List of Symbols
| a | distance front axle to c.g.; half of contact length |
| ax | longitudinal acceleration |
| ay | lateral acceleration |
| aμ | slip velocity dependency coefficient for friction |
| Ar | rolling resistance coefficient |
| b | distance rear axle to c.g.; half contact width |
| B | stiffness factor in ‘Magic Formula’ |
| B1 | brake force of rolling wheel |
| c | stiffness; factor |
| cc | lateral carcass stiffness per unit length |
| cgyr | non-dimensional gyroscopic coefficient |
| cpx,y | tread element stiffness per unit length of circumference |
 |
tread element longitudinal stiffness per unit area |
| C | cornering stiffness ; sum front and rear |
| Ci | cornering stiffness, sum left and right |
| C | contact centre (point of intersection) |
| C | shape factor in ‘Magic Formula’ |
| CdA | air drag coefficient |
| CFx | longitudinal stiffness of standing tyre |
| CFy | lateral stiffness of standing tyre |
| CFz | stiffness of tyre normal to the road |
| CFα | cornering stiffness |
| CFκ | longitudinal slip stiffness |
| CFγ | camber stiffness for side force |
| CFφ | spin stiffness for side force |
| Cgyr | tyre gyroscopic coefficient |
| CMα | aligning torque stiffness |
| CMγ | camber stiffness for aligning torque |
| CMφ | spin stiffness for aligning torque |
| CMψ | torsional yaw stiffness of standing tyre |
| CMxγ | overturning couple stiffness against camber |
| Ccx,y | carcass horizontal stiffness of standing tyre |
| Cgyr | gyroscopic coefficient |
| dfz | normalised change in normal load, Eq.(4.E2a) |
| dpi | normalised change in inflation pressure, Eq.(4.E2b) |
| dt | tread depth |
| D | peak factor in ‘Magic Formula’; dissipation function |
| E | curvature factor in ‘Magic Formula’ |
| e | caster length; tread element deflection |
| f | trail of c.g.; frequency [Hz] |
| fr | rolling resistance coefficient |
| Fax | force for forward acceleration |
| Fd | air drag force |
| Fx,tot | sum of longitudinal tyre forces |
| Fx | longitudinal tyre force |
| Fy | lateral tyre force |
| Fz | vertical (normal) tyre force (load) (>0), in Chap.9, 10: Fz <0 |
| Fr | rolling resistance force (>0) |
| FN | tyre normal force (>0) |
| FNo | reference vertical load, nominal load (= |Fzo|) |
| FV | tyre vertical force |
| FH | tyre longitudinal horizontal force |
| g | acceleration due to gravity; feedback rider control gain |
| G | weighting factor |
| h | height |
| H | height; sharpness factor in ‘Magic Formula’ |
| H | transform; Hurwitz determinant |
| i |  |
| iz | radius of inertia |
| I | moment of inertia |
| Iw | wheel polar moment of inertia |
| Ip | wheel polar moment of inertia |
| j | |
| k | radius of inertia; viscous damping coefficient |
| K | centrifugal force; force acting on belt, wheel centre |
| l | wheel base |
| ls | shift; two-point follower length |
| lb | length of basic curve |
| lf | offset |
| l | unit vector along line of intersection |
| m | mass; fraction of contact length 2a where adhesion occurs |
| mc | contact patch mass |
| mt | tyre mass |
| mm | mass of mainframe (including lower part of rider) |
| mmr | mass of mainframe plus rider |
| mr | mass of upper torso |
| MB,D | brake, drive torque |
| Mx | overturning couple |
| My | rolling resistance moment |
| Mz | (self) aligning torque |
 |
(self) aligning torque due to lateral deflections |
 |
aligning torque due to longitudinal deflections |
| Mz,gyr | gyroscopic couple |
| Mδ | steer torque |
| n | number of elements; frequency [Hz] |
| n | unit vector normal to the road =(0,0,‒1)T |
| nst | steer system ratio |
| p | Laplace variable [1/m] |
| pi | inflation pressure |
| q | average vehicle yaw resistance arm; generalised coordinate |
| q | contact force per unit length of circumference, vector |
| Q | generalised force |
| r | yaw rate; tyre (loaded) radius |
| rc | radius of carcass (belt), unloaded; cross section crown radius |
| ryo | free tyre radius varying along cross section contour, ryo = ryo(yco) |
| re | effective rolling radius of freely rolling wheel |
| rf | free unloaded tyre radius |
| rl | loaded radius |
| ro | free unloaded tyre radius (= Ro) |
| R | radius of curvature |
| Ro | free unloaded tyre radius (= ro) |
| s | forward position of neutral steer point; half track width |
| s | Laplace variable; travelled distance |
| ssx | κ (practical longitudinal slip component) |
| ssy | tanα (practical lateral slip component) |
| s | unit vector along wheel spin axis |
| S | wheel slip point; impulse; string tension force |
| SV,H | vertical, horizontal shift |
| t | pneumatic trail; time |
| tc | caster length |
| tr | rise time |
| t | unit vector in road plane perpendicular to line of intersection l |
| T | kinetic energy; moment acting on belt, wheel centre |
| u | forward velocity of c.g.; longitudinal deflection |
| U | potential energy |
| v | lateral velocity of c.g.; lateral deflection |
| V | speed of travel of c.g. (with x, y components) |
| V | speed of travel of wheel centre (with x, y components) |
| Vc | speed of contact centre C (with x, y components) |
| Vg | speed of sliding (with x, y components) |
| Vo | reference velocity =√(gRo) |
| Vr | wheel linear speed of rolling (= Vcx ‒Vsx) |
| Vs | wheel slip velocity of slip point S (with x, y components) |
| Vx | longitudinal speed component of wheel centre |
 |
velocity of contact patch mass (with x, y components) |
| w | vertical road (effective) profile (positive downwards) |
| W | work |
| x,y,z | longitudinal, lateral, vertical displacement |
| x,y,z | coordinates with respect to moving axes system, z axis vertical |
| xo,yo,zo | global coordinates |
 |
global coordinates |
| X | longitudinal horizontal tyre force |
| X,Y,Z | global coordinates |
| yco | distance from wheel centre plane |
| ymr | lateral offset of mmr c.g. |
| α | wheel (side) slip angle; axle (side) slip angle |
| α | road transverse slope angle |
 |
transient tyre slip angle |
| αa | virtual axle slip angle |
| β | vehicle side slip angle; tyre yaw torsion angle |
| βx,y | road transverse, forward (effective) slope angle |
| βgyr | gyroscopic wheel coupling coefficient, Eq.(6.35) |
| γ | camber (wheel inclination) angle |
 |
transient tyre camber angle |
| Γ | unit step response function |
| δ | steer angle of front wheels |
| δo | ≈l/R, steer angle at V→0 |
| Δ | increment |
| ε | roll steer coefficient; rake angle of steering axis |
| ε | string length ratio, Eq.(5.153); eff. roll. radius gradient ‒∂re/∂dt |
| ε | small quantity to avoid singularity |
| εγ | camber stiffness reduction factor |
| εNL | non-lagging part |
| ζ | damping ratio; spin factor (=1 if spin influence is disregarded) |
| ζh | height ratio, Eq.(6.36) |
| ζα | cornering stiffness load transfer coefficient |
| ζγ | camber stiffness load transfer coefficient |
| η | understeer coefficient; effective rolling radius gradient ‒∂re/∂ρz |
| ηy | c.g. offset steer coefficient |
| θ | tyre model parameter, Eqs.(3.6, 3.24, 3.46) |
| θ | angular displacement about η axis; pitch angle |
| θc | string model composite parameter, Eq.(5.160) |
| κ | longitudinal wheel slip |
 |
transient longitudinal tyre slip |
| κ∗ | damping coefficient due to tread width |
| λ | wavelength; root characteristic equation |
| λ | fraction of 2a where adhesion occurs; user scaling factor |
| μ | coefficient of friction |
| ρ | tyre radial (vertical) deflection |
| ρx,y,z | tyre longitudinal, lateral, normal deflection |
| σ | relaxation length; load transfer coefficient |
| σ | theoretical slip, vector, Eq.(3.34) |
| σ∗ | intersection length in string model with tread elements |
| σc | string model length parameter, Eq.(5.153) |
| σc | contact patch relaxation length |
| τ | roll camber coefficient |
| φ | body roll angle; spin slip |
 |
transient spin slip |
| φt | turn slip |
| ϕ | phase angle |
| ψ | yaw angle; steer angle |
| ψc1 | compliance steer angle |
| ψio | toe angle |
| ω | frequency [rad/s] |
| ωo | undamped natural frequency |
| ω1,2 | natural frequencies |
| ωn | damped natural frequency |
| ωs | path frequency [rad/m] |
| Ω | wheel speed of revolution |
| ξ;,η,ζ | moving axes system, η axis along spin axis, ξ; horizontal |
Subscripts and superscripts
| a | axle; from belt to wheel rim centre |
| b | belt; from belt centre to rim |
| c | compliance (steer) |
| c | contact patch; from contact patch centre to belt; crown; contour |
| D | drag |
| e | effective |
| eff | effective (cornering stiffness) |
| eq | equivalent |
| f | free, unloaded; of front frame |
| g | global |
| i | 1: front, 2: rear |
| L,R | left, right |
| m | of mainframe |
| mr | of mainframe plus rider |
| NL | non-lagging |
| o | original; initial; average; unloaded; nominal; at vanishing speed; natural |
| r | roll; rolling; rolling resistance; of residual spring; of rider |
| s | slip; from road surface to contact patch; of front sub-frame |
| sl | at verge of total sliding |
| sf | side force (steer) |
| ss | steady state |
| st | static |
| stw | steering wheel |
| t | transition from adhesion to sliding |
| w | wheel |
| x,y,z | forward (longitudinal), lateral (to the right), downward |
| zr | residual (torque) |
| ξ;,η,ζ | along, around ξ;,η,ζ axes |
| 0 | at zero condition |
| 1,2 | front, rear; leading, trailing edge |
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