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:

    “Camped on a tropic riverside,
    One day he missed his loving bride.
    She had, the guide informed him later,
    Been eaten by an alligator.
    Professor Twist could not but smile.
    “You mean,” he said, “a crocodile.””
    —Ogden Nash (1902–1971)

    “I believe in the total depravity of inanimate things ... the elusiveness of soap, the knottiness of strings, the transitory nature of buttons, the inclination of suspenders to twist and of hooks to forsake their lawful eyes, and cleave only unto the hairs of their hapless owner’s head.”
    —Katharine Walker (1840–1916)

    “When we were at school we were taught to sing the songs of the Europeans. How many of us were taught the songs of the Wanyamwezi or of the Wahehe? Many of us have learnt to dance the rumba, or the cha cha, to rock and roll and to twist and even to dance the waltz and foxtrot. But how many of us can dance, or have even heard of the gombe sugu, the mangala, nyang’umumi, kiduo, or lele mama?”
    —Julius K. Nyerere (b. 1922)