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| In [[physics]] the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of [[velocity]]. In this article the notion of spatial acceleration is explored, which entails looking at a fixed (unmoving) point in space and observing the changes of [[velocity]] of whatever particle/point happens to coincide with the observation point. This is similar to the acceleration definition fluid dynamics where typically one can measure velocity and/or accelerations on a fixed locate inside a testing apparatus.
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| == Definition ==
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| Consider a moving rigid body and the velocity of a particle/point ''P'' along the body being a function of the position and velocity of a center particle/point ''C'' and the angular velocity <math>\vec \omega</math>.
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| The linear velocity vector <math>\vec v_P</math> at ''P'' is expressed in terms of the velocity vector <math>\vec v_C</math> at ''C'' as:
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| <math>\vec v_P = \vec v_C + \vec \omega \times (\vec r_P-\vec r_C)</math>
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| where <math>\vec \omega</math> is the angular velocity vector.
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| The [[Material derivative|material acceleration]] at ''P'' is:
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| <math>\vec a_P = \frac{{\rm d} \vec v_P}{{\rm d} t}</math>
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| <math>\vec a_P = \vec a_C + \vec \alpha \times (\vec r_P-\vec r_C) + \vec \omega \times (\vec v_P-\vec v_C)</math>
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| where <math>\vec \alpha</math> is the angular acceleration vector.
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| The spatial acceleration <math>\vec \psi_P</math> at ''P'' is expressed in terms of the spatial acceleration <math>\vec \psi_C</math> at ''C'' as:
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| <math>\vec \psi_P = \frac{\partial \vec v_P}{\partial t}</math>
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| <math> \vec{\psi}_{P} = \vec{\psi}_{C}+\vec{\alpha}\times(\vec{r}_{P}-\vec{r}_{C}) </math>
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| which is similar to the velocity transformation above.
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| In general the spatial acceleration <math>\vec \psi_P</math> of a particle point ''P'' that is moving with linear velocity <math>\vec v_P</math> is derived from the material acceleration <math>\vec a_P</math> at ''P'' as:
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| <math> \vec{\psi}_{P}=\vec{a}_{P}-\vec{\omega}\times\vec{v}_{P} </math> | |
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| {{Expand section|date=April 2012}}
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| ==References==
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| *{{cite book|title=Fluid Mechanics|author=Frank M. White|publisher=McGraw-Hill Professional|year=2003|isbn=0-07-240217-2}}.
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| *{{cite book|title=Robot Dynamics Algorithms|author=Roy Featherstone|publisher=Springer|year=1987|isbn=0-89838-230-0}}. This reference effectively combines [[screw theory]] with rigid body [[dynamics (mechanics)|dynamics]] for robotic applications. The author also chooses to use spatial accelerations extensively in place of material accelerations as they simplify the equations and allows for compact notation.
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| *JPL DARTS page has a section on spatial operator algebra (link: [http://dshell.jpl.nasa.gov/SOA/index.php]) as well as an extensive list of references (link: [http://dshell.jpl.nasa.gov/References/index.php]).
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| *{{cite book|title=Springer Handbook of Robotics|author=Bruno Siciliano, [[Oussama Khatib]]|publisher=Springer|year=2008|isbn=}}. Page 41 (link: Google Books [http://books.google.com/books?id=Xpgi5gSuBxsC&printsec=frontcover#PPA40,M1]) defines spatial accelerations for use in rigid body mechanics.
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| [[Category:Rigid bodies]]
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| [[Category:Acceleration]]
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Hi there! :) My name is Marylou, I'm a student studying Biological Sciences from Bad Driburg, Germany.
my weblog :: wordpress backup plugin