Screw Theory - Twist

In order to define the twist of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements, D(t)=(,d(f)), where is a rotation matrix and d is a translation vector. This causes a point p that is fixed in moving body to trace a curve P(t) in the fixed frame given by,

\mathbf{P}(t) = \mathbf{p} + \mathbf{d}(t).

The velocity of P is

\mathbf{V}_P(t) = \left\mathbf{p} + \mathbf{v}(t),

where v is velocity of the origin of the moving frame, that is dd/dt. Now substitute p= (P-d) into this equation to obtain,

\mathbf{V}_P(t) = \mathbf{P} + \mathbf{v} - \mathbf{d}\quad\mbox{or}\quad\mathbf{V}_P(t) = \mathbf{\omega}\times\mathbf{P} + \mathbf{v} + \mathbf{d}\times\mathbf{\omega},

where = is the angular velocity matrix and ω is the angular velocity vector.

The screw

is the twist of the moving body. The vector V=v + d×ω is the velocity of the point in the body that corresponds with the origin of the fixed frame.

There are two important special cases: (i) when d is constant, that is v=0, then the twist is a pure rotation about a line, then the twist is

and (ii) when =0, that is the body does not rotate but only slides in the direction v, then the twist is a pure slide given by

Read more about this topic:  Screw Theory

Famous quotes containing the word twist:

    “For often at Church I’ve seen the stained high glass
    Pour out the Virgin and Saints, twist and untwist
    The mortal youth of Christ astride an ass.”
    —Allen Tate (1899–1979)

    “The greatest blunders, like the thickest ropes, are often compounded of a multitude of strands. Take the rope apart, separate it into the small threads that compose it, and you can break them one by one. You think, “That is all there was!” But twist them all together and you have something tremendous.”
    —Victor Hugo (1802–1885)

    “The hearts of small children are delicate organs. A cruel beginning in this world can twist them into curious shapes. The heart of a hurt child can shrink so that forever afterward it is hard and pitted as the seed of a peach. Or, again, the heart of such a child may fester and swell until it is misery to carry within the body, easily chafed and hurt by the most ordinary things.”
    —Carson McCullers (1917–1967)