Rotational Motions Excited by Vertical Harmonic Motions

0 Introduction

The rotational motion excited by vibrations has been utilized to develop novel rotational devices that can be employed in place of electric motors.For example，rotary motors utilizing ultrasonic vibration are currently commercially available.The advantages of these rotary motors using ultrasonic vibration are high torque and high efficiency at low speeds［1］.

Several configurations for the rotational motions excited by the sonic－range vibrations have been examined.Mainly，there are two configurations of rotational axis and vibration direction，parallel configuration［2－4］ and perpendicular configuration［5－10］.As for parallel configuration，Nakano et al.［3］examined a motion transformation mechanism from oscillation to rotation experimentally and numerically，and Sato et al.［4］proposed the energy regeneration from vibration by an oscillatory rotor.As for perpendicular configuration，automatic balancer［5］，gari－gari dragonfly［6－7］，and hula－hoop［8－10］ have been examined.Gari－gari dragonfly is a Japanese traditional toy，which is that a propeller can be rotated by vertical harmonic motions of a pin inserted into the center hole of the propeller.This rotational motion looks similar to the motion of a hula－hoop.The rotational motion of the hula－hoop is well－known to be driven by the periodical reaction of the hulahoop against a gymnast′s waist.Caughey［8］proposed a model considered the periodic motion of the athlete′s waist along one axis，which is same as the model of parametric resonance［11－13］，and found that the pendulum rotation with an average angular velocity equal to the excitation frequency.Belyakov and Seyranian［9］proposed a model considered the excitation along two axes correspond－ing to an elliptic trajectory of the motion of the athlete′s waist and obtained the solutions by the averaging method［14］corresponding to the rotation with a constant angular velocity equal to the excitation frequency.Also，Seyranian and Belyakov［10］proposed approximate analytical solutions.In the above－mentioned models，it is assumed that the rotation of the hula－hoop is driven by the periodical reaction of the hula－hoop against the periodical motion of the gymnast′s waist.However，the rotational motion driven by non－periodical reaction has received little attention.Here，using an apparatus of the similar configuration to the gari－gari dragonfly，the rotational motion driven by non－periodical reaction of the propeller against the pin occurred.To examine the generation conditions and the mechanism of the rotational motion excited by vertical motion，the relationship between the propeller wing and the inserted pin was examined by tracking their relative motions.

1 Experiment

1.1 Apparatus

Fig.1shows the schematic diagram of the apparatus used in this study.The main part of the apparatus consisted of a wing （acrylic，size 20mm×100mm×1mm，mass 3g），apin（diameter 1mm），and a vibration exciter.The wing had a hole in its center（diameter，φ，of 2or 7mm）as shown in Fig.2.The distance，e，between the centers of the hole and the wing was either 0or 10mm.Four marks for analyzing the motion of the wing were made in the wing.The pin was inserted into the hole in the wing.

The end of the pin was bent into an L－shape so as not to drop the wing.The pin was mounted on a block （wood，size 10mm×10mm×10mm），which was placed atop the vibration exciter.The wing was rotated by the vertical harmonic motion of the pin.The amplitude，A，and the frequency，fv，of the vertical harmonic motion of the pin were adjusted by a function generator.The vibration frequency，fv，of the pin and the rotational frequency，fr，of the wing were evaluated by two separate displacement sensors.The phase difference between the pin and the wing were measured by a high－speed camera and motion analysis software.

Fig.1 Schematic of apparatus（unit：mm）

Fig.2 Specifications of wing（unit：mm）

1.2 Procedure

The wing was installed on the pin and was rotated by means of the vertical harmonic motion of the pin.The rotational frequency of the wing was measured at values of e＝0and 10mm，φ＝2 and 7mm，A＝1.0and 1.5mm，and fv＝5—45Hz.Additionally，the motion of the pin and the wing was captured by a high－speed camera，and the centers of the wing and the pin were tracked by motion analysis software，where the position of the wing′s center of gravity was calculated by the position of the four marks.

2 Results

2.1 Effect of fvon fr

Fig.6shows the effect of the vertical harmonic motion frequency （fv）of the pin on the phase difference（θ）of the y－axis wing motion in relation to the vertical harmonic motion of the pin，which was calculated by 2πτ/Tv，whereτ and Tvare a period from the wing′s top dead posi－tion to the pin′s top dead position as shown in Fig.5and a period of the vertical harmonic motion of the pin，respectively.In the rolling mode and the slipping mode，the phase difference（θ）had a negative value，which increased with increasing frequency of the pin vertical harmonic motion（fv）.The qualitative change inθwas largely independent of the amplitude （A）of the vertical harmonic motion of the pin.In the jumping mode without eccentricity（e＝0mm），the values ofθ were approximately zero and did not change with fv.In the jumping mode with eccentricity（e＝10 mm），the values ofθvaried inconsistently with fv.

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Fig.3 Effect of fvon frat e＝0mm，φ＝7mm，and A＝1.5mm （solid line：Linear trendline）

Fig.4 Effect of fvon frat e＝10mm，φ＝7mm，and A＝1.5mm （solid line：Linear trendline）

Fig.4shows the effect of the vertical harmonic motion frequency of the pin，fv，on the rotational frequency of the wing，fr，at e＝10mm，φ＝7mm，and A＝1.5mm.In the case where e＝10mm，the hole of the wing is located at a distance of 10mm from the center of the wing.When fv＜10Hz，the wing was not rotated by the vertical harmonic motion of the pin.When fv＞20Hz and the initial rotational speed of the wing was low，the jumping mode occurred but did not shift to any other motion modes.When fv≥10Hz and the initial rotational speed of the wing was sufficiently high，the slipping mode occurred.The value of frin the slipping mode increased linearly with increasing fv.The solid line shows the linear trendline fit to the data：fr＝afv，where a＝1.00 （R2＝1.00）.

Fig.3shows the effect of the vertical harmonic motion frequency，fv，of the pin on the rotational frequency，fr，of the wing at e＝0mm，φ＝7mm，and A＝1.5mm.In the cases where e＝0mm，the placement of the hole of the wing corresponded to the center of the wing.When fv＜15Hz，the wing was not rotated by the vertical harmonic motion of the pin.When fv＞15Hz and the initial rotational speed of the wing was low，the jumping mode of motion occurred.The value of frin the jumping mode increased qualitatively with increasing fv.The jumping mode occasionally shifted into the rolling mode.However，the rolling mode did not shift back to the jumping mode.The value of frin the rolling mode increased linearly with increasing fv.The solid line shows the linear trendline fit to the data：fr＝afv，where a＝0.86（R2＝1.00）.When the pin was vibrated through one cycle，the wing rotated 360°around the pin with a rolling contact.The rotational speed，v，is described by v＝πφfr.The circumference，L，of the trajectory of the wing′s center is L＝π（φ－d）.The period，T，of one cycle of the pin is T＝1/fv＝L/v＝（φ－d）/φfr.Therefore，the relationship between frand fvis fr＝（1－d/φ）fv.When d＝1mm andφ＝7mm，the resulting relationship is fr＝6/7 fv＝0.86 fv，which corresponds to the experimental results.

2.2 Trajectory of wing motion

Fig.5 Temporal change in displacement of the wing′s center of gravity along the y－axis（left）and trajectory of the wing′s center of gravity in the x－y plane（right）.

Fig.5shows the temporal change in wing displacement of the wing′s center of gravity along the y－axis（left）and the trajectory of wing′s center of gravity in the x－y plane（right）under each mode of motion.In Fig.5，the closed circle symbols represent the position of the wing′s center of gravity as calculated by the position of the four marks in the wing.The open circle，diamond，square，and triangle symbols represent the points in time when the pin is located at particular positions in the vertical harmonic motion cycle：the top dead point，the bottom dead point，the mid－dle point（upward），and the middle point（downward），respectively.

Temporal change in displacement of the wing along the y－axis（left）and trajectory of wing motion in the x－y plane（right）.The open circle，diamond，square，and triangle symbols represent the times when the position of the pin in the cycle of vertical harmonic motion is at the top dead point，bottom dead point，middle point（upward），and middle point（downward），respectively.

Five distinct motion modes occurred in this study：Slipping，rolling，jumping （without eccentricity），jumping （with eccentricity），and non－rotation.The relationship between frand fv and the relationship between fyand fvfor each mode is summarized in Table 1.

In the jumping mode，two different wing motion trajectories occurred.When e＝0mm，the trajectory looked elliptical.The y－axis motion of the wing was equivalent to that of the pin in both frequency and phase.In contrast，when e＝10mm，the wing trajectory exhibited irregular pattern.The relationship of both the frequency and the phase between the motion of wing and the pin was not readily apparent.

In the jumping mode（with eccentricity），the values ofθvaried inconsistently with fvas shown in Fig.6（d）.It was observed that sometimes the motion of the wing was ahead of the motion of the pin.It means that the motion of the pin brakes the motion of the wing.The rotation mechanism in jumping mode（with eccentricity）is different from simple forced－vibration.The observation results imply that the rotational motion of the wing was driven by the non－periodical force with the collision between the wing hole and the pin.

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2.3 Phase difference between wing and pin

Fig.6 Effect of fvon the phase difference（θ）of y－axis wing motion in relation to the vertical harmonic motion of the pin

The motion modes observed during the measurements of frand fv，were of four types：Slipping，rolling，jumping，and non－rotation.The slipping mode is that the wing was rotated with fr＝fv.The rolling mode is that the wing was rotated with fr＜fvand the wing hole was kept in continuous contact with the pin.The jumping mode is that the wing was rotated with fr＜fvand the wing hole allowed to jump against the pin.The non－rotation mode is that the wing was not rotated while the pin was vibrated at fv.

3 Discussion

3.1 Characteristics of rotation mode

In both the rolling mode and the slipping mode，the trajectory of the wing motion was elliptical in shape.The frequency （fy）of the y－axis motion of the wing was same as the frequency（fv）of the vertical harmonic motion of the pin.However，the y－axis motion of the wing lagged behind that of the pin.The phase difference in the slipping mode was larger than that in the rolling mode.

Table 1 Characteristics of rotation mode

Rotation mode（wing type）frversus fvfyversus fv Rolling（without eccentricity）fr＝ （1－d/φ）fv fy＝fv Slipping（with eccentricity） fr＝fvfy＝fv Jumping（without eccentricity） fr≠fvfy＝fv Jumping（with eccentricity） fr≠fvfy≠fv Non－rotation

3.2 Rotational mechanism of jumping （with eccentricity）

The rotation motion excited by vertical vibration is considered as forced vibration［2］.In a system of forced vibration，the reaction of the object lags a constant phase behind the forced displacement.It means that the energy of forced displacement translates to the object motion.The results of phase difference shown in Fig.6show that slipping，rolling，and jumping（without eccentricity）are well－known mode，which are driven by the periodical reaction of the wing against the periodical motion of the pin such as forced vibration.

Therefore，these results imply that the mechanism of the wing rotation in the jumping mode is different with and without eccentricity.

4 Conclusions

The relationship between the wing and the pin was examined experimentally by tracking their respective motions.Here，five distinct motion modes occurred：slipping，rolling，jumping（without eccentricity），jumping（with eccentricity），and non－rotation.In the case where the hole of the wing was located at a distance from the center of the wing，jumping mode（with eccentricity）occurred.In jumping mode（with eccentricity），the phase difference between fyand fvvaried inconsistently.The rotational motion of the wing was driven by the non－periodical force with the collision between the wing hole and the pin.

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