don't recall this being previously posted, but here goes:
"The equations of motion of an idealized bike, consisting of
a rigid frame,
a rigid fork,
two knife-edged, rigid wheels,
all connected with frictionless bearings and rolling without friction or slip on a smooth horizontal surface and
operating at or near the upright and straight ahead unstable equilibrium
can be represented by two linearized second-order ordinary differential equations,[9] the lean equation
and the steer equation
where
θr is the lean angle of the rear assembly,
ψ is the steer angle of the front assembly relative to the rear assembly and
Mθ and Mψ are the moments (torques) applied at the rear assembly and the steering axis, respectively. For the analysis of an uncontrolled bike, both are taken to be zero.
These can be represented in matrix form as where
M is the symmetrical mass matrix which contains terms that include only the mass and geometry of the bike,
C is the so-called damping matrix, even though an idealized bike has no dissipation, which contains terms that include the forward speed V and is asymmetric,
K is the so-called stiffness matrix which contains terms that include the gravitational constant g and V2 and is symmetric in g and asymmetric in V2,
q is a vector of lean angle and steer angle, and
f is a vector of external forces, the moments mentioned above.
In this idealized and linearized model, there are many parameters (wheelbase, head angle, mass of each body, wheel radius, etc.), but only four significant variables: lean angle, lean rate, steer angle, and steer rate. These equations have been verified by comparison with multiple numeric models derived completely independently."
:tard: :laughing6: now that you understand these dynamics, you can read the rest at wikipedia:
https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics#Equations_of_motion
"The equations of motion of an idealized bike, consisting of
a rigid frame,
a rigid fork,
two knife-edged, rigid wheels,
all connected with frictionless bearings and rolling without friction or slip on a smooth horizontal surface and
operating at or near the upright and straight ahead unstable equilibrium
can be represented by two linearized second-order ordinary differential equations,[9] the lean equation
and the steer equation
where
θr is the lean angle of the rear assembly,
ψ is the steer angle of the front assembly relative to the rear assembly and
Mθ and Mψ are the moments (torques) applied at the rear assembly and the steering axis, respectively. For the analysis of an uncontrolled bike, both are taken to be zero.
These can be represented in matrix form as where
M is the symmetrical mass matrix which contains terms that include only the mass and geometry of the bike,
C is the so-called damping matrix, even though an idealized bike has no dissipation, which contains terms that include the forward speed V and is asymmetric,
K is the so-called stiffness matrix which contains terms that include the gravitational constant g and V2 and is symmetric in g and asymmetric in V2,
q is a vector of lean angle and steer angle, and
f is a vector of external forces, the moments mentioned above.
In this idealized and linearized model, there are many parameters (wheelbase, head angle, mass of each body, wheel radius, etc.), but only four significant variables: lean angle, lean rate, steer angle, and steer rate. These equations have been verified by comparison with multiple numeric models derived completely independently."
:tard: :laughing6: now that you understand these dynamics, you can read the rest at wikipedia:
https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics#Equations_of_motion
