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 :

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.

(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 c_{f} (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/s^{2}

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, c_{f} > 0, l_{f} < 0, d > 0.

__References of CBRP, published:
__

**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)

c_{Ni} coefficient of
normal-force

A_{i} 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) _{}

dc_{N}/da = c_{N}_{a} slope of lift-angle of attack-curve

a angle
of attack (angle between the centerplane of the fairing

and the
relative wind)

c_{L} 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)

N_{2} normal force
acting in the wheel-CP (N_{2} = 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:

*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) _{}

h_{2} 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 c_{N}a1=c_{N}a3.

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:

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) :

**‘Faired single-track vehicle trim equation’** 8) :

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 (n_{1}, d_{1}, n_{2}, d_{2}, n_{3}, d_{3}) 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 c_{N}_{a}_{2} and c_{N}_{a}_{3} 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:

9) _{}

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 A_{2} 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

h_{2} (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.

The variables
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

B)
Optimize the
bicycle geometry for good handling also in no-wind situations

(Patterson 1997)

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[1]

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 x_{i}:

10) Sensitivity of n3 with respect to design variable: x_{i} = _{}

The subscript ‘Vehicle’ signifies: Only x_{i} 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):

According to Joachim Fuchs (Fuchs Joachim, 1996) his fully faired single-track streamliner

*„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 n_{3} = +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.

8. __Conclusions__

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[2] (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.

__Acknowledgements
__

‘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.

equil5c.doc

[1] 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!

[2] 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.