Spatial acceleration

Consider a moving rigid body and the velocity of a point P on the body being a function of the position and velocity of a center-point C and the angular velocity ${\displaystyle {\boldsymbol {\omega }}}$.

The linear velocity vector ${\displaystyle \mathbf {v} _{P}}$ at P is expressed in terms of the velocity vector ${\displaystyle \mathbf {v} _{C}}$ at C as:

$${\displaystyle \mathbf {v} _{P}=\mathbf {v} _{C}+{\boldsymbol {\omega }}\times (\mathbf {r} _{P}-\mathbf {r} _{C})}$$

where ${\displaystyle {\boldsymbol {\omega }}}$ is the angular velocity vector.

The material acceleration at P is:

$${\displaystyle \mathbf {a} _{P}={\frac {d\mathbf {v} _{P}}{dt}}=\mathbf {a} _{C}+{\boldsymbol {\alpha }}\times (\mathbf {r} _{P}-\mathbf {r} _{C})+{\boldsymbol {\omega }}\times (\mathbf {v} _{P}-\mathbf {v} _{C})}$$

where ${\displaystyle {\boldsymbol {\alpha }}}$ is the angular acceleration vector.

The spatial acceleration ${\displaystyle {\boldsymbol {\psi }}_{P}}$ at P is expressed in terms of the spatial acceleration ${\displaystyle {\boldsymbol {\psi }}_{C}}$ at C as:

$${\displaystyle {\begin{aligned}{\boldsymbol {\psi }}_{P}&={\frac {\partial \mathbf {v} _{P}}{\partial t}}\\[1ex]&={\boldsymbol {\psi }}_{C}+{\boldsymbol {\alpha }}\times (\mathbf {r} _{P}-\mathbf {r} _{C})\end{aligned}}}$$

which is similar to the velocity transformation above.

In general the spatial acceleration ${\displaystyle {\boldsymbol {\psi }}_{P}}$ of a particle point P that is moving with linear velocity ${\displaystyle \mathbf {v} _{P}}$ is derived from the material acceleration ${\displaystyle \mathbf {a} _{P}}$ at P as:

$${\displaystyle {\boldsymbol {\psi }}_{P}=\mathbf {a} _{P}-{\boldsymbol {\omega }}\times \mathbf {v} _{P}}$$