Free Response of a Rotational Nonlinear Energy Sink: Complete Dissipation of Initial Energy for Small Initial Rectilinear Displacements

Inertial coupling of rectilinear motion of a linearly sprung primary mass to viscously damped rotation of a second mass about a vertical axis can lead to complicated dynamics, and “targeted energy transfer” in which kinetic energy of the rectilinear motion is transferred to rotational motion, which is then dissipated [1–3]. In such a rotational “nonlinear energy sink” (NES), the second mass can rotate at any angular speed, with no “preferred” frequency, and so a properly designed system can be expected to extract and dissipate energy from the rectilinear motion of the primary structure over a range of natural frequency of the latter. The approach has been demonstrated in experiments [1], and in simulations of flow-induced vibration (FIV) of a circular cylinder [4–6] and impulsively driven structural vibration [7].

With λ = 0, Eqs. (2a,b) have also been shown to apply if the rotating mass is distributed [4], with the distance from its center of mass to the axis (R_cm) allowed to differ from the radius of gyration (R_g). In this case, σ = m(R_cm/R_g)²/(m + M), and $α=γ/[mRg2C/(m+M)]$⁠. As shown in Sec. A available in the Supplemental Materials, Eqs. (2a,b) also describe the situation in which a single linearly sprung, rigid, distributed mass undergoes viscously damped rotation about a vertical axis which does not pass through its center of mass, and which moves horizontally with the rectilinear vibration of the body. Because Eqs. (2a,b) apply equally well to these other cases (for which σ is not a mass ratio), we hereafter refer to σ as a dimensionless coupling parameter.

For (α, σ, λ) = (0.1, 0.1, 0), Gendelman et al. [1] presented numerical solutions of Eqs. (2a,b) for six different motionless initial conditions (MICs, with v_i = Ω_i = 0), and in each case found that the long-time solution was one for which the primary mass oscillates harmonically with a nonzero amplitude, with the angular position an integer multiple of π. Sigalov et al. [2,3] subsequently considered σ = 0.1 with α = 0.05 and 0.1, again with MICs, and for λ = 0 reported a range of dynamical behavior, including a sequence of transitions between regular motion and chaos. They also stated [3] that computations with σ = 0.03 and α = 0.2 produced similar results. They presented no results for λ > 0, but stated that computations for λ > 0 were “similar” to those for λ = 0 and were “omitted for the sake of brevity”. Here, we reconsider the same dynamical equations studied previously [1–3], with three main objectives.

Second, for λ = 0, we show that the semi-trivial solutions, with harmonic rectilinear motion and no rotation, are orbitally stable only in limited ranges of amplitude and are unstable for all other amplitudes.

Third, we add damping of the primary mass, corresponding to λ > 0, in which case the only long-time solutions are zero-energy, regardless of the initial condition. We show that the time required to dissipate 99% of the initial energy correlates well with the predictions of the λ = 0 analysis, provided that the damping of the primary mass is weak.

The remainder of the paper is organized as follows. For λ = 0, we identify two classes of long-time solutions in Sec. 2, where we briefly describe the numerical methods used, and present results of a Floquet-based stability analysis, showing that for each α, the semi-trivial solutions are orbitally stable only in finite ranges of amplitude and are unstable for all other amplitudes. In Sec. 3, we present representative trajectories for several MICs and delineate the boundaries between three distinct regions in the MIC space, including one in which every initial condition leads to a zero-energy solution, with the initial energy completely dissipated by the NES. This is followed by a discussion of trajectories that are (a) asymptotically motionless (Sec. 4), or (b) asymptotically semi-trivial (Sec. 5). In Sec. 6, we consider the effects of viscous damping of the rectilinear motion of the primary mass, followed by a discussion in Sec. 7, and conclusions in Sec. 8.

All solutions of Eqs. (2a,b) reported previously [1–3] were computed for λ = 0 and are of the form shown in Eq. (4). In addition to those solutions, it is evident that Eqs. (2a,b) have zero-energy solutions with η(τ) = 0 and θ(τ) = θ_∞, where θ_∞ is arbitrary.

It is clear that the long-time solutions must correspond to constant values of θ (denoted by θ_∞). (In the alternative, the RHS of Eq. (6) shows that E must decrease, showing that no long-time solution is possible unless θ_∞ is constant.) For λ > 0, every long-time solution must have zero energy, with (η(τ), θ(τ)) = (0, θ_∞), again with θ_∞ arbitrary.

For numerical integration, we used the matlab code ode45, which implements a (4,5) Runge–Kutta method, with a step size adjustable to satisfy specified error tolerances. The absolute and relative error tolerances, taken to be equal here, and referred to as δ, are set to 10⁻⁸. For the values of η_i considered here, the results are insensitive to small changes in initial conditions, and this tolerance yields results sufficiently converged for the intended purpose.

The nature of the repeated root at 1 means that no semi-trivial solution is asymptotically stable, even for infinitesimal disturbances. From a physical standpoint, we see that any initial condition with $[θimodπ]2+Ωi2≠0$ leads to dissipation, so that the final amplitude of the rectilinear motion will always be less than the initial value. If one or both eigenvalues of H lie outside the unit circle, then the semi-trivial solution is unstable, whereas if both eigenvalues lie inside the unit circle, the semi-trivial solution is orbitally stable, in the sense that every infinitesimal initial disturbance leads to only an infinitesimal departure from the nominal solution. The case of instability is separated from the case of orbital stability by what we will call the stability boundary, corresponding to the situation where the eigenvalues of H lie on the unit circle.

Figure 1 shows a multi-valued stability boundary, where critical values lie on branches (denoted by C_i) terminating on the A-axis and connected at turning points (denoted by F_i), with the first two turning points (F₁ and F₂) and first five branches (C₀, C₁, C₂, C₃, and C₄) shown in Fig. 1. Consecutive branches (C_2k−1 and C_2k, k ≥ 1) bound “tongues” (denoted by T_k), each extending as a cusp to the A-axis, reminiscent of the tongues of instability for the Mathieu equation [10]. That is not surprising, in light of the fact that our boundaries correspond to the boundaries for which the Mathieu equation is unstable with a (positive) growth rate of α/2.

For λ = 0, the stability boundary separates combinations of α and A (and hence B_∞) for which semi-trivial solutions of Eqs. (2a,b) are orbitally stable (denoted by “S”), from those for which disturbances grow (denoted by “U”). For α > 0, the number of critical values lying below any A is a nonincreasing function of α, decreasing by two at each passage of a constant-α line beyond a turning point. Table 1 shows the first five critical values of A_k(α) for the values of α considered here. We note that A_2k − A_2k−1 (i.e., the tongue “width”) decreases rapidly with increasing k and with decreasing α.

These results are consistent with the numerical simulations of Gendelman et al. [1], in which all reported long-time solutions corresponded to harmonic rectilinear motion of the primary mass, with the angular position of the rotating mass being an integer multiple of π. (The zero-energy alternative B_∞ = 0, discussed in Secs. 3 and 4, was not recognized in Refs. [1–3].)

For λ = 0, we show here that different MICs can lead to qualitatively different long-time solutions, including solutions from which all initial energy is dissipated, a result not reported in the previous work [1–3].

For (α, σ, λ) = (0.1, 0.1, 0) [1–3] and (1, 0.01, 0), we restrict consideration to 0 < η_i < A₀(α). (For larger initial displacements, the dynamics and the dependence of solutions on initial conditions are considerably more complicated and will be considered separately.) For the range of initial rectilinear displacements considered, we show that the MIC space is divided into three simply connected regions, in which (a) B_∞ = 0 and all initial energy is dissipated by the NES (Region I), (b) B_∞ > 0 and θ_∞ = 0 (Region IIA), and (c) B_∞ > 0 and θ_∞ = π (Region IIB).

For (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0), the values of the coefficient ασ in the energy dissipation rate (Eq. (6)) are equal, but the coupling (represented by σ) and damping (represented by α) differ by factors of ten. For both combinations of α and σ, we illustrate the solutions in Regions I, IIA, and IIB by discussing specific trajectories, followed by a detailed delineation of the boundaries between regions. Dependence of the long-time solutions (characterized by B_∞ and θ_∞) on η_i and θ_i within the three regions is discussed in Secs. 4 and 5.

For (η_i, θ_i) = (0.03, 0.7π), all initial energy is again dissipated, with the time series (Figs. S2(a–f), available in the Supplemental Materials) being similar to those for (η_i, θ_i) = (0.03, 0.3π) (Figs. 2(a)–2(f)), except that the final orientation is θ_∞ = 0.8618π. For both MICs, θ_∞ is approximately 0.16π closer to the nearest integer multiple of π than is the initial orientation. Approximate symmetry of the behavior for these two cases is a consequence of the equal proximity of the values of θ_i to an integer multiple of π (leading to small values of the RHS of Eq. (2b)), as well as to the small value of η_i, for which linear behavior is a good approximation.

For (η_i, θ_i) = (0.06, 0.5π), Figs. 3(a)–3(f) show that the response is similar to the cases discussed above, except that θ undergoes noticeably asymmetric oscillation about θ_∞ = 0.4866π, as opposed to the essentially one-sided approach of θ to its final value for the previous cases.

Figures 4(a)–4(f) show time series for (η_i, θ_i) = (0.06, 0.4π). We see that B initially decreases rapidly, and then asymptotically approaches B_∞ = 0.03127 (Fig. 4(e)). The time series for θ undergoes oscillatory decay to zero (Fig. 4(c)). After an initial decrease from zero to −0.05397 (at τ = 1.42), and an increase to 0.05886 (at τ = 4.427), dθ/dτ undergoes oscillatory decay to zero (Fig. 4(d)). Wavelet transformations and power spectra (Figs. S3(a–f), available in the Supplemental Materials) again affirm that the dominant frequencies for the rectilinear and rotational responses are very close to 1/(2π).

Figures 5(a)–5(f) show the response for (η_i, θ_i) = (0.06, 0.8π), with long-time solution (B_∞, θ_∞) = (0.05509, π). The time series are qualitatively similar to those shown in Figs. 4(a)–4(f), except that the approach to the semi-trivial solution is faster than for θ_i = 0.4π. This is because for θ_i = 0.4π, (a) the relative closeness of the initial angular displacement to 0.5π gives rise to considerably more “leverage” so that more initial energy is dissipated, and (b) θ remains closer to 0.5π, so that rotation persists longer than when the initial angular displacement is 0.8π.

Here, we delineate the boundaries in the MIC space between the three regions discussed above. Figures 6(a) and 6(b) show B_∞ and θ_∞ for 0 < η_i ≤ 0.46 and 0 < θ_i < π, obtained from 14,268 trajectories (using nonuniform increments of η_i and θ_i). (Values of B_∞ and θ_∞ for many of these initial conditions are provided in Tables S1(a,b), available in the Supplemental Materials) Region I extends over the entire range of θ_i for η_i = 0, narrowing rapidly as η_i increases, and becoming exceedingly narrow for η_i greater than about 0.14. For η_i ≤ 0.10, the boundaries between Regions I and IIA and between Regions I and IIB (in both cases denoted by η_i,crit) rapidly approach and were determined as follows. For the boundary between Regions I and IIA, we rewrote the local approximation B_∞ = h(η_i − η_i,crit)^1/2 as $B∞2=h2ηi−h2ηi,crit$⁠, and then at each θ_i performed a least-squares fit (using the fitting parameters h² and h²η_i,crit) to the computed values of $B∞2$ at the six or so smallest values of η_i for which B_∞ is between 10⁻³ and 10⁻². The lower branch of the boundary between Regions I and IIB was determined by the same procedure. The two branches of the boundary separating Region I from IIB meet at a turning point near (η_i, θ_i) = (0.10, 0.5128π). For 0.10 < η_i ≤ 0.14, we computed the boundaries θ_i,crit,min (between Regions I and IIA) and θ_i,crit,max (between Regions I and IIB) by distinguishing trajectories for which B_∞ = 0 (Region I) from those for which either θ_∞ = 0 (Region IIA) or θ_∞ = π (Region IIB). For larger η_i, Δθ = θ_i,crit,max − θ_i,crit,min < 2 × 10⁻⁴π, with Δθ narrowing rapidly as η_i increases. In this part of Region I, which we refer to as the “depression,” the existence and location of the range for which B_∞ = 0 is inferred from the transition from θ_∞ = 0 to π as θ_i increases. As η_i increases, the dependence of B_∞ on θ_i shows a pronounced depression in Region I, which shifts from near θ_i = 0.5π for η_i < 0.10 to approximately 0.60π at η_i = 0.46, as indicated in Fig. 6(a). Behavior of the long-time solutions near this depression is discussed in more detail in Sec. 5.1.

For (α, σ, λ) = (1, 0.01, 0), trajectories emanating from (η_i, θ_i) = (0.2, 0.5π) (in Region I), (0.2, 0.3π) (in Region IIA), and (0.2, 0.7π) (in Region IIB) are shown in Figs. S4(a–f), S5(a–f), and S6(a–f), respectively, available in the Supplemental Materials. By comparing Figs. S4, S5, and S6, available in the Supplemental Materials, to Figs. 3, 4, and 5, we note that the time series for (α, σ, λ) = (1, 0.01, 0) are qualitatively similar to those for (α, σ, λ) = (0.1, 0.1, 0) in each region. For these values of η_i and θ_i, the approximate symmetry discussed in Sec. 3.1 obtains. This is illustrated by time series for (η_i, θ_i) = (0.05, 0.2π) and (0.05, 0.8π) (shown in Figs. S7(a–f) and S8(a–f), respectively, available in the Supplemental Materials), in which the dynamics are approximately linear, owing to the small value of σ and the proximity of θ_i to an integer multiple of π.

For (α, σ, λ) = (1, 0.01, 0), in the MIC space covering 0 < η_i ≤ 1.18 and 0 < θ_i < π, we established the boundaries separating Region I from Regions IIA and IIB using the same procedure as for (α, σ, λ) = (0.1, 0.1, 0), except that the “cutoff” value of η_i is now 0.36, below which the fit to $B∞2=h2ηi−h2ηi,crit$ was used. Figures 7(a) and 7(b) show the distributions of B_∞ and θ_∞ (interpolated based on trajectories computed using 4730 MICs), with the turning point of the boundary between Regions I and IIB being located near (η_i, θ_i) = (0.36, 0.5024π). Values of B_∞ and θ_∞ are shown in Tables S2(a) and S2(b), available in the Supplemental Materials, respectively, for all 4730 trajectories. For η_i > 0.36, the gap between the left and right branches is Δθ < 9 × 10⁻⁴π. Similar to results for (α, σ, λ) = (0.1, 0.1, 0), as η_i increases, the depression of Region I shifts from near θ_i = 0.5π to approximately 0.55π as η_i approaches 1.18. Behavior of the long-time solutions near this depression is discussed in more detail in Sec. 5.1.

For (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0), the asymptotic approach of B(τ) to zero illustrated in Sec. 3 shows that the initial energy can be completely dissipated in Region I, a possibility not recognized in previous work [1–3]. This is of particular interest when the objective of design is to completely suppress vibration.

The dependence of B_∞ and θ_∞ on η_i for several values of θ_i is shown in Figs. 8(a)–8(c) and Figs. S9(a–c), available in the Supplemental Materials, for (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0), respectively. Results for both sets of dimensionless parameters are qualitatively similar, with increases in η_i leading to re-entry into Region I at θ_i = 0.52π and 0.55π for (α, σ, λ) = (0.1, 0.1, 0), and at θ_i = 0.5025π, 0.503π, and 0.504π for (α, σ, λ) = (1, 0.01, 0).

For trajectories leading to B_∞ = 0, a key question is how long it takes to dissipate, say, 99% of the initial energy. In Region I, we define τ_0.01 as the time at which E(τ_0.01) = 0.01 E(0). Note that τ_0.01 is finite in Region I and is not defined in most of Regions IIA and IIB, because less than 99% of the initial energy is dissipated in most of those regions. In each of those regions, there is, however, a very small zone adjacent to Region I, in which τ_0.01 is defined and grows very rapidly as one moves away from the boundary with Region I, as shown in Fig. S10, available in the Supplemental Materials, for (α, σ, λ) = (0.1, 0.1, 0). On the scale of Figs. 9(a) and 9(b), that zone is too small to be apparent.

Results in Figs. 9(a) and 9(b) for (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0) show that values of τ_0.01 for (α, σ, λ) = (1, 0.01, 0) are generally greater than those for (α, σ, λ) = (0.1, 0.1, 0). For these two sets of parameters with identical MICs [(η_i, θ_i) = (0.04, 0.4π)] lying within Region I, Figs. 10(a)–10(c) show time series of dE/dτ, D, and B. It is clear that coupling in the (α, σ, λ) = (0.1, 0.1, 0) case is stronger than for (α, σ, λ) = (1, 0.01, 0), giving rise to a greater initial dissipation rate, which ultimately leads to shorter τ_0.01.

Figures 9(a) and 9(b) show that for each η_i, the smallest value of τ_0.01 occurs near the center of Region I (near θ_i = 0.5π for sufficiently small η_i), and that τ_0.01 increases monotonically as θ_i departs from the center, consistent with the fact that Eq. (2b) shows that the strongest inertial coupling (and hence most efficient energy transfer) occurs when sin θ achieves its maximum value. For larger η_i (i.e., beyond values shown in Figs. 9(a) and 9(b)), the only values of θ_i in Region I for which all of the initial energy is dissipated lie in a very narrow range of θ_i (in the depression) greater than 0.5π. For these η_i, departure of the depression from θ_i = 0.5π occurs because initial rectilinear motion in the −η direction has the effect of immediately reducing θ, and it is only for θ_i in a narrow range greater than 0.5π that θ(τ) remains close enough to 0.5π for long enough that the initial energy can be completely dissipated. For (α, σ, λ) = (1, 0.01, 0) at θ_i = 0.503π, time series shown in Figs. 11(a)–11(f), 12(a)–12(f), and S11(a,b), available in the Supplemental Materials, illustrate this for η_i = 0.421 (in the depression) and 0.425 (above the depression, in Region IIA). For η_i = 0.421 (Figs. 11(a)–11(f) and S11(a), available in the Supplemental Materials), θ remains near 0.5π much longer than for the η_i = 0.425 trajectory (Figs. 12(a)–12(f) and S11(b), available in the Supplemental Materials). Similar behavior is observed for (α, σ, λ) = (0.1, 0.1, 0), where for η_i = 0.12, Figs. S12(a,b), available in the Supplemental Materials, show that the initial decay of the mean value of θ(τ) is considerably more rapid for θ_i = 0.511π (outside of the depression) than for θ_i = 0.512π (inside the depression). On the other hand, the “settling time,” expressed in terms of the time required for the energy to fall, say, 99% of the difference between its initial and final values, is considerably faster for θ_i = 0.512π. The dependence of τ_0.01 on η_i and θ_i follows accordingly.

For initial conditions outside of Region I, it is not possible to dissipate all initial energy, and the final state will be one for which B_∞ > 0, with θ_∞ an integer multiple of π. Here, for Regions IIA and IIB, we characterize the dependence of B_∞, θ_∞, and $χ=1−B∞2/ηi2$ (the fraction of the initial energy ultimately dissipated) on η_i and θ_i.

For (α, σ, λ) = (1, 0.01, 0), Fig. 7(a) and Table S2(a), available in the Supplemental Materials, show that for η_i < A₀(1), B_∞ increases monotonically with increasing η_i, except for values of θ_i for which there is more than one crossing from Region I into either Region IIA or IIB. Moreover, B_∞ generally varies unimodally with θ_i, having maximum values at θ_i = 0 and π, and falling to zero as Region I is approached from either side. The final angular position, θ_∞, is zero everywhere in Region IIA and π everywhere in Region IIB, with a continuous variation through the intervening Region I. For (α, σ, λ) = (0.1, 0.1, 0), Fig. 6(a) and Table S1(a), available in the Supplemental Materials, show that the variation of B_∞ with η_i is monotonic for η_i less than approximately 0.4, again with the exception of those values of θ_i for which there is more than one crossing from Region I into either Region IIA or IIB. The more complicated dependence of B_∞ on η_i and θ_i just below η_i = A₀(0.1) is discussed below.

For (α, σ, λ) = (0.1, 0.1, 0) and several θ_i, Figs. 8(a) and 8(b) show the local square-root dependence B_∞ = h(η_i − η_i,crit)^1/2 of B_∞ on η_i used in Sec. 3 to determine the boundary of Region I. Figure 8(c) shows that there is a range of θ_i for which θ_∞ depends nonmonotonically on η_i. For each θ_i considered, the dependence of B_∞ on η_i is nearly linear over 0.3 ≤ η_i ≤ 0.4, beyond which the dependence is complicated near θ_i = 0.5π. There is a narrow range of θ_i, between the turning point (near θ_i = 0.5128π) and the value of θ_i (near 0.60π) in the depression for η_i = 0.46, for which there are two disjoint ranges of η_i wherein B_∞ = 0. We illustrate this for θ_i = 0.52π. Increasing η_i brings the initial state from Region I (with B_∞ = 0) into Region IIB (with B_∞ > 0) near η_i = 0.0848. Further increases in η_i result in B_∞ decreasing to zero as the initial condition again approaches Region I (near η_i = 0.1541). After passage through Region I (with B_∞ = 0), the initial condition emerges into Region IIA (near η_i = 0.1545), with further increases in η_i leading again to square-root dependence of B_∞ on η_i. At still higher η_i, B_∞ increases nearly linearly with η_i for η_i < 0.4. For still larger η_i, B_∞ increases monotonically but sublinearly as η_i increases. For (α, σ, λ) = (1, 0.01, 0), a similar dependence of B_∞ and θ_∞ on η_i and θ_i is shown in Figs. S9(a–c), available in the Supplemental Materials. Note that the nonmonotonic dependence of B_∞ on η_i and θ_i above η_i = 0.4 in the (α, σ, λ) = (0.1, 0.1, 0) case has no counterpart for (α, σ, λ) = (1, 0.01, 0).

For (α, σ, λ) = (0.1, 0.1, 0) and η_i = 0.1, Fig. 13 shows that B_∞ decreases monotonically with θ_i from B_∞ = η_i at θ_i = 0 to B_∞ = 0 at the boundary between Regions I and IIA and remains zero through the depression. Emerging from the depression, B_∞ grows rapidly as θ_i increases, ultimately reaching η_i at θ_i = π. Similar unimodal dependence of B_∞ on θ_i is observed for η_i = 0.2 and 0.3, with the width of the depression in Region I being less than Δθ = 10⁻⁵π, compared with 5 × 10⁻³π for η_i = 0.1. For η_i = 0.4, dependence of B_∞ on θ_i is more complex than the unimodal behavior found for η_i = 0.1, 0.2, and 0.3. For η_i = 0.4, B_∞ = 0 in only a narrow range of θ_i (0.584857π < θ_i < 0.584858π), where the lower and upper bounds are ones for which we compute θ_∞ = 0 and π, respectively. Thus, MICs leading to values of B_∞ smaller than the minima shown near θ_i = 0.58π (Fig. 13) are difficult to locate. The corresponding variation of θ_∞ with θ_i is shown in Figs. 14(a) and 14(b) for η_i = 0.1 and 0.4, respectively, and is used to identify the transition at larger η_i. For η_i = 0.1, the transition from θ_∞ = 0 to θ_∞ = π (across the width of Region I) occurs over a narrow range of θ_i, for which intermediate values of θ_∞/π are shown at two values of θ_i. For η_i = 0.4, the range is exceedingly narrow, and no intermediate values of θ_∞/π were computed.

To better understand the more complicated dependence of B_∞ on θ_i described above for η_i = 0.4 (see Fig. 13), we show time series of θ and dθ/dτ in Figs. 15(a)–15(d) for θ_i = 0.563π and 0.5849π at η_i = 0.4. For θ_i = 0.563π (in Region IIA), Figs. 15(a) and 15(b) show that the orientation and angular velocity undergo three excursions, after which dθ/dτ settles into a multi-peak oscillation whose amplitude decays slowly to zero, while the angular displacement approaches θ_∞ = 0. On the other hand, for θ_i = 0.5849π (in Region IIB, very close to the boundary with Region I), Fig. 15(c) shows that θ undergoes seven oscillations about θ_i before (nonmonotonically) approaching π. Again following seven early oscillations, dθ/dτ also decays nonmonotonically to zero (Fig. 15(d)). That the amount of energy dissipated increases with the number of excursions explains why B_∞ is smaller for θ_i = 0.5849π than for θ_i = 0.563π. For (α, σ, λ) = (1, 0.01, 0), complicated dependence of B_∞ on θ_i near the depression of Region I also occurs (Fig. S13, available in the Supplemental Materials).

From Fig. 6(a), it is clear that for sufficiently large η_i, complete dissipation of initial energy is possible only in a very narrow range of θ_i. The η_i- and θ_i-dependence of the fraction of initial energy ultimately dissipated (⁠$χ=1−B∞2/ηi2$⁠) is shown in Figs. 16(a) and 16(b) for (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0), respectively. In each white area and in the associated depression extending upward from it (Region I in Figs. 8(a) and 8(b)), all initial energy is dissipated, and χ = 1. Over the entire range of θ_i in Regions IIA and IIB, χ generally decreases to a value less than 0.36 as η_i increases to 0.46. Complex dependence of χ on θ_i sets in near the depression (as reflected in the complex dependence of B_∞ on θ_i shown in Fig. 13 for η_i = 0.4). For (α, σ, λ) = (1, 0.01, 0), similar dependence of χ on MICs is shown in Fig. 16(b), with χ in Regions IIA and IIB being generally less than for (α, σ, λ) = (0.1, 0.1, 0).

From the standpoint of energy dissipation, Figs. 16(a) and 16(b) show that the contours of the “worst” parts of the MIC space (i.e., with smallest χ) are very different for the two combinations of α and σ. For both cases, there is essentially no dissipation for initial conditions in significant portions of Regions IIA and IIB, near whose θ_i = 0 and θ_i = π boundaries there is very little inertial coupling between the translational and rotational motions, and hence little energy transfer from the former to the latter. The (α, σ, λ) = (0.1, 0.1, 0) and (1, 0.01, 0) cases differ markedly, however, in the shape and extent of the portions of Regions IIA and IIB in which there is essentially no dissipation, and indeed in the distribution of χ more generally. It is particularly noteworthy that for (α, σ, λ) = (0.1, 0.1, 0), the portion of Region IIB in which there is essentially no dissipation narrows as η_i increases beyond about 0.2, whereas that portion of Region IIA widens continuously as η_i increases.

For λ > 0, it is clear that the long-time solution has zero energy, regardless of initial conditions. Thus, unlike the λ = 0 case discussed above, τ_0.01 (the time required to dissipate 99% of the initial energy) is defined for all initial conditions. Here, we focus on how τ_0.01 depends on η_i and θ_i for both combinations of α and σ.

Figures 17(a)–17(d) show τ_0.01 for (α, σ, λ) = (0.1, 0.1, λ) with λ = 10⁻⁴, 10⁻³, 3 × 10⁻³, and 10⁻². At any combination of η_i and θ_i, τ_0.01 decreases monotonically with increasing λ, as expected. A typical case is (η_i, θ_i) = (0.02, 0.3π), for which τ_0.01 = 837.3, 817.8, 677.2, 499.7, and 269.3 for λ = 0, 10⁻⁴, 10⁻³, 3 × 10⁻³, and 10⁻², respectively. We see that for combinations of η_i and θ_i lying within λ = 0's Region I, the values of τ_0.01 for λ = 10⁻⁴ and 10⁻³ are quite similar to the λ = 0 case. The very large values of τ_0.01 for λ = 0 in Region I near the boundaries with Regions IIA and IIB (red in Fig. 9(a)) have given way at λ = 10⁻⁴ to considerably smaller values of τ_0.01 for the same values of η_i and θ_i, a clear influence of the rectilinear damping. (Note that the false-color scale is different for Fig. 9(a) than for Figs. 17(a)–17(d). A more direct comparison of τ_0.01 is provided in Fig. S14, available in the Supplemental Materials) Even for λ = 3 × 10⁻³ and 10⁻², the largest values of τ_0.01 continue to be centered about θ_i/π = 0.5, although at the same values of η_i and θ_i, τ_0.01 is considerably smaller than it is for smaller λ. For each λ ≥ 0, the minimum value of τ_0.01 occurs at θ_i/π = 0.5 in the limit as η_i → 0. (This is because θ_i/π = 0.5 provides the most “leverage,” and η_i → 0 is the limit in which temporal deviations from θ_i/π = 0.5 are smallest.)

For (α, σ, λ) = (1, 0.01, λ), Figs. 19(a)–19(d) show τ_0.01 with the same values of λ. The results, and their relationship to the λ = 0 case (Fig. 9(b)), are qualitatively quite similar to those for (α, σ, λ) = (0.1, 0.1, λ), with the main difference being that the values of τ_0.01 are larger than for (α, σ, λ) = (0.1, 0.1, λ). This is a direct “carry-over” from the λ = 0 case and reflects the fact that the stronger inertial coupling (larger σ) plays a dominant role in transferring energy from the rectilinear mode to the rotational mode, where it is dissipated. Figures S15(a–d), available in the Supplemental Materials, show that, as for (α, σ, λ) = (1, 0.01, λ), rotational dissipation is dominant for small λ and remains important even at larger λ.

For the simple two-body point-mass system considered by Gendelman et al. [1] (i.e., with no direct damping of the primary mass), and for the equivalent distributed-mass and one-body systems governed by the same equations (see Ref. [4] and Sec. A available in the Supplemental Materials), we have shown that there is a part of the MIC space (Region I) in which every trajectory leads to a zero-energy final state. That result, not recognized in earlier work [1–3], along with the energy dissipation results in Secs. 4 and 5.2, are of particular interest from the standpoint of applications. Specifically, they show that advantageous selection of the dimensionless coupling parameter σ and dimensionless damping parameter α can significantly affect the portion of the initial condition space in which all initial energy is dissipated when there is no direct damping of the motion of the primary structure (Region I), and how fast that occurs. Such selection can also affect the fraction of the initial energy dissipated in other portions of the initial condition space (Regions IIA and IIB), as recognized by Saeed et al. [7] in a higher-dimensional structural application. Not surprisingly, the results depend significantly on the individual values of α and σ, and not simply on their product (see Eq. (6)). When direct damping of the motion of the primary structure is accounted for (Sec. 6), the λ = 0 results provide very good guidance as to how fast the zero-energy state is approached, especially when damping of the motion of the primary structure is relatively weak.

In some applications, the initial conditions are not motionless [7], and so it is important to know whether our analysis carries over to the more general case. While a complete exploration of the four-dimensional initial condition space for Eqs. (2a,b) is beyond the scope of this work, for (α, σ, λ) = (1, 0.01, 0) we have computed the range of initial rectilinear velocities v_i for which initial conditions (η_i, v_i, θ_i, Ω_i) = (0.2, v_i, 0.5π, 0) lead to trajectories with B_∞ = 0. We find that all initial potential and kinetic energy is dissipated for −0.0914 ≤ v_i ≤ 0.0903. For (η_i, v_i, θ_i, Ω_i) = (0.2, 0.05, 0.5π, Ω_i), similar computations over the range $−0.06≤Ωi≤0.17$ show that all of the initial potential and kinetic energy can be dissipated when all of the initial displacements and velocities are nonzero.

To put into context the initial dimensionless rectilinear displacements η_i considered here, we compare the ranges of initial energy corresponding to our initial rectilinear displacements of 0 < η_i ≤ 0.46 for (α, σ) = (0.1, 0.1), and 0 < η_i ≤ 1.18 for (α, σ) = (1, 0.01), to the ranges of initial energy considered by Saeed et al. [7]. In that computational study of damping by a rotational NES of impulsively driven structural vibration, the mass and stiffness matrices were taken from experiments with 51 kg two-story [11] and 10,000 kg nine-story [12] structures. In Sec. D available in the Supplemental Materials, we show that η_i = 0.46 corresponds to an initial energy nearly equal to the largest value considered by Saeed et al. for the nine-story structure, and to an energy approximately five times that at which their simulation predicted maximum dissipation (over a specified time interval) for that structure. For the two-story structure, maximum dissipation occurred at an initial energy only slightly greater than that corresponding to η_i = 0.46. On the other hand, η_i = 1.18 corresponds to an initial energy more than five times the largest considered by Saeed et al. for the nine-story structure, and slightly larger than the largest energy considered for the two-story structure. This shows that the initial energies considered here are comparable with, and in some cases considerably larger than, those relevant to structural vibration mitigation.

When design criteria allow for rotation of the primary structure about an axis not containing its center of mass, then as discussed in Sec. 1 and in Sec. A available in the Supplemental Materials, the dimensionless damping coefficient and the distance of the center of mass of the primary structure from the axis of rotation can be selected to achieve the same dynamical response that could have been effected by a second rotating mass, thus eliminating the second mass and its connection to the shaft about which it would rotate.

Finally, in addition to structural applications, we note that a rotational NES can both suppress and excite FIV in flow past a cylinder [4–6]. This suggests potential applications to enhancement of FIV of submerged cylinders in energy harvesting by vibrating cylinders submerged in riverine, estuarine, or marine currents [13], as well as in low Reynolds number mixing applications [14,15].

For a rotational nonlinear energy sink with no direct damping of the motion of the primary mass, we have examined part of the motionless projection of the initial condition space (i.e., in which the initial rectilinear and angular velocities are zero) for two combinations of the dimensionless coupling parameter and dimensionless rotational damping coefficient. We have identified a range of motionless initial conditions for which every trajectory leads to previously unrecognized zero-energy final states and characterized the time required for the initial energy to dissipate. We have also shown that the previously identified solutions, with harmonic rectilinear motion of the primary mass, and no motion of the rotatable mass, are (orbitally) stable only for limited ranges of the amplitude, and unstable outside of those ranges. Finally, when direct damping of the rectilinear motion of the primary mass is weak, we show that the characteristic time to dissipate the initial energy correlates well with predictions for the undamped case.

• The authors gratefully acknowledge support from National Science Foundation (NSF) Grant CMMI-1363231.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.