Screw theory

The screw theory is a theory mainly used in the mechanics of rigid bodies to describe static and kinematic systems. The central concept of the theory is screw the screw (Engl. Screw, double. Torseur ), a mathematical object that the modeling used by mechanical actions, speeds and other variables.

The screw theory was first published in 1876 by the Irish astronomer and mathematician Sir Robert Stawell Ball . After receiving little attention for a long time, it is now being used more frequently again, for example in robotics . In France , the screw theory was taken up by Paul Appell , where, together with an established notation, it has long formed the basis of university teaching in mechanics. At non-French universities, mechanics is very rarely taught according to the screw theory.

Screws of the first kind

The totality of the velocity vectors of all points of a rigid body (a vector field ) can be described by a rotation vector and a translation vector (unit m / s) after choosing a reference point . The rotation vector is aligned along the axis of rotation and has a length that corresponds to the speed of rotation around this axis (unit rad / s). ${\displaystyle {\overrightarrow {\Omega }}}$ ${\displaystyle {\overrightarrow {V}}}$

These two sizes are combined in a screw of the first kind (English twist, French torseur cinématique ); the German name goes back to Felix Klein . A screw of the first type defined at the point of a body can therefore be expressed in a reference system by six scalar quantities: ${\displaystyle A}$ ${\displaystyle S}$ ${\displaystyle R}$

${\displaystyle \{{\mathcal {V}}(S/R)\}_{A/R}={\begin{Bmatrix}{\overrightarrow {\Omega }}(S/R)\\{\overrightarrow {V}}(A\in S/R)\\\end{Bmatrix}}_{A/R}={\begin{Bmatrix}\omega _{x}&V_{A\ x}\\\omega _{y}&V_{A\ y}\\\omega _{z}&V_{A\ z}\\\end{Bmatrix}}_{/R}}$.

For a body 1 that moves relative to body 0, we write:

${\displaystyle \{{\mathcal {V}}_{1/0}\}_{A}={\begin{Bmatrix}{\overrightarrow {\Omega }}_{1/0}\\{\overrightarrow {V}}_{A\ 1/0}\end{Bmatrix}}_{A}={\begin{Bmatrix}\omega _{x}&V_{A\ x}\\\omega _{y}&V_{A\ y}\\\omega _{z}&V_{A\ z}\\\end{Bmatrix}}_{A}}$.

properties

• A screw defined at the point of a body in a reference system has the value at one point : ${\displaystyle A}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle B}$

${\displaystyle \{{\mathcal {V}}(S/R)\}_{B/R}={\begin{Bmatrix}{\overrightarrow {\Omega }}(S/R)\\{\overrightarrow {V}}(B\in S/R)\end{Bmatrix}}_{B/R}={\begin{Bmatrix}{\overrightarrow {\Omega }}(S/R)\\{\overrightarrow {V}}(A\in S/R)+{\overrightarrow {BA}}\times {\overrightarrow {\Omega }}(S/R)\end{Bmatrix}}_{B/R}}$.

• The sum of two screws of the first kind expressed at the same point is the screw composed of the sum of the respective rotation and translation vectors.
• Chaining of screws of the first kind: ${\displaystyle \{{\mathcal {V}}_{2/0}\}_{A}=\{{\mathcal {V}}_{2/1}\}_{A}+\{{\mathcal {V}}_{1/0}\}_{A}}$
• The following applies to the translation speeds at two different points and on the same body ; this property is known as equiprojectivity . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle {\overrightarrow {V}}(A/R)\cdot {\overrightarrow {AB}}={\overrightarrow {V}}(B/R)\cdot {\overrightarrow {AB}}}$

Results

The movement of two components connected by a mechanical joint can be described by a screw of the first type. The movements allowed depend on the type of joint, which can be characterized by a screw of the first type. For example, for a swivel joint aligned along the z-axis, the screw of the permitted movements is:

${\displaystyle {\begin{Bmatrix}0&0\\0&0\\\Omega _{z}&0\\\end{Bmatrix}}_{A}}$.

In clear terms, this means that the only possible relative movement of two components connected in this way is a rotation about the z-axis.

Screws of the second kind (Dynamen)

It can be shown that every system of forces can be described by a statically equivalent pair of resultant (unit N, Newton ) and torque (unit Nm, Newton meter) after choosing a reference point . This pair is called the second type of screw, dyname , power screw or power winder (English wrench, French torseur statique ). ${\displaystyle {\overrightarrow {\mathcal {R}}}}$ ${\displaystyle {\overrightarrow {\mathcal {M}}}}$

A spatial dyname defined at the point can therefore be expressed by six scalar quantities: ${\displaystyle A}$

${\displaystyle \{{\mathcal {T}}_{A}\}_{A}={\begin{Bmatrix}{\overrightarrow {\mathcal {R}}}\\{\overrightarrow {\mathcal {M}}}_{A}\end{Bmatrix}}_{A}={\begin{Bmatrix}X&L_{A}\\Y&M_{A}\\Z&N_{A}\\\end{Bmatrix}}_{A}}$.

For the action that a body 1 exerts on a body 2 at the point , we write: ${\displaystyle A}$

${\displaystyle \{{\mathcal {T}}_{1\rightarrow 2}\}_{A}={\begin{Bmatrix}{\overrightarrow {\mathcal {R}}}_{1\rightarrow 2}\\{\overrightarrow {\mathcal {M}}}_{A\ 1\rightarrow 2}\end{Bmatrix}}_{A}={\begin{Bmatrix}X_{1\rightarrow 2}&L_{A\ 1\rightarrow 2}\\Y_{1\rightarrow 2}&M_{A\ 1\rightarrow 2}\\Z_{1\rightarrow 2}&N_{A\ 1\rightarrow 2}\\\end{Bmatrix}}_{A}}$.

properties

A dyname has similar properties to a screw of the first kind:

• The torque of a Dyname defined at the point has the value at the point . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle {\overrightarrow {{\mathcal {M}}_{B}}}={\overrightarrow {{\mathcal {M}}_{A}}}+{\overrightarrow {BA}}\times {\overrightarrow {R}}}$
• The sum of two dynames expressed at the same point is the dyname composed of the sum of the respective resultants and moments.
• A pair of forces is a dyname, the resultant of which is the zero vector .
• A dyname whose moment is the zero vector is called a glisseur .
• For the moments of a dyname at two different points and applies (Équiprojectivité). ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle {\overrightarrow {{\mathcal {M}}_{A}}}\cdot {\overrightarrow {AB}}={\overrightarrow {{\mathcal {M}}_{B}}}\cdot {\overrightarrow {AB}}}$

Results

If actions described by dynames at the same point act on a body , then the equilibrium condition says that the sum of all dynames results in the zero dyname: ${\displaystyle n}$ ${\displaystyle I}$ ${\displaystyle S}$

${\displaystyle \{{\mathcal {T}}_{1\rightarrow S}\}_{I}+\{{\mathcal {T}}_{2\rightarrow S}\}_{I}+\ldots +\{{\mathcal {T}}_{n\rightarrow S}\}_{I}=\{0\}}$

A body can only pass a force on to another body at the point of contact if mutual movement is prevented. For the example of the swivel joint aligned along the z-axis, the dyname of the transferable actions is:

${\displaystyle {\begin{Bmatrix}X&L_{A}\\Y&M_{A}\\Z&0\\\end{Bmatrix}}_{A}}$

Torseur cinétique and Torseur dynamique

French mechanics also knows the Torseur cinétique, which is used to calculate the kinetic energy of a system, and the Torseur dynamique, with which Newton's second law can be expressed.

literature

• Pierre Agati and others: Mécanique du solide: applications industrielles. Dunod, Paris 2003, ISBN 2-10-007945-X
• Robert Stawell Ball: A Treatise on the Theory of Screws. Cambridge University Press, Cambridge 1998, ISBN 0-521-63650-7 ( online at Google Books )
• Jean-Louis Fanchon: Guide de mécanique. Nathan, Paris 2008, ISBN 978-2-09-160711-5

Individual evidence

1. ^ H. Lipkin, J. Duffy: Sir Robert Stawell Ball and Methodologies of Modern Screw Theory. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216, 1 (2002), ISSN  0954-4062
2. ^ Joseph Davidson, Kenneth Hunt: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press, Oxford 2004, ISBN 0-19-856245-4
3. ↑ Stefano Stramigioli and Herman Bruyninckx: Geometry and Screw Theory for Robotics. IEEE / RSJ International Conference on Robotics and Automation (ICRA), 2001, Tutorial ( PDF, 850 kB )
4. ^ Paul Émile appeal: Traité de mécanique rationnelle. Gauthier-Villars, Paris 1932 ( online at Gallica )
5. ^ Paul Germain et al: Continuum Thermomechanics: The Art and Science of Modeling Material Behavior, p. 15. Kluwer Academic Publishers, Dordrecht 2000, ISBN 0-7923-6407-4
6. ↑ Felix Klein: On the screw theory of Sir Robert Ball. In Felix Klein: Collected mathematical treatises. Vol. 1, pp. 503-532. Springer-Verlag, Berlin 1921, ISBN 3-540-05852-4 ( PDF, 147 kB (Engl.) )