Rigid body dynamics
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 reference frames attached to each body.
The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.
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.
Use the center of mass C as the reference point, so these equations for Newton's laws simplify to become
Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed.
"[3] Because Newton generally referred to mass times velocity as the "motion" of a particle, the phrase "change of motion" refers to the mass times acceleration of the particle, and so this law is usually written as
If Fi is the external force applied to particle Pi with mass mi, then
Newton's second law for a particle combines with these formulas for the resultant force and torque to yield,
The inertia matrix [IR] of the system relative to the reference point R is defined by
Using the center of mass and inertia matrix, the force and torque equations for a single rigid body take the form
The resultant of the external and interaction forces on each body, yields the force-torque equations
Newton's formulation yields 6M equations that define the dynamics of a system of M rigid bodies.
[4] A rotating object, whether under the influence of torques or not, may exhibit the behaviours of precession and nutation.
The angular velocity of precession ΩP is given by the cross product:[citation needed]
where θ is the angle between the vectors ΩP and L. Thus, if the top's spin slows down (for example, due to friction), its angular momentum decreases and so the rate of precession increases.
By convention, these three vectors - torque, spin, and precession - are all oriented with respect to each other according to the right-hand rule.
Work is computed from the dot product of each force with the displacement of its point of contact
If the trajectory of a rigid body is defined by a set of generalized coordinates qj, j = 1, ..., m, then the virtual displacements δri are given by
The virtual work of this system of forces acting on the body in terms of the generalized coordinates becomes
For simplicity consider a trajectory of a rigid body that is specified by a single generalized coordinate q, such as a rotation angle, then the formula becomes
The equations of motion for a mechanical system of rigid bodies can be determined using D'Alembert's form of the principle of virtual work.
The virtual work of the forces and torques, Fi and Ti, applied to this one degree of freedom system is given by
The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is
Consider a single rigid body which moves under the action of a resultant force F and torque T, with one degree of freedom defined by the generalized coordinate q.
Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that
The result is a set of m equations of motion that define the dynamics of the rigid body system.
If the generalized forces Qj are derivable from a potential energy V(q1, ..., qm), then these equations of motion take the form
Select a reference point R and compute the relative position and velocity vectors,
The total linear and angular momentum vectors relative to the reference point R are
Select a reference point R and compute the relative position and velocity vectors,
