In classical mechanics, rigid body dynamics studies the movement of systems of interconnected bodies under the action of external forces. Along with statics, it forms the field of rigid body mechanics. The assumption that the bodies are rigid (i.e. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of body-fixed frames. This excludes bodies that display fluid, highly elastic, and plastic behavior. The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law (kinetics) or their derivative form, Lagrangian mechanics. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.
Planar rigid body dynamics If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement. In this case, Newton's laws (kinetics) for a rigid system of N particles, Pi, i=1,...,N, simplify because there is no movement in the k direction. Determine the resultant force and torque at a reference point R, to obtain
F = ∑ i = 1 N m i A i , T = ∑ i = 1 N ( r i − R ) × m i A i , {\displaystyle \mathbf {F} =\sum _{i=1}^{N}m_{i}\mathbf {A} _{i},\quad \mathbf {T} =\sum _{i=1}^{N}(\mathbf {r} _{i}-\mathbf {R} )\times m_{i}\mathbf {A} _{i},}
where ri denotes the planar trajectory of each particle. The kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as,
A i = α × ( r i − R ) + ω × ( ω × ( r i − R ) ) + A . {\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times (\mathbf {r} _{i}-\mathbf {R} )+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times (\mathbf {r} _{i}-\mathbf {R} ))+\mathbf {A} .}
For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri and the unit vectors t i = k × e i {\textstyle \mathbf {t} _{i}=\mathbf {k} \times \mathbf {e} _{i}} , so
A i = α ( Δ r i t i ) − ω 2 ( Δ r i e i ) + A . {\displaystyle \mathbf {A} _{i}=\alpha (\Delta r_{i}\mathbf {t} _{i})-\omega ^{2}(\Delta r_{i}\mathbf {e} _{i})+\mathbf {A} .}
This yields the resultant force on the system as
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