Acceleration of a Rigid Body

The linear and angular accelerations are the time derivatives of the linear and angular velocity vectors at any instant:

${\displaystyle _{a}{\dot {v}}={\dfrac {d\,_{a}v}{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}v(t+\Delta t)-\,_{a}v(t)}{\Delta t}}}$,

and:

${\displaystyle _{a}{\dot {\omega }}={\dfrac {d\,_{a}\omega }{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}\omega (t+\Delta t)-\,_{a}\omega (t)}{\Delta t}}}$

The linear velocity, as seen from a reference frame ${\displaystyle \{a\}}$, of a vector ${\displaystyle q}$, relative to frame ${\displaystyle \{b\}}$ of which the origin coincides with ${\displaystyle \{a\}}$, is given by:

${\displaystyle _{a}v_{q}=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q}$

Differentiating the above expression gives the acceleration of the vector ${\displaystyle q}$:

${\displaystyle _{a}{\dot {v}}_{q}={\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times {\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}q}$

The equation for the linear velocity may also be written as:

${\displaystyle _{a}v_{q}={\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}q=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q}$

Applying this result to the acceleration leads to:

${\displaystyle _{a}{\dot {v}}_{q}=_{a}^{b}R\,_{b}{\dot {v}}_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(^{b}_{a}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)}$

In the case the origins of ${\displaystyle \{a\}}$ and ${\displaystyle \{b\}}$ do not coincide, a term for the linear acceleration of ${\displaystyle \{b\}}$, with respect to ${\displaystyle \{a\}}$, is added:

${\displaystyle _{a}{\dot {v}}_{q}=\,_{a}{\dot {v}}_{b,org}+\,_{a}^{b}R\,_{b}{\dot {v}}_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(^{b}_{a}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)}$

For rotational joints, ${\displaystyle _{b}q}$ is constant, and the above expression simplifies to:

${\displaystyle _{a}{\dot {v}}_{q}=\,_{a}{\dot {v}}_{b,org}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)}$

The angular velocity of a frame ${\displaystyle \{c\}}$, rotating relative to a frame ${\displaystyle \{b\}}$, which in itself is rotating relative to the reference frame ${\displaystyle \{a\}}$, with respect to ${\displaystyle \{a\}}$, is given by:

${\displaystyle _{a}\omega _{c}=\,_{a}\omega _{b}+\,_{a}^{b}R\,_{b}\omega _{c}}$

Differentiating leads to:

${\displaystyle _{a}{\dot {\omega }}_{c}=\,_{a}{\dot {\omega }}_{b}+{\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}\omega _{c}}$

Replacing the last term with one of the expressions derived earlier:

${\displaystyle _{a}{\dot {\omega }}_{c}=\,_{a}{\dot {\omega }}_{b}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}\omega _{c}}$

Inertia Tensor

The inertia tensor can be thought of as a generalization of the scalar moment of inertia:

${\displaystyle _{a}I={\begin{pmatrix}I_{xx}&-I_{xy}&-I_{xz}\\I_{xy}&I_{yy}&-I_{yz}\\I_{xz}&-I_{yz}&I_{zz}\\\end{pmatrix}}}$

Newton's and Euler's equation

The force ${\displaystyle F}$, acting at the center of mass of a rigid body with total mass${\displaystyle m}$, causing an acceleration ${\displaystyle {\dot {v}}_{com}}$, equals:

${\displaystyle F=m{\dot {v}}_{com}}$

In a similar way, the moment ${\displaystyle N}$, causing an angular acceleration ${\displaystyle {\dot {\omega }}}$, is given by:

${\displaystyle N=\,_{c}I{\dot {\omega }}+\omega \times \,_{c}I\omega }$,

where ${\displaystyle _{c}I}$ is the inertia tensor, expressed in a frame ${\displaystyle \{c\}}$ of which the origin coincides with the center of mass of the rigid body.