Trim of aerodynamically faired single-track vehicles in crosswinds
P: Waldheimstrasse 32, CH-3012 Bern, Switzerland
W: Hochschule für Technik und Architektur Bern, Morgartenstrasse 2c, CH-3014 Bern, Switzerland. email@example.com
This paper is about minimizing the disturbing effects of steady crosswinds on single-track vehicles (velomobiles and hpv / bicycles / motorcycles). A solution of the static problem ‘aerodynamically faired single-track vehicle in crosswind’ is presented.
The Cornell Bicycle Model (Cornell Bicycle Research Project) describes the physical behavior of an idealized bicycle (single-track vehicle) at no wind. Other equations in a previous paper describe the torques on fairings due to aerodynamic forces which induce lean of single-track vehicles and lead to steering-action. These equations are combined with those of the bicycle model to describe the conditions for equilibrium at some lean but zero steering angle. Parameters affecting equilibrium are mass distribution, vehicle- and fairing geometry and the relative position of fairing and vehicle structure. Faired single-track velomobiles whose parameters are such that the equilibrium-equation (‘trim equation’) is fullfilled could be easier to ride in steady crosswind than those designed at random.
Because the trim equation derived in this paper does not describe the dynamic behavior e.g. of a velomobile coming from a no-wind situation into one with steady, alternating or impulse-input crosswind, further investigations will be needed for even better hpv- or other single-track vehicle design.
Bicycles with disk wheels or other lifting surfaces and aerodynamically faired single- or multi-track human powered vehicles may be safely ridden in low and steady crosswind. But when the speed and direction of the wind change in unsteady patterns, today’s lightweight, aerodynamically faired vehicles become hard to control. Multi-track vehicles may remain rideable because mainly one degree of freedom, rotations about the yaw-axis, has to be controlled, whereas single-track vehicles need to be controlled in the two degrees of freedom roll and yaw (yaw-roll-coupling) and may become very difficult to handle.
It is therefore important to increase understanding of the statics and the dynamics of these latter vehicles. This paper is about the statics of single-track vehicles in steady crosswind; it presents conditions for equilibrium at some lean angle and at zero steer angle. Solutions of the dynamic problem, where angular velocities are not zero, may be derived in later work.
A single-track vehicle can be compared to a great extent with a sailing-boat. There, trim also needs to be achieved in order that the boat neither turns away from the wind nor very quickly turns into the wind. Yet, the comparability of single-track vehicles and boats is limited in that a boat at zero speed may return from high rolling angles whereas a single-track vehicle returns to vertical only when speed is not zero.
Riders of faired velomobiles (single- or multi-track) know that lift may help to compensate drag (of any form: due to slope, to rolling resistance or air-flow). In order to gain a lot of energy from the wind by maximizing lift as much as possible, it would be necessary to increase the lateral area. But this is in conflict with the wish to ride the velomobile on narrow and on public streets as safely as possible.
The main intention of this paper is not to explain sailing with velomobiles, but to describe the statics of single-track vehicles in crosswinds in the hope that safer velomobiles may be designed in the future. Therefore, here, minimization of the lateral area of the fairing is suggested.
The main problem in the static case is the location of the center of pressure relative to the center of mass and the wheels of the vehicle. The center of pressure is the center of the aerodynamic forces acting on the vehicle, whereas the center of mass is the center of the gravitational forces (Fuchs, 1993 and 1994).
If the vehicle was airborne and if the center of pressure lay behind the center of mass, the nose of the vehicle would turn into a lateral wind (airborne vehicles rotate around axis through the center of mass). If, conversely, the center of pressure was in front of the center of mass, the vehicle’s nose would point out of the crosswind.
Since land vehicles are (hopefully!) seldom airborne the behavior of the suspension has to be taken into account. Cooper (1974) explains : „ ... In terms of response to the wind, I don’t feel that you want a lot of weathercock stability (note by AF: far rear center of pressure location), that is, you should try to use as small a vertical tail as possible (note by AF: in contradiction to Bülk 1992 and 1994). To initiate a right turn on a motorcycle you have to initially steer left. Following this line of reasoning, if a sidewind from the right hits a motorcycle with weathercock stability, it will cause the motorcycle to turn right into the wind. But the aerodynamic rolling moment and the lateral acceleration due to path curvature will cause the motorcycle to lean left getting you into real trouble. I think what you want is a careful balance of aerodynamic roll and yaw moment such that the wind vector will tend to force the motorcycle out of the wind but this tendency will be balanced by the lateral acceleration produced by the curvature of the path which will roll the motorcycle into the sidewind. From these arguments it’s clear that you don’t want to follow the dictates of aircraft or of four-wheel cars. You must consider the aero sufaces and the chassis together because a surface vehicle has tires which are doing things at the same time the aerodynamic forces are acting.“
When the angle of attack (angle between the lifting body and the relative wind) increases due to rotation of the vehicle, the center of pressure location changes and moves towards the tail. Therefore, one should not simply talk about ‘the center of pressure’. But since velomobiles are often faster than the wind, the relative wind mainly comes from ahead (see below, angle of attack). For this case, and when the fairing is made from thin airfoil sections, ‘the center of pressure’ is usually the one at small angles between the relative wind and the vehicle main plane and then the center of pressure position is fairly constant.
Gloger conducted crosswind-experiments in real scale (Gloger, 1996). From the results he concluded that for good handling of a single-track vehicle the center of lateral area of a fairing should be far front and low. This finding is compatible with the low weathercock stability suggested by Cooper (see above). Low weathercock-stability is equal to a center of pressure location near the nose, possibly in front of the center of mass.
Up to now, no analytical proof or simulation (numerical solution) existed that demonstrates mathematically what was guessed by Cooper and what was suggested by Gloger after the interpretation of the results from his crosswind experiments. In this paper, the first analytical approach known to the author was tried in order to find a solution to the statics of the crosswind problem and to derive the location of the center of pressure that would require minimal rider action (‘equilibrium center of pressure location’).
A bicycle model (by members of the Cornell Bicycle Research Project, see below) was modified by the author with the terms for the aerodynamic forces. The resulting equation allows the designer to trim a single-track vehicle so that it keeps its course in a steady field of crosswind. In order for trimming to be possible, a designer needs to know the position of the center of pressure in dependence of the angle of attack. With a method given in Fuchs 1993, two extreme positions may be estimated (at small angles, and at about 90 degrees). In wind tunnel experiments the center of pressure locations of between 0 and 90 degrees angle of attack could be determined.
So far no experimental validations - e.g. in a way Gloger performed his crosswind-studies - of the aerodynamically modified Cornell bicycle model exist. But there is qualitative evidence for the correctness of the model (see further below).
2. Bicycle model
Extracts from the Cornell Bicycle Model
Cornell Bicycle Research Project
(Summary by Andreas Fuchs)
Some readers may know about the Cornell Bicycle Research Project (CBRP) due to a paper by Olsen and Papadopoulos in Bike Tech (Olsen and Papadopoulos, 1988), a journal which is no longer being published. There, the equations of motions of a bike model having rigid knife-edge wheels („ideal tires“), rigid rear frame with rider being immobile relative to it, and a rigid front assembly consisting of a steerable front fork with front-wheel, stem and handlebar, were published.
Thanks to personal communication with Andy Ruina the author of this paper has access to a unpublished report (Papadopoulos 1987) that contains sections about sidewind-effects (p. 10 and p. 19). Andreas Fuchs combined equations from Papadopoulos’s 1987 report with equations about the aerodynamic torques and found the results of interest for the hpv community. Therefore, below there follows a short summary of the relevant equations of the Cornell Bicycle Model with reference to the unpublished report (personal communication with Papadopoulos, starting October 1996).
According to Hand
(1988), cited by Papadopoulos (1987), the unmodified Cornell bicycle model was
compared to bicycle models by earlier authors (see ref. cited at the end of
this box 1) and the Cornell bicycle model was found to be consistent with some,
whereas it was inconsistent with others. But confidence in its correctness is
increased by the fact that the equations of motion were derived using two
diverse approaches. The equations of the Cornell bicycle model are consistent
with those by Whipple (1899, with typographical corrections), Carvallo (1901),
Sommerfeld & Klein (1910), Döhring (1955), Neimark & Fufaev (1967,
potential energy corrected), Sharp (not the paper, but the dissertation 1971,
minor algebraic correction) and Weir (Dissertation, 1972).
The Structure of the Cornell Bicycle Model
A bicycle model consist of a set of equations describing the dynamics of this single-track vehicle, the equations of motion. In the Cornell Bicycle Model, their derivation starts with the formulation of the following four equations for :
F1) The total lateral forces that lead
to the lateral acceleration of all mass points, that is the whole bicycle
(total x-force; originally, the bicycle moves along the y-axis)
F2a) The total moment about the heading line of the rear assembly required for the acceleration of all mass-points in a general lateral motion (total c-moment by external forces; see figure A below)
F2b) The total moment about a vertical axis through the rear wheel contact point required for the acceleration of all mass points in a general lateral motion (total q-moment by external forces)
F3) The total moment exerted by external forces about the steering axis (total y-moment)
In the case of the Cornell bicycle model, the equations of motion are linearized and therefore are valid only for small angular deflections from the upright state of the single-track vehicle.
‘Reduced Equations of Motion’
To study the motion of the bicycle itself, if one is not interested in the position of the vehicle in the x-y-plane (positive directions: x to the right, y forward, z up) and the heading q, two equations to solve for the lean angle c and the steering angle y would suffice. By using relations between all the angles and the lateral acceleration (acceleration in the x-direction), x and q and their time derivatives may be eliminated. The side force in the front-wheel contact point may be eliminated from the set of four equations also by combining F2b) and F3). Three equations, equation F1) and the two ‘reduced equations of motion’ (lean- and steer-equation), remain:
(I) (p. 17, Papadopoulos 1987)
c lean angle of the rear assembly, to the right
y leftwards steer angle of the front assembly relative to the rear assembly
Mc tipping (or supporting) moment (usually 0)
My steering moment exerted by rider
All terms on the left side in the equations consist of indexed coefficients M, K and C, dependent on physical parameters of the rider- bicycle-system, multiplied with the lean- or the steer-angle or their time-derivatives.
Both the lean- and steer-equations are to be found in other letters also in Olsen and Papadopoulos (1988).
A sidewind creates forces acting on some point of the front assembly and on some point on the rear assembly (Remark by Fuchs: the respective centers of pressure).
These forces create moments (Mc)w and (My)w (‘w’ for wind): Mc tends to tip the bicycle, whereas My steers due to the forces acting in the points on the front and rear assembly.
If at zero lean the center of pressure of the rear assembly (rear frame and rider) would be vertically above the rear wheel ground contact point, lean occurs, but there is no steering. If the center of pressure of the front assembly (fork, wheel, stem, handlebar) lies on the line between the front-wheel ground contact point and the intersection of the steering axis with the vertical line through the rear wheel ground contact point, no steering occurs, but bicycle-tipping results.
Condition for equilibrium in steady crosswind
For a steady response to the wind with
steering angle y=0, the equation
(III) (p. 10, Papadopoulos 1987)
has to be fulfilled. This equation results from dividing (I) and (II) and setting all angular accelerations, angular velocities and the steer angle to zero.
The indexed coefficients of the equations of motion are as following (p. 16, Papadopoulos 1987):
According to lists and figures in Papadopoulos (1987) and Hand (1988) the parameters are (Abbreviations, see below):
1988, p. 29)
VIa) (Hand 1988, p. 27)
If the center of mass of the front assembly is in front of the steering axis, then d > 0.
VIb) (Hand 1988, p. 29)
VIc) (Hand 1988, p. 29)
The parameters and have the following physical significance:
is the sum of two terms, and . Both terms are due to mass-forces acting on the front assembly when the bike is leaned, when c ¹ 0: is a vertical mass-force acting on the lever d and is another vertical mass-force acting on the lever cf (proportional to trail). For equilibrium, the steering-torque induced by these gravitational forces needs to be counterbalanced by lift acting in the center of pressure.
is due to the total mass-force acting on the lever [sin(c) » c for small angles] and is the tipping moment due to gravitation.
Abbreviations in the
Cornell Bicycle Model
constant, » 9.81 m/s2
mf mass of front assembly (Hand 1988, p. 29)
cf measure for trail, . cf > 0, also in
case of mirrored front-wheel geometry. (Hand 1988, p. 51)
cw wheelbase (Hand 1988, p. 21)
ht height of center of mass of rider-bicycle-system (Hand 1988, p. 53)
hf height of center of mass of front assembly (Hand 1988, p. 51)
l steering axis tilt (from vertical) (Hand 1988, p. 21)
lf horizontal position of front assembly center of mass,
forward of front-wheel ground contact point (Hand 1988, p. 51)
mr mass of rear assembly (Hand 1988, p. 29)
lt horizontal position of system center of mass,
forward of rear wheel ground contact point (Hand 1988, p. 53)
lr horizontal position of rear assembly center of mass,
forward of rear wheel ground contact point (Hand 1988, p. 49)
Table a Parameter designations
FIGURE A Main dimensions on a bicycle according to the Cornell Bicycle Model
Here, cf > 0, lf < 0, d > 0.
References of CBRP, published:
Olsen, John, and Papadopoulos, Jim. Bicycle Dynamics - The Meaning Behind the Math. Bike Tech December 1988
References of CBRP, unpublished:
Hand, Richard Scott. Comparisons and Stability Analysis of Linearized Equations of Motion for a Basic Bicycle Model. Thesis. Cornell University, May 1988 (Received by author due to personal communication with Andy Ruina.)
Papadopoulos, Jim. Bicycle Steering Dynamics and Self-Stability: A Summary Report on Work in Progress. Preliminary Draft. Cornell Bicycle Research Project Report, December-15 1987 (Received by author due to personal communication with Andy Ruina.)
References which support the correctness of the Cornell Bicycle Model
Carvallo. Théorie Du Mouvement Du Monocycle. Part 2: Theorie de la Bicyclette. Journal de L’Ecole Polytechnique, Series 2, Volume 6, 1901
Döhring, E. Stability of Single-Track Vehicles. Forschung Ing.-Wes. Vol 21, No. 2, pp. 50-62, 1955
Neimark J.I., and Fufaev N.A. Dynamics of Nonholonomic Systems. American Mathematical Society Translations of Mathematical Monographs, Vol. 33 (1972), pp. 330-374, 1967
Sommerfeld, A. and Klein, F. Ueber die Theorie des Kreisels. Die Technischen Anwendungen der Kreiseltheorie, Vol. IV, ch. IX-8, pp. 863-884, Leipzig (Teubner) 1910
Sharp R.S. The Stability and Control of Motorcycles. Journal on Mechanical Engineering Science, Vol. 13, No. 5, pp. 316-329, 1971
D.H. Motorcycle Handling Dynamics and Rider Control and the
Effect of Design Configuration on Response and Performance, Ph.D. Dissertation,
Dept. of Engineering, UCLA, June 1972
Whipple, F.J.W. The Stability of the Motion of a Bicycle. Quarterly Journal of Pure and Applied Mathematics, Vol. 30, pp. 312-348, 1899
(End of Box 1)
3. Normal force and true angle of attack
Lift is usually defined as acting against gravity. On faired velomobiles, the lifting surface is vertical so that when the word ‘lift’ is used here, actually a predominantly sidewards force is denominated.
Lift and drag
combine to the total aerodynamic force. The component perpendicular (normal) to
the centerplane of the fairing is called normal force:
of the air
v airspeed of the relative wind (vectorial sum of vehicle-groundspeed
cNi coefficient of normal-force
Ai reference area
For lift: Often, the lateral area of
fairing is used (for drag: often the
cross-section of the fairing is used). On how to transform coefficients: See
[The equation for the normal-force is formally similar to the one for drag.]
wind is the vectorial sum of the vehicle groundspeed and the speed of the
crosswind relative to the ground. The faster the hpv (the higer its propelling
power and the smaller its drag), the smaller the angle of attack. Yaw-, roll-
and pitch-rate also influence the angle of attack (Cooper 1974): Upon taking
these rates into account, we arrive at the ‘true angle of attack’. In this
paper, however, only the static case is considered, all angular velocities are
zero and therefore the true angle of attack is the angle with which the
relative wind approaches the fairing.
angles of attack, the coefficient of the normal-force is approximately the same
as the coefficient of lift:
dcN/da = cNa slope of lift-angle of attack-curve
a angle of attack (angle between the centerplane of the fairing
and the relative wind)
cL coefficient of lift
In Fuchs (1993) the magnitude of the lift-alpha-slope is given in dependence of the airfoil / fairing thickness. For thin, symmetrical airfoils, the linear approximation may be used in an interval of -10 »< a »< +10 deg.
4. The center of pressure locations
Most single-track velomobiles can be described as consisting of two lifting surfaces (see Fig. 2):
1. The vehicle body including main fairing, top (covering the rider’s head) and possibly faired rear wheel
2. The steered front-wheel which produces considerable lift if faired or if the wheel is a trispoke
Fuchs (1993) is on how to estimate the location at which the normal force (‘lift’) acts for small angles of attack (0 < a < 15 degrees) and large angles of attack (a » 90 degrees). This location is called center of pressure CP.
4.1 Center of pressure (CP) of the vehicle excluding the
front-wheel but including the possibly faired rear wheel (fairing, body)
According to Fuchs (1993), for common fairing geometries of supine recumbents, the CP is located at about 1/3 body length from the nose of the fairing for small angles of attack and slightly more than ½ body length for large angles of attack. If the top covering the head is fairly small relative to the rest of the fairing, then the CP lies just below half the vertical extension of the fairing (not the whole vehicle!).
Fairing center of pressure relative to the rear wheel
With the definitions of a bicycle’s geometry (® Box 1, Fig. 1), the position of the CP of the fairing relative to the rear wheel, rh, is (See also Fig. 2):
w distance between the nose of the
the front-wheel ground contact point
4.2 CP of the faired front-wheel
- If the front-wheel is not faired and does not produce lift, then A2 = 0
- If the faired front-wheel is only partially exposed to the airstream, then: , R: front-wheel radius
Wheel center of pressure relative to the rear wheel
With the definitions of a bicycle’s geometry (® Box 1), the position of the CP of the wheel relative to the rear wheel, rhw, is (See also Fig. 2) :
w distance between the nose of the
the front-wheel ground contact point
Assuming the front-wheel to be a flat plate, the CP-location can be estimated using an integration method in Fuchs (1993).
Figure 1 CP-location of a faired front-wheel with radius R. The wheel may be hidden in the fairing by the distance T. When the CP is in front of the steering axis (as in this figure) then h2 > 0.
Distance between the front-wheel CP and the steering axis
The distance between the steering axis and the front-wheel-CP is given by:
h2 distance of the faired-wheel CP
to the steering axis
(positive if CP in front of the steering axis)
w distance between the nose of the vehicle and
the front-wheel ground contact point
Figure 2 Definitions of the
position of the body center of pressure, the position of the faired wheel
center of pressure and the position of the fairing relative to the bicycle
Here, the symbol ‘circle with dot’ does not represent a vector pointed towards the reader, but is simply the symbol for center of pressure.
4.3 Center of pressure location of the whole vehicle
The CP of the vehicle is determined by the CP of the fairing and the CP of the faired front-wheel:
The lift-curve-slope of the whole
vehicle can be derived from
If the front-wheel is unfaired, A2=0, then A1=A3, N1=N3, n1=n3, d1=d3 and cNa1=cNa3.
5. Introduction of the aerodynamic terms into the Cornell Bicycle Model
Aerodynamic terms using the formulaes of the two preceding chapters were combined with the equations F1, F2a, F2b and F3 of the Cornell Bicycle Model (See box 1); the aerodynamic terms were added on the side of the external forces of the formulaes F1 to F3:
Formula F1, X-force (lateral forces) :
Force due to ‘lift’ on fairing:
Force due to ‘lift’ on faired front-wheel:
Formula F2a, c-moment (moment relative to a horizontal forward axis) :
Moment due to ‘lift’ on the fairing:
Moment due to ‘lift’ on the faired front-wheel:
(Since for small lean angles d and N are approximately perpendicular, we do not use the cross-product: sin(90 deg) = 1!)
(moment relative to a vertical axis through the rear wheel ground contact) :
Moment due to ‘lift’ on the fairing: (negative sign!)
Moment due to ‘lift’ on the faired front-wheel: (negative sign!)
Formula F3, y-moment (moment relative to the steering axis) :
External forces do not directly create moments at the steering axis, but indirectly due to the forces in the front-wheel contact. Therefore N3 does not appear in the equation for y.
Moment due to ‘lift’ on the faired front-wheel: (negative sign if h2 > 0)
From the four modified equations the ‘modified reduced equations of motion’, now including the aerodynamic terms, were derived as described in box 1. From these modified reduced equations of motion, in a further step an aerodynamically modified equation III was derived (Condition for equilibrium in steady crosswind):
6. Trim equation
Equation (III), now modified with the aerodynamic terms, is as follows:
Kyc: Equation Iva (Box 1)
Kcc: Equation Ivb (Box 1)
My: Steering moment by rider
Mc: Supporting moment (e.g. supporting side-wheels)
The terms with N2 describe the moments due to ‘lift’ on the front-wheel and those with N3 the moments on the main fairing, the top and the eventually faired rear wheel.
equation 7) all the detailed geometrical and aerodynamic relations yields the ‘faired single-track vehicle trim equation’
‘Faired single-track vehicle trim equation’ 8) :
This trim equation aplies to states of single-track vehicle near upright (small lean angles) with zero steer angle (See box 1 for details about the bicycle model). If the trim equation is true, then equilibrium is established: As long as no disturbance alters the state of the vehicle, it will go straight, with some lean but zero steer angle. The trim equation is valid only if the speed of sideslip is much smaller than the component of the crosswind perpendicular to the vehicle heading.
The significance of the abbreviations may be found elsewhere
in this paper:
a) Influence of rider (My) ® Box 1
b) Variables describing the mass distribution ® Box 1
c) Variables describing the ‘bicycle geometry’ ® Box 1
d) Variables describing the lift-distribution on the body and the faired front-wheel: Chapters 3 and 4
e) Possible moment Mc by e.g. supporting wheels or fins which produce roll-moments ® Box 1
6.1 The trim equation and effects that affect trim
- The trim equation (8) is arranged similar to equation III of box 1 to show the resemblance.
The velomobile designer may rearrange it according to his wishes.
- The equilibrium is independent of the gravitational acceleration and, if the steering-moment My and the supporting moment Mc are zero, the equilibrium is also independent of velocity : (the dynamic pressure) then cancels on the right side of the equilibrium equation.
- At equilibrium, body-CP-locations near the center of mass are only reached at the far extremes of the parameter ranges. Generally, the equilibrium center of pressure locations lie front of the center of mass.
- In the case where Mc and My are zero, in formula 8) the angle of attack a factors in such that the result is 1. If the center of pressure did not move in dependence of angle of attack, trim would thus be independent of a.
For small angles of attack (typically: 0 < a < 10 degrees), the center of pressure of thin symmetrical airfoils does indeed not move much. Therefore, for that small region of angle of attack the above expression can be considered valid on the whole range of about 10 degrees so that the parameters describing the CP-locations (n1, d1, n2, d2, n3, d3) do not have to be varied. But since outside the range 0 < a < 10 degrees even for thin airfoils the CP-location depends on angle of attack a, it is factored in to remind us of that fact.
How insensitive the CP-location is with respect to a for thicker symmetrical airfoils common on hpv fairings should / could be measured in wind tunnel experiments.
- Trim depends on the lift-curve-slopes cNa2 and cNa3 which in turn depend mainly on airfoil thickness. Faired wheels are relatively thinner airfoils than bodies: They therefore produce more lift per degree angle of attack than bodies. See Fuchs, 1993.
- The supporting moment Mc is in the case of single-track vehicles without side-wheels non-existent and thus equal to zero.
But a moment exerted by the rider on the handlebars, My, may exist, although the rider is unnecessarlily stressed by a non-zero steering moment. Trim should be established so that My = 0 for the most common angles of attack.
If My > 0 (the rider pulls on the left side on the handlebar and pushes on the right side), then n3 reqired for equilibrium becomes smaller and the equilibrium center of pressure location moves forward compared to the case where My = 0 (remember: The sign convention is such that the wind blowing from left to right blows in the positive lateral-x-direction). Conversely, if the rider turns the handlebar to the right (My < 0) the equilibrium center of pressure location moves back, n3 needs to become larger.
- In the case of an unfaired
front-wheel (and vanishing moments), A2=0 and d2 as well as
n2 are not defined. All terms including A2 and n2 vanish and the trim equation
becomes much simpler and shorter:
In the case of a lifting wheel (A2 ¹ 0 and d2, n2 defined), two cases have to be distinguished:
a) Standard front-wheel geometry
Since the CP is in front of the steering axis (h2 > 0, see Fig. 1) the front-wheel is steered out of the wind (e.g. on a triathlon bike with a trispoke front-wheel or a track bike with a disk front-wheel). Trail exerts a moment due to lift on the main fairing N3 that also pushes the steering axis out of the wind. Therefore, the moment by lift required for equilibrium could be smaller; with a constant N3, the lever between the main body center of pressure and the center of mass may become smaller and the cp may therefore be further away from the nose of the fairing (bigger n3).
An elegant way to get rid of the disturbing moments on a faired front-wheel is to fair the whole wheel with the main fairing. This is what Brichet did when designing ‘Nilgo III’ (Fehlau 1996, p. 128).
Another way to minimize the moments on the front-wheel is to add tail fins at the rear-side of the fork, that is, to redistribute the lift on the front-wheel assembly such that the center of pressure lies on or behind the steering axis. But this method yields more lift - due to increased lifting area - on the front-wheel alone.
Note: When the steer angle y is not zero, the angle of attack of the exposed front-wheel area A2 is different from the angle of attack of the body. Thus, lift on the front-wheel alone may increase or decrease. This alters the lift distribution on the vehicle and therefore the CP moves (mainly up or down). Putting the whole front-wheel inside a fairing fixed to the body avoids this vertical shift in the vehicle-CP-location.
b) Mirrored front-wheel geometry
h2 (see Fig. 1) may be minimized by choosing a mirrored front-wheel geometry, where the steering axis tilts forward instead of rearwards. Read more on this in Gloger 1996.
6.2 Application of the the ‘trim equation’:
The trim equation (8) is basic for the lateral stability of faired single-track vehicles in crosswind (at small lean angles, at zero steer angles). The trim equation may be used in numerous ways: E.g. for a given faired single-track vehicle, if the location of the center of pressure in dependence of the angle of attack a is known, the trim equation may be rearranged to calculate the moment My in dependence of a required on the handlebar to keep the vehicle moving with zero steer angle.
in the ‘trim equation’ can be classified in five groups:
a) Influence of rider (My)
b) Variables describing the mass distribution
c) Variables describing the ‘bicycle geometry’
d) Variables describing the lift distribution on the body and the faired front-wheel
e) Possible moment Mc by e.g. supporting wheels or fins which produce roll-moments
The trim equation may be true after any set of variables from these five groups has been varied. A designer of velomobiles, wishing to establish equilibrium at certain conditions (that is at a certain angle of attack a) could now optimize his vehicle e.g. according to the following steps :
A) Define the position of the rider. Since the rider is the main mass, this defines the mass distribution of the vehicle to a high extent
bicycle geometry for good handling also in no-wind situations
C) Tune the lift distribution in such a way that the ‘trim equation’ 8) is true. Then trim is established at a certain angle of attack
The lift distribution may be varied by the following actions :
· redistribution of the lifting area (for minimum sidewind sensitivity, the lateral area A1 = [A2+A3] needs to be as minimal as possible)
· add lift-producing devices as small as possible (e.g. small fins or a fairing around a front-wheel that steers inside) to achieve a desired CP-location. Even though total sideforce will increase, handling may improve
· add devices that create disturbance to the airstream so that lift and sideforce vanish
To tune a velomobile the designer wants to know what changes will have what influence. The important parameters may be identified by calculating e.g. the sensitivities of the horizontal fairing CP-location n3 with respect to any other parameter / design variable xi:
10) Sensitivity of n3 with respect to design variable: xi =
The subscript ‘Vehicle’ signifies: Only xi is varied; all other parameters are held constant at the vehicle’s nominal values.
(An example of this variation is shown in Fig. 3. There d3 has been varied to get the curve ‘cp-locations’. The sensitivity of n3 with respect to d3 is the slope of this curve.)
Increasing the wheelbase, the rear center of mass height, or the trail, moves the equilibrium CP-location backwards. Moving the rear center of mass location forward relative to the rear wheel ground contact or moving the fairing CP up shifts the equilibrium body-CP-location forward.
Changes of the steering tilt angle do not have dramatic effects on the equilibrium CP-location.
Trail and rear center of mass height have strong influence on the equilibrium center of pressure postion in a positive way and the fairing CP height and rear center of mass forward position have strong influence in a negative way.
One may even think of applications of the trim equation (8) that at first sound like Science Fiction: Measure the angle of attack of the relative wind using a vane and automatically adjust e.g. the bicycle geometry or the lift distribution while riding such that trim is always established !
7. Evidence for the validity of the trim equation
No extensive validation of the aerodynamically modified Cornell Bicycle Model has been made, e.g. in an experiment similar to the crosswind-experiment by Gloger (1996). But a simple, illustrative experiment and the single-track velomobile ‘Aeolos’ by Joachim Fuchs demonstrate the (qualitative) validity of the trim equation (8):
7.1 A Simple Experiment (See also Milliken 1989)
Instructions: Lean a bicycle towards yourself to simulate lean against a crosswind. In this case, the crosswind would come from your side of the symmetry plane of the bicycle. Push the frame away from you at various horizontal locations such that the front-wheel aligns with the frame (steering angle zero degrees).
Question: Where do you have to push: more toward the back or more on the front of the frame?
Results: Most likely, you will find it easier to align the front-wheel with the vehicle’s main plane when you push somewhere near the steering axis (usually near the front of the bicycle).
Interpretation: The push by your fingertip is similar to a lift-force by the crosswind. The location where you push is near the equilibrium location of the center of pressure.
7.2 Aeolos - an example for a minimal crosswind-sensitive hpv
According to Joachim Fuchs (Fuchs Joachim, 1996) his fully faired single-track streamliner Aeolos can easily be ridden in crosswinds. The following text by Joachim Fuchs originates from the HPV CD 1997 (pictures of Aeolos may also be found on the HPV CD 1997):
„The construction is focused on little side wind sensitivity because it is known that race recumbents are difficult to control. Nevertheless, Stefan Gloger (DESIRA II) showed in his PhD work that this problem can be overcome. My plan was to reduce the lateral area of the rear part of the fairing. The centre of pressure is then closer to the front of the vehicle. The wind takes influence on the steering in a way that the rider must not react actively. The only thing he has to do is to hold the handle with a slack grip. Even in gusty and strong winds, the vehicle finds its own way and leans into the wind by itself.“
In order to check the state of trim of Aeolos, the longitudinal position of the center of pressure of the fairing, n3, was calculated using the trim equation (8). Those parameters that are not known were estimated from reasonable assumptions (Aeolos front-wheel is unfaired). In Fig. 3 the lateral area of the fairing and the wheels are shown. The locations of the center of pressure CP (small angles of attack) and the center of lateral area CLA (center of pressure at angles of attack near 90 deg) were estimated by using methods in Fuchs (1993). Known masses and a foto of the frame permitted an estimation of the position of the center of mass.
Fig. 3 Aeolos sideview shows the relative position of the center of mass, the estimated centers of pressure for small and large angles of attack (the latter being probably near the center of lateral area) and the possible equilibrium positions of the center of pressure (cp-locations).
If the height above the ground of the fairing center of pressure is d3 = 0.60 m, for equilibrium a body-CP-longitudinal location of n3 = +0.81 m behind the nose of the fairing is calculated. This longitudinal position is in front of the center of mass.
The center of pressure height d3 was then varied, and by using the solver of a worksheet, corresponding longitudinal positions were calculated (or one may use equation 9). The results are plotted in Fig. 3 as the line ‘cp-locations’. This line shows that a vertical displacement in up-direction of a fairing needs to be accompanied by a horizontal displacement towards the front of the vehicle. Otherwise, equilibrium is no longer assured.
The line of ‘cp-locations’ crosses the shortest line between the estimated centers of pressure (CP and CLA) for small and large angles of attack. This is an indication that the Aeolos-fairing is already positioned quite optimally relative to the person and the recumbent inside. Joachim Fuchs arrived at this technical solution after having considered the results of the crosswind experiment by Stefan Gloger (Fuchs Joachim 1996).
The line ‘cp-locations’ crosses the line between the estimated centers of pressure CP and CLA near the center of pressure at small angles of attack CP. This indicates that Aeolos will need no steering input when the relative wind comes from angles of attack of less than 45 degrees (about 15 to 30 degrees).
If the exact locations of the center of pressure and corresponding angles of attack were known (e.g. by wind-tunnel experiments), the parameters of Aeolos could, with the help of the trim equation, be varied such that Aeolos is trimmed at a defined angle of attack.
Very probably the design of Aeolos would be worsened by adding a tail or a long nose. In both cases, the centers of pressure would move away from the line of statically optimal center of pressure locations (line ‘cp-locations’) and the vehicle would turn into the wind or out of the wind. The rider of the worsened Aeolos would have to compensate for that by steering action.
Matt Weaver’s ‘Cutting Edge’ ultra-streamliner has a very long nose (Weaver, 1991). Matt states that in sidewinds he had to gently steer out of a lean into the wind. This can be interpreted in the following way: The nose is so long that the actual center of pressure lies in front of the line of possible equilibrium center of pressure locations (derived by using the parameters of the mass distribution and the parameters of the suspension geometry). This leads to steering action and to rolling into the wind. If the actual center of pressure was higher above ground, that is nearer to the line of the cp-locations, then the tipping moment due to sidewind would also be higher and thus would work stronger against the lean. As a consequence, Matt would not have to roll out of the wind as much by steering.
The equation to calculate the equilibrium location of the center of pressure for zero steering angle in crosswinds - the ‘trim equation’ - has been derived. Using it, a single-track velomobile designer may trim his vehicle to achieve good handling characteristics under certain conditions (angle of attack); the torque that has to be exerted by the rider onto the handlebar may be minimized. But the fact that a vehicle is in trim at certain angles of attack does not assure safe handling in any situation that may be encountered in windy conditions on the street.
For the first time it was mathematically shown that static stability of single-track vehicles in crosswinds is achieved when the center of pressure is in front of the center of mass (Hucho 1994).
What has not been discussed in this paper are the dynamics of the transition from one state of crosswind-influence to another state of crosswind-influence. This would require further research.
9. Suggestions for further research
· Wind tunnel experiments to determine the wandering of the CP with angle of attack so that estimations are no longer needed.
· Investigations concerning the importance of the human as a controller. Gloger’s crosswind-experiment (Gloger 1996) shows that there is a difference in vehicle reaction to crosswind dependent on the familiarity of the rider with a certain vehicle. Joachim Fuchs text about how to handle Aeolos is another hint to the importance of the rider.
· Theoretical investigations of the dynamics, since it is not only important to establish equilibrium at certain conditions, but (for safety) it is even more important how equilibrium is approached from any state the vehicle is in!
· Simulations of fully-faired (single-track) vehicles to study the dynamics in dependence of various forcings by crosswind: Impulse input (wind gusts), step input (coming from a no-wind region into a region with steady crosswind), and periodical as well as variable crosswind-patterns. Rigid-body simulation tools such as ‘Mechanica’ could be used.
· Estimations to determine the relevance of aerodynamic damping: hpv’s are lightweight and their fairing lateral area may be huge. It is therefore possible that aerodynamic damping is important for mainly the yaw movement.
· Validations of simulations by measuring vehicle behavior in different crosswind-patterns.
‘Many thanks’ go to Andy Ruina for providing copies of documents about the Cornell Bicycle Model.
Jim Papadopoulos’s comments were essential for the derivation of the trim equation as can be followed in this paper - Thank you very much, Jim ! Having copies only of the final result, Jim also derived the trim equation and arrived at a result similar to that of the author.
Thanks go also to Joachim Fuchs for providing Aeolos data (no, Joachim is not my brother!).
In an early stage of this work, Joachim Fuchs, Stefan Gloger and Bill Patterson contributed many important ideas. With Theo Schmidt, the author was able to conduct valuable discussion about sailing with faired hpv’s.
Thanks go to David Picken for checking the text.
(See also references at the end of box 1)
Bülk, Eggert „Aerodynamik an HPV-Fahrzeugen (Teil 1)“. Pro Velo 31, 1992
Bülk, Eggert „Aerodynamik an HPV-Fahrzeugen (Teil 4)“. Pro Velo 36, 1994
Cooper, K. R. „The Effect of Aerodynamics on the Performance and Stability of High Speed Motorcycles“. Proceedings ot the second AIAA symposium on aerodynamics of sports & competition automobiles, Los Angeles, 1974
Fehlau, Gunnar „Das Liegerad“. Moby Dick 1996
Fuchs, Andreas, Leserbrief betreffend Bülk’s Artikel „Aerodynamik an HPV-Fahrzeugen (Teil 1)“ Pro Velo 32, 1993
Fuchs, Andreas „Towards the Understanding of (dynamic) Stability of Velomobiles: The Forces, their Distributions and associated Torques“. Proceedings of the First European Seminar on Velomobile Design, Technical University of Denmark, July 8th 1993
Fuchs, Andreas „The Link between Stability and Performance“. Human Power 11/2, Spring-Summer 1994
Fuchs, Joachim „Aeolos-Verkleidung“. Pro Velo 44, 1996
Gloger, Stefan „Entwicklung muskelkraftgetriebener Leichtfahrzeuge“. VDI Reihe 12 Nr. 263. VDI Verlag 1996
Hucho, Wolf-Heinrich „Aerodynamik des Automobils“. VDI Verlag 1994
Meyer Christian, Zechlin Oliver, Zerbst Carsten (Editors) „HPV CD 1997“
Milliken, Doug „Stability? or Control?“, Human Power, Vol. 7 No. 3, Spring 1989
Patterson, Bill „Designing a Single Track Vehicle for Handling Qualities“. Recumbent Cyclist News No. 41, 1997
Weaver, Matt „The Cutting Edge streamlined bicycle“, Cycling Science, September/December 1991
 At angles of attack of about 10 to 15 degrees, thrust due to lift is maximal; a velomobile could be tuned, for example so that a zero steer angle results at this region of angle of attack!
 Further investigations, not reported here, indicate that the height of the center of pressure above ground should not be much more than the height of the center of mass.