A reappraisal of Petrus van Musschenbroek's study of lubricated sliding friction: Did he really observe a Stribeck curve?

Abstract

Petrus van Musschenbroek (1692–1761) was an influential Dutch natural philosopher who carried out many measurements of friction in the early eighteenth century. For hard steel sliding on copper, he found a rapid increase in friction with sliding speed for both lubricated and unlubricated conditions, and it has been suggested that this may represent the first observation of Stribeck-like behaviour. By modelling the dynamic behaviour of his tribometer, we are able to infer the methods he used to perform his experiments, and show that his results were dominated by the inertia of the apparatus rather than by a true increase in friction with velocity.

Introduction

Petrus (Pieter) van Musschenbroek (1692–1761) was a Dutch natural philosopher who played an important role in the promotion of Newtonian physics in continental Europe in the eighteenth century. Born and educated in Leiden where his family achieved fame as scientific instrument makers, he became a doctor of medicine and subsequently practiced as a physician in Leiden. In 1717 he travelled to London, met Newton and attended the lectures of John Theophilus Desaguliers; after returning to Leiden, in 1719 he took up a professorship in mathematics and philosophy at the University of Duisburg in the Rhineland. He moved to a professorship at the University of Utrecht in 1723 and stayed there, despite several invitations to take up chairs at other universities, until in 1739 he returned to Leiden, where he remained for the rest of his life. Petrus van Musschenbroek was elected a Fellow of the Royal Society in 1734, with Desaguliers as one of his proposers (for biographical information see De Pater ¹ and Van Leeuwen ² .).

Van Musschenbroek's lectures on natural philosophy were widely read and highly influential, as reflected by the large number of editions published in Latin, Dutch, French and German during his lifetime^3–10 as well as posthumously,^11,12 with a single English translation appearing in 1744. ¹³ Latin editions also appeared in Venice and Naples.^14–16 This series of textbooks can be regarded as a single work that he continuously expanded and refined, with the final version appearing in 1762 just after his death. ¹¹ Van Musschenbroek was well aware of the importance of friction in the operation of mechanical devices and all editions of his work contained a chapter on this subject entitled ‘On the friction of machines’ (De attritu machinarum).

Harry van Leeuwen has provided a detailed study of van Musschenbroek's work on tribology, specifically his investigation of friction. ² Van Leeuwen suggests with good justification that van Musschenbroek's extensive work, which included many measurements of friction, has not received the attention it deserves. For example, Dowson in his History of Tribology refers briefly only to van Musschenbroek's ideas on the origin of friction at asperity contacts, and makes no mention at all of his experimental measurements of friction. ¹⁷

Here we shall focus on a small set of results reported by van Musschenbroek that relate to friction under conditions of lubricated sliding. He found a very marked increase in friction with sliding speed which van Leeuwen has compared with the behaviour noted by Richard Stribeck in 1902 (and earlier by Adolf Martens in 1888 ¹⁸ ): the well-known ‘Stribeck curve’ associated with progressive changes in lubrication conditions with sliding speed or load, as well as viscous drag at higher speeds. Van Leeuwen comments that ‘it appears that more than 150 years earlier van Musschenbroek measured Stribeck-like behaviour as well’. We shall examine van Musschenbroek's experiments and his results critically.

Van Musschenbroek's measurements of friction

Van Musschenbroek's measurements of friction at various sliding speeds are absent from his earliest discussion of friction in 1726 which is largely qualitative, ³ but are reported for the first time in 1734, together with data for static friction of different woods and metals, dry and lubricated. ⁴ The same friction data appear with more or less detail in all subsequent editions of his work, in some cases accompanied by additional comments.

The apparatus that van Musschenbroek used in his study, shown in Figure 1, consisted of a cylindrical rotor fixed on a horizontal axle which was supported at its two ends by simple plain bearing surfaces (probably half-journal bearings although he provides no details). The apparatus is very similar in concept to a design described by Jacob Leupold of Leipzig in 1724, ¹⁹ although in that version the rotor was a simple cylinder which acted as its own axle, and was less massive.

Figure 1.

Apparatus used by van Musschenbroek to investigate friction between a rotating axle and a half-bearing. Weights attached to cords were used to load the rotor. (a) General view and (b) detail of the rotor. The difference between the weights P and Q needed to cause the rotor to move provided a measure of the static friction. ⁴ (c) Later drawing in which the additional weight R is shown separately. ¹¹

Van Musschenbroek used his apparatus, which he called (in Latin) a ‘tribometrum’, to investigate static friction, varying the applied load and the materials of the bearing surfaces by which the hard steel axle shaft was supported. With the application of his results to practical machines clearly in mind, he chose lignum vitae, hard steel, red copper, tin, lead and brass as bearing materials. The large wooden cylinder with its shaft weighed 3 pounds and was 4 inches in diameter, while the narrow ends of the axle (marked D) had a diameter of ¼ inch. He attached various weights (P and Q) to both ends of the cord wrapped around the rotor in order to vary the load on the shaft, and used an additional weight to determine the frictional force needed to cause the shaft to rotate. Only two weights P and Q are shown in Figure 1(a), the small difference between them providing the torque needed to overcome the friction on the axle, whereas in Figure 1(c) which appeared in later editions of the text an additional weight (marked R) is shown explicitly, hanging from a separate cord.

While van Musschenbroek's ‘tribometer’ may have been essentially identical to the apparatus described by Leupold before him, he used it to make a much more systematic and thorough series of experiments on static friction. As well as using six different bearing materials, he varied the load on the shaft from three pounds (corresponding to the weight of the rotor alone) to nine pounds (with weights each of three pounds hanging at P and Q); he measured the static friction under dry conditions and also with the journals lubricated with olive oil. He listed in tabular form the additional weight needed to cause the cylinder to rotate under the different conditions, and in later editions (from 1741) he also included the ratio between the friction force and the load, a value that would later be known as the coefficient of friction.

Figure 2 shows his results for the static friction of hard steel against copper, both dry and lubricated with olive oil. For units of mass and weight van Musschenbroek used pounds, ounces and drachms, where 1 pound = 16 ounces and 1 ounce = 8 drachms (as shown by Van Leeuwen ² , the use of these measures, rather than the avoirdupois measure in which 1 ounce = 16 drams, is consistent with Van Musschenbroek's data.). In order to calculate the friction ratio (i.e the ratio between the friction force and the normal load) he correctly took account of the fact that the radius of the cylinder was 16 times that of the axle shaft, so that the moment of the additional weight was 16 times that of the friction force. Under lubricated conditions, he found a friction ratio of 1 to 8 (i.e., a coefficient of friction of 0.125) for the steel shaft in copper bearings when subjected to a total load of three pounds, with values between 0.175 and 0.181 for the higher loads. These values are unexceptional for boundary lubrication which would be expected under static conditions.

Figure 2.

Extract from van Musschenbroek's table of results for static friction between a hard steel axle and copper bearings, under dry (unubricated) conditions and lubricated with olive oil. The second and third columns show the additional weight needed to cause the rotor to move, when loaded with the weights shown in the first column (from the English translation of 1744). ¹³

The results for kinetic friction which are the main topic of this paper were included within van Musschenbroek's text rather than in a table. It is these data that require further scrutiny. The text from 1734 reads (pp. 102–3) ⁴ :

‘At lower velocities of bodies, friction follows the velocity in some proportion, though not exactly; but at higher velocities the ratio of friction increases greatly, and this holds whether the bodies move dry or lubricated with oil. For when the steel axle of the tribometer rotated in red copper lubricated with oil, and the velocities were as 4, 6, 7, 8, 10, the friction was as 1, 1½, 2, 3, 4. And when the friction was the same, the velocities in dry copper were 1⅔, 3, 5, 7, 8. When the velocity was greatest in these experiments, namely equal to 10, the axle DD made 25 revolutions in a time of 2′′ 24′′′. The friction which is taken as equal to 1 is to the moving weight as 16 to 95.’ (In minoribus corporum velocitatibus sequitur Attritus utcunque rationem velocitatis, non tamen accurate: Verum in maioribus velocitatibus multum increscit ratio Attritus, idque obtinet, sive corpora sicca, sive oleo uncta supra se moventur. Cum enim axis Tribometri chalybeus volvebatur in Cupro rubro, oleo uncto, atque velocitates erant uti 4, 6, 7, 8, 10 fuit Attritus uti 1, 1½, 2, 3, 4. Atque Attritu existente eodem, fuerunt velocitates in Cupro sicco 1⅔, 3, 5, 7, 8. Cum velocitas erat maxima in his Experimentis, sive æqualis 10, fiebant 25 revolutiones axeos DD intra tempus 2", 24′′′. Attritus, qui ponitur æqualis 1, est ad pondus motum, uti 16 ad 95.)

These data, plotted in Figure 3, show a very sharp increase in friction with sliding speed for steel sliding on copper, both dry and lubricated with olive oil. The data are plotted in terms of relative speed and relative friction. Van Musschenbroek defines these quantities so that a relative speed of 10 corresponds to 25 rotations of the drum in a period of 2′′ 24′′′, and a relative friction ratio of 1 corresponds to a ratio between the friction and the ‘moving weight’ of 16/95. Van Leeuwen ² comments that the rapid increase of friction with sliding speed is ‘remarkable’ as indeed it is. It is also remarkable that a similar dependence is seen even for the case of dry, unlubricated sliding. Such a strong dependence of friction on sliding speed would not be expected for dry sliding of these materials, nor for conditions of boundary, mixed or full-film lubrication, and presents us with a puzzle.

Figure 3.

Van Musschenbroek's results for the friction of steel sliding against copper, both unlubricated and lubricated with olive oil, plotted against sliding speed. Both axes use relative units as described in the text.

While Van Musschenbroek provides enough information about his methods of measuring static friction for us to have some confidence in those results, he gives almost no description of his experimental method for measuring kinetic friction. Indeed, in the account published in 1739 ⁷ he adds after reporting his results:

‘It would be too long to explain here the manner in which I carried out these laborious experiments. I have drawn the preceding conclusions only after having performed a very large number of experiments, and I do not doubt that they are of great utility in mechanics.’ (Il seroit trop long d'exposer ici, de quelle manière j'ai fait ces Expériences pénibles. Je n'ai tiré les conclusions précédentes, qu'après avoir fait un très grand nombre d'Expériences, & je ne doute pas qu'elles ne soient d'une grande utilité dans la Mécanique.)

We therefore have only very limited evidence from which to deduce the methods he used.

Van Leeuwen has interpreted the time of 2′′ 24′′′ in which the axle rotated 25 times at the highest speed, to mean 2 min and 24 s. ² That would imply a rather slow sliding speed at the surface of the axle of 3.6 mm s⁻¹. He argues that it is ‘very unlikely that in the early eighteenth century 25 revolutions could be determined in about 2 s’. However, there are good reasons to suppose that this time was indeed between two and three seconds. First, van Musschenbroek uses exactly the same symbols elsewhere in his books to denote seconds and fractions of a second (which can be taken to be sixtieths by analogy with the subdivisions of hours into minutes and minutes into seconds): for example in his edition of 1762 in referring to the orbital period of the moons of Jupiter. ¹¹ It would seem perverse to assign different meanings to the same symbols within the same book. Second, he writes in both his Dutch and French editions of 1739^6,7:

‘When I caused the disk to turn 25 times in the time of 2′′ 24′′′, this was the greatest speed that I could give to the disk and still measure; I call this speed 10.’

This would be consistent with a very short time for the 25 rotations and not with almost 2½ minutes; he would surely not have made the remark that it was the shortest time that he could accurately measure.

If we therefore read 2′′ 24′′′ as 2 s and 24 sixtieths (i.e., 2.4 s), it is reasonable to suppose that van Musschenbroek may have been working to a precision of 12/60 = 1/5 s. How he actually measured this short time interval is not clear, but it is worth noting that by simply counting rapidly and repeatedly ‘one; two; three; four; five’ (or in Dutch ‘één; twee; drie; vier; vijf’) it is easy to achieve a periodicity of one second with some accuracy. For an experimenter, with practice, to determine short periods of a few seconds with an apparent resolution of 1/5 s is therefore not at all implausible.

We next need to examine how these high speeds of rotation of the drum might have been achieved, and how the frictional drag could have been measured. A modern-day approach would be to rotate the shaft at a constant speed with a motor, and to measure the torque acting on the bearings. But van Musschenbroek clearly indicates that he used the same apparatus that he had used in his static friction experiments, the device shown in Figure 1. By using the imbalance of the weights on the cords to rotate the drum, and with a long cord and sufficient height of fall for the weights, it would be straightforward to cause the drum to rotate 25 times in a short period. 25 times the circumference of the drum would be about 8 metres, a distance easily achieved in a multi-storey building.

The phrasing that van Musschenbroek uses to describe his results for unlubricated sliding is curious (‘and when the friction was the same, the velocities in dry copper were 1⅔, 3, 5, 7, 8’) since it suggests that in his experiments he somehow used the friction as the independent variable, and then measured the resulting velocity. This would be a highly unusual approach, but one that would be consistent with treating the added weight on his tribometer as a measure of the friction force. In his static friction experiments the added weight was indeed a measure of the friction force, and it is the value of this weight that van Musschenbroek lists in his tabulated results (as shown in Figure 2).

One can therefore envisage an experiment in which the fall of a certain weight is used to drive the rotor. The weight is attached to a cord initially wound 25 times around the rotor, and the experimenter then records the time for the cord to unwind, quite probably being the time taken for the falling weight to hit the floor some 8 metres below. The mean sliding velocity is then taken to be proportional to the reciprocal of the time.

While it may be clear to us that the action of the additional weight needs not only to overcome the frictional torque acting on the axle but also to accelerate the weights themselves as well as the rotating cylinder, this would not have been clear to van Musschenbroek, who although familiar with Newton's (then novel) second law for linear motion would not have appreciated its application to rotating bodies. At that time the mechanics of rotational acceleration was still being developed, the concept of a moment of inertia being first introduced by Leonhard Euler in 1750. ²⁰

Modelling the dynamics of van Musschenbroek's tribometer

By analysing the motion of the system depicted in Figure 4 which shows the forces acting on the rotor, we can examine how van Musschenbroek's apparatus would have behaved with different weights on the cords and with different amounts of friction acting on the axle, and compare this with his experimental results.

Figure 4.

Free-body diagram showing the forces acting on the rotor in van Musschenbroek's tribometer.

We assume the rotor to have radius R and the axle radius to be r. The mass of the rotor is M and equal weights each with mass P hang from the cords at each side, which are attached to the rotor. On the right-hand side an additional mass m is added and the rotor starts to rotate clockwise. The tensions in the two cords are T₁ and T₂. After time t the masses are each moving with velocity v and the rotor has an angular velocity ω = v/R. For simplicity we assume that the coefficient of friction μ acting on the axle as it rotates is constant and independent of sliding speed. The reaction force from the bearing on the axle is vertical, with components N in the radial direction and μN tangentially. The angle θ is the friction angle given by μ = tan θ.

Resolving the forces on the rotor along the direction of N gives:

$$N=(T_1+T_2+Mg)\cos \theta$$

(1)

where g is the acceleration due to gravity.

If the linear acceleration of the masses is a (= dv/dt), by applying Newton's second law to the weights we find:

$$T_1-Pg=Pa$$

(2)

and

$$(P+m)g-T_2=(P+m)a$$

(3)

From Newton's second law for rotational motion of the rotor with moment of inertia I:

$$(T_2-T_1)R-\mu Nr={\displaystyle \frac{Ia}{R}}$$

(4)

Treating the rotor as a uniform solid cylinder for which I = MR²/2, and combining equations (1) to (4), we find the acceleration to be:

$$a={\displaystyle \frac{m-{\displaystyle \frac{r}{R}(2P+M+m)\sin \theta }}{\left(2P+{\displaystyle \frac{M}{2}+m}\right)-{\displaystyle \frac{r}{R}m\sin \theta }}g}$$

(5)

The time T for the right-hand mass to fall a distance H from rest is:

$$T=\sqrt{{\displaystyle \frac{2H}{a}}}$$

(6)

and the mean velocity of fall V is given by:

$$V={\displaystyle \frac{H}{T}=\sqrt{{\displaystyle \frac{aH}{2}}}}$$

(7)

The model assumes sliding friction at the axle and is therefore not valid for static conditions, nor indeed when the numerator in equation (5) becomes negative, i.e., when

$$m<{\displaystyle \frac{r}{R}(2P+M+m)\sin \theta }$$

(8)

This simple model assumes a constant coefficient of friction at the axle surface, and the cords to be perfectly flexible and without mass, but it provides a valuable basis for comparison with van Musschenbroek's data.

Van Musschenbroek used a rotor with a mass of 3 pounds and a diameter of 4 inches, or 104.6 mm, so that we can take R = 52.3 mm (Van Musschenbroek specified the dimensions in Rhenish (Rhineland) inches, for which 1 inch = 1/12 Rhineland foot = 26.154 mm. ²¹ Van Leeuwen assumed a slightly different conversion but provides no source ² .) and H = 8.22 m (corresponding to 25 times the circumference of the rotor). The diameter of the axle was ¼ inch, so that R/r = 16. For easy comparison with van Musschenbroek's data we shall state masses in pounds, so that in these units M = 3. We assume g the acceleration due to gravity to be 9.81 m s⁻².

Figure 5(a) shows how, for different values of μ between zero and 0.8, the mean velocity of the falling mass varies with the additional mass m (in pounds) with no other weights on the cords (i.e., for P = 0). The predictions of the model are also plotted in Figure 5(b) for values of P between 0 and 3 pounds, for μ = 0.

Figure 5.

Results from the model, showing the relationship between the additional mass m and the mean velocity V of the falling weight over a distance corresponding to 25 turns of the rotor: (a) for values of μ from 0 to 0.8 with P = 0; (b) for values of P from 0 to 3 pounds with μ = 0.

It is immediately clear that even with no friction acting on the axle (i.e., for μ = 0) the relationship between the additional mass m and the mean velocity of the falling weight V shows a behaviour very similar to that reported by van Musschenbroek for the velocity-dependence of friction. At high speeds the behaviour is dominated by the inertia of the system, with friction playing a progressively less important role as the speed increases.

Comparison with van Musschenbroek's results

In order to make a quantitative comparison with van Musschenbroek's data, we need to infer the velocities and masses he employed in his experiments. He makes it clear that the highest velocity, a value of 10 on his relative scale, corresponded to 25 revolutions of the rotor in a time of 2.4 s, which equates to a mean velocity V of the descending weight of 3.425 m s⁻¹.

The values of ‘friction’ that he reports follow a simple pattern of numbers for both lubricated and dry conditions: 1, 1½, 2, 3, 4. It seems likely that if he indeed attached different weights to the cord and treated them as measures of ‘friction’, he would have used standard weights readily available to him with masses in these ratios. Inspection of Figure 5(a) (or use of equations (5) and (7)) shows that for μ = 0 and P = 0 a mass m of 0.5 pounds (= 8 ounces) would give a mean velocity of 3.17 m s⁻¹, and suggests that van Musschenbroek may therefore have used weights of 2, 3, 4, 6 and 8 ounces in his experiments. These values have been used to plot his results in Figure 6.

Figure 6.

Comparison between van Musschenbroek's data and the predictions from the model, for P = 0.

Figure 6 shows a comparison between the predictions of the model for P = 0 and van Musschenbroek's data for both lubricated and dry conditions, scaled so that a relative velocity of 10 corresponds to a velocity V of 3.425 m s⁻¹, and a relative ‘friction’ of 4 corresponds to a mass m of 0.5 pounds.

Discussion

The correlation between van Musschenbroek's data and the model, as shown in Figure 6, is remarkably good, bearing in mind the very low precision with which his data are reported, and the inaccuracies he would have faced in measuring the short time intervals involved. His data for lubricated sliding are consistent with values of μ below about 0.2, which would be reasonable for a poorly lubricated contact, probably operating in the boundary regime. For dry sliding the values suggest μ to lie between about 0.6 and 0.8, again very reasonable values for steel sliding on copper. Although the data point corresponding to the highest velocity appears to be an outlier, it does correspond to the shortest time measurement (of 2.4 s) and would therefore have been subject to the greatest percentage error. While it is possible that he in fact carried out his experiments with a greater load on the rotor (i.e., with P > 0), the good agreement shown in Figure 6 suggests that he probably used a single weight, on a single cord initially wrapped around the rotor.

Van Musschenbroek would have found that small values of m, below the threshold given by equation (8), were insufficient to overcome friction and cause the rotor to move; for values above the threshold but close to it, the motion of the rotor would be very sensitive to small changes in friction and it would be difficult or even impossible to obtain reproducible measurements of velocity. This would explain the absence of any data for lubricated conditions at low velocities, below 4 on his relative scale.

The good correlation between his data and the predictions of a simple model suggests strongly that van Musschenbroek, making an analogy with his measurements of static friction, mistakenly assumed that the additional weight m could be used as a measure of the friction acting under dynamic conditions. However, we still need to understand his statement that ‘the friction which is taken as equal to 1 is to the moving weight as 16 to 95’.

There is no doubt that van Musschenbroek used the correct method to determine the friction ratio from the value of m in his static experiments, as illustrated in his table in Figure 2. But applying the same method to his dynamic results does not yield a value at all close to 16/95 (= 0.168). For example, if a mass m of 0.125 pounds corresponds to a ‘friction value’ of 1 for the rotor with a total mass of 3 pounds (with no added weights; i.e., P = 0), his method of calculation would give a friction ratio of 2/3.

Examination of his results for static friction, however, suggests that faced with the inevitable inconsistencies in the friction ratios he could derive from his dynamic experiments, he simply took the average of the values from his static friction experiments and assumed that it would also represent the value for kinetic friction at low speed. If we take the values for m for lubricated sliding on copper in the third column of Figure 2, which apply to four different loads on the axle, and add them we obtain: 3 + 7 + 10 + 13 = 33 drachms = 33/8 ounces. The sum of the axle loads is 3 + 5 + 7 + 9 = 24 pounds = 384 ounces. Using his method (and allowing for the ratio of 16 between the radius of the rotor and that of the axle) we find a mean friction ratio of (33 × 16)/(8 × 384) = 33/192 = 0.172.

Although van Musschenbroek continued to include his results for kinetic friction in successive editions of his text, in the final edition of his book ¹¹ (although the book was published after his death in 1762, the preface makes clear that van Musschenbroek had himself edited and corrected the relevant part) he added after the results the following cautionary note:

‘Yet I did not fully satisfy myself in these experiments: at the time when I was making them, almost nothing had become known to philosophers about the mechanics of motion, which absolutely must be considered here. Moreover, an error is easily committed in the distance travelled when a body is moved with excessively rapid speed, and likewise in the moving power, which is either the spring of a large table clock, or a descending and winding weight, but producing motion along a tautochronous curve. In old age I can scarcely hope that strength will remain to me, so that I might revise all these things; rather I exhort other philosophers to refine and correct these things, which are of great use in mechanics.’ (In hisce experimentis tamen mihi non plane satisfeci: quo tempore capiebam, fere nihil innotuerat Philosophis de Mechanica Motus, quæ omnino hic est consideranda: præterea error facile committitur in via percursa, quando prærapida celeritate corpus movetur, tum in potentia movente, quæ vel est elater magni horologii tabularii, vel pondus descendens & volubile, sed motum in curva Tautochrona: Vix in hac senectute mihi vires superfuturas sperare licet, ut hæc omnia denuo ad incudem revocem, potius alios Philosophos adhortor, ut hæc, quæ magni in Mechanica sunt usus, liment, emendentque.)

By then he clearly appreciated that, quite apart from the uncertainty in his time measurements, the behaviour of his tribometer needed re-appraisal in the light of recent advances in the understanding of mechanics.

Conclusions

Petrus van Musschenbroek's measurements of friction as a function of speed pose a puzzle, since they appear to show a very strong increase in friction with sliding speed under both lubricated and unlubricated conditions. His measurements of static friction made with the same apparatus are in contrast unremarkable, and for them he provides ample information about his experimental technique. But for kinetic friction we are forced to infer his methods from very sparse clues.

By examining the dynamic behaviour of a simple model of his apparatus, we can conclude that he used a falling weight attached to a cord to turn the rotor of his tribometer, and that by analogy with the static case, he assumed that the magnitude of the weight was proportional to the friction acting on the axle of the rotor. He used the time of fall of the weight to calculate the mean velocity of sliding. The behaviour of his tribometer was in fact dominated by the inertia of the rotor, and the method could only provide a very inaccurate measure of friction.

Although his data appear to show Stribeck-like behaviour, the dominant cause is the rotational inertia of the apparatus rather than any true velocity-dependence of friction. Early in the eighteenth century Van Musschenbroek was understandably unaware of these issues as knowledge of the mechanics of rotating bodies was then in its infancy, although by the end of his life he began to express doubts about the validity of his results.

Inertial effects remain a concern even in modern designs of tribometer.