1.4. Relative Kinematics#

Relative kinematics is discussed here using two Cartesian reference frames.

PO0=xP/O00ie^i0
O1O0=xO1/O00ie^i0
PO1=xP/O11ie^i1
e^i1=e^i1e^k0e^k0=e^j1e^k0e^k0e^j0e^i0==Rkj01e^k0e^j0e^i0=R01e^i0 .

1.4.1. Points#

Position. Given two reference frames Oxi, Oxi, for the position of a point P reads

(1.7)#(PO0)=(O1O0)+(PO1) ,
xP/O0,i0e^i0=xO1/O0,i0e^i0+xP/O1,k1e^k1 ,

i.e. the position vector PO of the point P w.r.t. point O - origin of the reference frame Oxi - is the sum of the position vector PO of the point P w.r.t. to the point O - origin of the reference frame Oxi - and the position vector OO, of the origin O w.r.t. to O.

Velocity. Time derivative of relative position relation (1.7) w.r.t. to reference frame 0 is performed keeping e^i0 constant.

0ddt(PO0)=0ddt[(O1O0)+(PO1)]==0ddt(xO1/O0,i0e^i0)+0ddt(xP/O1,k1e^k1)==0ddtxO1/O0,i0e^i0+0ddtxP/O1,k1e^k1+xP/O1,k10ddte^k1==vO1/O00+vP/O11+ω1/0×e^k1xP/O1,k1==vO1/O00+vP/O11+ω1/0×(PO1) ,

so that

(1.8)#vP/O0=vO1/O00+vP/O11+ω1/0×(PO1) .

Acceleration. Time derivative of relative velocity relation (1.8) w.r.t. reference frame 0 reads

0ddtvP/O00=0ddt[vO1/O00+vP/O11+ω1/0×(PO1)]===aO1/O00+aP/O11+2ω1/0×vP/O11+α1/0×(PO1)+ω1/0×[ω1/0×(PO1)]

so that

(1.9)#aP/O00=aO1/O00+aP/O11+α1/0×(PO1)tangential+2ω1/0×vP/O11Coriolis+ω1/0×[ω1/0×(PO1)]centripetal .

where:

  • the “tangential component” is orthogonal to the instantaneous angular acceleration and radius,

  • the “centripetal component” is orthogonal w.r.t. the instantaneous angular velocity

todo tangent to what, centripetal w.r.t. what? state it clearly, otherwise delete this

1.4.2. Rigid bodies#

Orientation.

Angular velocity.

Angular acceleration.