Kinematic torque

The kinematic torque is, like the torques static, kinetic and dynamic, a mathematical tool used usually in traditional Mécanique.

The kinematic torque is used to describe behaviors of translation and rotation of an indeformable solid, in general in a direct orthonormé reference mark.

Definition

The kinematic torque of a solid compared to an unspecified reference frame R is entirely defined by two vectors:

the first, characteristic of the field speeds and independent of the point of form of the torque, described the rotary behavior of the solid.

\ vec \ Omega (S/R)

One must read Oméga of S (the studied solid) compared to R.

the second, expressed in a point P of the reference mark corresponds at the speed of the point P pertaining to the solid compared to R.

$\ vec V (\ in S/R has)$

One must read V has some (not pertaining to the solid S in R) of S compared to R.

Finally the unit is written:

$\ {\ mathcal {V} (S/R) \} _ {A/R} =

\begin{Bmatrix}
\ \ vec \ Omega (S/R) \ \
\ \ vec V (\ in S/R has)
\end{Bmatrix}_{A/R}$

One must read torque V has S compared to R of it is equal to Oméga of S compared to R and V has S compared to R. of it.

The field speeds of an indeformable solid is representable by a torque because of the équiprojectif character of this field, character which is closely related to the indeformability of the solid.

The vector rotation

The vector rotation $\ vec \ Omega (S/R)$ has in a three-dimensional reference mark {R, X, there, Z} three components noted $\ Omega _x$ , $\ omega_y$ , $\ omega_z$ .

The vector rotation is written then $\ vec \ Omega (S/R) = \ Omega _x \ vec X + \ Omega _y \ vec there + \ Omega _z \ vec Z$ with $\ vec x$ , $\ vec y$ , $\ vec z$ vectors forming the reference mark orthonormé R.

The vector rotation can also be noted $\ vec \ Omega (S/R) =

\begin{Bmatrix}
\ \ Omega _x \ \
\ Omega _y \ \
\ Omega _z \ \
\end{Bmatrix}_{/R}$

If the vector rotation is null the movement of the solid is a simple translation. All the points of the solid have the same Flight Path Vector then.

The Flight Path Vector

The Flight Path Vector $\ vec V (\ in S/R has)$ has in a three-dimensional reference mark {R, X, there, Z} three components noted $\ nu_x$ , $\ nu_y$ , $\ nu_z$ .

The Flight Path Vector is written $\ vec V then (\ in S/R has) = \ nu_x \ vec X + \ nu_y \ vec there + \ nu_z \ vec Z$ with $\ vec x$ , $\ vec y$ , $\ vec z$ vectors forming the reference mark orthonormé R.

The Flight Path Vector can also be noted $\ vec V (\ in S/R has) =

\begin{Bmatrix}
\ \nu_x \\
\ nu_y \ \
\ nu_z \ \
\ end {Bmatrix} _ {has \ in S/R}$

If two points distinct from the solid have a null Flight Path Vector, the solid is motionless in R.

If two points distinct from the solid have the same Flight Path Vector (not no one), it is that the mobile is in translation and that the vector rotation is null.

If not, only one point, at one moment T, can have a null Flight Path Vector. This point is then called Instantaneous Center of Rotation (CIR).

Calculation of the torque in another point of the solid

Knowing the complete kinematic torque in a point has reference mark R and knowing the distance between the point has and a point B, one can calculate the complete torque at the point B.

one notes the torque at the point has $\ {\ mathcal {V} (S/R) \} _ {A/R} =

\begin{Bmatrix}
\ \ vec \ Omega (S/R) \ \
\ \ vec V (\ in S/R has)
\end{Bmatrix}_{A/R}$

one notes the distance between BA $\ vec {BA} =
\begin{Bmatrix}
\ X_ {Ba} \ \
\ Y_ {Ba} \ \
\ Z_ {Ba}
\end{Bmatrix}$

the vector rotation is identical in each point of the reference mark, it simply remains to calculate the vector translation.

$\ vec V _B = \ vec V _A + \ vec {BA} \ wedge \ vec \ Omega$

It is of course that $\ vec V _A$ , $\ vec V _B$ and $\ vec \ Omega$ belongs to S and are all reference mark like $\ vec {BA}$ dans R.

Finally $\ {\ mathcal {V} (S/R) \} _ {B/R} =

\begin{Bmatrix}
\ \ vec \ Omega (S/R) \ \
\ \ vec V (B \ in S/R)
\ end {Bmatrix} _ {B/R} =

\begin{Bmatrix}
\ \ vec \ Omega (S/R) \ \
\ \ vec V (\ in S/R has) + \ vec {BA} \ wedge \ vec \ Omega (S/R)
\end{Bmatrix}_{B/R}$

Various information

The speeds of traverses are normally expressed in meters per seconds (m/s). The number of revolutions is normally expressed in radians per seconds (rad/s).

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