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Leonardo in Your Toothbrush
by Francis Moon

Kinematic Mechanisms in Daily Life

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Introduction

Two popular products today are the 'Spin-Brush' or motorized toothbrush and the 'iPOD'. In the latter, wires in the listener's ears lead to a small box with an electronic chip that stores music from the web. The 'iPOD' is an icon of the 21st century IT technology. In contrast are the wireless, motorized toothbrushes initially costing $30-50 that can now be bought for a few dollars. Some of these dental devices also embody modern mechatronics and electronic chips, but all possess kinematic mechanisms that have roots in the Italian Renaissance 1450-1600 AD. This period can be called the 1st Machine Age and Leonardo da Vinci one of the early machine engineers. Kinematics is the study of the mathematics of motion in machines and is an active field of research governing the design of everything from space robots and DVD players to the modern toothbrush.


Leonardo da Vinci: Engineer

Leonardo da Vinci was born in 1452 in Vinci, a village near Florence, Italy. He was trained as an artist and sculptor in the studio of the well known renaissance artist Andrea del Verrocchio and worked for the Duke of Milan for many years. Leonardo died in 1519 at the palace of the French King Francis I at Chambois.

Leonardo has been called an artist, painter, and architect by art historians, but in fact his surviving artistic works consist of less than a dozen paintings, including the Mona Lisa in Paris and the Last Supper in Milan. A good part of his career (1482-1500) was spent as an engineer designing civil and military structures for "Il Moro", Duke of Milan. After Leonardo's death, he left hundreds of drawings of machines, machine elements, and kinematic mechanisms. Leonardo designed machines for the technical problems of his day, including textile machines, clocks, metalworking devices, and most certainly military machines. Even his scientific work in hydrodynamics and flight were closely related to engineering design. Leonardo's drawings for human flight machines contained complicated linkages similar to those in his other machine drawings.

Leonardo was not the only artist-engineer in this endeavor. There were other Italian architects making a living as machine designers such as Mariana Taccola (1382-c.1460) and Francesco di Giorgio Martini (c.1439-1501) both of Siena. There is evidence that Leonardo had access to the machine drawings of these and other artist-engineers. By around 1500, Leonardo was credited with attempting to catalog the basic mechanisms that all machines are made of: linkages, gear pairs, ratchets, escapements, and pumps, some of which are used today in your motorized toothbrush. Many of these mechanisms were studied and cataloged by machine engineers of the 19th century like the German machine theorist Franz Reuleaux, who built hundreds of models of these basic mechanisms, many of which can be found in our cars, washing machines, garage door openers, cameras, and VCRs, as well as in your electric toothbrush.

Kinematic Mechanisms in Your Toothbrush

An example of one of the first motorized toothbrushes is the upscale model by the German company Braun, circa 2001, shown in Figure 2. This is a marvel of miniature mechatronic design. The brush sits in a holder (not shown) that picks up electrical power from an alternating voltage outlet in the wall. A coil in the brush converts the ac power to dc current that charges the batteries using an electronic circuit board in the handle. This is the '-tronic' part of the mecha-tronic machine. The battery drives a small continuous speed electric motor. The goal of the machine designer is to convert the motor motion into oscillating motion in the small brush at the end that cleans your teeth. The mechanical parts consist of three kinematic mechanisms:

  1. a brass gear-pinion mechanism (Fig. 2, 3)
  2. a four-bar linkage (Fig. 2, 3)
  3. a 3-dimensional ball joint mechanism (Fig.2, 4)

These mechanisms produce the following change of motions in your toothbrush:

  1. change of high speed continuous rotation into lower speed rotation;
  2. the crank in the &bar linkage converts continuous rotary motion into oscillating motion in the follower link;
  3. the 3D ball linkage converts oscillating motion about the vertical axis in the handle into oscillating motion about a transverse axis that 'spins' the brush; the final motion that helps clean the teeth.

Another dental product costing about one tenth the price is the Crest 'Spin Brush Pro' shown in Figure 5, 6. Here there is no on-board rechargeable circuit: just a AA battery. In this device there are again three kinematic mechanisms:

  1. a plastic crown wheel gear and pinion mechanism (Fig. 5,6)
  2. a slider-crank mechanism with the crank and the drive link (Fig.5)
  3. another slider-crank mechanism with the slider and the drive link. (Fig. 5)

These mechanisms provide the following change of motions in your toothbrush:

  1. the crown-wheel and pinion change high-speed motor rotation about the vertical axis into lower-speed motion about a transverse axis;
  2. the crank in the slider crank mechanism converts continuous rotary motion into oscillating motion of the slider along the vertical axis;
  3. the upper slider-crank converts the oscillating motion of the slider into oscillating motion of the brush.

It should be noted that the brush is really oscillating about a horizontal axis and is not spinning with constant speed as the name would imply.

Leonardo's Machines and Mechanisms

Although several manufacturers have secured patents on these devices, the submechanisms that change the constant rotary motion of the motor into the oscillating brush motion can all be found in the machine books of the Renaissance, notably in the drawings of Leonardo da Vinci. When he died in France, he left his student in charge of the thousands of drawings which had never been published. These drawings in turn became dispersed throughout Europe and were collected into several manuscripts called 'Codices'. Two of the principal manuscripts with drawings of machines and mechanisms are the Codex Atlanticus in the Ambrosiana Library in Milan, Italy and the Codex Madrid in Spain. In these manuscripts one can find hundreds of drawings of machines, geometric exercises, architectural designs and textural descriptions and notes intermingled with the drawings.Although Leonardo's writings were thought by some scholars to have been designed for formal books on painting, machine design or bird flight, what survives is more or less the first sketches, ideas and musings of one of the principal icons of the Renaissance man.

There are many drawings by Leonardo of complete machines. But there are many other drawings of machine components that are used in all machines. This represented one of the first attempts to deconstruct machines into a set of basic mechanisms. The Greek philosopher Aristotle and his students attempted to classify machines in how they converted forces using the so-called 'simple machines'; the screw, lever, pulley, wedge, wheel, etc. In Leonardo's machine elements, the focus is often on how the mechanism converts motion from one form to another, like the continuous rotation of a water wheel converted into the oscillating motion of a linkage in a textile machine.

The geometric principle of conversion of motion from one form to another, without regard to the forces, is called kinematics or the study of pure motion. This name was given by the French mathematician, A.M. Ampere, in the early 19th century. Kinematics was further developed by Robert Willis of Cambridge University in England around 1841 and by Franz Reuleaux of Berlin Technical University in Germany from 1867 to 1896.

Returning to Leonardo and the mechanisms of the Renaissance Machine Age, several of the toothbrush mechanisms can been seen in Leonardo's drawings from his Codex Madrid I as shown in Figures 7- 10. In Figure 7 is a drawing of a 4-bar mechanism, and Figure 8 shows a slider crank device. Another drawing is a pinion-gear pair in Figure 10, and a crown-wheel and pinion set in Figure 9. The difference between these two gear pairs is that the axes of rotation of the pinion-gear set of Figure 10 are parallel, in contrast to the crown wheel-pinion gears, whose axes of rotation are set at a 90-degree angle to each other. Another motorized toothbrush with a slider-crank and crown wheel gear is shown in Figure 11.

A three dimensional slider-crank mechanism of Leonardo da Vinci is shown in Figure 12. A novel use of a similar mechanism is in the Oral-B 'Cross-Action' toothbrush design, shown in Figures 13, 14. The motor spins a cup with an offset slot for a rod that undergoes a coning motion. The mid-point of the rod is supported by a soft bearing. The upper circular motion of the steel rod drives the brush in an oscillating motion.

Dynamic Forces in Your Toothbrush

Another toothbrush design with a true dynamic action using an offset flywheel is the Homedics 'SoniDent' toothbrush shown in Figures 15 and 16. An eccentric steel mass spins on the elastic plastic rod driven by the electric motor. The centrifugal 'force' of the spinning mass excites vibrations in the brush. The design of this machine incorporates the dynamic laws of motion of Isaac Newton (1686) in contrast to the pure geometric motions of kinematic mechanisms. However, Leonardo made several drawings of flywheels that he saw as necessary in many machines to sustain motions against the damping forces of friction. One such drawing of a flywheel (Figure 17) shows an unbalanced flywheel similar to that in the toothbrush. However, it is not certain that he understood the idea of centrifugal force of rotation of unbalanced masses.

The Cornell Collection of Reuleaux Kinematic Mechanisms

Many of these mechanisms were constructed in models for teaching machine design in the 18th and 19th centuries. One of the largest collections of kinematic models (800) was made by Franz Reuleaux of Germany, from 1867- 1900. Cornell University purchased a set of around 230 reproductions of these models from Professor Reuleaux in 1882. Reuleaux was famous for developing a formal system of basic mechanisms or a language of machine invention. Two models of mechanisms found in your toothbrush are the slider crank and gear-pinion mechanisms.

Another toothbrush with a simple spatial crank mechanism not found in the drawings of Leonardo is the Colgate Motion brush shown in Figure 20. The circular motion of the crank moves the slot back and forth which rotates the brush about its axis. This kinematic action is similar but not quite identical to the double slider or Scotch yoke mechanism shown in the 1882 Reuleaux model.

A complete set of 230 Reuleaux mechanisms in the Cornell Collection of Reuleaux Kinematic Models may be found on the KMODDL website:
http://kmoddl.library.cornell.edu.


Fig. 2: Click for sketch of Braun toothbrush.


Fig. 3: Braun toothbrush: View of pinion and gear and four-bar linkage.


Fig. 4: Braun toothbrush: Close-up view of oscillating brush mechanism.


Fig. 5: Click for sketch or movie of Crest Spin Brush Pro.


Fig. 6: Crest toothbrush: Close-up view of crown wheel, pinion, and slider-crank.


Fig. 7: Leonardo da Vinci; Codex Madrid I: Drawing of a 4-bar linkage.


Fig. 8: Leonardo da Vinci; Codex Madrid I: Drawing of a slider-crank mechanism.


Fig. 9: Leonardo da Vinci; Codex Madrid I: Drawing of a crown wheel and pinion.


Fig. 10: Leonardo da Vinci; Codex Madrid I: Drawing of two meshed gears.


Fig. 11: Another motorized toothbrush with crown wheel and slider-crank mechanisms.


Fig. 12: Leonardo da Vinci; Codex Madrid I: Drawing of spatial slider-crank mechanism.


Fig. 13: Photograph of Oral B Cross Action toothbrush.


Fig. 14: Click for sketch of Oral B Cross Action toothbrush.


Fig. 15: Photograph of SoniDent toothbrush: Close-up of rotating eccentric mass.


Fig. 16: Click for sketch of SoniDent Toothbrush.


Fig. 17: Leonardo da Vinci; Codex Madrid I: Eccentric (top) and circular flywheels.


Fig. 20: Click for sketch of Colgate Motion toothbrush.

Leonardo in Your Toothbrush
References

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