## Abstract

For two combinations of a dimensionless rotational damping parameter and a dimensionless inertial coupling parameter, we consider free response of a rectilinearly vibrating linearly sprung primary mass inertially coupled to damped rotation of a second mass, for which Gendelman et al. (2012, “Dynamics of an Eccentric Rotational Nonlinear Energy Sink,” ASME J. Appl. Mech. 79(1), 011012) developed equations of motion in the context of a rotational nonlinear energy sink (NES) with no direct damping of the rectilinear motion. For dimensionless initial rectilinear displacements comparable with those considered by Gendelman et al., we identify a region in the motionless projection of the initial condition space (i.e., for zero values of the initial rectilinear and rotational velocities) in which every initial condition leads to a previously unrecognized zero-energy solution, with all initial energy dissipated by rotation. We also show that the long-time nonrotating, rectilinear solutions of the type found by Gendelman et al. are (orbitally) stable only in limited ranges of amplitude. Finally, we show how direct viscous damping of rectilinear motion of the primary mass affects dissipation, and that results with no direct rectilinear dissipation provide excellent guidance for performance when direct rectilinear dissipation occurs. Some applications are discussed.

## 1 Introduction

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].

*a*)

*b*)

*x*is the rectilinear displacement of the primary mass

*M*in the lab frame relative to its equilibrium position,

*θ*is the angular position of the point mass

*m*rotating about the vertical axis at a radius

*r*

_{0},

*C*is a linear spring constant, and

*γ*is a viscous damping coefficient. On the right-hand side (RHS) of Eq. (1

*a*), Sigalov et al. [2,3] introduced a term

*ζdx*/

*dt*to account for (direct) viscous damping of the rectilinear motion of the primary mass, and nondimensionalized the resulting equations as

*a*)

*b*)

*η*=

*x*/

*r*

_{0}and $\tau =tC/(M+m)$, and the dimensionless parameters are a mass ratio

*σ*=

*m*/(

*M*+

*m*), a rotational damping coefficient $\alpha =\gamma /[mr02C/(m+M)]$, and a rectilinear damping coefficient $\lambda =\zeta /C(M+m)$. We specify the initial conditions as

*a*–

*d*)

With *λ* = 0, Eqs. (2*a*,*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 $\alpha =\gamma /[mRg2C/(m+M)]$. As shown in Sec. A available in the Supplemental Materials, Eqs. (2*a*,*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. (2*a*,*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. (2*a*,*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.

*λ*= 0, the long-time solutions include not only the “semi-trivial” [8] solutions

*B*

_{∞}> 0 found previously [1–3], but also a previously unrecognized class of zero-energy solutions in which the initial energy has been completely dissipated. For two combinations of

*α*and

*σ*, including

*α*=

*σ*= 0.1 [1–3], we then establish a range of initial conditions in the MIC space for which complete dissipation of initial energy actually occurs. Notwithstanding the statement [2] that the initial angular position “

*θ*

_{0}is expected to have little effect on the dynamics as long as

*θ*

_{0}≠

*nπ*for any integer

*n*,” we show that the range of initial energy for which complete dissipation occurs depends strongly on the initial angular position.

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.

## 2 Preliminary Considerations

### 2.1 Nature of the Long-Time Solutions.

All solutions of Eqs. (2*a*,*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. (2*a*,*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.

### 2.2 Numerical Methods.

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.

*η*

_{asymp}(

*τ*) =

*B*

_{∞}sin(

*τ*+

*φ*) (sometimes with

*B*

_{∞}= 0) and estimate the constants

*φ*and

*η*(

*τ*) (with asymptotic period 2

*π*). Except where otherwise specified in Sec. 3, we used 50 Levenberg–Marquardt iterations.

### 2.3 Floquet Analysis of the Stability of Semi-trivial Solutions for *λ* = 0.

*λ*= 0, the stability of a semi-trivial solution with respect to infinitesimal disturbances can be assessed by substituting $\theta (\tau )=n\pi +\theta *(\tau )$ and $\eta (\tau )=B\u221esin(\tau +\phi )+\eta *(\tau )$ into Eqs. (2

*a*,

*b*) and linearizing to get

*a*)

*b*)

*B*for which infinitesimal disturbances to the semi-trivial solution grow or decay. We define

_{∞}*s*=

*τ*+

*φ*+ [1 − (−1)

^{n}]

*π*/2, and rewrite Eqs. (8

*a*,

*b*) as

*a*)

*b*)

*A*=

*B*

_{∞}, and note that Eqs. (9

*a*,

*b*) are uncoupled, with no dependence on

*σ*.

*b*) can be transformed into the Mathieu equation using $\theta *=e\u2212\alpha s/2H(s)$ and $s=2s^+3\pi /2$, it is not possible to determine the stability of solutions of Eqs. (2

*a*,

*b*) by reference to the stability boundary for the Mathieu equation. To use results from the Mathieu equation, one would need to know the locus in the

*α*−

*A*plane on which the growth rate is

*α*/2. While this can certainly be done, we instead conduct a Floquet analysis, writing Eqs. (9

*a*,

*b*) as a system of four first-order ordinary differential equations (ODEs)

*a*–

*d*)

*a*–

*d*)

*s*≤ 2

*π*, where

*δ*

_{lm}is the Kronecker delta, to form the matrix

**G**are 1, 1, and the eigenvalues of

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 $[\theta imod\pi ]2+\Omega i2\u22600$ 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 (*F*_{1} and *F*_{2}) and first five branches (*C*_{0}, *C*_{1}, *C*_{2}, *C*_{3}, and *C*_{4}) 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. (2*a*,*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*.

α | ||
---|---|---|

0.1 | 1 | |

A_{0} | 0.4638 | 1.18341 |

A_{1} | 3.7617 | 4.2030 |

A_{2} | 3.7961 | 4.406423 |

A_{3} | 10.6548 | 11.1843 |

A_{4} | 10.6571 | 11.2021 |

α | ||
---|---|---|

0.1 | 1 | |

A_{0} | 0.4638 | 1.18341 |

A_{1} | 3.7617 | 4.2030 |

A_{2} | 3.7961 | 4.406423 |

A_{3} | 10.6548 | 11.1843 |

A_{4} | 10.6571 | 11.2021 |

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].)

## 3 Distinct Regions in the Motionless Initial Condition Space Without Rectilinear Dissipation

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} < *A*_{0}(*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.

### 3.1 (*α*, *σ*, *λ*) = (0.1, 0.1, 0)

#### 3.1.1 Region I Trajectories (*B*_{∞} = 0).

*η*

_{i},

*θ*

_{i}) = (0.03, 0.3

*π*), Figs. 2(a)–2(f) show

*η*,

*dη*/

*dτ*,

*θ*,

*dθ*/

*dτ*,

*B*, and

*D*=

*σ*[

*d*

^{2}

*θ*/

*dτ*

^{2}sin

*θ*+ (

*dθ*/

*dτ*)

^{2}cos

*θ*] (an inertial coupling term). All initial energy is dissipated, and

*B*decays to zero nearly monotonically with very small superimposed decaying oscillations. The asymptotic angular position is 0.1357

*π*. To characterize the decay, we found the local maxima of the very small superimposed oscillations in

*B*(

*τ*). [The small time-step size (imposed by the small integration tolerances) compared with the characteristic oscillation time (2

*π*) means that this can be done very accurately without interpolation.] We then used 500 Levenberg–Marquardt iterations to directly perform a nonlinear least-squares fit (i.e., without a logarithmic transformation of the

*B*values) of 642 local maxima to the exponential model

*B*

_{exp}(

*τ*) =

*g*

_{1}+

*g*

_{2}

*e*

^{−γτ}over the interval 2000 ≤

*τ*≤ 4000. This gave

*g*

_{1}= 1.484 × 10

^{−5},

*g*

_{2}= 2.384 × 10

^{−3}, and

*γ*= 8.816 × 10

^{−4}, with mean absolute values of the absolute and relative errors of 6.235 × 10

^{−7}and 6.644 × 10

^{−4}, respectively. The decay rate is in excellent agreement with linear theory for decay of an infinitesimal disturbance to a zero-energy solution (see Sec. B available in the Supplemental Materials). We also fitted the local maxima, over the same range of

*τ*, to a Landau model

*dy*/

*dτ*=

*ay*−

*by*

^{3}and found

*a*= −8.623 × 10

^{−4},

*b*= 4.078, and

*c*= −153.6, with mean absolute values of the absolute and relative errors of 4.594 × 10

^{−8}and 5.426 × 10

^{−5}, respectively. Since both fits have three parameters, it is clear that the Landau model is superior. The orientation

*θ*and inertial coupling term

*D*also undergo oscillations having the same dominant frequency, and decaying similarly in time. For the entire response, power spectra were calculated, and wavelet transformations were performed according to the procedure described in Sec. C available in the Supplemental Materials, showing (Figs. S1(a,b)) that the dominant frequencies for the rectilinear and rotational responses are 0.1622 and 0.1617, respectively, both very close to the natural frequency of the primary mass, namely, 1/(2

*π*).

For (*η*_{i}, *θ*_{i}) = (0.03, 0.7*π*), 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, 0.3*π*) (Figs. 2(a)–2(f)), except that the final orientation is *θ*_{∞} = 0.8618*π*. For both MICs, *θ*_{∞} is approximately 0.16*π* 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. (2*b*)), as well as to the small value of *η*_{i}, for which linear behavior is a good approximation.

##### 3.1.2 Region IIA Trajectories (*B*_{∞} > 0, *θ*_{∞} = 0).

Figures 4(a)–4(f) show time series for (*η*_{i}, *θ*_{i}) = (0.06, 0.4*π*). 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/(2*π*).

##### 3.1.3 Region IIB Trajectories (*B*_{∞} > 0, *θ*_{∞} = *π*).

Figures 5(a)–5(f) show the response for (*η*_{i}, *θ*_{i}) = (0.06, 0.8*π*), 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} = 0.4*π*. This is because for *θ*_{i} = 0.4*π*, (a) the relative closeness of the initial angular displacement to 0.5*π* gives rise to considerably more “leverage” so that more initial energy is dissipated, and (b) *θ* remains closer to 0.5*π*, so that rotation persists longer than when the initial angular displacement is 0.8*π*.

###### 3.1.4 Boundaries Between Regions.

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 $B\u221e2=h2\eta i\u2212h2\eta i,crit$, and then at each *θ*_{i} performed a least-squares fit (using the fitting parameters *h*^{2} and *h*^{2}*η*_{i,crit}) to the computed values of $B\u221e2$ 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, 0.5128*π*). 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}, Δ*θ* = *θ*_{i,crit,max} − *θ*_{i,crit,min} < 2 × 10^{−4}*π*, with Δ*θ* 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} = 0.5*π* for *η*_{i} < 0.10 to approximately 0.60*π* 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.

### 3.2 (*α*, *σ*, *λ*) = (1, 0.01, 0)

#### 3.2.1 Trajectories in Region I, IIA, and IIB.

For (*α*, *σ*, *λ*) = (1, 0.01, 0), trajectories emanating from (*η*_{i}, *θ*_{i}) = (0.2, 0.5*π*) (in Region I), (0.2, 0.3*π*) (in Region IIA), and (0.2, 0.7*π*) (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, 0.2*π*) and (0.05, 0.8*π*) (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*.

##### 3.2.2 Boundaries Between Regions.

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 $B\u221e2=h2\eta i\u2212h2\eta i,crit$ 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, 0.5024*π*). 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 Δ*θ* < 9 × 10^{−4}*π*. 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} = 0.5*π* to approximately 0.55*π* as *η*_{i} approaches 1.18. Behavior of the long-time solutions near this depression is discussed in more detail in Sec. 5.1.

## 4 Complete Dissipation of Initial Energy Without Rectilinear Dissipation

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} = 0.52*π* and 0.55*π* for (*α*, *σ*, *λ*) = (0.1, 0.1, 0), and at *θ*_{i} = 0.5025*π*, 0.503*π*, and 0.504*π* 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, 0.4*π*)] 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} = 0.5*π* 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. (2*b*) 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 0.5*π*. For these *η*_{i}, departure of the depression from *θ*_{i} = 0.5*π* 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 0.5*π* that *θ*(*τ*) remains close enough to 0.5*π* for long enough that the initial energy can be completely dissipated. For (*α*, *σ*, *λ*) = (1, 0.01, 0) at *θ*_{i} = 0.503*π*, 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 0.5*π* 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} = 0.511*π* (outside of the depression) than for *θ*_{i} = 0.512*π* (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} = 0.512*π*. The dependence of *τ*_{0.01} on *η*_{i} and *θ*_{i} follows accordingly.

## 5 Asymptotically Rectilinear Harmonic Motion

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 $\chi =1\u2212B\u221e2/\eta i2$ (the fraction of the initial energy ultimately dissipated) on *η*_{i} and *θ*_{i}.

### 5.1 Dependence of Asymptotic Rectilinear Amplitude on Initial Conditions.

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} < *A*_{0}(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} = *A*_{0}(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} = 0.5*π*. There is a narrow range of *θ*_{i}, between the turning point (near *θ*_{i} = 0.5128*π*) and the value of *θ*_{i} (near 0.60*π*) 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} = 0.52*π*. 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 Δ*θ* = 10^{−5}*π*, compared with 5 × 10^{−3}*π* 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} (0.584857*π* < *θ*_{i} < 0.584858*π*), 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} = 0.58*π* (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} = 0.563*π* and 0.5849*π* at *η*_{i} = 0.4. For *θ*_{i} = 0.563*π* (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} = 0.5849*π* (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} = 0.5849*π* than for *θ*_{i} = 0.563*π*. 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).

### 5.2 Dependence of Energy Dissipation on Initial Conditions.

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 ($\chi =1\u2212B\u221e2/\eta i2$) 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.

## 6 Damping of the Rectilinear Motion of the Primary Mass

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, 0.3*π*), 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.)

*β*=

*G*

_{rot}/(

*G*

_{rect}+

*G*

_{rot}), where

*λ*= 0's Region I for

*λ*= 10

^{−4}and 10

^{−3}is made clear in Figs. 18(a) and 18(b). Even at larger values of

*λ*, Figs. 18(c) and 18(d) show that the effect of rotational dissipation remains important for these initial conditions.

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*.

## 7 Discussion

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. (2*a*,*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}, 0.5*π*, 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, 0.5*π*, Ω_{i}), similar computations over the range $\u22120.06\u2264\Omega i\u22640.17$ 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].

## 8 Conclusions

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.

## Funding Data

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

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

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.