Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
You have accessResearch articles

Understanding targeted energy transfer from a symmetry breaking perspective

,
Thomas L. Hill

Thomas L. Hill

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

Google Scholar

Find this author on PubMed

and
Simon A. Neild

Simon A. Neild

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

Google Scholar

Find this author on PubMed

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 [13]; 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 [46]. 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 mass1 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 [911]. 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 m1, has displacement x1, and is grounded by a linear spring, a linear damper and a cubic nonlinear spring with coefficients k1, c1 and α1, respectively (shown in blue in figure 1). The NES is represented as the second mass with mass value, m2, where m2m1, and displacement x2. This device is attached to the primary system via a linear spring, a linear damper and a cubic nonlinear spring with coefficients k2, c2 and α2, respectively (shown in red in figure 1). The dynamics of the system can be described by the equations of motion

m1x¨1+c1x˙1+c2(x˙1x˙2)+k1x1+k2(x1x2)+α1x13+α2(x1x2) 3=0 2.1a
and
m2x¨2+c2(x˙2x˙1)+k2(x2x1)+α2(x2x1)3=0, 2.1b
where ˙ and ¨ represent the first and second derivatives with respect to time. Many studies of TET consider a linear primary system, with α1=0, and an NES with k2=0 [9,11,23]. This classic case is first considered here and, in later sections, will be extended to more general cases, accounting for a primary system with nonlinearity, α10, and an NES with a linear stiffness, k20. This reflects more practical application scenarios, e.g. a nonlinear beam with an NES [29], a drill-string system with an NES [30], as well as in experimental set-ups [24,31]. Here, the example system has parameters given in table 1.
Figure 1.

Figure 1. A schematic diagram ofa primary system, shown in blue, with a nonlinear energy sink (NES), shown in red. The primary system, with mass value m1, has displacement x1 and is grounded by a linear spring, a cubic nonlinear spring and a damper, with coefficients k1, α1 and c1, respectively. The NES, with mass value m2, has displacement x2 and is attached to the primary system by a linear spring, a cubic nonlinear spring and a damper, with coefficients k2, α2 and c2, respectively. (Online version in colour.)

Table 1. Parameters of the example system, schematically shown in figure 1.

m1 m2 k1 k2 α1 α2 c1 c2
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 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,

Eins, NES=ENESEP+ENES=(1/2)m2x˙22+(1/2)k2(x2x1) 2+(1/4)α2(x2x1) 4(1/2)m1x˙12+(1/2)k1x12+(1/4)α1x14+(1/2)m2x˙22+(1/2)k2(x2x1) 2+(1/4)α2(x2x1) 4, 2.2
which is shown in the bottom panel of figure 2a, where only a very limited amount of energy may be transferred to the NES over time. If the initial velocity is increased from x˙1(0)=0.1 to x˙1(0)=0.2, the corresponding response histories of the two masses and the instantaneous energy ratio carried by the NES are shown in figure 2b. Significant differences may be seen when compared with the low-initial-energy case, shown in figure 2a. For this high-initial-energy case, the majority of the response is carried by the NES, and a more complex phenomenon can be seen in the bottom panel of figure 2b, which may be divided into the following stages [23]:

nonlinear beating (050s): for the first stage, the energy is transferring back and forth between the primary system and the NES;

resonant capture, or resonant decay (50120s): following the nonlinear beating, the energy is irreversibly transferred to the NES, until almost all energy is localized in the NES;

escape (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.

Figure 2.

Figure 2. Responses and energy transfer in the time domain for the example system, represented in figure 1, with parameters listed in table 1. (a) Response histories for the primary system and the NES (top panel), and instantaneous energy carried by the NES (bottom panel) with an initial velocity of x˙1(0)=0.1, in the primary system. (b) Response histories for the primary system and the NES (top panel), and instantaneous energy carried by the NES (bottom panel) with an initial velocity of x˙1(0)=0.2, in the primary system. (Online version in colour.)

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,

Ed,NES=c20T(x˙2x˙1)2dtc20T(x˙2x˙1) 2dt+c10Tx˙12dt|T, 2.3
with respect to a varied initial velocity, x˙1(0). Note that, to compute Ed,NES, T is approximated to a finite value such that the total energy in the system is decayed to less than 1% of the initial value. The critical energy level is related to the steep transition seen around x˙1(0)=0.11, and the low-initial-energy and high-initial-energy cases, studied above, are located on either side of this critical energy level. Generally, when the initial energy in the primary system is below the critical level, the majority of the energy remains in the primary system with little being transferred to the NES; by contrast, when the initial energy in the primary system is above the critical level, the majority of energy may be irreversibly transferred to the NES. The above-critical-energy case is, indeed, related to TET, which may be defined as irreversible energy transfer from the energy-imparted component to another component if the provided initial energy level is above a critical value [9,11].
Figure 3.

Figure 3. Energy ratio dissipated by the NES with respect to a varied initial velocity in the primary system for the example system, represented in figure 1, with parameters listed in table 1. (Online version in colour.)

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, (x˙2(0),x˙1(0)), are shown as solid curves, where one in-phase backbone curve, S1,2+, and one anti-phase backbone curve, S1,2, 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,nZ+) 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.

Einitial,NES=[1/2m2x˙221/2m1x˙12+1/2m2x˙22]t=0. 2.4
In this initial velocity space, (x˙2(0),x˙1(0)), the fundamental backbone curve topology, for the example system, can be characterized, considering the energy localization:

S1,2: 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).

S1,2+: 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).

Figure 4.

Figure 4. Critical energy level, in the targeted energy transfer, captured by the backbone curves of the example system, shown in figure 1, where all parameters aside from α2 are given in table 1, and where α2=1 and α2=0.5 for the solid and dotted lines, respectively. (a) Backbone curves in the initial velocity space (x˙2(0),x˙1(0)). The colour scale, representing the backbone curves, shows the energy in the NES, scaled by total energy in the system. (b) Energy ratio dissipated by the NES with a varied initial energy, indicated by an initial velocity, x˙1(0), in the primary system. (Online version in colour.)

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 S1,2. Besides the example system, whose backbone curves are shown as solid lines in figure 4, another case where α2=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 x1(t)=0 (when 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 S1,2 (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, S1,2+, 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 S1,2+ in figure 4a, resonant decay down S1,2+ 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 S1,2+, exhibits irreversible energy transfer from the primary system to the NES, as observed in figures 2b.

Figure 5.

Figure 5. Resonant decay captured by the backbone curves for the example system, shownin figure 1, where all parameters are given in table 1. The backbone curves of the system are shown as blue solid lines; the initial conditions with non-zero velocity in the primary system are marked by ‘×’; and resonant decays are represented by extreme values, where x1=0 (when x˙1>0), as red dots, and connected by thin-red lines to indicate the decaying path. (a) Resonant decay of the system when an initial velocity, x˙1(0)=0.1, is in the primary system, where the time histories are shown in figure 2a. (b) Resonant decay of the system when an initial velocity, x˙1(0)=0.2, is in the primary system, where the time histories are shown in figure 2b. (Online version in colour.)

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 [21,3234]. 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.

x=Φq=[x1x2]=[ϕ11ϕ12ϕ21ϕ22][q1q2], 2.5
where q1 and q2 denote the first and second modal coordinates respectively, and ϕij are the modeshape coefficients. As such, equations (2.1) can be transformed into modal equations of motion
q¨1+d1q˙1+d12q˙2+ωn12q1+Ψ4q13+3Ψ1q12q2+Ψ3q1q22+Ψ2q23=0 2.6a
and
q¨2+d12q˙1+d2q˙2+ωn22q2+Ψ1q13+Ψ3q12q2+3Ψ2q1q22+Ψ5q23=0, 2.6b
where ωn1 and ωn2 are the first and second linear natural frequencies, d1,d2 and d12 denote modal damping coefficients, and Ψi represent coefficients of the nonlinear terms given by expressions (4) in appendix A. For the example system with parameters given in table 1, where k2=0, the modeshape matrix is an anti-diagonal matrix, i.e. ϕ11=0 and ϕ22=0. As such, x1=ϕ12q2 and x2=ϕ21q1, which means the responses in the primary system and the NES, x1 and x2, are exactly captured by modal coordinates, q2 and q1, respectively. This indicates that, in initial modal velocity space (q˙1(0),q˙2(0)), the backbone curves are topologically equivalent to those shown in figure 4a, and may be described

S1,2: at low energy levels, the energy is localized in q2 (the primary system), while at high energy levels, the energy is localized in q1 (the NES).

S1,2+: at low energy levels, the energy is localized in q1, while at high energy levels, the energy is localized in q2,

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 α1=0 and k2=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.

q¨1+ωn12q1+Ψ4q13+3μΨ1q12q2+Ψ3q1q22+μΨ2q23=0 3.1a
and
q¨2+ωn22q2+μΨ1q13+Ψ3q12q2+3μΨ2q1q22+Ψ5q23=0, 3.1b
where coefficients of nonlinear terms are given by
Ψ1=ϕ213ϕ12α2,Ψ2=ϕ21ϕ123α2,Ψ3=3ϕ212ϕ122α2,Ψ4=ϕ214α2,Ψ5=ϕ124α2, 3.2
and where Ψ1 and Ψ2 are multiplied by μ, i.e. a ’symmetry breaking’ parameter with 0μ1. While introducing parameter μ is artificial, and cannot be realized in the original system, it allows the symmetry to be continued directly in a single parameter. As shown in [35], a system with dynamically symmetric behaviour has Ψ1=Ψ2=0;2 Instead, a dynamically asymmetric system possesses non-zero Ψ1 and Ψ2. No combination of parameters in the example system can lead to dynamic symmetry, i.e. Ψ1=Ψ2=0 (see expressions (3.2)); therefore, to relate this model to symmetry breaking, a corresponding symmetrized model is introduced by enforcing μ=0 while the other modal parameters remain unchanged. The case where μ=1, is termed the original model. As such, the original model may be seen as an evolution from the symmetrized model due to symmetry breaking, (i.e. where Ψ1 and Ψ2 become non-zero).

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

qiui=Uicos(ωrit), 3.3
where ui denotes the fundamental component of qi, and where Ui and ωri represent the amplitude and response frequency of ui, respectively. In addition, as this paper considers the realization of TET via fundamental resonant capture, the response frequencies of the two modal coordinates are assumed to be equal, i.e. ωr1=ωr2=Ω. With modal responses in expression (3.3) substituted into the equations of motion (3.1), and the non-resonant terms removed, the expressions for computing backbone curves may be obtained after some algebraic manipulations, given by
4(ωn12Ω2)U1+3Ψ4U13+3Ψ3U1U22+3μp(Ψ2U23+3Ψ1U12U2)=0 3.4a
and
4(ωn22Ω2)U2+3Ψ5U23+3Ψ3U12U2+3μp(Ψ1U13+3Ψ2U1U22)=0, 3.4b
where p=+1 and 1, denote in-phase and anti-phase relationships between two modal coordinates, respectively.

(a) Backbone curves of the symmetrized model

With μ=0, the equations describing the backbone curves of the symmetrized model are

[4(ωn12Ω2)+3Ψ4U12+3Ψ3U22]U1=0 3.5a
and
[4(ωn22Ω2)+3Ψ5U22+3Ψ3U12]U2=0. 3.5b
For this case, two non-trivial solution sets may be found with respect to either
(a)S1:U10 and U2=0, and S2:U1=0 and U20;or (b)S1,2±:U10 and U20,
where S1 and S2 denote single-mode backbone curves, whereas S1,2+ and S1,2 represent in-phase and anti-phase backbone curves, respectively (also known as mixed-mode backbone curves). Analytical frequency–amplitude expressions for these backbone curves are given by equations (9) and (10) in appendix A.

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 S1 (denoted by BP1), S1,2± are degenerated to U2=0; likewise, at the BPs on S2 (denoted by BP2), S1,2± are degenerated to U1=0. Using these conditions, the frequency–amplitude relationships for these BPs may be obtained via equations (10)

BP1:U12=4(ωn22ωn12)3(Ψ4Ψ3),Ω2=Ψ4ωn22Ψ3ωn12Ψ4Ψ3, 3.6a
BP2:U22=4(ωn22ωn12)3(Ψ3Ψ5),Ω2=Ψ3ωn22Ψ5ωn12Ψ3Ψ5. 3.6b
Note that, these branch points can exist when the relationships are associated with real positive amplitude and frequency, i.e.
existence of BP1:Ψ4Ψ3>0, 3.7a
existence of BP2:Ψ3Ψ5>0. 3.7b
Using conditions (3.7), the bifurcation scenarios and the backbone curve topologies of the example system can be considered by substituting modeshape expressions (5a) into coefficients of the nonlinear terms (3.2). After some algebraic manipulation, it is revealed that both BP1 and BP2 exist when m2<m1/3, which can usually be satisfied for the application of an NES, where the NES is a small mass compared to the primary system (m2m1). The backbone curves for the symmetrized model of the example system, whose parameters are given in table 1, are shown in figure 6a—two single-mode backbone curves, S1 and S2, are connected by mixed-mode backbone curves, S1,2±, via branch points, BP1 and BP2.
Figure 6.

Figure 6. Symmetry breaking interpretation of the backbone curves for the example system with parameters given in table 1. (a) Backbone curves for the symmetrized model in the initial modal velocity space (q˙1(0),q˙2(0)). The single-mode backbone curves, S1 and S2, are denoted by grey lines. The in-phase and anti-phase backbone curves, S1,2+ and S1,2, are denoted by blue and red curves, respectively. The branch points are denoted by black dots and labelled by BP1 and BP2. (b) Evolutions of backbone curves due to symmetry breaking in the initial modal velocity space (q˙1(0),q˙2(0)). The backbone curves for the example system are shown by solid curves. Two intermediate asymmetric cases, with μ=0.05 and μ=0.5, are shown by dotted and dot-dashed curves respectively. (Online version in colour.)

(b) Backbone curves for an asymmetric case

For the asymmetric case, with non-zero Ψ1 and Ψ2, 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 μ=0.05 and μ=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, S1,2+ and S1,2. 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 S1,2: the critical energy level, shown in figure 4b, is captured by the fold on S1,2. 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, S1,2+: the fundamental resonant decay follows the in-phase backbone curve, S1,2+, and exhibits energy localization in q2 at high energy levels, and localization in q1 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 μ, 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. α1=0 and k2=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 k2 and α1, the modeshape matrix, Φ, is no longer anti-diagonal,4 and the coefficients of nonlinear terms, Ψi, 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 μ=0, using conditions (3.7). Combining expressions (3.7), (4) and (5a), two critical boundaries, defining the existence of the BPs, can be obtained

critical boundary,f1(m1,m2,k1,k2,α1,α2), for the existence of BP1Ψ4Ψ3=0:α2,critα1,crit=P^1=(P1P2+P3)P4(P5P2+P6)(P2+P7) 2, 4.1a
critical boundary,f2(m1,m2,k1,k2,α1,α2), for the existence of BP2 4.1b
Ψ3Ψ5=0:α2,critα1,crit=P^2=(P1P2+P3)P4(P5P2+P6)(P2+P7) 2, 4.1c
where parameters, Pi, are given by expressions (B13) in appendix B. These boundaries, defined by expressions (4.1), together divide the conservative parameter space, (m1,m2,k1,k2,α1,α2), into regions, distinguishing the existence of BPs, and hence the backbone curve topologies, for the symmetrized model. These divisions are shown in figure 7a, where the axes represent these boundaries; the corresponding backbone curve topology within each region for the general cases are computed via equations (9) and (10). Besides the backbone curve topology seen for the example system in figure 6a (also shown in the first quadrant of figure 7a for the general cases), where both BP1 and BP2 can be seen, three other cases, i.e. with BP1 and without BP2 (the second quadrant), without BP1 and with BP2 (the fourth quadrant), and without both BP1 and BP2 (the third quadrant), may also be identified. For backbone curves in panels (a2) and (a3), the mixed-mode backbone curves, S1,2±, bifurcate from the single-mode backbone curve, S1 and S2, respectively, and extend to higher energy levels. In panel (a4), only two single-mode backbone curves, S1 and S2, are observed, as neither BP1 nor BP2 exist. As parameters of the system cross the axes (boundaries), for example when parameters cross the axis from the first quadrant to the second quadrant, the bifurcations denoted BP2 degenerate to infinite frequency and amplitude, leading to a change of backbone curve topology from panels (a1) to (a2).
Figure 7.

Figure 7. Evolutions of backbone curve topologies due to symmetry breaking. (a) Backbone curve topologies in initial modal velocity space, (q˙1(0),q˙2(0)), for the symmetrized model with μ=0. The axes denote the critical boundaries for the existence of branch points defined by equations (3.7). The single-mode backbone curves, S1 and S2, are shown as grey curves, the in-phase and anti-phase backbone curves, S1,2+ and S1,2, are shown as blue and red curves, respectively. Branch points on S1 and S2 are denoted by black dots and labelled with BP1 and BP2, respectively. (a1) (the first quadrant) S1 and S2 are connected by S1,2+ and S1,2 via BP1 and BP2. (a2) (the second quadrant) S1,2+ and S1,2 bifurcate from S1 via BP1. (a3) (the fourth quadrant) S1,2+ and S1,2 bifurcate from S2 via BP2. (a4) (the third quadrant) Two single-mode backbone curves, S1 and S2, without branch points. (b) Backbone curve topologies in initial modal velocity space, (q˙1(0),q˙2(0)), for asymmetric cases perturbed from symmetrized model with μ=0.05. Backbone curve topologies in panels (b1)(b4) may be seen as evolutions from those in (a), due to symmetry breaking. (Online version in colour.)

A non-zero μ 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 μ=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 (b1) shows the same topology as that for the example system in figure 6b. For those in panels (b2) and (b3), 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 (b4), the two single-mode backbone curves, S1 and S2, in panel (a4), evolve to mixed-mode backbone curves, S1,2.

For these asymmetric cases in figure 7b, aside from the backbone curves in panel (b1), which show the same topology as those seen in the example system considered in previous sections, other backbone curve topologies, in panels (b2)(b4), exhibit fundamentally different features in energy localization. With an initial energy in q2 (i.e. the primary system), systems with backbone curves shown in panels (b2)(b4) do not show the necessary energy localization features in backbone curves to exhibit TET, and hence energy remains in q2 without being transferred to q1 (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 (b1) 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 (a1) from those in figures 7a.

To demonstrate this, it is useful to project the boundaries, f1 and f2, defined by expressions (4.1), to the subspace, (k2,α2), 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 m1=1, k1=1 and α1=1 is considered, and the NES has a mass m2=0.05. Using expressions (4.1), the NES-parameter space (k2,α2) may be divided into several major regions,5 labelled by (a), (b1), (b2), (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 f1 and f2 (projections of the axes in figure 7), respectively; the thin purple and brown dotted lines represent asymptotic lines of f1 and f2 respectively, which are given by

asymptotic line of f1:k2,crit1=[3(m1m2)23m1m2]k1m23(m1+m2) 2, 4.2a
asymptotic line of f2:k2,crit2=[3(m1m2)+23m1m2]k1m23(m1+m2) 2, 4.2b
Figure 8.

Figure 8. Divisions of the NES-parameter space, (k2,α2), considering the existence of branch points on backbone curves for the symmetrized model with m1=1, k1=1 and α1=1. Purple and brown solid curves denote the critical boundaries for the existence ofbranch points, defined by expressions (4.1). Asymptotic lines of the boundaries are defined by expressions (4.2) and denoted via dotted lines. These boundaries divide the space into five major regions, labelled with (a), (b1), (b2), (c) and (d). Backbone curve topologies for the symmetrized model in these regions are correspondingly described in table 2 and shown in panels of figure 7a. Backbone curves and numerical simulations for the five example system, labelled by cross signs, are shown in figure 9. (Online version in colour.)

Table 2. Backbone curve topologies for the symmetrized model in regions of figure 8.

region symmetric backbone curve topology exhibits TET
(a) with both BP1 and BP2, schematically shown in figure 7a1
(b1) and (b2) with BP1 and without BP2, schematically shown in figure 7a2
(c) with BP2 and without BP1, schematically shown in figure 7a3
(d) without both BP1 and BP2, schematically shown in figure 7a4
denoting critical linear stiffness values in the NES. The backbone curve topologies for the symmetrized model in these regions are given in table 2 and related to these shown in figure 7a. Using these divisions, the backbone curve topologies for the original model may be approximately classified, along with which the existence of TET is evaluated:

region (a): an example system with k2=0.02 and α2=0.02 is first considered. The backbone curves of this system are shown in figure 9a1. The energy ratio dissipated by the NES, with respect to the initial velocity in the primary system, is shown in figure 9a2, which is similar to that in figure 4b and a critical energy level may be seen. As α2 increased to α2=1, this critical energy level is decreased from x˙1(0)0.5 to 0.06, see figure 9a4. 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 α2. The instantaneous energy carried by the NES with 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 (b1): with k2=0.02 and α2 decreased to 0.005, the system crosses f2 and moves from region (a) to region (b1). The corresponding backbone curves are shown in figure 9b1, 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 7a2. When applied the primary system with an initial velocity, the energy dissipated by the NES is presented in figure 9b2, 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 α2=1 and k2=0.05 lies in region (c) and is labelled by a cross in figure 8. Its backbone curves are shown in figure 9c1, which are asymmetric evolutions of those shown in figure 7a3. 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 9c2. For systems in this region, more energy may be dissipated by the NES for a low 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 x˙1(0)=0.02. While, as 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 k2=0.07, the system is in region (d). The backbone curves of the system are shown in figure 9d1, while the energy ratio dissipated by the NES is presented in panel (d2). In this region, like those in region (b1) 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 k2 is further increased.

Figure 9.

Figure 9. Backbone curves and energy transfer characteristics of the example systems with m1=1, m2=0.05, k1=1,α1=1 and c1=c2=0.005, in figure 8. (a) Example systems in region (a) with k2=0.02 and α2=0.02 (panels (1) and (2)); with k2=0.02 and α2=1 (panels (3), (4), (i) and (ii)). (b) Example system in region (b1) with k2=0.02 and α2=0.005. (c) Example system in region (c) with k2=0.04 and α2=1. (d) Example system in region (d) with k2=0.07 and α2=1. (Online version in colour.)

Comparing the examples in region (a) and (b1), the existence of region (b1) characterizes the minimum degree of nonlinearity necessary for the system to exhibit TET, shown as boundary f2 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 f1, i.e. k2,crit1, 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

m1x¨1+k1x1+k2(x1x2)+α1x13+α2(x1x2) 3=0 A 1a
and
m2x¨2+k2(x2x1)+α2(x2x1) 3=0, A 1b
where m1 and m2 denote the mass values of the two masses, whose displacements are x1 and x2, respectively, and k1 and α1 are the coefficients of the linear and nonlinear springs ground the first mass, while k2 and α2 are the coupling linear and nonlinear spring coefficients. The system is translated into the modal space by substitution of
(x1x2)=[ϕ11ϕ12ϕ21ϕ22](q1q2). A 2
After linear modal transform, the modal equations of motion can be obtained
q¨1+ωn12q1+Ψ4q13+3Ψ1q12q2+Ψ3q1q22+Ψ2q23=0 A 3a
and
q¨2+ωn22q2+Ψ1q13+Ψ3q12q2+3Ψ2q1q22+Ψ5q23=0, A 3b
where ωni denote the ith natural frequency and Ψi are coefficients of the nonlinear terms, given by
Ψ1=ϕ113ϕ12α1+(ϕ11ϕ21) 3(ϕ12ϕ22)α2,Ψ2=ϕ11ϕ123α1+(ϕ11ϕ21)(ϕ12ϕ22) 3α2,Ψ3=3[ϕ112ϕ122α1+(ϕ11ϕ21) 2(ϕ12ϕ22) 2α2],Ψ4=ϕ114α1+(ϕ11ϕ21) 4α2andΨ5=ϕ124α1+(ϕ12ϕ22) 4α2.} A 4
The modal parameters, i.e. ωni and ϕij can be obtained by eigenvalue and eigenvector analysis, and are given by
ωni2=(k2m2+k1+k2m1)±(k2m2k1+k2m1) 2+4k2m1k2m22, A 5a
ϕ112=k22(k1+k2m1ωn12) 2m2+m1k22,ϕ122=k22(k1+k2m1ωn22) 2m2+m1k22 A 5b
andϕ212=(k1+k2m1ωn12) 2(k1+k2m1ωn12) 2m2+m1k22,ϕ222=(k1+k2m1ωn22) 2(k1+k2m1ωn22) 2m2+m1k22. A 5c

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

qiui=Uicos(ωrit), A 6
where ui denotes the fundamental component of qi, and where Ui and ωri represent the amplitude and response frequency of ui respectively. In addition, as this paper considers the realization of TET via fundamental resonant capture, the response frequencies of the two modal coordinates are assumed to be equal, i.e. ωr1=ωr2=Ω. With modal responses in expression (6) substituted into equations of motion (3), and the non-resonant terms removed, the expressions for computing backbone curves may be obtained after some algebraic manipulations, given by
4(ωn12Ω2)U1+3Ψ4U13+3Ψ3U1U22+3p(Ψ2U23+3Ψ1U12U2)=0 A 7a
and
4(ωn22Ω2)U2+3Ψ5U23+3Ψ3U12U2+3p(Ψ1U13+3Ψ2U1U22)=0, A 7b
where p=+1 and 1, denoting in-phase and anti-phase relationships between two modal coordinates, respectively.

For symmetric case, Ψ1=Ψ2=0, the expressions for backbone curves may be reduced to

[4(ωn12Ω2)+3Ψ4U12+3Ψ3U22]U1=0 A 8a
and
[4(ωn22Ω2)+3Ψ5U22+3Ψ3U12]U2=0. A 8b
Solutions with respect to U1=0 and U2=0 are trivial where the system is stationary. Besides this trivial solution set, single-mode backbone curves can be found with respect to either U10 and U2=0, or U1=0 and U20. The amplitude–frequency relationships for these single-mode backbone curves are given by
S1:U2=0,Ω2=ωn12+34Ψ4U12 A 9a
and
S2:U1=0,Ω2=ωn22+34Ψ5U22, A 9b
where the subscripts of S1 and S2 denote the modal components on these backbone curves. Another solution set is related to U10 and U20, where the NNMs on backbone curves contain both modal coordinates, q1 and q2. Combining equations (8a) and (8b), the frequency–amplitude relationships for the mixed-mode backbone curves are
S1,2+,S1,2{U12=4(ωn22ωn12)+3(Ψ5Ψ3)U223(Ψ4Ψ3)Ω2=4(Ψ4ωn22Ψ3ωn12)+3(Ψ4Ψ5Ψ32)U224(Ψ4Ψ3), A 10
where S1,2+ represents mixed-mode in-phase backbone curve, and S1,2 denotes mixed-mode anti-phase backbone curves.

While for asymmetric case, with non-zero Ψ1 and Ψ2, 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

Ω2=ωn12+34[Ψ4U13+Ψ3U22U1+p(Ψ2U23+3Ψ1U12U2)]U11 A 11a
and
0=(3pΨ2U11)U24+3(Ψ5Ψ3)U23+[9p(Ψ2Ψ1)U1]U22+[4ωn224ωn12+3(Ψ3Ψ4)U12]U2+3pΨ1U13. A 11b
Using expressions (11), the backbone curves for an asymmetric system may be computed.

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

critical boundary,f1(m1,m2,k1,k2,α1,α2), for the existence of BP1α2,critα1,crit=P^1=(P1P2+P3)P4(P5P2+P6)(P2+P7) 2, B 1a
critical boundary,f2(m1,m2,k1,k2,α1,α2), for the existence of BP2
α2,critα1,crit=P^2=(P1P2+P3)P4(P5P2+P6)(P2+P7) 2, B 1b
where
P1=2(k2m2+k1m2k2m1),P2=k22(m1+m2) 22k1k2(m1m2)m2+k12m22,P3=(k1m2+k2m2k2m1) 2+4k22m1m2,P4=4k22m23,P5=2[(k1+k2)m22+k2m12(k12k2)m1m2],P6=P3(m1+m2)andP7=(k1k2)m2k2m1.} B 2
and where two asymptotic lines of these boundaries are given by
asymptotic line of f1:k2,crit1=[3(m1m2)23m1m2]k1m23(m1+m2) 2 B 3a
asymptotic line of f2:k2,crit2=[3(m1m2)+23m1m2]k1m23(m1+m2) 2. B 3b
These boundaries and asymptotic lines together divide the conservative parameter space, (m1,m2,k1,k2,α1,α2), into regions, distinguishing the backbone curve topologies for the symmetrized model.

Footnotes

1 The NES is seen as a small mass in comparison to the primary system.

2 Note that, this definition of symmetry is equivalent to having Z2Z2 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, x1, is no longer exactly represented by the second modal coordinate, q2; likewise, the displacement of the NES, x2, is no longer exactly captured by q1. Nonetheless, an accurate approximation to this is still achieved if k2 is sufficiently small. In addition, it will be demonstrated in the following that a sufficiently small k2 is necessary in order to exhibit TET. Thus, the following discussions continue to consider the backbone curve topology in the modal domain.

5 Some small regions, and regions related to negative nonlinearity do not exhibit the topology of interest and are not considered here.

Published by the Royal Society. All rights reserved.

References