what are the equations of motion in the x, y and z direction
- sum X = m(u_dot - vΩ)
- sum Y = m(v_dot + uΩ)
- sum N = I_z*Ω_dot - [sum(r) x (X + Y)]
a car going at speed u has rear wheels and front wheels with its center of gravity in in between. The distance between the rear and the center is b and between the center and the front its a. the car is turning with angular velocity Ω and the front wheels have a yaw angle of δ. what are the front and rear slip angles α_f and α_r
- α_f = (v + aΩ / u) - δ
- α_r = (v - bΩ / u)
if the car has side force coefficients for the front and rear axles C_f and C_r, what is the equation of motion Y
- Y = m(v_dot + uΩ) + C_fα_f + C_rα_r
what is the equation of motion N
- N = IΩ_dot + aC_fα_f - bC_r*α_r
what is total cornering stiffness c
- c = C_f + C_r
what is the square of the yaw stiffness radius q^2
- q^2 = (a^2C_f + b^2C_r) / (C_f + C_r)
what is the length of the vehicle l
l = a + b
what is the yaw moment of inertia I
= I = mk^2
what is the condition for the vehicle to always be stable
- bC_r > aC_f
if the vehicle doesnt meet the condition to always be stable, what is the condition for it to be stable
- u^2 < (C_fC_rl^2) / m(aC_f - bC_r)
the static margin is defined as -s/l, but what is the expression for s
- s = (aC_f - bC_r) / (C_f + C_r)
what is the simple condition for the vehicle to reach steady state
- v_dot = Ω_dot = 0
- v_ss and Ω_ss are what satisfy this
what is the formula for β
- β = v/u
what are the neutral steer conditions
- s = 0
- Ω_ss = 0
- β > 0 for all speeds
what are the understeer conditions
- s < 0
- Ω_ss > 0
- β > 0 for all speeds
what are the oversteer conditions
- s > 0 for low speeds
- Ω_ss < 0
- β > 0 when U reaches sqrt(C_fC_rl^2 / csm) = U_c
what is the neutral steer point
- the point on the vehicle for which a side force produces no steady state yaw
what is the static margin SM = -s/l, measuring
- the static margin is a measure of handling behaviour
- its the distance that the neutral steer point lies behind G, normalized by l
- when NS is at G, SM = 0 so s = 0 so you get neutral steer
- when NS is behind G, SM > 0 so s < 0 so you get understeer
- when NS is forward of G, SM < 0 so s > 0 so you get oversteer
in oversteer, what is the condition for the vehicle to become unstable and spin
- when U > sqrt(C_fC_rl^2 / csm) = U_c
