Trim of aerodynamically faired single-track vehicles in crosswinds
Andreas Fuchs
P: Waldheimstrasse 32, CH-3012 Bern, Switzerland
W: Hochschule für Technik und Architektur Bern, Morgartenstrasse 2c, CH-3014
Bern, Switzerland. fuchs@isbe.ch
ABSTRACT
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.
1. Introduction
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
(Box 1)
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:
Lean equation:
(I) 
(p. 17,
Papadopoulos 1987)
Steer equation:
(II) 
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).
Crosswind
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):
(Iva) 
(Ivb) 
According to lists and figures in Papadopoulos (1987) and Hand (1988) the
parameters are (Abbreviations, see below):
V) 
(Hand
1988, p. 29)
In detail:
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
g gravitational
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
Weir
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:
1a)
i
= 1,2,3
r density
of the air
v airspeed of the relative wind
(vectorial sum of vehicle-groundspeed
and windspeed)
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
Fuchs 1993
[The equation for the normal-force is formally similar to the one for drag.]
The relative
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.
For small
angles of attack, the coefficient of the normal-force is approximately the same
as the coefficient of lift:
1b) 
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)
n3 distance between the nose of the
fairing and the CP of the fairing
(excluding the front-wheel)
d3 height
of the CP of the fairing (excluding the front-wheel)
A3 reference area of the
vehicle (for lift: the lateral area) excluding the
wetted area of the front-wheel
N3 normal force on the
fairing (excluding the front-wheel)
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):
2) 
w distance between the nose of the
vehicle and
the front-wheel ground
contact point
4.2 CP of the faired front-wheel
n2 horizontal
distance between the nose of the vehicle and
the CP of the faired
front-wheel
d2 height
of the CP of the faired front weel
A2 ‘wetted area’, area of the
faired front-wheel exposed to wind (for lift: the
lateral area)
N2 normal force
acting in the wheel-CP (N2 = 0 if wheel unfaired)
Remarks:
-
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) :
3) 
w distance between the nose of the
vehicle and
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:
4) 
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
inside.
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
n1 distance between the nose of the
fairing and the CP of the vehicle
d1 height
of the CP of the vehicle
A1 reference area of the
whole vehicle (for lift: the lateral area)
N1 Total normal force on
vehicle
The CP of the vehicle is determined by the CP of the fairing and the CP of the
faired front-wheel:
5) 
The lift-curve-slope of the whole
vehicle can be derived from
6) 
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!)
Formula
F2b, q-moment
(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:
7) 
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.
Putting into
equation 7) all the detailed geometrical and aerodynamic relations yields the ‘faired single-track vehicle trim equation’
8) :
