood using backbone curves, i.e.
— | When the initial energy in the primary system is below the critical energy level, the response of the system decays down ${S}_{1,2}^{-}$ (figure 5a), whose energy is localized in the primary system at low energy levels, shown in figure 4a. As such, the majority of the energy remains in the primary system without being transferred to the NES, seen in figure 2a. | ||||
— | When the initial energy in the primary system is above the critical energy level, triggered by nonlinear beating, the resonant capture represents response decaying down the backbone curve, ${S}_{1,2}^{+}$, shown in figure 5b. Note that, the response decays following the fundamental branch, exhibiting the feature of $1:1$ resonant capture, without triggering subharmonic resonant responses. As indicated by the energy localization characteristics of ${S}_{1,2}^{+}$ in figure 4a, resonant decay down ${S}_{1,2}^{+}$ relates to change of energy from the primary system, at high energy levels, to the NES, at low energy levels. As such, the resonant decay, down ${S}_{1,2}^{+}$, exhibits irreversible energy transfer from the primary system to the NES, as observed in figures 2b. |
To assist in understanding this behaviour through the use of backbone curves, it is useful to consider the system in its modal representation as this domain is normally used in backbone curve analysis [21,32–34]. In addition, in the modal domain, the symmetry of system configurations may be identified via coefficients of the nonlinear terms [35], which assists the analysis of TET from a symmetry breaking perspective. In the modal domain, the corresponding equations of motion for the system in figure 1 can be obtained by introducing linear modal transform, i.e.
— | ${S}_{1,2}^{-}$: at low energy levels, the energy is localized in ${q}_{2}$ (the primary system), while at high energy levels, the energy is localized in ${q}_{1}$ (the NES). | ||||
— | ${S}_{1,2}^{+}$: at low energy levels, the energy is localized in ${q}_{1}$, while at high energy levels, the energy is localized in ${q}_{2}$, |
and, again, such backbone curve topology captures the phenomenon of TET in the modal domain.
In the geometric sense, mass and potential asymmetries bring about backbone curves with such energy localization properties, and hence lead to TET [10,27,28]. Nonetheless, how such backbone curve topology evolves from a symmetric case, due to symmetry breaking, has not yet been considered. In addition, a similar backbone curve topology in the modal domain is reported in a nonlinear beam system [36], which is achieved through an asymmetrically attached cross-beam. As demonstrated above, this backbone curve topology captures the characteristics of TET, namely the critical energy level and the irreversible energy transfer via fundamental resonant decay. If such a backbone curve topology is essential for the realization of TET, then symmetry breaking offers one method of examining how it is achieved. To demonstrate this, in the following sections, we study the realization of TET in the example system shown in figure 1 from a symmetry breaking perspective.
3. Relating targeted energy transfer to symmetry breaking
In this section, the example system, where ${\alpha}_{1}=0$ and ${k}_{2}=0$, is used to demonstrate the TET phenomenon from a symmetry breaking perspective. This is achieved by introducing a symmetrized model—a degenerated model of the example system. This section provides a geometric perspective to interpret TET and, in addition, offers a quantitative method to distinguish the systems that exhibit TET from general nonlinear systems, detailed in §4.
As the concept of backbone curves, considered in this paper, denote undamped and unforced periodic solution branches, the damping terms in equations (2.6) are first removed to give the equations of motion for the underlying conservative system, i.e.
To find the backbone curves, the harmonic balance method is used. As the fundamental backbone curves for the two-mass oscillator in figure 1 only exhibit synchronous responses [24,35], where the phase relationships between modal coordinates are either in-phase or anti-phase,^{3} the modal responses may be approximated as
(a) Backbone curves of the symmetrized model
With $\mu =0$, the equations describing the backbone curves of the symmetrized model are
For symmetric cases, these mixed-mode backbone curves may be seen as solution branches bifurcating from the single-mode backbone curves via branch points (BPs). At the BPs on ${S}_{1}$ (denoted by BP1), ${S}_{1,2}^{\pm}$ are degenerated to ${U}_{2}=0$; likewise, at the BPs on ${S}_{2}$ (denoted by BP2), ${S}_{1,2}^{\pm}$ are degenerated to ${U}_{1}=0$. Using these conditions, the frequency–amplitude relationships for these BPs may be obtained via equations (10)
(b) Backbone curves for an asymmetric case
For the asymmetric case, with non-zero ${\Psi}_{1}$ and ${\Psi}_{2}$, the backbone curves are described by equations (3.4). In contrast to the symmetric case, the system no longer exhibits single-mode backbone curves, instead, only mixed-mode backbone curves may be found, using frequency–amplitude relationships (11).
In figure 6b, the backbone curves for the original system are represented by solid curves, while two intermediate asymmetric cases, with $\mu =0.05$ and $\mu =0.5$, are represented by dotted and dot-dashed lines. Comparing figures 6a and 6b shows that the backbone curve topology for the original model is an evolution from that in figure 6a due to symmetry breaking—it splits the branch points, BP1 and BP2, on single-mode backbone curves, and results in one in-phase and one anti-phase backbone curve, ${S}_{1,2}^{+}$ and ${S}_{1,2}^{-}$. As demonstrated in §2., the backbone curve topology captures the key features of TET. From a symmetry breaking perspective, the generation of the TET-related backbone curve topology may be understood as:
— | generation of the critical-energy-level-related fold on ${S}_{1,2}^{-}$: the critical energy level, shown in figure 4b, is captured by the fold on ${S}_{1,2}^{-}$. The generation of this fold may be seen as the result of the branch point, BP2, splitting due to symmetry breaking, shown in figure 6; | ||||
— | generation of the irreversible-energy-transfer-related backbone curve, ${S}_{1,2}^{+}$: the fundamental resonant decay follows the in-phase backbone curve, ${S}_{1,2}^{+}$, and exhibits energy localization in ${q}_{2}$ at high energy levels, and localization in ${q}_{1}$ at low energy levels. This transition of energy localization may be seen as an evolution from the symmetrized case through symmetry breaking, leading to BP1 and BP2 splitting. |
By introducing a symmetrized model through parameter $\mu $, the phenomenon of TET is interpreted from a symmetry breaking perspective. This provides a mechanism to understand how the classic example system, which exhibits TET, may evolve from the symmetric case. By understanding how the simpler, symmetric case is related to the asymmetric case exhibiting TET, the fundamental properties leading to TET may be studied in greater detail and, therefore, it can provide a method to differentiate a system that exhibits TET from others. In the following section, the example system is extended to more general cases, among which the systems that exhibit TET are identified.
4. Identifying systems that exhibit TET
To identify systems that exhibit TET, the parameter restrictions, i.e. ${\alpha}_{1}=0$ and ${k}_{2}=0$, on the example model are first removed to account for a more general case of systems in figure 1. This generically asymmetric case is explored by considering its evolution from the symmetrized case, via symmetry breaking. Using this technique, the special backbone curve topology, observed in the example system (figure 6), can be identified, which allows the mechanism, leading to TET, to be distinguished from general cases.
Considering non-zero ${k}_{2}$ and ${\alpha}_{1}$, the modeshape matrix, $\mathit{\Phi}$, is no longer anti-diagonal,^{4} and the coefficients of nonlinear terms, ${\Psi}_{i}$, are given by expressions (4). Compared with the example system considered in previous sections, where the symmetrized model exhibits only one backbone curve topology, shown in figure 6a, such a general case exhibits more complex scenarios. To explore these, the existence of BPs is again considered for the symmetrized model, where $\mu =0$, using conditions (3.7). Combining expressions (3.7), (4) and (5a), two critical boundaries, defining the existence of the BPs, can be obtained
A non-zero $\mu $ is then introduced to the symmetrized model to study the effect of symmetry breaking. The backbone curves for an asymmetric case may be computed via equations (11). With $\mu =0.05$, figure 7b presents the backbone curves for asymmetric cases corresponding to the symmetric examples in figure 7a. Note that, for asymmetric cases, the axes are used to approximately demonstrate the evolution of backbone curve topologies, rather than the exact boundaries. Panel $({b}_{1})$ shows the same topology as that for the example system in figure 6b. For those in panels $({b}_{2})$ and $({b}_{3})$, the split of branch points results in two primary backbone curves (passing through the origin) and two isolated backbone curves. Note that, these isolated backbone curves may vanish with infinite amplitudes when special parameter conditions are satisfied, see ref. [35]. In panel $({b}_{4})$, the two single-mode backbone curves, ${S}_{1}$ and ${S}_{2}$, in panel $({a}_{4})$, evolve to mixed-mode backbone curves, ${S}_{1,2}^{-}$.
For these asymmetric cases in figure 7b, aside from the backbone curves in panel $({b}_{1})$, which show the same topology as those seen in the example system considered in previous sections, other backbone curve topologies, in panels $({b}_{2})\sim ({b}_{4})$, exhibit fundamentally different features in energy localization. With an initial energy in ${q}_{2}$ (i.e. the primary system), systems with backbone curves shown in panels $({b}_{2})\sim ({b}_{4})$ do not show the necessary energy localization features in backbone curves to exhibit TET, and hence energy remains in ${q}_{2}$ without being transferred to ${q}_{1}$ (the NES). From this symmetry breaking perspective, to identify the system that exhibits TET from general systems is to distinguish the backbone curve topology shown in panel $({b}_{1})$ from all cases in figure 7b. In addition, as these asymmetric cases are evolved from the corresponding symmetric cases, it can be understood by distinguishing the symmetric topology in panel $({a}_{1})$ from those in figures 7a.
To demonstrate this, it is useful to project the boundaries, ${f}_{1}$ and ${f}_{2}$, defined by expressions (4.1), to the subspace, $({k}_{2},{\alpha}_{2})$, i.e. considering the NES-parameter projection of figure 7. Therefore, it allows one to consider the parameter conditions on the NES in order to exhibit TET. Here, a primary system with ${m}_{1}=1$, ${k}_{1}=1$ and ${\alpha}_{1}=1$ is considered, and the NES has a mass ${m}_{2}=0.05$. Using expressions (4.1), the NES-parameter space $({k}_{2},{\alpha}_{2})$ may be divided into several major regions,^{5} labelled by (a), $({b}_{1})$, $({b}_{2})$, (c) and (d), see figure 8, where these regions are shaded using the same colour schema as that in figure 7. In this figure, the purple and brown solid lines denote ${f}_{1}$ and ${f}_{2}$ (projections of the axes in figure 7), respectively; the thin purple and brown dotted lines represent asymptotic lines of ${f}_{1}$ and ${f}_{2}$ respectively, which are given by
region | symmetric backbone curve topology | exhibits TET |
---|---|---|
(a) | with both BP1 and BP2, schematically shown in figure 7${a}_{1}$ | ✓ |
$({b}_{1})$ and $({b}_{2})$ | with BP1 and without BP2, schematically shown in figure 7${a}_{2}$ | ✗ |
(c) | with BP2 and without BP1, schematically shown in figure 7${a}_{3}$ | ✗ |
(d) | without both BP1 and BP2, schematically shown in figure 7${a}_{4}$ | ✗ |
— | region (a): an example system with ${k}_{2}=0.02$ and ${\alpha}_{2}=0.02$ is first considered. The backbone curves of this system are shown in figure 9${a}_{1}$. The energy ratio dissipated by the NES, with respect to the initial velocity in the primary system, is shown in figure 9${a}_{2}$, which is similar to that in figure 4b and a critical energy level may be seen. As ${\alpha}_{2}$ increased to ${\alpha}_{2}=1$, this critical energy level is decreased from ${\dot{x}}_{1}(0)\approx 0.5$ to $0.06$, see figure 9${a}_{4}$. This can be explained by the symmetrized backbone curves where the amplitude of BP2 (associated with the critical energy level) is decreased with an increasing ${\alpha}_{2}$. The instantaneous energy carried by the NES with ${\dot{x}}_{1}(0)$ below the critical energy level (panel (i)) and above the critical energy level (panel (ii)), again, show the same features as those seen in the example system (figure 2). Capturing the key features of TET, this region, shaded in green in figure 8, represents the parameter conditions to exhibits TET. | ||||
— | region $({b}_{1})$: with ${k}_{2}=0.02$ and ${\alpha}_{2}$ decreased to $0.005$, the system crosses ${f}_{2}$ and moves from region (a) to region $({b}_{1})$. The corresponding backbone curves are shown in figure 9${b}_{1}$, whose topology is fundamentally different to that in region (a). Besides two primary backbone curves, one isolated backbone curve may be observed. Such a backbone curve topology, indeed, can be seen as evolved from that shown in figure 7${a}_{2}$. When applied the primary system with an initial velocity, the energy dissipated by the NES is presented in figure 9${b}_{2}$, where no critical energy level may be seen. In addition, the instantaneous energy in the NES (panels (i) and (ii)) indicates that very limited amount of energy can be transferred to the NES as initial energy level varies. | ||||
— | region (c): the point ${\alpha}_{2}=1$ and ${k}_{2}=0.05$ lies in region (c) and is labelled by a cross in figure 8. Its backbone curves are shown in figure 9${c}_{1}$, which are asymmetric evolutions of those shown in figure 7${a}_{3}$. Note that the isolated backbone curves, in this case, are vanished with infinite frequency and amplitude [35]. As the system moves from region (a) to region (c), the critical energy level is less clear, depicted in figure 9${c}_{2}$. For systems in this region, more energy may be dissipated by the NES for a low ${\dot{x}}_{1}(0)$ than a high value, and a strong oscillation of energy between the primary system and the NES may be seen, see panel (i) for ${\dot{x}}_{1}(0)=0.02$. While, as ${\dot{x}}_{1}(0)$ increased to $0.2$, the instantaneous energy in the NES (panel $(ii)$) indicates a trend of energy transferring from the primary system to the NES, but it is less efficient than the example system in region (a). Even though the system in this region exhibits some similarity to that in region (a), key features of TET are not present, e.g. a critical energy level and the energy localization in the primary system for low initial velocity cases. | ||||
— | region (d): further increasing the linear stiffness of the NES to ${k}_{2}=0.07$, the system is in region (d). The backbone curves of the system are shown in figure 9${d}_{1}$, while the energy ratio dissipated by the NES is presented in panel $({d}_{2})$. In this region, like those in region $({b}_{1})$ and (c), no critical energy level can be seen. Shown in panels (i) and (ii) are the instantaneous energy carried by the NES over time for a low and a high initial energy cases, respectively. It is observed again that limited amount of energy can be transferred to the NES, and less is transferred if ${k}_{2}$ is further increased. |
Comparing the examples in region (a) and $({b}_{1})$, the existence of region $({b}_{1})$ characterizes the minimum degree of nonlinearity necessary for the system to exhibit TET, shown as boundary ${f}_{2}$ and described by expression (4.1c). While comparing the examples in region (a), (c) and (d), region (c) may be seen as a transition region which connects the regions whether TET can be seen (region (a)) or not (region (d)). It also demonstrates that a small linear stiffness in the NES is another requirement, quantified by the asymptotic line of ${f}_{1}$, i.e. ${k}_{2,\text{crit}1}$, and described by expression (4.2a). Even though these boundaries are obtained based on the symmetrized model, they classify the behaviours of the system and allow one to identify the regions where system may exhibit TET.
5. Conclusion
TET has been extensively considered in vibration suppression to realize irreversible energy transfer from the primary system to the NES when the energy in the primary system is above a critical level. The realization of TET is related to an essentially asymmetric configuration, leading to localized NNM branches, or backbone curves. To further explore the significance and necessity of asymmetry, this paper interprets the realization of TET from a symmetry breaking perspective. To this end, a symmetrized model is introduced to study the asymmetric evolutions of backbone curves for the generically asymmetric original system. A classic example system—a linear primary system with a nonlinearizable NES, is first considered. It is found that the strongly localized backbone curves may be seen as an asymmetric evolution from a symmetric case where two branch points can be seen. It is the symmetry breaking that splits these two branch points and leads to a backbone curve topology capturing the key features of TET—a critical energy level and the irreversible energy transfer. This example system is then extended to more general cases, from which the systems that exhibit TET are identified by distinguishing the characterized backbone curves, observed in the example system, from others. Based on the symmetrized model, this technique is applied to quantitatively identify the NES parameter conditions required in order to exhibit TET.
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 an EPSRC Programme 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. Backbone curves for the example model
The equations of motion for the underlying conservative system shown in figure 1 are
To find the backbone curves, the harmonic balance technique is used. Considering the case where synchronous NNMs exist, the phase relationships between modal coordinates are either in-phase or anti-phase, and hence the modal responses may be assumed
For symmetric case, ${\Psi}_{1}={\Psi}_{2}=0$, the expressions for backbone curves may be reduced to
While for asymmetric case, with non-zero ${\Psi}_{1}$ and ${\Psi}_{2}$, single-mode backbone curves are no longer exist, and only mixed-mode backbone curves may be found. The amplitude–frequency relationship for these mixed-mode backbone curves may be obtained by rearranging equations (7a) and (7b), which are given by
Appendix B. Boundaries of the conservative parameter space
Considering the existence of branch points on backbone curves for the symmetric system, there are four topological cases, namely (a) with both BP1 and BP2; (b) with BP1 and without BP2; (c) without BP1 and with BP2; and (d) without both BP1 and BP2. Combining expressions (3.7), (4) and (5a), two critical boundaries, defining the existence of BPs, can be obtained
Footnotes
2 Note that, this definition of symmetry is equivalent to having ${Z}_{2}\u2a01{Z}_{2}$ symmetry as discussions in [37,38].
3 For a general case, besides these synchronous NNMs, asynchronous NNMs, where the phase relationship may be assumed any value, can also be seen on fundamental backbone curves [39].
4 In this case, the physical displacement of the primary system, ${x}_{1}$, is no longer exactly represented by the second modal coordinate, ${q}_{2}$; likewise, the displacement of the NES, ${x}_{2}$, is no longer exactly captured by ${q}_{1}$. Nonetheless, an accurate approximation to this is still achieved if ${k}_{2}$ is sufficiently small. In addition, it will be demonstrated in the following that a sufficiently small ${k}_{2}$ is necessary in order to exhibit TET. Thus, the following discussions continue to consider the backbone curve topology in the modal domain.
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