To Understand and to Model

But no inventions neither without science.

Kinematics (from greek "kinema", movement) is the "Science in which the movements are considered in itself (independant from the forces which produce them), as we observe in the solid bodies all around us, ans specially in the assemblies called machines" according André-Marie Ampère'definition in 1834 in his "Essay on sciences philosophy".

The two stars about mecanism study are Léonard de Vinci already cited, and the Belgian Franz Reuleaux born in 1829, the modern kinematics father.

His studies published then in Germany, Swiss and England and his models are collected at the Cornell US University USA and in Germany.

The numerical models are on line, sometimes in interactive graphics and some videos show the computation results :



We can see where Felix Wankel sourced some inspiration.

understand and model the vehicule dynamics.

Jean-Pierre Brossard claims that ships led to the first reflections on stability: The analyses made by Archimedges, Euler, Bouquier and Dupin resolved the problem.

In 1835, Gustave Coriolis wrote the first text on road vehicle dynamics.On the Stability of Vehicles, with respect to France’s .

Messengers, Journal de l'École Polytechnique. He analyzed the conditions causing imperial mail-carrying carriages to roll over on slopes.

In the modern period, theories have been formalised for the vehicules whose problems are obvious, the bicycle and the planes :

  • Following the Wright Brothers first flight in 1903, theoricians analyse and the first flight theory formalization was done by H. Bryan in 1910.
  • Theory are also interested by the wheel shimmy, for whom the tire side slip is essential.
  • For the bicycle, the lateral fall-down is part of the human being. Its analysis is detailed herunder.


Cyprien Chateau published in "la Nature, revue des Sciences et de leurs applications aux arts et industries" dated 5th of Novembre, 1892, an analysis of cycle stability entitled "Vélocipédie, de l'aplomb dans les bicycles".

The Fourneyron prize, whose value is 1000 francs (equivalent to a 2011 4000¤), is awareded two times a year by the French Sciences Academy. In 1897 the subject was "movement theory and in particular the velocipede stability conditions". This will challenge for 10 years the most brilliant brains : Carlo Bourlet, Joseph Boussinesq, Henri Bouasse, Léauté, Paul Appell, Archibald Sharp, Jacob and Francis Whipple.

Emmanuel Carvallo published "Theory of monocycle and bicycle movement" in 200 pages, derserving the 2nd prize, in which he formulated the 2-wheeler steering golden rule (hereunder), about we still talk one century later!

Knowledge thirst vanished for some decades, till the computer allows easier and more in-depth analysis than the discussion on literal stability condition.

We shall find a history precise and informed on the dynamic models of Two Wheels.

We shall find here a scientific comparative degree of the equations of these dynamic models.

For cars, understanding the tire behavior is fondamental.

The understanding of the tire sideslip concept is due to Brouilhet at Peugeot in 1925, an analysis of its role to Becker, Fromm and Maruhn in 1931.

Maurice Olley at General Motors invented words "understeer" and "oversteer" and the critical speed notion in 1937, while Evans at Goodyear published the first tire cornering properties on a test bench in 1935.

Second degree models of drift-yah freedom ,poorly named the "model bicycle, culminated in an equation typical of second degree, analogous to the one for the longitudinal stability of airplanes. The cornering stiffness of the tires’ sideslip replace the wings lift gradients. In 1937, Yves Rocard invented the first models; De Sèze, Gratzmuller (1942) and Julien (1948) in France.Evans and Olley in the US; … in Germany, and Sharp in England.

The three degrees of freedom models introduced the body roll. William (Bill) Milliken, Léonard Segel and David Whitcomb, working at Cornell Aeronautical Laboratory in Buffalo, New York, in collaboration with GM, designed models with three degrees of freedom that introduced body roll. Their ideas were presented in three papers in London in 1956, because the American Society of Automotive Engineer had rejected them! The calculations were done on a computer, and included qualitative and quantitative aspects and validations of real tests (to 0.3g): modern automobile dynamics was born.

The models were subsequently perfected, but above all, integrated fundamental non-linearity or adherence limits: no, you can’t go twice as fast on a turn by turning the steering wheel 4 times harder!

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