Notation and Notation List

Table 1.3 shows notation of parameters, variables and subscripts used in this compendium. The inten- tion of this compendium is to follow International Standards (ISO 8855), but deviations from this is sometimes used, especially motivated by conflicts with areas not covered by (ISO 8855). Some alterna- tive notations are also shown, to prepare the reader for other frequently used notation in other litera- ture.

The list does not show the order of subscripting. For example, it does not show whether longitudinal (𝑥) force ( ) on rear axle (𝑟) should be denoted or . The intention in this compendium is to or-

der subscript with the physical vehicle part (here rear axle) as 1st _{subscript and the specification (lon-} gitudinal, x) as 2nd_{, leading to}

. If there are further detailed specifications, such as coordinate sys-

tem, e.g. wheel coordinate ( ), it will be the 3rd _{subscript, leading to}

. Further additional specifica-

tions, such as case, e.g. without (0) or with (1) a certain technical solution, will be the 4th _subscript, leading to ₀ and . If subscripts have >1 token, it can sometimes be good to use comma, e.g.

𝑒𝑎 ℎ𝑒𝑒𝑙 𝑖𝑡ℎ.

Naming of variables (signals), parameters and function components (blocks) is often important in large products as vehicles. Some naming needs to be readable for many engineers, such as CAN signals. Hence, companies developing their own naming standards. But also, the standardization organisation AUTOSAR has released a naming standard, with intention to be accepted by both vehicle manufactur- ers and system suppliers.

Table 1.3: Notation

Alternatives

Subject for notation in code by hand Unit Description / Note

Basic (Shaft) Torque T T M Nm

Basic Coefficient of friction mu μ mue, muh 1=N/N

Basic Damping coefficient d d c,k,D _Nm/(rad/s)N/(m/s) or

Basic Density roh ρ

Basic Efficiency eta h η 1=W/W

Ratio between useful/output power and used/input power

Basic Energy E E W Nm=J

Basic Force F F f N

Basic Gravity g g m/s2 9.80665 average on _Earth

Basic Height h h H

Basic Imaginary unit j j i - j*j=-1

Basic Mass m m M kg

Basic Mass moment of inertia J J I kg*m2

Subscript can be given to denote around which axis

Basic Moment (of forces) M M T Nm Not for shaft torque

Basic Power P P W=Nm/s=J/s Basic Ratio r r u, i (ISO8855) Nm/(Nm) or (rad/s)/(rad/s )

Ratio between input and output rotational speed, or output and input torque, in a transmission

Basic Rotational speed w ω w rad/s

Basic Shear stress tau τ N/m2

Basic Stiffness coefficient c c C, k, K _Nm/radN/m or

Basic Strain eps ε m/m

Basic Translational speed v v V m/s

Basic Wave length lambda λ m

Basic Width W W w m

Can be track width, vehicle width, tyre width, …

Dynamics (Time) Frequency f f 1/s=periods/s

Dynamics Angular (time) frequency w ω w rad/s

Dynamics Angular spatial frequency W Ω rad/m

Dynamics Dependent variables in a dynamic

system z z <various>

Both state variables and output variables Dynamics Input variables in a dynamic

system u u <various>

Dynamics Mean Square value MS MS

Dynamics Output variables in a dynamic

system y y <various>

Dynamics Power spectral density G Φ PSD

Dynamics Root Mean Square value RMS RMS

Dynamics Spacial frequency as radians per

travelled distance W Ω rad/m

Dynamics Spatial frequency fs fs 1/m=periods/_m

Dynamics States variables in a dynamic

system x x <various>

Dynamics Time t t t, T s

The independent variable in a dynamic system

Operators Transfer function H H

1. General Air resistance coefficient cd cd Cd 1

1. General Cornering stiffness or lateral tyre

stiffness Cy Cy Ca N/rad or N/1 dF_a=sy/da = dFy/dsy at y=0

1. General Longitudinal tyre stiffness Cx Cx N/1 dfyw/dsx at sx=0

2. V eh ic le Notation Recommended Categorization 1. G en er al

Alternatives

Subject for notation in code by hand Unit Description / Note

1. General Mass m m M kg

1. General Rolling resistance coefficient f f 1=N/N

1. General Track width W W m

1. General Tyre stiffness C C c N/1 dF/ds at s=0

1. General Understeer gradient Ku Ku Kus, kus,_{U (ISO8855)} N/(N/rad) = _rad/(m/s2₎

1. General Vehicle side slip angle b β b rad If no subscript, undefined or CoG

1. General Wheel base L L l, lf+lr, WB m

1. General Wheel radius R R r, Rw m

2. Road Curvature kappa κ roh, r rad/m=1/m road or path

2. Road Curve radius R R r, roh, r m road or path

2. Road Road bank angle pxr φxr

Positive when right side of ground is lower than left side ground 2. Road Road inclination angle pyr φyr rad Positive downhill

2. Road Vertical position of road zr zr z, Z m

3. Motion Vehicle acceleration, in interial _system ax, ay,

az ax, ay, az m/(s

2₎ decomposed in vehicle

coordinates direction 3. Motion Time derivatives of each _{component v}

x, vy, vz dervx, dervy, dervz der(vx), der(vy), der(vz)

m/(s2) decomposed in vehicle _{coordinates direction} 3. Motion Vehicle position x,y,z x, y, z r=[rx,ry,rz]

or [X,Y,Z] m

often position of Center of Gravity, CoG 3. Motion Vehicle velocity, in interial system vx,vy,vz vx, vy, vz u,v,w m/s decomposed in vehicle _{coordinates direction}

4. Forces Forces and moments on vehicle

Fx, Fy, Fz, Mx, My, Mz Fx, Fy, Fz, Mx, My, Mz [N, N, N, Nm, Nm, Nm]

From ground, air, towed units, colliding objects, etc. May appear also for in road plane: F=[Fx, Fy, Mz]

4. Forces Forces on one wheel, axle or side from ground

Fix,Fiy,F

iz Fix, Fiy, Fiz lowercase f N

Subscript "i" is

placeholder for particular wheel, axle or side. May appear also extended with moments: [Fx, Fy, Fz, Mx, My, Mz]

4. Forces Forces on one wheel, axle or side _{from ground} Fixv, Fiyv,Fizv

Fixv, Fiyv,

Fizv lowercase f N

in vehicle coordinate system

4. Forces Forces on one wheel, axle or side from ground

Fixw,Fiyw ,Fizw

Fixw, Fiyw, Fizw

lowercase f N in wheel coordinate system

5. Angles Camber angle g γ g rad

Sign convention positive when either of "rotated as wheel roll angle" or "top leaning outward vs vehicle body".

5. Angles Low speed or Ackermann speed _{steering wheel angle} dswA δswA rad

The angle required for a certain vehicle path curvature at low speeds

5. Angles Euler rotations ψ, θ, φ rad Order: yaw, pitch, roll

5. Angles Roll, pitch, yaw angle px, py,

pz φx, φy, φz j, q, y rad

Angles, not rotations. Roll and Pitch normally small 5. Angles Vehicle angular velocity, in interial _system wx, wy, _wz ωx, ωy, ωz rad/s

decomposed in vehicle coordinates direction (roll, pitch, yaw) 5. Angles Steering angle d δ delta rad May refer to steering _{wheel, road wheel or axle} 5. Angles Steering angle of road wheels drw δrw RWA rad

Normally an average of angles on front axle

2. V eh ic le Categorization Recommended Notation 𝑦

In document Vehicle Dynamics Compendium (Page 50-54)