Ball-bearing motor effect with rolling cylinders

D.B.Watson, S.M.Patel and N.F!Sedcole

Abstract: A force in the direction of motion is exerted on a metal cylinder when rolling on metal rails connected to a current source. The force is almost directly proportional to the current and to the mass of the rolling cylinder. A model involving thermal expansion is put forward to explain these results.

1 Introduction

When an electric current is passed between the outer and inner races of a rotating ball bearing a torque is developed in the direction of motion [l]. This ball bearing motor phe- nomenon has been the subject of some sophisticated elec- tromagnetic theoretical analysis [2, 31 and several experimental investigations [MI. There appears to be some conflict between the results of these experiments and the theoretical predictions.

It is shown in [3] that when a current I flows in a ball- bearing motor a torque results from the interaction between the magnetic field of the central axis and the induced currents and magnetic fields developed withn the balls. The theory predicts that the driving torque z of the ball-bearing motor can be expressed in the form

r = K I 1 ~ p 2 a w (1) where is the magnetic permeability of the ball, (r is its electrical conductivity, w is the angular velocity and Kl is a constant for a particular ball. Under certain conditions and with appropriate assumptions this theoretical relationship has yielded a value of z which is close to that measured experimentally [2, 31. However, the only experimental study of the ball-bearing motor in which the torque was directly measured over a wide range of currents has shown that z is not proportional to I2 but actually almost directly propor- tional to I [4].

A similar result has been obtained from an experiment in which a steel ball was rolled along parallel steel rails, an electric current passing from rail to rail through the ball. A driving force taken as acting through the centre of the ball was exerted in the direction of motion, and over a wide current range the force was almost directly proportional to the current [5]. Subsequently in an experiment of a similar nature in which a nonmagnetic metal ball was rolled along nonmagnetic metal rails, a driving force proportional to the current was recorded [6]. Those experimental results have highlighted two inadequacies of eqn. 1 in that the force was not found to be proportional to the square of the current, neither was there any significant dependency on magnetic permeability although the force does depend to some extent on the materials in contact [6].

0 IEE, 1999 IEE Proceedings online no. 19990289 DOL 10. 1049/ip-smt:19990289 Paper first received 6th May and in revised form 4th August 1998 The authors are with the Department of Electrical & Electronic Engineer- ing, University of Canterbury, Private Bag 4800, Christchurch, New Zea- land

One further significant observation arising from the experiment in whch a ball was rolled along parallel cylin- drical rails is that as the separation of the rails is increased so the electrical driving force is raised [6]. The reason for this change in the driving force is not clear. The force may be directly related to the distance L between the contact points. Alternatively it may be related to diameter D of the rolling circle, or it may be related to the radial load P at the contact points.

Eqn. 1 apparently does not involve the dimensions of the ball or the loading at the contact points. However, these are involved in the constant K, which is proportional to the cube of the diameter and to F(B,). Here F(B,) is a function of the flux density B,, within the ball at the contact points. Mechanical loading determines the contact area and the angle subtended by the contact cap. For a given current the value of B, is inversely proportional to the angle of contact [2] so that the value of K, is expected to fall as the loading is raised. It is therefore of some interest to investigate experimentally how the physical dimensions and the mechanical loading affect the driving force.

4 L -

Fig. 1 Distance between contact points L = 2R sin 0; rolling diameter D = 2R cos 0; load at contact points P = mg/? cos 0

Sphere q f r d i w R rollirg on purull.1 cyli?ukicul mils

The situation with the rolling ball is illustrated in Fig. 1. If the rail separation is increased, the diameter D of the rolling circle falls and the radial contact load P rises. For a sphere rolling on parallel cylindrical rails the distance between contact points, the diameter of the rolling circle, and the radial contact loading are closely coupled.

83 ZEE Proc.-Sei. Mea.r Technol., Vol. 146, No. 2, March 1999

Independent control of these parameters is difficult to achieve: hence a different technique has been used. To determine how the electrical driving force is related to L, D and P the following investigations have been carried out with cylinders rolling on parallel cylindrical rails. Ths ena- bles independent variation of these parameters to be made. The results are discussed in terms of forces arising at the contact points.

2 Experimental results

The electrical driving forces exhibited by various readily available materials have recently been compared [7]. The ranking of these materials from best to worst was stainless steel, carbon, mild steel, brass, copper, aluminium and lead. The material giving the highest force was chosen for ths investigation: nonferromagnetic stainless steel grade 304 (containing 18%0 chromium and 10% nickel) in the form of solid cylindrical rods. Two 12mm diameter rods each of 600mm length were mounted parallel on a board which was tilted at a small angle a = 0.57" as shown in Fig. 2. A third rod of mass m was allowed to roll down the incline. A constant-current source was connected to the rails at the foot of the incline. An oscilloscope was connected across the rails at the top of the incline to display the voltage across the rolling cylinder. Measurement of the time T taken to roll down a distance il = 500mm of the rails was taken from the oscilloscope, together with the contact volt- age V.

odjustabk current source

0-LOA

ongk of incbnatio

\ I insuiation tape'

I 1 oscilloscope

I t l o ~

Fig.2 , Experkntal setup The oscilloscope records time and potential between rails

When the cylinder rolls down the slope there are three forces to consider. The first is the gravitational force mg sin a in the direction of motion. The second is the electrical driving force F in the direction of motion. The third is a retarding force that occurs whenever there is rolling motion: it arises partly from molecular adhesion between the rolling surfaces and partly from the local deformation of the cylinder and the rails at the contact zones, which results in a loss of energy owing to mechanical hysteresis and local slip [8, 91. This retarding force, or rolling fric- tional force, is represented in the following analysis by a force F0 taken as acting through the centre of the cylinder, opposing its motion. Assuming that each of these forces is constant, the driving force is related to the experimental rolling time T by

3mX T2

F E - + Fo - rngsincu

Now if the driving force is proportional to the current as found in previous work [5-7], say

84

F = h'2I (3) then I should be related to T2 in the same way as F but with a change in scale i.e.

3mX I = - Fo - mg sin a K2T2 + h'2

The experimental results such as those illustrated in Fig. 3 were therefore plotted in the form of T2 against I . The vir- tually linear relationship over a wide range of current justi- fies the assumption that over this range the force is directly proportional to current. The slopes of the characteristics shown in Fig. 3 when multiplied by 3rd. give the constant K2 in newton per amp. It is this factor K2, or the force per amp, which is used to assess the influence of the parameters under investigation.

In the first experiment the effect of rail separation was investigated using a 32mm diameter roller of mass m = 0.9kg. The stainless steel rod was rolled down the incline five times at each current setting over a range of currents from 0.5 to 20A. Results are given in Fig. 3 for rail separa- tion 35, 50 and 95mm. The corresponding K2 values of 4.2, 4.3 and 4.3mN/A show that the driving force is not related to the distance between the contact points.

0.12y

N I

I-

N I

t-

N I

0 0 5 10 15 20

a

O l I

(1 5 10 15 20 b

O l I

0 5 10 15 20 current, A

C

Fig.3 Roller diameter 32mm: roller mass 0.9kg: rail diameter 12mm n 35mm h 50mm c 95mm

Results of experhmts with variable rail separation

The next experiment was performed to investigate the relationship between the driving force and the diameter of the rolling cylinder, using cylinders of various diameters but of constant mass m = 0.9kg. The force per amp for these cylinders shown in Fig. 4 falls from 5.1 to 4.1 mN/A over a diameter range from 16 to 38mm. The 19.6% reduc- tion in force over the 140% increase in diameter is a rela- tively small fall in force as the diameter is increased. The theoretical points on Fig. 4 were calculated using eqn. 9 developed in Section 3.

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The influence of the contact load P was investigated using 32mm diameter rollers of different lengths hence dif- ferent mass. The experimentally derived force per amp from the T2 against I characteristics, plotted against the mass of the roller in Fig. 5, clearly shows that the electrical driving force is directly related to the radial mechanical load P at the contact points. The driving force consistently increases with loading at the rate of 2.3mNiA per kg.

0.5

0.2 N3 ; :I , , , * , 10.1 - 0 0

10 15 20 25 30 35 Lo ro l le r d iometer,mm

Force per amp us function of roller diameter, for roller mass of 0.9kg Fig.4 A experimental points 0 theoretical points

12r

01 I

0 0.5 1 1.5 2 2.5 3 3.5 L roNer moss, kg

Fig.5 Force per amp rn function of muss of rolling cylinder

0.61

0 . 5 ~ 0 5 10 15 20 25

current,A Fig.6 Contuct voltuge f o r rolling cylinder diameter 32mnz, m s 0.9kg

During the progress of these investigations the voltage V across the ends of the rails was recorded. This is plotted against current in Fig. 6 for a rolling cylinder of mass 0.9kg and 32mm diameter. Very similar results were obtained for other cylinders. This gives the voltage dropped across the two contact points plus the voltage drop across the rolling cylinder. In all cases this voltage had a constant component of about 0.8V and a small ohmic component of the order of 0.01VlA.

3 Discussion

The experimental results have established that over a wide range of current the electrical driving force on the rolling cylinder is almost directly proportional to the current. This is in agreement with previously published results [MI but is in conflict with the electromagnetic theory [2, 31. It has been found that although the force does not vary rapidly with the diameter of the rolling cylinder, it tends to fall as

IEE Proc.-Sci. Meas. Technol.. Vol. 146, No. 2, Murch 1999

the diameter is increased, again in conflict with the electro- magnetic theory which predicts that the force is propor- tional to the square of the diameter. Also the force has been found to be directly proportional to the load at the contacts whereas the electromagnetic theory appears to pre- dict an inverse relationship.

A previous investigation [7] has shown that both solid and hollow cylinders of the same weight and diameter experience the same electrical driving force. It therefore seems unlikely that the force is generated in a distributed way over the whole volume of the cylinder. Taken together with the present results, and others [6, 71, it is suggested that the electrical driving force is exerted within the mechanical contact zones between the roller and the rails. Now the contact area between crossed cylinders of steel is an ellipse of major and minor axes 213 and 2b, respectively, where in mm

(4) 2a=O.lOlp ~ [AI (5)

Here P is the load in kg, the sum of the curvatures is Cp = RI-' + R2-' with the cylinder radii R, and R2mm, the fac- tors p and v determined from the ratio (RI - R2)/(R1 + R2) being taken from tables [SI. Within this mechanical contact zone there are several smaller areas of good electrical con- tact [lo]. Heat produced by current flowing through the small contact regions raises the local temperature. The tem- perature may rise to the melting point [lo].

When the cylinder is stationary on the rails there will be a symmetrical temperature profile across the mechanical contact zone, but when the cylinder rolls on the rails cool material enters the contact zone at the leading edge while hot material leaves at the t r a h g edge producing an asym- metrical temperature profile across the contact zone. Also when the cylinder is rolling, the current distribution may be asymmetrical because electrical contact is most likely to be first made near the region of higher pressure at the centre of the contact zone, and once made the electrical contact is broken only near the trailing edge. Hence both electrical and thermal imbalance can occur when the cylinder rolls.

The imbalanced situation within the mechanical contact zone implies the possibility of torque-producing mecha- nisms other than the one which leads to eqn. 1. They include the electrodynamic force of repulsion and the mechanical force of repulsion due to thermal expansion. Such forces can only produce a rolling torque when their distribution within the contact zone is asymmetrical, result- ing in a differential force Fd operating at an effective dis- tance d from the centre of the contact zone. T h s occurs when the cylinder is rolling, as discussed. The correspond- ing force F considered as acting through the centre of the cylinder is then F = d&#Rl where RI is the radius of the rolling cylinder.

The required magnitude of Fd can now be determined. Taking the case of a 0.9kg cylinder of 32mm diameter roll- ing on 12mm diameter rails, the driving force from Fig. 3 at midrange current of 10A is 43mN. Each contact zone has a major axis 2u = 0.182mm and minor axis 2b = 0.092mm. The corresponding differential force acting at a nominal distance from the centre d = a12 = 0.046mm is Fd = 15 or 7.5N at each contact when the current is IOA. Now consider whether either of the mechanisms mentioned could exert this differential force across the face of the con- tact zone.

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First consider the electrodynamic force, or pinch effect, due to the flow of current from rail to roller. This can pro- duce a driving torque when the current flowing through the trailing region of the contact zone exceeds the current flow- ing through the leading region. The electrodynamic repul- sion force between identical crossed cylinders in contact when carrying a current 1 has been analysed by Holm [IO]. The force is

where R is the cylinder radius and U is the radius of the contact area. At the midrange current 1 = 10A and with ln(Nu) = 10 this force is F, = 104N, clearly several orders of magnitude below the required 7.5N. Likewise at 10A the Lorentz force on the rolling cylinder is negligible in com- parison with the actual driving force, in fact Fig. 2 implies that the Lorentz force is in the opposite direction to the driving force.

Before considering the effects of thermal expansion we would mention that in the mid-1800s it was known that a hollow metal ball placed on parallel metal rails vibrates and may roll back and forth when an electric current is passed through it from rail to rail. It was found that when the rails take the form of a circular track, continuous motion of the ball is possible. This phenomenon was explained by Tyn- dall [l 11 in terms of thermal expansion producing local ele- vation of the rails at the contact points. A similar explanation for the ball-bearing motor has been put for- ward by Marinov and others [l]. However, there has been no theoretical analysis of the thermal expansion model. No relationship between the thermal driving force and other parameters has been found in the literature, but it is possi- ble to estimate the mechanical elastic force F, associated with thermal expansion. With a linear temperature coefi- cient of expansion a and a temperature rise 8 the prospec- tive local expansion at an electrical contact spot is a 8 per unit. The mechanical elastic force associated with this expansion is

F, = AEaO (7 ) where A is the area of the heated region and E is the mod- ulus of elasticity. Assuming now that 8 is the mean temper- ature difference between the hotter lagging and the cooler leading halves of the contact zone, the corresponding ther- mal differential force is

For the 0.9kg stainless steel cylinder rolling on 12mm diameter rails the contact area is A = n ab = 1.3 x 1e8m2, E = 200 GPa and a = 8. By equating this to the calculated mid range 10A differential force of 7.5N we find that the required temperature differ- ence between the leading and lagging halves of the contact zone is 8 = 577°C. This is well withn the range for stamless steel which has a melting temperature of around 1400°C. The drive force F could therefore well be due to thermal expansion.

Now taking the out-of-balance thermal expansion force to act nominally at a distance d = u/2 from the centre

of the contact zone, the thermal driving force can be expressed as

giving Fd(m, = 1.3 x

or

(9) 7rEaO(O. 101)"2 V P

32R1 P F =

The thermal force F taken as acting through the centre of the cylinder is thus predicted to be directly proportional to loading E', as found experimentally. Eqn. 9 also includes the radii of the rails and the roller. Taking for the rails R2 = 6nun the values of p2v/(RICp) have been calculated and plotted in Fig. 4. These values follow the expenmental trend very closely, falling from 0.479 to 0.390 over the roller diameter range from 16 to 38mm, a fall of 18.5'%, compared with the experimental fall of 19.6%. These results provide further support for the proposed thermal model.

In the model the force due to thermal expansion is pro- portional to the local temperature rise in the contact region: this is related to the electrical power absorbed at the con- tacts. The measured contact voltage illustrated in Fig. 6 is related to current above 3A by

V = 0.830 + 0.00751 There is a voltage of 0.415 volt per contact, and an ohmic component which may be due to current flow external to the contact zones amounting to 9% of the contact voltage at a current of 10A. The electrical power absorbed at the contacts can therefore be considered to be almost directly proportional to the current. The overall effective tempera- ture rise is consequently proportional to current, producing a force due to thermal expansion which, as found experi- mentally, is virtually proportional to current.

While a complete theoretical analysis of the thermal expansion model when the roller is in motion is beyond the scope of this paper, the experimental results together with the theoretical analysis provide strong evidence in favour of a thermal mechanism. This is not inconsistent with the results of [7] where the higher driving forces were associated with the metals having the higher melting temperatures. However, as previously stated, there may be other force- producing mechanisms at work simultaneously. Possibly a thermal mechanism dominates over the range of our exper- iments but no single mechanism is adequate to describe the characteristics over the whole range of possible currents. At higher currents other factors may predominate, for instance sparking at the trailing edges and local welding occurs which tends to reduce the driving force. Many of the mate- rials investigated in [7] have protective surface layers: stain- less- steel has a very effective protective film about 10 to 20A thick [lo]. Electron tunnelling, and electrical break- down in this layer when the voltage is high enough, may also affect the rolling torque. At very low currents the force/current characteristic deviates considerably from line- arity; t h s may simply reflect the more o h c relationshp between the current and the contact voltage in this region.

4 Conclusions

(10)

These investigations have shown that over a wide operating range the ball-bearing motor force for rolling cylinders is proportional to the current and to the loading. It falls slowly as the rolling diameter is increased. Electromagnetic theories do not explain these results.

A thermal expansion model successfully predicts the cor- rect order of magnitude of the driving force, descnbes the influence of loading and rolling diameter, and explains the relationship between force and current.

Further work is required to establish the precise relation- ship between the thermal driving force, the current and the parameters of the roller when in motion.

IEE Pro< -SCI MeuJ Techno1 V"J1 I46 N o 2 b l o i c h I999 86

5 Acknowledgments 4

The authors gratefully acknowledge the technical assistance of K.G. Smart. The work was supported by research grant D3123. 6

7 References

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MARINOV, S.: ‘The intriguing ball-bearing motor’, Electron. Wireless World, 1989. vv. 356357

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GRUENBERG, H.: ‘The ballbearing as a motor’, Am. J. Phys., 1978, 46, (12), pp. 1213-1219 HATZIKONSTANTINOU, P., and MOYSSIDES, P.G.: ‘Explana- tion of the ballbearing motor and exact solutions of related Maxwell equations’, J. Phys. A , Math. Gen., 1990, 23, pp. 3183-3197

9

10 HOLM, R.: ‘Electric contacts handbook’ (Springer-Verlag, 19538, 3rd

11 TYNDALL, J.: ‘Heat: a mode of motion’ (Longman, London, 1870) edn.)

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