Forced oscillations of the string under conditions of ‘sonic vacuum’

Abstract

We present the results of analytical study of the significant regularities which are inherent to forced nonlinear oscillations of a string with uniformly distributed discrete masses, without its preliminary stretching. It was found recently that a corresponding autonomous system admits a series of nonlinear normal modes with a lot of possible intermodal resonances and that similar synchronized solutions can exist in the presence of a periodic external field also. The paper is devoted to theoretical explanation of numerical data relating to one of possible scenarios of intermodal interaction which was numerically revealed earlier. This is unidirectional energy flow from unstable nonlinear normal mode to nonlinear normal modes with higher wavenumbers under the conditions of sonic vacuum. The mechanism of such a scenario has not yet been clarified contrary to alternative mechanisms consisting in almost simultaneous energy flow to all nonlinear normal modes with breaking the above-mentioned conditions of sonic vacuum. We begin with a description of single-mode manifolds and then show that consideration of arbitrary double mode manifolds is sufficient for solution of the problem. Because of this, the two-modal equations of motion can be reduced to a linear equation which describes a perturbation of initially excited nonlinear normal mode of the forced system in the conditions of sonic vacuum. We have found analytical representation (in the parametric space) of the thresholds for all possible energy transfers corresponding to unidirectional energy flow from unstable nonlinear normal modes. The analytical results are in a good agreement with previous numerical calculations.

This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

1. Introduction

The problems of energy transport, localization and equipartition in nonlinear dynamical systems appear in the various fields of mechanics, physics, chemistry, biology, etc. [1]. Owing to complexity of nonlinear dynamical equations and the diversity of behaviours, there is no general theory that can claim to be exhaustive in the description of nonlinear phenomena arising in the fields of knowledge mentioned above [2]. Therefore, the majority of the studies are based on the quasi-linear approximation [3,4]. The latter allows us to refer to a lot of the results that have been obtained for linear dynamics. However, such an approach cannot be applied to systems which do not have any linear elastic forces initially and the equations of motion which cannot be reduced to a linearized problem. The specific peculiarity of such systems is that the oscillation frequency is determined by the initial conditions or external factors (intensity and frequency of loading), because they do not have their own frequency in the usual interpretation. In particular, they are perceived as the sensible elements which effectively respond to the external actions in the wide band of the frequencies.

It was shown recently [5–7] that a string with uniformly distributed discrete masses but without a preliminary stretching can demonstrate important regularities in its dynamical behaviour: the existence of well-defined nonlinear normal modes (NNMs) and a lot of possible resonances between them. The latter property which is strongly connected with providing the conditions of acoustic vacuum [5] allows one to consider such system as efficient energy sink for defence of different structural elements from intensive external excitations by means of almost irreversible targeted energy transfer (TET). The main attention in this field has focused on the case of impulse excitation [8,9] using the simplest single-particle sink. As for periodic external loading, the efficiency of such a simple energy sink is much less studied [10,11]. The mentioned properties of multi-particle sinks impel estimation of a possibility to use them for efficient energy transfer or harvesting under the conditions of periodic loading. There are two starting points for this study which were clarified recently [12]. First is the existence of well-defined nonlinear normal modes and a lot of intermodal resonances in a corresponding autonomous system. Second is numerical evidence of two possible scenarios of intermodal interaction in the presence of a periodic external excitation. One of them is unidirectional energy flow from unstable nonlinear normal modes to nonlinear normal modes with higher wavenumbers under the conditions of sonic vacuum. Another scenario consists in almost simultaneous energy flow to all nonlinear normal modes with breaking of the above-mentioned conditions. While the possible mechanism of such ‘explosion’ was briefly discussed previously the mechanism of the unidirectional energy flow has not yet been clarified. Its clarification is the objective of the paper. Beginning from detailed description of unimodal manifolds, we have shown that consideration of arbitrary two-modal manifolds is sufficient for solution of the problem. On this basis, the two-modal equations of motion are reduced to a linear equation which describes a perturbation of initially excited nonlinear normal mode of the forced system in the conditions of sonic vacuum. It has allowed one to find the analytical estimate of all possible energy transfers from unstable nonlinear normal modes, including unidirectional energy flow.

2. The model and equations of motion

The first problem relates to the properties of the sink itself in these conditions. This model has already been described several times [5–7,12,13].

The system consists of several masses coupled by elastic string without a preliminary stretching which can undergo a planar motion (figure 1).

Figure 1.

The ‘sonic vacuum’ system. The masses coupled by non-stretched elastic string move in the x- and z-directions. (Online version in colour.)

The equations of motion can be written as follows:

2.1

where U_n and V_n are the dimensionless (in the units of the string length) longitudinal and transversal displacements of nth mass, respectively; θ_n is the angle between nth string and x-direction. The dimensionless time t is measured in . m is the mass, L and κ are the length and the elastic modulus of the string, respectively. The dimensionless force F is measured in units of the string elastic modulus.

In the framework of the small-amplitude approximation, one can write the tension of the string coupling the (n − 1)th and nth masses as follows:

2.2

and

There are two asymptotic limits for the oscillations of the system under consideration, which are distinguished by the predominant displacement direction: (i) axial and (ii) transversal. In the first case, we assume U_n≫V_n and the typical time has the same order as unity (τ∼1). So, the first of equation (2.1) is written as follows:

2.3

The transversal motion is the driven one:

2.4

The second asymptotic case occurs when V_n≫U_N. Taking into account that V_n+1 − V_n∼ε, we should propose U_n+1 − U_n∼ε². Substituting these estimations into the second of equations (2.1), we get that d²V_n/dt²∼ε³. It means that the typical time is a ‘slow’ one and the inertial term in the first of equations (2.1) is negligible. In such a case

2.5

where we have defined the ‘average’ tension

2.6

The presence of small longitudinal displacements leads to an effective non-local stretching, which is proportional to the second power of the transversal displacements. The detailed discussion of this approximation can be found in [5]. Further, we will consider the second asymptotic regime, which supposes that the transversal displacements of masses essentially exceed the longitudinal ones.

3. Dynamics on the single-mode manifold

The conditions mentioned above lead to the ‘sonic vacuum’ equations [5] where the longitudinal tension is replaced by the term which is proportional :

3.1

under the boundary conditions

It is easy to show that the function

3.2

is the solution of equation (3.1) for the free oscillations (F(n, t) = 0), if the amplitude v_k satisfies the equation

3.3

with

3.4

Function (3.2) describes the single mode manifold (SMM). This section contains the analysis of the SMM under the external force according to equation (3.1). One should note that SMMs make up the orthogonal set, but in contrast to the linear case any combination of SMMs is not a solution for the free oscillations equation. In order to analyse the behaviour of the system under an action of the external field, we consider the unimodal excitation. In such a case, we assume that the force in the right-hand side of equation (3.1) is distributed accordingly to the mode with wavenumber k (see equation (3.2)):

where F_k = fcosΩt does not depend on the wavenumber or the number of particles. So, the equation of motion of the forced oscillations can be written as follows:

3.5

In order to analyse equation (3.5), it is useful to introduce new variables, which are the classic analogue of the quantum creation and annihilation operators [14]:

3.6

where ω is an undefined (yet) frequency and the asterisk denotes the complex conjugated value.

Putting the complex variables Ψ_k into operation allows us to reduce the order of the differential equation and, in quite a number of cases, to get the additional integral of motion—the occupation number [9]:

3.7

Substituting expressions (3.6) into equation (3.5), we get

3.8

We will show that equation (3.8) is extremely useful for the analysis of both stationary and non-stationary dynamics. We will consider the solution of equation (3.8) in the form of single-frequency oscillations:

3.9

where ψ_k is a slowly changed function.

Varying the parameter ω one can study the different regimes of the forced nonlinear oscillations [15,16].

(a). Stationary dynamics on the single-mode manifold

The stationary forced oscillations are realized when the parameter ω is equal to the external force frequency:

and ψ_k = const.

Substituting expression (3.9) into equation (3.8) and averaging it over the period 2π/ω, and considering the resonant (secular) terms only, we can rewrite equation (3.8) as follows:

3.10

The relationship between the modulus of complex function and the oscillation amplitude A_k follows immediately from expressions (3.6):

3.11

So, under the conditions ω = Ω;ψ_k = const. equation (3.10) becomes

3.12

The latter can be solved with respect to Ω, which leads to the relation between the characteristics of the stationary forced oscillations—frequency (Ω) and amplitude (f) of the external force, and the amplitude of the oscillations (A_k):

3.13

Figure 2a,b shows examples of amplitude–frequency relations (3.13) for the 3-d and 6th modes for a string with 10 particles.

Figure 2.

The amplitude–frequency relations for the 3d (a) and 6th (b) modes for a string with 10 masses. The key in (a) shows the force amplitudes (f). (b) The comparison between analytically predicted amplitudes (solid curves) and the data of the numerical integration of full system equations (2.1) (dots). The red, blue and black dots correspond to the different initial conditions, which are the roots of equation (3.13). The black dashed line shows the backbone curve. The external force f = 1 × 10⁻⁴. (Online version in colour.)

In figure 2, the black lines show the backbone curve, which corresponds to the free oscillations (f = 0). One should note that the total range of the frequencies can be separated into two domains: Ω < Ω_b and Ω > Ω_b for every value of the exciting force amplitude. There is only one ‘high’ amplitude branch of the oscillations in the first domain, while the number of oscillation branches in the second one is equal to three. The frequency corresponding to the bifurcation of the oscillations can be determined from relation (3.13) as the root of the equation (figure 3a)

3.14

The bifurcation frequencies Ω_b for different modes are shown in figure 3b.

Figure 3.

(a) The amplitude–frequency relation for the 10-masses string under exciting force f = 0.002 (dark red curve); black line shows the backbone curve. The vertical red line denotes the bifurcation frequency Ω_b, and the dashed blue one corresponds to switching different regimes (see (b)). (b) The dependence of the bifurcation frequency Ω_b on the force amplitude for different modes (k). (Online version in colour.)

Figure 2b shows the comparison between amplitude–frequency relation (3.13) and the data of the numerical simulation of equations (2.1). The points marked by red, blue and black correspond to different initial conditions, which are the roots of equation (3.13).

Analysis of figure 2b shows that the low-amplitude branch (black curve) demonstrates the good accordance between the data of the numerical simulations and the analytically predicted values. However, the discrepancies of the high-amplitude branches (red and blue curves) with the numerical simulations data turn out to be essential in the frequency range Ω > Ω_b. It will be shown in the next section that the reason of such divergence is coupled with the stability of the nonlinear normal modes.

(b). Non-stationary dynamics on the single-mode manifold

The non-stationary dynamics of the SMM can be due to various reasons, the most evident of which is an imbalance between SMM stationary characteristics (3.13) caused by initial conditions. The non-stationary dynamics in the SMM becomes apparent as a modulation of the carrier wave but the parameters of modulation depend on the closeness to the stationary solution. In such a case, we assume that the carrier wave frequency is the same as for the stationary case, but the amplitude of the complex function Ψ_k turns out to be a function of the time with specific scale, which is essentially larger than the period 2π/Ω. So, we propose that the time derivative of Ψ_k in equation (3.8) has to be written as follows:

where τ is the ‘slow’ time.

Now we can formulate equation (3.8) in the form

3.15

Let us rewrite the variable ψ_k in the polar representation: ψ_k = a_k(τ)e^−iδ_k(τ). Separating equation (3.15) into the real and imaginary parts, we get the following:

3.16

In order to analyse the non-stationary dynamics of the SMM, one should note that equations (3.16) permit the energy integral as follows:

3.17

The variables (a²_k, δ_k) form the canonical set, and equations (3.16) can be obtained following the rules:

3.18

Owing to the presence of energy integral (3.17), we can consider the full set of the non-stationary trajectories on the phase plane (δ_k, a_k), fixing the frequency Ω. One should note that the stationary solutions are represented by the points on the phase plane of the system since energy integral (3.17) is the function of slow time τ. First of all, equations (3.16) show that the stationary points for δ_k = ± nπ with n = 0, 1, 2… correspond to the stationary solution obtained in the previous section. For frequencies that are smaller than the value Ω_b, which is defined by equation (3.14), there is only one stationary state at δ_k = 0. There are two kinds of trajectories in figure 4a. The first of them is specified by the limited variation of phase δ_k and such the trajectories surround the stationary point at δ_k = 0. The trajectories of the second kind are called the ‘transit-time’ and they are characterized by the unlimited growth of the phase. The trajectories of different kinds are separated by the limiting phase trajectory (LPT), which passes through the state a_k = 0. It is shown in figure 4a by the red dashed curve. It is important to note that the LPT corresponds to the most effective energy exchange between the system and the exciting force.

Figure 4.

Phase plane of the system (3.17) at various values of the frequency Ω: (a) Ω = 0.95 Ω_b, (b) Ω = 1.025Ω_b, (c) Ω = Ω_t, (d) Ω = 1.05 Ω_t. The LPT and the separatrix are marked by red and blue dashed lines, respectively. Parameters: N = 10, f = 0.1, k = 6. (Online version in colour.)

The frequency Ω_b corresponds to the bifurcation which leads to the formation of two additional stationary states with the phase δ_k = π. Figure 4b shows the phase plane topology for the frequency Ω = 1.025Ω_b. One can see that the LPT and its inner domain are weakly changed, but the separatrix passing through the unstable stationary point divides the transit-time trajectories into two flows. The separatrix grows while the frequency Ω increases up to the value when the separatrix touches the state a_k = 0. At this moment, the LPT and the separatrix coincide and no transit-time trajectories in the gap between them are observed. The more important change is that the separatrix becomes heteroclinous and the trajectory which passes through zero state cannot surround the stationary state at δ_k = 0. This bifurcation occurs when the energy of the unstable state becomes equal to zero. One can show that the respective frequency (it is shown in figure 3a by the blue dashed line). The further growth of the exciting frequency leads to the LPT becomes closed around the small-amplitude stable state at δ_k = π and the efficiency of the energy exchange between the system and the external field is essentially decreased (figure 4d).

4. Dynamics on the double-mode manifold

The nonlinear modes interaction is one more exciting problem in the dynamics of the system under conditions of ‘sonic vacuum’. It is important from the viewpoint of the system excitation by a broad-band external excitation as well as the problem of the normal mode stability. Owing to the possibility of many resonances for the modes with strongly different wavenumbers, the system under consideration can demonstrate effective intermodal energy exchange and fast energy spreading. In this section, we formulate the exact resonant conditions for free oscillations and consider the energy exchange between two nonlinear normal modes.

(a). Stationary dynamics on the double-mode manifold

Let us consider the solution of equation (3.1) as a combination of two modes:

4.1

This expression describes the double mode manifold (DMM). Substituting expression (4.1) into equations (3.1), one can get the equation of motion for the modes. Owing to the orthogonality of the normal modes, the sum in equations (3.1) is clearly calculated:

Taking into account this relation, one should substitute expression (4.1) into equations (3.1) and make the projection onto the modes. As a result, we get the equations for the modes' amplitudes:

4.2

and

4.3

where ω_k, ω_m and f_k, f_m are defined by expressions (3.4) and the second of equations (3.5), respectively. Introducing the complex variables similar to equations (3.6), and considering the stationary oscillations with the frequency of the exciting force Ω, we can write in the analogue of previous section:

4.4

Taking into account expression (3.11), one can solve equations (4.4) with respect to frequency Ω:

4.5

Fixing the frequency Ω and the ‘partial’ forces f_k and f_m, we can solve equations (4.5) to find the ‘partial’ amplitudes A_k and A_m. With A_m = 0, the first of equations (4.5) transforms to relation (3.13) for SMM. Because the left-hand sides of equations (4.5) should be equal, we can obtain the resonant condition for the stationary free oscillation of DMM:

4.6

So one can conclude that for free oscillations with wavenumbers k and m with a positive right-hand side of expression (4.6), the resonant relation of the modes' amplitude always occurs.

Figure 5 shows the ‘map’ of resonantly interacting modes for strings with 30 masses. The variables k and m are the wavenumbers of the interacting modes. Each dark point in the (k − m) plane corresponds to such a combination of the wavenumbers, that the right-hand side of equation (4.6) is positive. So, if is it true, two values A_k and A_m, which satisfy equation (4.6) can always be found. And vice versa, the white domains in figure 5 correspond to the combinations of the wavenumbers, which cannot be under resonant conditions.

Figure 5.

The map of the resonances in the string with 30 masses. Here m and k are the wavenumbers of the modes.

(b). Non-stationary dynamics of the double-mode manifold

In order to pass to non-stationary dynamics, we should apply the same reasoning as for SMM dynamics. Omitting the details, one can write the energy in the terms of complex functions (ψ_k, ψ_m).

4.7

The non-stationary equations of DMM dynamics can be obtained using the standard rules:

Let us consider the modes interaction. In particular, we want to analyse the energy transfer from one mode to another one, i.e. we describe the behaviour of the mth mode under its excitation by kth mode. We will call the kth mode the ‘leading’ one (LM) and the mth mode as the ‘driven’ mode (DM). In such a case, we assume that ‘partial’ force f_m is equal to zero (f_m = 0), while the LM oscillates without any feedback from the DM. It means that we propose the smallness of the DM amplitude, but the amplitude of the LM is determined by stationary relation (3.13).

Introducing the polar representation of the functions ψ_k and ψ_m in compliance with the previous section, one can obtain the dynamical equations from the Hamilton function (4.7). Omitting some tedious details, we write

4.8

and

4.9

for the LM and

4.10

for the DM. Here the prime denotes the derivative with respect to time τ. The part of (4.7) which corresponds to equations (4.10) can be written as follows:

4.11

Taking the initial conditions for the DM amplitude as a_m(τ = 0) = 0, one can formulate the equations of motion in the form

4.12

Generally speaking, equation (4.12) allows us to find the evolution of the DM amplitude under stationary excitation by the LM with wavenumber k. It will be valid for the range of DM amplitudes a_m, till its effect on the dynamics of the LM is negligible. The linearized version of equation (4.12) allows us to make a conclusion about the stability of the DM under excitation by the LM:

4.13

where

4.14

If the parameter Λ is positive, equation (4.13) describes the periodic energy exchange between the LM and DM. If it is negative, the respective eigenvalues of problem (4.13) are imaginary and the growth of DM is unlimited. However, the magnitude of the parameter Λ at a fixed value of exciting frequency Ω depends on which branches the LM amplitude A_k relates to. Solving equation (3.13) with respect to A_k, we obtain three roots:

4.15

Expressions (4.15)(i) and (4.15)(iii) correspond to stable branches and expression (4.15)(ii) describes the unstable one. One should estimate the parameter Λ for each of solutions (4.15). The possible values of this parameter depend on the LM and DM wavenumbers as well as on the frequency and amplitude of the exciting force. Fixing the latter, we can determine the stable and unstable combinations of the LMs and DMs. Figure 7a–f shows the instability domains on the (LM–DM) maps for the different oscillation branches (4.15)(i)–(iii) at two frequencies of the LM. The horizontal and vertical axes of the maps correspond to the wavenumbers of the LM and DM, respectively. If the parameter Λ (4.14) for the pair (n_LM, n_DM) is positive, such a pair is declared ‘stable’ and the respective domain on the map is coloured yellow. Vice versa, if Λ(n_LM, n_DM) is negative, this combination is assumed ‘unstable’ and it is coloured by blue. The red lines separate the stable and unstable domains.

Figure 7.

The stability maps for the LM–DM interactions at two frequencies of the LM. (a–c) Ω = 1.05Ω_b; (d–f) Ω = 1.5Ω_b. The combinations with the periodic (Λ > 0) and aperiodic (Λ < 0) energy exchange are coloured yellow and blue, respectively. The amplitude of the exciting force f_k = 0.05. (Online version in colour.)

One should note that the frequencies of the exciting force which correspond to the bifurcation (3.14) depend on the LM's wavenumber. They can be calculated as follows:

4.16

In figure 7, we considered the fixed frequency for the LM, and its value exceeds bifurcation frequency (4.16) (figure 6). Therefore, the frequency Ω is determined by the wavenumber of the LM, and it is constant for different DMs at fixed LM.

Figure 6.

The amplitude–frequency dependence for the system with 30 masses under forcing f = 0.05. The indexes (i)–(iii) correspond to equations (4.15). The red and blue dashed lines show the frequencies Ω corresponding to ‘stability maps’ in figure 7a–c and d–f , respectively. (Online version in colour.)

Figure 7a–c shows the stability maps at the exciting frequency Ω = 1.05Ω_b. The yellow domains correspond to stability of DM (Λ > 0) and blue regions are the domains of instability (Λ < 0). The indexes above the figures correspond to different oscillation branches (4.15).

One can see that the maps in figure 7a,c show that the domains of instability are predominantly in the region with low wavenumbers LM and high wavenumbers DM. The instability domain in figure 7b is more symmetric and its boundaries are closely correlated with those of the resonant domain of the free oscillations, which is determined by relation (4.6) (figure 5). It is natural enough, but one can see that the instability domain is more narrow due to the effect of the external force.

Figure 7d–f shows the stability maps at the frequency that is higher than previous one. It exceeds Ω_b by 50% and is shown in figure 6 by the blue dashed line. One can see that the instability domains in figure 7a,b have been changed weakly, while they in fact disappear in figure 7c. It can be understood from this viewpoint that the oscillations corresponding to branch (4.15)(iii) cannot excite another mode since their amplitudes are small themselves (figure 6).

In order to illustrate intermodal interactions, we performed numerical integration of the original system (2.1) for chains with 20 massive elements. The amplitude f, the wavenumber n and the frequency Ω of the external force have been fixed and the initial displacements have been chosen according to the mode with wavenumber n. Such a numerical simulation correlates with the numerical experiments that have been performed with the full system and discussed in the recent paper [12].

Figure 8(i)–(iii) shows the result of numerical integration of the original system equations (2.1). Each of the panels represents the distribution of the modal kinetic energy. The differences between the panels are the initial conditions, which correspond to the different branches of stationary solutions (4.15) at the same values of the external force and its frequency. One can see that in accordance with figure 7, the first (i) and second (ii) branches on the amplitude–frequency relation (figure 6) effectively interact with modes with larger wavenumbers, while the low-amplitude branch (iii) is stable with respect to energy exchange with another modes. The further evolution of the oscillations, which correspond to branches (i) and (ii), is very complicated and was discussed in [12].

Figure 8.

Numerical integration of original system equations (2.1). (a–c) Time evolution of the modal kinetic energy for the initial conditions which correspond to the modal amplitudes (4.15) for the mode with wavenumber n = 6. k and t are the mode number and time, respectively. The frequency and the amplitude of the external force are Ω = 0.2ω_n, f = 0.001. The blue background corresponds to the non-excited modes, while the bright domains show the excited modes. (Online version in colour.)

5. Conclusion

The obtained analytical results allow one to predict the dynamic behaviour of the non-stretched string with uniformly distributed identical masses under harmonic excitation provided that the conditions of sonic vacuum are satisfied. As this takes place, the instability of any nonlinear normal mode with respect to inter-modal perturbation leads to unidirectional energy flow to nonlinear normal modes with higher wavenumbers in full agreement with previous numerical study. We have shown that such a conclusion can be made on the basis of a two-modal manifold consideration. Owing to such a unique possibility which is a consequence of the sonic vacuum conditions, we could obtain simple analytical thresholds corresponding to the scenario mentioned above.

Data accessibility

This article has no additional data.

Competing interests

We declare we have no competing interests.

Funding

The authors are grateful the Russian Foundation for Basic Research (grant nos. 17-01-00582, 16-02-00400 for the financial support.