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Reverse rotation of soft ferromagnetic ball in rotating magnetic field
Accepted Manuscript Reverse Rotation of Soft Ferromagnetic Ball in Rotating Magnetic Field Yeung Yeung Chau, Ruo-Yang Zhang, Weijia Wen PII: DOI: Reference:
S0304-8853(18)31368-4 https://doi.org/10.1016/j.jmmm.2018.12.073 MAGMA 64764
To appear in:
Journal of Magnetism and Magnetic Materials
Please cite this article as: Y.Y. Chau, R-Y. Zhang, W. Wen, Reverse Rotation of Soft Ferromagnetic Ball in Rotating Magnetic Field, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm. 2018.12.073
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Reverse Rotation of Soft Ferromagnetic Ball in Rotating Magnetic Field Yeung Yeung Chau∗, Ruo-Yang Zhang∗, Weijia Wen∗∗ Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
Abstract In this paper, we exhaustively investigated the motion of a soft ferromagnetic ball driven by two rotating magnets with either same or opposite polarity. The experimental observations exhibit plentiful unexpected phenomena in this system, including the counterintuitive reverse rotation of the ball against the direction of the high-speed-rotating magnetic field. For the case of same polarity, the motion of the ball can be classified into several distinct phases, namely, the first direct rotation phase (1st DRP), the unstable phase (UP), the second direct rotation phase (2nd DRP), and the reverse rotation phase (RRP). At the transitional points between 2nd DRP and RRP, the ball can nearly stop on a random point of the orbit with small vibrations. When the height of the ball raises to a certain value, it will eventually be trapped at the center of the field. For the case of opposite polarity, the classification of the motion is roughly the same except some typical distinctions, for example, the rotating diameter of the ball increases with the height in the mass and there exists an anomalous phase. We drew the phase diagrams of the motions and characterized the trajectories for both cases. Keywords: Ferromagnetic particles, Rotating magnetic field, Nonlinear dynamics
1. Introduction
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The classical dynamics of periodically driven system is 30 a widely studied field in physics. There are many counterintuitive phenomena in this kind of systems, such as the stable upside-down pendulum with high-frequency vibrating suspension point [1, 2], and the extraordinary motion of particles on a rotating saddle including both the un- 35 expected trap state [3, 4] and the precessional motion induced by the virtual Coriolis-like force [5, 6]. The rotating saddle trap has inspired the idea of Paul trap and rotatingelectric traps for ions [7, 8], and has been extended to quantum case for trapping Dirac particles [9]. Another 40 specific class of examples is on the diverse motion and deformation of magnetic materials, including point particles and chains [10–12], suspended micro- or nano-scaled objects [13–20], ferrofluid droplets [21–23], and ferrofluid bulk flow [24–26], in time-periodic magnetic fields. And 45 the method of periodic magnetic drive has been used to control the self-assembly of magnetic particles [27–29] and to achieve microrobots [30–32]. However, as a more readily observed case, the planar motion of macroscopic solid particles in time-dependent magnetic fields has rarely been 50 investigated, but we shall see that plentiful interesting dynamic phenomena are hidden in this kind of systems. Consider a millimeter-scale iron ball moving on a plane under the action of magnetic fields produced by two rotating magnets below the plane. At first sight, the ball would 55 ∗ These
authors contributed equally to this work. author Email address: [email protected] (Weijia Wen)
∗∗ Corresponding
Preprint submitted to Journal of Magnetism and Magnetic Materials
be trapped on the top of either one magnet and rotate with it synchronously. If the speed of the magnets is slow and the ball is sufficiently close to the magnets, this anticipation does coincide with the observation. However, the motion of the ball becomes complicated, when the magnetic rotor whirls faster and faster. When the rotating speed is fast enough, the ball will surprisingly begin to rotate in the opposite direction to that of the rotor. Actually, this phenomenon is so prevalent in this system that it can always be observed as long as the height of ball is above a critical value from the magnets for a certain rotating speed. Moreover, the reverse rotation is very stable and insensitive to the initial conditions. We discovered this effect by chance in experiment, while we noticed that a similar phenomenon had been reported recently [33]. Apart from Ref. [33], the asynchronous and partly reverse rotating motion of magnetic field driven objects are also reported in previous works [14–16, 19, 20], however, for those phenomena, the objects only moves reversely against the external field in a fraction of a period, but cannot counter rotate uniformly against the driving field. In this letter, we will investigate the motion of an iron ball driven by two rotating magnets with either same or opposite polarity in detail, and will report a group of unexpected and fascinating motion states of the ball under the drive of the rotating magnets. The dynamics of the ball can be characterized by distinct settled states which are all insensitive to the initial conditions, and are only determined by the external parameters, i.e. the height of the ball above the magnets and the rotor speed spanning a 2D parameter space. These settled states can be classified November 14, 2018
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Figure 2: Radial profiles of (a) the vertical magnetic field Bz of a single magnet, and the profiles of the time-averaged square of the magnetic field hB 2 i produced by the two rotating magnets with (b) the same polarity and (c) the opposite polarity. The red dashed curves denote the loci of the maximum of hB 2 i changing with height h, while the vertical black lines denote the distance (6 mm) between the magnets and the rotating axis. In (a), the fields are measured at the heights from 2 mm to 14 mm with 2 mm intervals. In (b)(c), the heights h corresponding to the curves change from 0 mm to 14 mm at 1 mm intervals.
into different phases, and the phase diagrams for both the same-polarity condition (SPC) and opposite-polarity condition (OPC) are drawn and compared. Apart from the 95 synchronous direct rotation phase and the reverse rotation phase, our system with two driven magnets manifests much more unexpected states of motion of the ball, which are not shown in the the system of 16 circularly arranged magnets studied in Ref. [33], and cannot be explained by the simplified 1D model given in Ref. [33]. For instance, there exists an unstable phase in our system, where the ball is always flung out. And we found an asynchronous direct rotation phase for SPC, where the ball can rotate stably with the same direction but much lower speed than the rotor. Through tuning the parameters, the ball can even nearly stop on the orbit before entering reverse rotation state. Moreover, we also analyzed the trajectories of the ball in different phases quantitatively, which exhibit rich diversity depending on the parameters. 2. Experimental setup
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Figure 1: (a) Schematic diagram for the experimental setup. Two magnets (same polarity or opposite polarity) symmetric with respect to the rotating axis (the orange dashed line) are inserted in a rotor and rotate with it at a speed Ω. An iron ball moves on a plane (wafer) at a height of h above the magnets. For a certain Ω, the ball will rotate reversely to the rotor, when h is higher than a critical height. (b) The magnetic hysteresis loop of the iron ball, magnetic moment m versus magnetic field strength H.
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The experimental setup is illustrated in fig. 1(a). Two columnar magnets of 5 mm in diameter and 18 mm in height with either the same polarity or opposite polarity are symmetrically inserted into a rotor with respect to its rotating axis (the orange dashed line shown in fig. 1(a)), and the center-to-center distance between the two magnets is 12 mm. The rotor rotates anticlockwise around the axis,100 whose speed Ω can be adjusted from 200 rpm (rounds per minute) to 3200 rpm. An iron ball with 1 mm in diameter, driven by the rotating magnetic field, moves on a plane (smooth silicon wafer) suspended above the rotor. The vertical distance h between the wafer and the top105 surface of the magnets can be adjusted from 0.5 mm onwards, consequently, the magnetic field acting on the ball 2
can be modulated by changing the height of the wafer. The iron ball is soft ferromagnetic, and its magnetic moment m is approximately in linear response to the external magnetic field strength H with a small remanence in the working range as shown by the hysteresis loop in fig. 1(b). The detailed element analysis of the ball based on X-ray fluorescence spectrum is supplied in the supplemental materials [34]. We used a Hirst GM08 Gaussmeter with a magnetic field probe for measuring the z-component of the magnetic field (magnetic induction) generated by a single static magnet, at various distances from the magnets. Then we fitted the measured data with a theoretical model (see the supplemental materials [34] for details) and extracted the parameters of the magnet so as to rebuild the complete field distribution generated by two rotating magnets theoretically. The radial profiles of the magnetic field Bz of the static magnet for both experimental measurement and theoretical fitting are shown in fig. 2(a). On account of remanence, the magnetic momentum of the iron ball are assumed to be composed by a linear magnetization part and a remanent moment m0 : m = αB + m0 . In particular, if the remanent moment is always polarized along the external field (this assumption is discussed in Section 4), ˆ Accordingly, the ball perceives we also have m0 = m0 B the external magnetic field as an effective potential [35] U = −m · B = −αB 2 − m0 |B|.
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Since the rotor speed is much faster than the ball’s in most conditions, the time-averaged potential hU i = −αhB 2 i − p 2 m0 hB i, which is circularly symmetric with respect to the central axis, dominates the motion of the ball, and the peak of hB 2 i corresponds to the potential well. As shown in fig. 2(b) and (c), we computed the profiles of hB 2 i(r) for different heights h in terms of the fitted parameters of the magnet from the data in fig. 2(a). The red curves in
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Figure 3: (a) Phase diagram of the motion of the ball in the case of same polarity. The areas of black and blue dots represent 1st and 2nd direct rotation phases respectively; the area of red squares represents the unstable phase; the curve of green stars represents the critical states between 2nd DRP and RRP, where the ball stops on its orbit with slight vibration; the area of blue diamonds is the reverse rotation phase; and the orange triangles denote the critical heights of each Ω at which the ball is trapped at the center. Four representative trajectories of the ball with the parameters of (b) Ω = 500 rpm, h = 2.5 mm in 1st DRP, (c) Ω = 1200 rpm, h = 5.4 mm in 2nd DRP, (d) Ω = 2000 rpm, h = 4.05 mm in RRP, and (e) Ω = 3000 rpm, h = 6.25 mm in RRP. The points of parameters corresponding to (b)–(e) are marked in (a).
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fig. 2(b) and (c) exhibit the peaks of hB 2 i(r) changing with height h for SPC and OPC respectively. For each case, when the height is close to the magnets, the potential well is formed by a ring with the radius roughly corresponding 135 to the locations of the two magnets. However, as the height increases to about 8 mm for SPC and 5 mm for OPC, each ring shrinks into a single well at the center. This feature of the magnetic fields are relevant to the orbital radius of the ball, which will be described in the subsequent text. 140 3. Experimental Results
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Figure 4: (a) Phase diagram of the motion of the ball in the case of opposite polarity. The area of black dots represents the direct rotation phase; the area of red squares represents the unstable phase; the curve of green stars represents the critical states that the ball nearly stops on its orbit; the area of purple triangles represents the anomalous phase; and the area of blue diamonds is the phase of reverse rotation. Four representative trajectories of the ball with the parameters of (b) Ω = 500 rpm, h = 3.5 mm in 1st DRP, (c) Ω = 1300 rpm, h = 3.5 mm in RRP, (d) Ω = 1900 rpm, h = 3.7 mm in RRP, and (e) Ω = 2500 rpm, h = 11 mm in RRP. The points of parameters corresponding to (b)–(e) are marked in (a).
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In experiments, we reset the ball onto the plane and recorded its motion via high-speed digital video, whenever the parameters (h, Ω etc.) were changed. And the mo-145 tions of the ball were analyzed quantitatively via tracking its trajectories from the videos. According to our observations, although the motion of ball depends on its initial conditions in the very beginning stage, the ball will always gradually settle into a particular state, which is only deter-150 mined by the parameters of the system but insensitive to the initial conditions of the ball, namely the settled state of a ball is almost irrelevant to its initial position and velocity as long as the ball is initially placed not so far away 3
from the rotating center with a relatively small velocity. As a result, these initial-condition-insensitive (ICI) states of a certain ball can be directly labeled with the height h and the rotor speed Ω, if all other factors of the experimental system are fixed. Indeed, according to the characters of motion, the ICI states of the ball can be classified into different phases with clear phase transition boundaries in the parameter space (Ω, h). We discretized the parameter space, identified what phase the observed ICI state at each discrete point belongs to, and then drew the two phase diagrams of SPC and OPC separately. In the following sections, we are going to analyze the iron ball behaviors in different kinds of ICI States in details based on the two phase diagrams. 3.1. Same polarity case The phase diagram of the ball subject to the magnetic field of the same polarity is shown in fig. 3(a), where the recorded ICI states are marked with different colors and shapes to denote different phases. The area of black dots represents the first direct rotation phase (1st DRP). In this phase, the ball is dragged by the magnets and rotates synchronously with them. Accordingly, the averaged angular velocities of the ball are identical with the corresponding rotor speed Ω but are irrelevant to the height, as shown
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3.2. Opposite polarity case In OPC, the behavior of the ball is quite similar to SPC, i.e. there also exist 1st DRP, UP, and RRP as shown in fig. 4(a). However, the phase diagram of OPC also has distinct differences from that of SPC. First, there is no 2nd DRP for OPC. After traversing UP, the ball will directly enter the phase of reverse rotation (Ω > 1300 rpm),
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in the upper panel of fig. 5(c). However, there is a boundary of 1st DRP, outside which the ball can no longer catch up with the rotor and will fall into the unstable phase (UP) (area of red squares). Within UP, the ball will be flung out immediately after placing it onto the wafer. The phase boundary between 1st DRP and UP exhibits that the critical height of DRP decreases with increasing Ω, and Ω = 1500 rpm is the maximum speed that can support synchronously rotating states in our system. If we further increase the height at a particular rotor speed, the ICI states will regain stability and enter the second direct rotation phase (2nd DRP) (blue dots in fig. 3(a)). However, 2nd DRP is significantly different from 1st DRP. In 2nd DRP, the ball rotates asynchronously with respect to the rotor, and its angular velocity of orbital revolution is about one order of magnitude smaller than the rotor speed, and decreases rapidly with h raising, eventually reduces to zero (as long as Ω > 1000 rpm). In these extraordinary critical states (green stars in fig. 3(a)), the ball can nearly stop at a random point on the orbit with a definite radius, and rapidly oscillates along a small loop around the “stop point” . Actually, these critical states form the phase transition boundary (green stars) between 2nd DRP and the reverse rotation phase (RRP) (light blue diamonds in fig. 3(a)). Traversing this boundary, the ICI states vary continuously from direct rotation to reverse rotation against the direction of the rotor. Despite the counterintuitiveness, the reverse rotation can be easily observed and is robust as long as the rotating speed of the rotor is high enough. Moreover, the critical height of reverse rotation decreases with Ω increasing, and DRP becomes the210 most prevalent phase of the ball in extremely high speed of the rotor. In both 2nd DRP and RRP, the averaged orbital radius of the ball also decreases with h increasing, and will always eventually reduce to zero. When Ω < 1000 rpm, the or-215 bit radius will decrease to zero before reaching RRP, as a result the phenomenon of reverse rotation cannot be observed as the rotor speed is less than this critical speed. As the orbit radius reduces to zero, the ball is stably trapped at the center of the rotating field but autorotates around220 its vertical symmetric axis, the corresponding states are marked by orange inverted triangles in fig. 3(a). Although the center of the magnetic field is an unstable equilibrium point (a saddle point of the potential acting on the ball) when the magnets do not rotate, these trapped states in225 the fast rotating field are quite stable against perturbations. This phenomenon is very similar to the rotating saddle trap [3, 4].
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or will firstly enter the transition states (Ω > 2000 rpm), namely stopping on a point of the orbit with a definite radius. Besides, in these transition states, the ball oscillates more acutely on a conspicuously larger loop around the “stop point” than the “stop states” in SPC. Second, the orbital radii of ICI states will in general suddenly increase beyond a certain distance in RRP, as a result, the ball will not be trapped eventually at high distance differentiating from the situation in SPC. Third, there is a new “anomalous phase”(AP) (purple triangles in fig. 4(a)) between 1st DRP, UP, and RRP in OPC, in which the radii of trajectories decrease rapidly to zero as h increases, but will suddenly jump to a large value as h traverses the boundary between AP and RRP. In addition, both direct and reverse rotations exist in AP, even the ball will frequently interchange its moving direction in some states. 3.3. Trajectories of the ball Except in UP and AP, the settled trajectories of the ball can be characterized by centered trochoids with the parametric equations xc (ϑ) = a cos ϑ ± b cos (k ϑ) ,
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and epitrochoid respectively, and a, b (a > b) denotes the radii of deferent and epicycle respectively. The choice of ± in eq. (2a) and the parameters a, b, k of the trajectories rely on the strength and pattern of the driven magnetic field as well as the rotor speed. When k is rational, the corresponding trajectories are closed, which have been observed experimentally via fine-tuning of h and Ω, otherwise the trajectories would tend to be dense in the annulus (a − b) ≤ r ≤ (a + b). As examples, several tracked trajectories are shown in fig. 3 and fig. 4 for both the same- and opposite-polarity cases, and more examples can be found in the supplemental materials [34]. In 1st DRP, the radius of epicycle b is comparable with the radius of deferent a caused by the strong magnetic field, thus the trajectories exhibit conspicuous petaloid patterns. However, since the driven field decays with height, the amplitude of petals also decreases accordingly. In 2nd DRP of SPC and RRP of both SPC and OPC, the vestigial petals have become much smaller than the radius of the revolution orbit, thereupon the trajectories approach circles as height increases. In addition, for SPC, all trajectories reduce their radii uniformly as the distance from the rotor increases (fig. 5(a)), this trend is well consistent with the radius variation of the effective potential ring (red dashed curve in fig. 5(a)). This fact indicates that the ball is trapped by the potential ring and moves along it. On the contrary, for OPC, the radii of trajectories match the potential ring only when h < 4 mm. When h > 4 mm, only under relatively low rotor speed, the ball will follow the locus of the potential well, and will fall into a small-radius stage in a certain height range, as illustrated by the curves of Ω = 700 and 1200 rpm in fig. 5(b), which just corresponds to the anomalous phase mentioned above. However, majority of trajectories significantly deviate from the locus of the potential well, and will abruptly increase during the height increase, after that, they resettle on a plane with slow decrease (fig. 5(b)). Figures 5(c) and (d) exhibit the time-averaged orbital angular velocity ω = 2π/T of the ball changing with height under different rotor speeds, where T is the time-averaged orbital period for sufficient revolutions. As shown in the290 upper panels of figs. 5(c) and (d), the ball rotates synchronously with the rotor in 1st DRP for both SPC and OPC. In the lower panels, the section of ω = 0 corresponds to the “stop states” on the phase transition boundary, the ω > 0 half space in fig. 5(c) correspond to 2nd DRP, and the ω < 0 half space of the lower panels in both fig. 5(c)295 and (d) correspond to RRP for SPC and OPC respectively. We can see that the angular velocity (typically |ω| < 20 rad/s) of the ball is much smaller than the rotor speed (> 70 rad/s for the shown data) in 2nd DRP and RRP for SPC and in RRP for OPC. For SPC, ω changes smoothly from > 0 to < 0 at a critical height h0 (Ω), and it tends to an asymptotic frequency ωasy (Ω) as h increases, and the the upper bound |ωasy | of reverse rotation increases with Ω. The relation between ω and h in 2nd DRP and RRP is well characterized by the function 5
Figure 6: Averaged rotating angular velocity ω versus time-averaged orbital radius under 6 different driven frequencies Ω of the rotor in the cases of (a) same polarity and (b) opposite polarity. The data in (a) correspond to the states in 2nd DRP and RRP, while the data in (b) are entirely from RRP. The red curve denotes the expression of the 1D model given by Eq. (4). In each case, an typical examples of the static magnetic field distribution on a ISI circular orbits is plotted, the parameters of the two corresponding ISI states framed by black squares are (h = 5 mm, rc ≈ 4.5 mm) for (a) and (h = 10 mm, rc ≈ 14 mm) for (b).
ω(h) = ωasy 1 − ek(h0 −h) , as shown by the fitted dashed curves in fig. 5(c). However, for OPC, the change of ω in RRP is greatly distinct from SPC. For different rotor speeds Ω, the ball reaches the maximum reverse angular velocities at a nearly fixed height hmra ≈ 3.4mm. In addition, ω also tends to an asymptotic frequency ωasy when h increases, but differentiating form SPC, ωasy in OPC is very close to zero, and is nearly independent of the rotor speed Ω. 4. Discussions about the mechanism In Ref. [33], a 1D model is proposed to interpret the reverse rotation. In this model, the magnetic field is simplified as a traveling harmonic wave with rotating polarization: B = B0 e−kz [cos(Ωt − kx)ex + sin(Ωt − kx)ez ],
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where Ω is the the angular frequency, k = 2π/L is the wave number, and L is the spatial period of the field.
And the remanence of the ferromagnetic ball is crucial in this model, indeed, it turns out that the reverse rotation corresponds to the stationary state that the direction of the remanent magnetic momentum of the ball locks to the345 magnetic field at its location in any time. Assuming the ball rolls on the plane without sliding, the model leads to a simple expression of the orbital angular velocity, ω, of the ball [33] k rb ω=− Ω, (4) 1 − k rb
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where rb is the radius of the ball. For a certain circular350 orbit, the effective 1D wave number k of the driven field is determined by the radius, r, of the orbit. As rb is fixed in our experiments, the ratio ω/Ω is expected to depend only on the orbit radius r. In Fig. 6, we plot anew the data in Fig. 5 to check355 the validity of the 1D model for SPC and OPC respectively. As shown in Fig. 6(a), the data of different driven frequencies Ω do not overlap on a unique curve, as a result, the 1D model falls for explaining SPC, so much as any relations like ω = f (r)Ω. Moreover, Fig. 6(a) shows360 that the angular velocity of the ball tumbles from positive to negative at exactly the radius of the magnets’ location (r = 6 mm). By contrast, most of the data with different driven frequencies uniformly fall on the curve of the 1D model with k = 1/r as illustrated in Fig. 6(b). Thus365 the 1D model in Ref. [33] is still valid for explaining the reverse rotation effect in the systems driven by two magnets with opposite polarity. The validity of this model for OPC can be supported by examining the distribution of the magnetic fields. As shown by the inset of Fig. 6(b), the370 magnetic fields on a typical circular orbit form a quasi-1D harmonic rotating pattern along the azimuthal direction, which is satisfactorily consistent with Eq. (3). While the magnetic field distribution on the orbit (see the inset in Fig. 6(a) are always upturned, which is rather different375 from the assumption of Eq. (4) in the 1D model. The incapability of Eq. (4) for SPC implies that the spatial distribution of the driven magnetic field is a crucial factor to describe the asynchronous and reverse motion. As the magnetic field distribution on an circular orbit cannot be simplifed to a sinusoidal function in SPC, the motion of ball on the plane would have to be solved380 by a truly 3D rigid body dynamics. Even in OPC, it is shown that a portion of the data (near r = 5 mm) deviates from the prediction of the 1D model, which also indicates more complicated mechanism of the reverse rotation states. Nevertheless, the reverse rotation states are surely the most prevalent kind of states, which would appear in the systems with nearly arbitrary arrangement of385 the magnets, therefore, a more general interpretation is needed to reveal the mechanism of such prevalent reverse motion states. In addition, Ref. [33] asserts that the eddy current in the ball produces a similar effect to the magneti-390 zation. However, we found that silver balls do not present orbital motions but merely spun at the center of the rotat6
ing field. Consequently, the eddy current induced by the time-varying magnetic field cannot solely cause the reverse rotation. As to how the eddy current influence the motion of the ball invites further studies. 5. Conclusion To summarize, we have comprehensively investigated the motion of an iron ball in rotating magnetic fields generated by two rotating magnets with same polarity and opposite polarity respectively. Apart from observing the reverse rotation states against the direction of the rotor, we discovered several new kinds of peculiar ICI motion states of the ball, including the asynchronous direct rotation states with stable but much lower speed than the rotor, the “stop states” on a random point of a definite orbit, and the “trapped states” at the center of the field in SPC, all of which are contrary to one’s natural intuition. We drew the phase diagrams for both SPC and OPC, and found that the trajectories of the ball exhibited varied centered-trochoid-like patterns that can be finely tuned by height and rotor speed. After checking the 1D model introduced in Ref. [33], we found that although the model can explain a majority of the reverse motion states satisfactorily in OPC, it is entirely invalid in SPC. Hence, the distribution of the driven field can significantly affect the model. These novel kinds of peculiar states in rotating magnetic field are not only of theoretical interest as new counterintuitive examples in periodically driven dynamic systems, but also have potential value for applications. For instance, they may inspire new trap design for magnetic particles or ions with variable stages, including an orbital trap stage with either direct or reverse rotation, and a centered trap stage corresponding to the trapped states in SPC, controlled by the rotating speed of external fields. Acknowledgments We are grateful to Yun Su for helping us code the program of trajectory tracking. And we thank Chuandeng Hu, Xiping Zeng, Cong Wang, and Li Wang for helpful discussions. This work was supported by Hong Kong RGC Grant (No. HKUST 605411). References [1] J. A. Blackburn, H. J. T. Smith, N. Grønbech-Jensen, Stability and hopf bifurcations in an inverted pendulum, Am. J. Phys. 60 (10) (1992) 903–908. doi:http://dx.doi.org/10.1119/1. 17011. [2] M. Levi, W. Weckesser, Stabilization of the inverted linearized pendulum by high frequency vibrations, SIAM review 37 (2) (1995) 219–223. doi:10.1137/1037044. [3] R. I. Thompson, T. J. Harmon, M. G. Ball, The rotatingsaddle trap: a mechanical analogy to rf-electric-quadrupole ion trapping?, Can. J. Phys. 80 (12) (2002) 1433–1448. doi: 10.1139/p02-110.
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S0304-8853(18)31368-4 https://doi.org/10.1016/j.jmmm.2018.12.073 MAGMA 64764
To appear in:
Journal of Magnetism and Magnetic Materials
Please cite this article as: Y.Y. Chau, R-Y. Zhang, W. Wen, Reverse Rotation of Soft Ferromagnetic Ball in Rotating Magnetic Field, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm. 2018.12.073
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Reverse Rotation of Soft Ferromagnetic Ball in Rotating Magnetic Field Yeung Yeung Chau∗, Ruo-Yang Zhang∗, Weijia Wen∗∗ Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
Abstract In this paper, we exhaustively investigated the motion of a soft ferromagnetic ball driven by two rotating magnets with either same or opposite polarity. The experimental observations exhibit plentiful unexpected phenomena in this system, including the counterintuitive reverse rotation of the ball against the direction of the high-speed-rotating magnetic field. For the case of same polarity, the motion of the ball can be classified into several distinct phases, namely, the first direct rotation phase (1st DRP), the unstable phase (UP), the second direct rotation phase (2nd DRP), and the reverse rotation phase (RRP). At the transitional points between 2nd DRP and RRP, the ball can nearly stop on a random point of the orbit with small vibrations. When the height of the ball raises to a certain value, it will eventually be trapped at the center of the field. For the case of opposite polarity, the classification of the motion is roughly the same except some typical distinctions, for example, the rotating diameter of the ball increases with the height in the mass and there exists an anomalous phase. We drew the phase diagrams of the motions and characterized the trajectories for both cases. Keywords: Ferromagnetic particles, Rotating magnetic field, Nonlinear dynamics
1. Introduction
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The classical dynamics of periodically driven system is 30 a widely studied field in physics. There are many counterintuitive phenomena in this kind of systems, such as the stable upside-down pendulum with high-frequency vibrating suspension point [1, 2], and the extraordinary motion of particles on a rotating saddle including both the un- 35 expected trap state [3, 4] and the precessional motion induced by the virtual Coriolis-like force [5, 6]. The rotating saddle trap has inspired the idea of Paul trap and rotatingelectric traps for ions [7, 8], and has been extended to quantum case for trapping Dirac particles [9]. Another 40 specific class of examples is on the diverse motion and deformation of magnetic materials, including point particles and chains [10–12], suspended micro- or nano-scaled objects [13–20], ferrofluid droplets [21–23], and ferrofluid bulk flow [24–26], in time-periodic magnetic fields. And 45 the method of periodic magnetic drive has been used to control the self-assembly of magnetic particles [27–29] and to achieve microrobots [30–32]. However, as a more readily observed case, the planar motion of macroscopic solid particles in time-dependent magnetic fields has rarely been 50 investigated, but we shall see that plentiful interesting dynamic phenomena are hidden in this kind of systems. Consider a millimeter-scale iron ball moving on a plane under the action of magnetic fields produced by two rotating magnets below the plane. At first sight, the ball would 55 ∗ These
authors contributed equally to this work. author Email address: [email protected] (Weijia Wen)
∗∗ Corresponding
Preprint submitted to Journal of Magnetism and Magnetic Materials
be trapped on the top of either one magnet and rotate with it synchronously. If the speed of the magnets is slow and the ball is sufficiently close to the magnets, this anticipation does coincide with the observation. However, the motion of the ball becomes complicated, when the magnetic rotor whirls faster and faster. When the rotating speed is fast enough, the ball will surprisingly begin to rotate in the opposite direction to that of the rotor. Actually, this phenomenon is so prevalent in this system that it can always be observed as long as the height of ball is above a critical value from the magnets for a certain rotating speed. Moreover, the reverse rotation is very stable and insensitive to the initial conditions. We discovered this effect by chance in experiment, while we noticed that a similar phenomenon had been reported recently [33]. Apart from Ref. [33], the asynchronous and partly reverse rotating motion of magnetic field driven objects are also reported in previous works [14–16, 19, 20], however, for those phenomena, the objects only moves reversely against the external field in a fraction of a period, but cannot counter rotate uniformly against the driving field. In this letter, we will investigate the motion of an iron ball driven by two rotating magnets with either same or opposite polarity in detail, and will report a group of unexpected and fascinating motion states of the ball under the drive of the rotating magnets. The dynamics of the ball can be characterized by distinct settled states which are all insensitive to the initial conditions, and are only determined by the external parameters, i.e. the height of the ball above the magnets and the rotor speed spanning a 2D parameter space. These settled states can be classified November 14, 2018
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into different phases, and the phase diagrams for both the same-polarity condition (SPC) and opposite-polarity condition (OPC) are drawn and compared. Apart from the 95 synchronous direct rotation phase and the reverse rotation phase, our system with two driven magnets manifests much more unexpected states of motion of the ball, which are not shown in the the system of 16 circularly arranged magnets studied in Ref. [33], and cannot be explained by the simplified 1D model given in Ref. [33]. For instance, there exists an unstable phase in our system, where the ball is always flung out. And we found an asynchronous direct rotation phase for SPC, where the ball can rotate stably with the same direction but much lower speed than the rotor. Through tuning the parameters, the ball can even nearly stop on the orbit before entering reverse rotation state. Moreover, we also analyzed the trajectories of the ball in different phases quantitatively, which exhibit rich diversity depending on the parameters. 2. Experimental setup
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Figure 1: (a) Schematic diagram for the experimental setup. Two magnets (same polarity or opposite polarity) symmetric with respect to the rotating axis (the orange dashed line) are inserted in a rotor and rotate with it at a speed Ω. An iron ball moves on a plane (wafer) at a height of h above the magnets. For a certain Ω, the ball will rotate reversely to the rotor, when h is higher than a critical height. (b) The magnetic hysteresis loop of the iron ball, magnetic moment m versus magnetic field strength H.
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The experimental setup is illustrated in fig. 1(a). Two columnar magnets of 5 mm in diameter and 18 mm in height with either the same polarity or opposite polarity are symmetrically inserted into a rotor with respect to its rotating axis (the orange dashed line shown in fig. 1(a)), and the center-to-center distance between the two magnets is 12 mm. The rotor rotates anticlockwise around the axis,100 whose speed Ω can be adjusted from 200 rpm (rounds per minute) to 3200 rpm. An iron ball with 1 mm in diameter, driven by the rotating magnetic field, moves on a plane (smooth silicon wafer) suspended above the rotor. The vertical distance h between the wafer and the top105 surface of the magnets can be adjusted from 0.5 mm onwards, consequently, the magnetic field acting on the ball 2
can be modulated by changing the height of the wafer. The iron ball is soft ferromagnetic, and its magnetic moment m is approximately in linear response to the external magnetic field strength H with a small remanence in the working range as shown by the hysteresis loop in fig. 1(b). The detailed element analysis of the ball based on X-ray fluorescence spectrum is supplied in the supplemental materials [34]. We used a Hirst GM08 Gaussmeter with a magnetic field probe for measuring the z-component of the magnetic field (magnetic induction) generated by a single static magnet, at various distances from the magnets. Then we fitted the measured data with a theoretical model (see the supplemental materials [34] for details) and extracted the parameters of the magnet so as to rebuild the complete field distribution generated by two rotating magnets theoretically. The radial profiles of the magnetic field Bz of the static magnet for both experimental measurement and theoretical fitting are shown in fig. 2(a). On account of remanence, the magnetic momentum of the iron ball are assumed to be composed by a linear magnetization part and a remanent moment m0 : m = αB + m0 . In particular, if the remanent moment is always polarized along the external field (this assumption is discussed in Section 4), ˆ Accordingly, the ball perceives we also have m0 = m0 B the external magnetic field as an effective potential [35] U = −m · B = −αB 2 − m0 |B|.
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Since the rotor speed is much faster than the ball’s in most conditions, the time-averaged potential hU i = −αhB 2 i − p 2 m0 hB i, which is circularly symmetric with respect to the central axis, dominates the motion of the ball, and the peak of hB 2 i corresponds to the potential well. As shown in fig. 2(b) and (c), we computed the profiles of hB 2 i(r) for different heights h in terms of the fitted parameters of the magnet from the data in fig. 2(a). The red curves in
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Figure 3: (a) Phase diagram of the motion of the ball in the case of same polarity. The areas of black and blue dots represent 1st and 2nd direct rotation phases respectively; the area of red squares represents the unstable phase; the curve of green stars represents the critical states between 2nd DRP and RRP, where the ball stops on its orbit with slight vibration; the area of blue diamonds is the reverse rotation phase; and the orange triangles denote the critical heights of each Ω at which the ball is trapped at the center. Four representative trajectories of the ball with the parameters of (b) Ω = 500 rpm, h = 2.5 mm in 1st DRP, (c) Ω = 1200 rpm, h = 5.4 mm in 2nd DRP, (d) Ω = 2000 rpm, h = 4.05 mm in RRP, and (e) Ω = 3000 rpm, h = 6.25 mm in RRP. The points of parameters corresponding to (b)–(e) are marked in (a).
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fig. 2(b) and (c) exhibit the peaks of hB 2 i(r) changing with height h for SPC and OPC respectively. For each case, when the height is close to the magnets, the potential well is formed by a ring with the radius roughly corresponding 135 to the locations of the two magnets. However, as the height increases to about 8 mm for SPC and 5 mm for OPC, each ring shrinks into a single well at the center. This feature of the magnetic fields are relevant to the orbital radius of the ball, which will be described in the subsequent text. 140 3. Experimental Results
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Figure 4: (a) Phase diagram of the motion of the ball in the case of opposite polarity. The area of black dots represents the direct rotation phase; the area of red squares represents the unstable phase; the curve of green stars represents the critical states that the ball nearly stops on its orbit; the area of purple triangles represents the anomalous phase; and the area of blue diamonds is the phase of reverse rotation. Four representative trajectories of the ball with the parameters of (b) Ω = 500 rpm, h = 3.5 mm in 1st DRP, (c) Ω = 1300 rpm, h = 3.5 mm in RRP, (d) Ω = 1900 rpm, h = 3.7 mm in RRP, and (e) Ω = 2500 rpm, h = 11 mm in RRP. The points of parameters corresponding to (b)–(e) are marked in (a).
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In experiments, we reset the ball onto the plane and recorded its motion via high-speed digital video, whenever the parameters (h, Ω etc.) were changed. And the mo-145 tions of the ball were analyzed quantitatively via tracking its trajectories from the videos. According to our observations, although the motion of ball depends on its initial conditions in the very beginning stage, the ball will always gradually settle into a particular state, which is only deter-150 mined by the parameters of the system but insensitive to the initial conditions of the ball, namely the settled state of a ball is almost irrelevant to its initial position and velocity as long as the ball is initially placed not so far away 3
from the rotating center with a relatively small velocity. As a result, these initial-condition-insensitive (ICI) states of a certain ball can be directly labeled with the height h and the rotor speed Ω, if all other factors of the experimental system are fixed. Indeed, according to the characters of motion, the ICI states of the ball can be classified into different phases with clear phase transition boundaries in the parameter space (Ω, h). We discretized the parameter space, identified what phase the observed ICI state at each discrete point belongs to, and then drew the two phase diagrams of SPC and OPC separately. In the following sections, we are going to analyze the iron ball behaviors in different kinds of ICI States in details based on the two phase diagrams. 3.1. Same polarity case The phase diagram of the ball subject to the magnetic field of the same polarity is shown in fig. 3(a), where the recorded ICI states are marked with different colors and shapes to denote different phases. The area of black dots represents the first direct rotation phase (1st DRP). In this phase, the ball is dragged by the magnets and rotates synchronously with them. Accordingly, the averaged angular velocities of the ball are identical with the corresponding rotor speed Ω but are irrelevant to the height, as shown
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3.2. Opposite polarity case In OPC, the behavior of the ball is quite similar to SPC, i.e. there also exist 1st DRP, UP, and RRP as shown in fig. 4(a). However, the phase diagram of OPC also has distinct differences from that of SPC. First, there is no 2nd DRP for OPC. After traversing UP, the ball will directly enter the phase of reverse rotation (Ω > 1300 rpm),
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in the upper panel of fig. 5(c). However, there is a boundary of 1st DRP, outside which the ball can no longer catch up with the rotor and will fall into the unstable phase (UP) (area of red squares). Within UP, the ball will be flung out immediately after placing it onto the wafer. The phase boundary between 1st DRP and UP exhibits that the critical height of DRP decreases with increasing Ω, and Ω = 1500 rpm is the maximum speed that can support synchronously rotating states in our system. If we further increase the height at a particular rotor speed, the ICI states will regain stability and enter the second direct rotation phase (2nd DRP) (blue dots in fig. 3(a)). However, 2nd DRP is significantly different from 1st DRP. In 2nd DRP, the ball rotates asynchronously with respect to the rotor, and its angular velocity of orbital revolution is about one order of magnitude smaller than the rotor speed, and decreases rapidly with h raising, eventually reduces to zero (as long as Ω > 1000 rpm). In these extraordinary critical states (green stars in fig. 3(a)), the ball can nearly stop at a random point on the orbit with a definite radius, and rapidly oscillates along a small loop around the “stop point” . Actually, these critical states form the phase transition boundary (green stars) between 2nd DRP and the reverse rotation phase (RRP) (light blue diamonds in fig. 3(a)). Traversing this boundary, the ICI states vary continuously from direct rotation to reverse rotation against the direction of the rotor. Despite the counterintuitiveness, the reverse rotation can be easily observed and is robust as long as the rotating speed of the rotor is high enough. Moreover, the critical height of reverse rotation decreases with Ω increasing, and DRP becomes the210 most prevalent phase of the ball in extremely high speed of the rotor. In both 2nd DRP and RRP, the averaged orbital radius of the ball also decreases with h increasing, and will always eventually reduce to zero. When Ω < 1000 rpm, the or-215 bit radius will decrease to zero before reaching RRP, as a result the phenomenon of reverse rotation cannot be observed as the rotor speed is less than this critical speed. As the orbit radius reduces to zero, the ball is stably trapped at the center of the rotating field but autorotates around220 its vertical symmetric axis, the corresponding states are marked by orange inverted triangles in fig. 3(a). Although the center of the magnetic field is an unstable equilibrium point (a saddle point of the potential acting on the ball) when the magnets do not rotate, these trapped states in225 the fast rotating field are quite stable against perturbations. This phenomenon is very similar to the rotating saddle trap [3, 4].
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or will firstly enter the transition states (Ω > 2000 rpm), namely stopping on a point of the orbit with a definite radius. Besides, in these transition states, the ball oscillates more acutely on a conspicuously larger loop around the “stop point” than the “stop states” in SPC. Second, the orbital radii of ICI states will in general suddenly increase beyond a certain distance in RRP, as a result, the ball will not be trapped eventually at high distance differentiating from the situation in SPC. Third, there is a new “anomalous phase”(AP) (purple triangles in fig. 4(a)) between 1st DRP, UP, and RRP in OPC, in which the radii of trajectories decrease rapidly to zero as h increases, but will suddenly jump to a large value as h traverses the boundary between AP and RRP. In addition, both direct and reverse rotations exist in AP, even the ball will frequently interchange its moving direction in some states. 3.3. Trajectories of the ball Except in UP and AP, the settled trajectories of the ball can be characterized by centered trochoids with the parametric equations xc (ϑ) = a cos ϑ ± b cos (k ϑ) ,
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and epitrochoid respectively, and a, b (a > b) denotes the radii of deferent and epicycle respectively. The choice of ± in eq. (2a) and the parameters a, b, k of the trajectories rely on the strength and pattern of the driven magnetic field as well as the rotor speed. When k is rational, the corresponding trajectories are closed, which have been observed experimentally via fine-tuning of h and Ω, otherwise the trajectories would tend to be dense in the annulus (a − b) ≤ r ≤ (a + b). As examples, several tracked trajectories are shown in fig. 3 and fig. 4 for both the same- and opposite-polarity cases, and more examples can be found in the supplemental materials [34]. In 1st DRP, the radius of epicycle b is comparable with the radius of deferent a caused by the strong magnetic field, thus the trajectories exhibit conspicuous petaloid patterns. However, since the driven field decays with height, the amplitude of petals also decreases accordingly. In 2nd DRP of SPC and RRP of both SPC and OPC, the vestigial petals have become much smaller than the radius of the revolution orbit, thereupon the trajectories approach circles as height increases. In addition, for SPC, all trajectories reduce their radii uniformly as the distance from the rotor increases (fig. 5(a)), this trend is well consistent with the radius variation of the effective potential ring (red dashed curve in fig. 5(a)). This fact indicates that the ball is trapped by the potential ring and moves along it. On the contrary, for OPC, the radii of trajectories match the potential ring only when h < 4 mm. When h > 4 mm, only under relatively low rotor speed, the ball will follow the locus of the potential well, and will fall into a small-radius stage in a certain height range, as illustrated by the curves of Ω = 700 and 1200 rpm in fig. 5(b), which just corresponds to the anomalous phase mentioned above. However, majority of trajectories significantly deviate from the locus of the potential well, and will abruptly increase during the height increase, after that, they resettle on a plane with slow decrease (fig. 5(b)). Figures 5(c) and (d) exhibit the time-averaged orbital angular velocity ω = 2π/T of the ball changing with height under different rotor speeds, where T is the time-averaged orbital period for sufficient revolutions. As shown in the290 upper panels of figs. 5(c) and (d), the ball rotates synchronously with the rotor in 1st DRP for both SPC and OPC. In the lower panels, the section of ω = 0 corresponds to the “stop states” on the phase transition boundary, the ω > 0 half space in fig. 5(c) correspond to 2nd DRP, and the ω < 0 half space of the lower panels in both fig. 5(c)295 and (d) correspond to RRP for SPC and OPC respectively. We can see that the angular velocity (typically |ω| < 20 rad/s) of the ball is much smaller than the rotor speed (> 70 rad/s for the shown data) in 2nd DRP and RRP for SPC and in RRP for OPC. For SPC, ω changes smoothly from > 0 to < 0 at a critical height h0 (Ω), and it tends to an asymptotic frequency ωasy (Ω) as h increases, and the the upper bound |ωasy | of reverse rotation increases with Ω. The relation between ω and h in 2nd DRP and RRP is well characterized by the function 5
Figure 6: Averaged rotating angular velocity ω versus time-averaged orbital radius under 6 different driven frequencies Ω of the rotor in the cases of (a) same polarity and (b) opposite polarity. The data in (a) correspond to the states in 2nd DRP and RRP, while the data in (b) are entirely from RRP. The red curve denotes the expression of the 1D model given by Eq. (4). In each case, an typical examples of the static magnetic field distribution on a ISI circular orbits is plotted, the parameters of the two corresponding ISI states framed by black squares are (h = 5 mm, rc ≈ 4.5 mm) for (a) and (h = 10 mm, rc ≈ 14 mm) for (b).
ω(h) = ωasy 1 − ek(h0 −h) , as shown by the fitted dashed curves in fig. 5(c). However, for OPC, the change of ω in RRP is greatly distinct from SPC. For different rotor speeds Ω, the ball reaches the maximum reverse angular velocities at a nearly fixed height hmra ≈ 3.4mm. In addition, ω also tends to an asymptotic frequency ωasy when h increases, but differentiating form SPC, ωasy in OPC is very close to zero, and is nearly independent of the rotor speed Ω. 4. Discussions about the mechanism In Ref. [33], a 1D model is proposed to interpret the reverse rotation. In this model, the magnetic field is simplified as a traveling harmonic wave with rotating polarization: B = B0 e−kz [cos(Ωt − kx)ex + sin(Ωt − kx)ez ],
(3)
where Ω is the the angular frequency, k = 2π/L is the wave number, and L is the spatial period of the field.
And the remanence of the ferromagnetic ball is crucial in this model, indeed, it turns out that the reverse rotation corresponds to the stationary state that the direction of the remanent magnetic momentum of the ball locks to the345 magnetic field at its location in any time. Assuming the ball rolls on the plane without sliding, the model leads to a simple expression of the orbital angular velocity, ω, of the ball [33] k rb ω=− Ω, (4) 1 − k rb
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where rb is the radius of the ball. For a certain circular350 orbit, the effective 1D wave number k of the driven field is determined by the radius, r, of the orbit. As rb is fixed in our experiments, the ratio ω/Ω is expected to depend only on the orbit radius r. In Fig. 6, we plot anew the data in Fig. 5 to check355 the validity of the 1D model for SPC and OPC respectively. As shown in Fig. 6(a), the data of different driven frequencies Ω do not overlap on a unique curve, as a result, the 1D model falls for explaining SPC, so much as any relations like ω = f (r)Ω. Moreover, Fig. 6(a) shows360 that the angular velocity of the ball tumbles from positive to negative at exactly the radius of the magnets’ location (r = 6 mm). By contrast, most of the data with different driven frequencies uniformly fall on the curve of the 1D model with k = 1/r as illustrated in Fig. 6(b). Thus365 the 1D model in Ref. [33] is still valid for explaining the reverse rotation effect in the systems driven by two magnets with opposite polarity. The validity of this model for OPC can be supported by examining the distribution of the magnetic fields. As shown by the inset of Fig. 6(b), the370 magnetic fields on a typical circular orbit form a quasi-1D harmonic rotating pattern along the azimuthal direction, which is satisfactorily consistent with Eq. (3). While the magnetic field distribution on the orbit (see the inset in Fig. 6(a) are always upturned, which is rather different375 from the assumption of Eq. (4) in the 1D model. The incapability of Eq. (4) for SPC implies that the spatial distribution of the driven magnetic field is a crucial factor to describe the asynchronous and reverse motion. As the magnetic field distribution on an circular orbit cannot be simplifed to a sinusoidal function in SPC, the motion of ball on the plane would have to be solved380 by a truly 3D rigid body dynamics. Even in OPC, it is shown that a portion of the data (near r = 5 mm) deviates from the prediction of the 1D model, which also indicates more complicated mechanism of the reverse rotation states. Nevertheless, the reverse rotation states are surely the most prevalent kind of states, which would appear in the systems with nearly arbitrary arrangement of385 the magnets, therefore, a more general interpretation is needed to reveal the mechanism of such prevalent reverse motion states. In addition, Ref. [33] asserts that the eddy current in the ball produces a similar effect to the magneti-390 zation. However, we found that silver balls do not present orbital motions but merely spun at the center of the rotat6
ing field. Consequently, the eddy current induced by the time-varying magnetic field cannot solely cause the reverse rotation. As to how the eddy current influence the motion of the ball invites further studies. 5. Conclusion To summarize, we have comprehensively investigated the motion of an iron ball in rotating magnetic fields generated by two rotating magnets with same polarity and opposite polarity respectively. Apart from observing the reverse rotation states against the direction of the rotor, we discovered several new kinds of peculiar ICI motion states of the ball, including the asynchronous direct rotation states with stable but much lower speed than the rotor, the “stop states” on a random point of a definite orbit, and the “trapped states” at the center of the field in SPC, all of which are contrary to one’s natural intuition. We drew the phase diagrams for both SPC and OPC, and found that the trajectories of the ball exhibited varied centered-trochoid-like patterns that can be finely tuned by height and rotor speed. After checking the 1D model introduced in Ref. [33], we found that although the model can explain a majority of the reverse motion states satisfactorily in OPC, it is entirely invalid in SPC. Hence, the distribution of the driven field can significantly affect the model. These novel kinds of peculiar states in rotating magnetic field are not only of theoretical interest as new counterintuitive examples in periodically driven dynamic systems, but also have potential value for applications. For instance, they may inspire new trap design for magnetic particles or ions with variable stages, including an orbital trap stage with either direct or reverse rotation, and a centered trap stage corresponding to the trapped states in SPC, controlled by the rotating speed of external fields. Acknowledgments We are grateful to Yun Su for helping us code the program of trajectory tracking. And we thank Chuandeng Hu, Xiping Zeng, Cong Wang, and Li Wang for helpful discussions. This work was supported by Hong Kong RGC Grant (No. HKUST 605411). References [1] J. A. Blackburn, H. J. T. Smith, N. Grønbech-Jensen, Stability and hopf bifurcations in an inverted pendulum, Am. J. Phys. 60 (10) (1992) 903–908. doi:http://dx.doi.org/10.1119/1. 17011. [2] M. Levi, W. Weckesser, Stabilization of the inverted linearized pendulum by high frequency vibrations, SIAM review 37 (2) (1995) 219–223. doi:10.1137/1037044. [3] R. I. Thompson, T. J. Harmon, M. G. Ball, The rotatingsaddle trap: a mechanical analogy to rf-electric-quadrupole ion trapping?, Can. J. Phys. 80 (12) (2002) 1433–1448. doi: 10.1139/p02-110.
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