1. Introduction and synopsis
Bicycle riding is a very popular activity. The bicycle is one of the most important inventions of mankind. Everybody who rides a bicycle at a brisk speed is impressed by its apparent stability. This has intrigued people and has lead to a large number of attempts to explain the stability based on laws of physics. Many prominent physicists addressed bicycle stability already in the very early days of the bicycle. A notable example is a paper by Rankine of 1869 (1). A comprehensive collection of early references can be found in the article by Meijnard et al. (2). This paper gives an excellent review of the physics of the autonomous bicycle with a large collection of references. The first realistic linear equations are probably due to Whipple (3) in the year 1899. The salient features of bicycle geometry, such as trail or angle of the steering axis were correctly included in his model. A very important cornerstone was the section on bicycles in the book on the theory of the gyroscope by Klein and Sommerfeld in 1910 (4). Klein and Sommerfeld realized that gyroscopic torques are negligible compared to the other torques acting in the lean axis. They also concluded that gyroscopic torques contribute to the capsize instability of an autonomous bicycle above 20 km/h. The autonomous bicycle without any friction or damping has a strong tendency towards oscillatory instabilities. Klein and Sommerfeld showed that gyroscopic torques can provide a small stability island below v = 20 km/h. If realistic damping constants for the steering system are included, the oscillatory instabilites are no longer an issue, even in the absence of gyroscopic torques.
Already in 1970 David Jones (5) has carried out a series of extraordinary experiments with specially designed bicycles. His approach was to understand bicycle stability by designing an unridable bicycle. Eliminating gyroscopic torques by replacing the wheels by tiny ball bearings or by inverting the sign of the gyroscopic torques by having a dummy wheel turning at high speed in the opposite direction had no effect on stability. Stability was destroyed by designing a bicycle with anti-trail, i.e. with a contact point of the tire in front of the steering axis.
In this contribution we present a consistent picture on the forces and torques which act on the system rider - bicycle and on the handle bar at various speeds and riding conditions. This allows to understand the stability of no-hands riding. The level is such that it should close the gap between the scientific literature which requires high level mathematics and popular presentations which are often outright incorrect. In particular we use a point mass model for the system rider - bicycle.
The dominant torques acting in the lean axis are the torque due to gravity and the centrifugal torque. In a stationary turn, the two torques balance each other. The gyroscopic torque is negligible, because the mass of the system rider and bicycle is about two orders of magnitude larger than the mass of both wheels and because the center of mass is about a factor of three higher than the hub of the wheel.
In a nonstationary situation an additional force, the kink force, comes into action. The kink force is related to the centrifugal force. It is due to the fact that the trajectory of the front wheel exhibits a kink, if the handle bar is suddenly turned. The kink force varies as the product of velocity and handle bar turn velocity (first derivative of the hadle bar angle). Since the centrifugal force scales with the square of the velocity and the kink force is linear, the importance of the kink force is largest in the low velocity regime.
Under normal riding conditions, i.e. with the hands at the handle bar, the rider is an active part in a control loop. The output of the control loop is the lean angle, the input the handle bar turn angle. The rider is the controller which connects output to input. As will be shown in the no-hands riding section, there exists an intrinsic control system, which assists the rider in keeping the equilibrium against capsizing. However, this intrinsic control system is not required for safely riding a bicycle. The intrinsic torques on the handle bar which support the rider in stabilizing the bicycle can be easily overpowered by attaching a heavy weight to the end of a handle bar. The bicycle is still ridable. The physics of standard bicycle riding is thus quite simple. Much more complex is the physics of no-hands riding and the physics of the autonomous bicycle (which are completely different problems).
In order to understand no-hands riding, the torques acting on the handle bar have to be discussed. Modern bicycles have a non-vertical steering axis with an angle of about 700 with respect to horizontal. This has important consequences. It causes the contact point of the front tire to be behind the steering axis (trail) and also has the effect that the stationary gyroscopic torque on the front wheel has a component in the steering axis. Both effects are important. Due to the trail, gravity and centrifugal forces exert in general a torque on the steering axis. Gravity tends to turn the handle bar into the lean, centrifugal forces and the about an order of magnitude smaller gyroscopic torque have the opposite effect. The torque due to gravity is independent on velocity, the two other torques scale with the square of the velocity. In the absence of external torques, the equilibrium turn angle of the handle bar is given by the condition that the sum of all torques is zero. For zero lean angle the zero handle bar turn angle position is an equilibrium position. However, the equilibrium is not stable at small speed. A stable equilibrium is only possible above a critical velocity vcrit, the self alignment velocity, which is in the range of 6 - 8 km/h. Above this velocity, the relation between turn angle and lean angle is quasilinear. This is a prerequisite for no-hands riding. Below the self alignment velocity the handle bar has no small angle equilibrium position. Nevertheless standard riding is still possible.
At a given speed above vcrit and at a given stationary lean angle, the handle bar turn angle has an equilibrium position σeq. If σeq is larger than the value required for compensating the gravitational torque acting on the center of mass, then the bicycle will gradually reduce the lean angle to zero. If σeq is too small, then the bicycle will capsize. For the autonomous bicycle σeq is large enough to provide stability below v = 20 km/h. Above 20 km/h the autonomous bicycle capsizes.
To understand no-hands riding, it is very helpful to use concepts of linear control theory (6). Under stationary conditions the handle bar turn angle is a simple function of the frame lean angle θf. Under dynamic conditions the relation is given by a transfer function
Tf-σ(s). In a similar way the static equilibrium relation between the handle bar turn angle σ and the center of mass lean angle θcm is substituted for the dynamic case by the transfer function Tσ-cm(s). For the autonomous bicycle the open loop gain Gloop(s) is simply the product of the two transfer functions. Under no-hands riding conditions the rider will continuously adjust the frame lean angle θf with respect to the center of mass lean angle to keep equilibrium. This is described by a transfer function Trider(s). The open loop gain Gloop(s) is now the product of all three transfer functions.
The stability of the system can be discussed in terms of the open loop gain by using Nyquist formalism. Above vcrit Tf-σ(s) is stable. Tσ-cm(s) on the other hand is instable. This is because the bicycle is basically an inverted pendulum. Without any negative feedback the bicycle will capsize. For such conditions Nyquist theorem says that the system is stable if F(iω) = 1 - Gloop(iω) encircles the origin once when ω runs from - infinity to + infinity. As will be shown, a necessary condition for stability is that the static negative feedback Gloop(s) at s = 0 is larger than one. Intuitively this means that the static negative feedback has to be sufficiently strong to counteract the instability of the center of mass. This means that stability against capsizing is given, if Gloop( 0) > 1, i.e. if the static equilibrium condition is fulfilled.
Even if static equilibrium is given, the system may still be unstable with respect to exponentially growing oscillations. Klein and Sommerfeld (4) discussed the autonomous bicycle in terms of two coupled differential equations, each representing an undamped oscillator. The dynamics of the handle bar is discussed in terms of a pendulum, the dynamics of the center of mass in terms of an inverted pendulum. The system is instable with respect to growing oscillations if G(iωres) > 1, where ωres is the resonance frequency of the system. The dominant effect in reducing G(iωres) is the friction in the handle bar system (neglected by Klein and Sommerfeld) and to lesser extend by a gyroscopic torque. Since this torque is the only stabilizing element in the model of Klein and Sommerfeld, its importance has been grossly overstated. Stability against growing oscillations is predominantly provided by dissipative effects and not by gyroscopic dynamic effects.
In real bicycle riding capsizing and oscillatory instabilities are of no practical importance. Capsizing is an accident and not a property of the system. The only relevant intrinsic dynamical property is the response function, i.e. the response of the system to a transient perturbation. The desired response function is a fast and smooth return to the stable equilibrium without any overshoot or oscillations. Surprisingly the response function has never been addressed in the bicycle physics literature. As it turns out, the effect of the gyroscopic dΘ/dt term has been completely misunderstood. It has a small effect in suppressing oscilatory instabilities, but has a pronounced positive effect on the respnse function.
The autonomous bicycle is a poor model for no-hands riding. True no-hands riding exhibits none of the instabilities of the autonomous bicycle. No-hands riding above 20 km/h is easily possible. This is a result of gyroscopic effects. Without such effects the required response time of a rider to react to a perturbation is of the order of 0.1 sec. Including gyroscopic effects the admissible reaction time increases to above 2 sec.
Also it is possible to steer the bicycle in the no-hands riding mode. For the autonomous bicycle the only stable solution is the upright bicycle moving in a straight line. The physics of true no-hands riding will obtain a strong focus in this contribution.
It is generally believed that a trail is absolutely necessary for the stability of an autonomous bicycle. This was recently shown to be incorrect (Kooijman et al. 6,7). Stability requires a sufficient gain in the negative feedback loop. The negative feedback is basically provided by the handle bar transfer function Tf-σ(s). For a standard bicycle geometry, the trail is instrumental for the negative feedback. If, hovever, another effect causes a negative feedback, i.e. causes the handle bar to turn into the lean with a sufficient amplitude, the bicycle remains stable against capsizing. A recently reported (7,8) strangely designed "bicycle" showed stability at all speeds without trail and in the complete absence of gyroscopic torques. The discussion of such designs is outside the scope of this contribution.
Table 1 Effect of major frame geometry parameters on riding properties
Table 2 Stability and steering
(1) M.W. J. Rankine. On the dynamical principles of the motion of velocipedes.The Engineer, 28, pp. 79, 129, 153, 157 (1869)
(2) J. P. Meijnaard, Jim M. Papadopoulos, Andy Ruina and A. L. Schwab, Proc. R. Soc. A 463, 2007 (2084): 1955–1982. Download
(3) F. J. W. Whipple. The stability of the motion of a bicycle. Quarterly Journal of Pure and Applied Math., 30, pp. 312 -348 (1899)
(4) F. Klein und A. Sommerfeld. Über die Theorie des Kreisels, Heft IV.Teubner, Leipzig, 1910. pp. 863 - 884
Download Paper of Klein und Sommerfeld (Caution: 17 MB!!!)
(5) D. E. H. Jones. The stability of the bicycle. Physics Today, April 1970, pp. 34-40. Download
(6) Karl J. Åström, Richard L. Klein and Anders Lennartsson, IEEE Control Systems Magazine, August 2005, pp. 26 - 47
(7) http://bicycle.tudelft.nl/stablebicycle/
(8) http://ruina.tam.cornell.edu/research/topics/bicycle_mechanics/