Rigid body 455769 225955175 2008-07-16T04:59:30Z Chobot 259798 robot Adding: be, ko, pt Modifying: he [[Image:Flight dynamics with text.png|right|thumb| The position of a rigid body is determined by the position of its center of mass and by its [[Orientation (geometry)|orientation]] (at least six parameters in total).]] In [[physics]], a '''rigid body''' is an idealization of a solid [[Physical body|body]] of finite size in which [[deformation]] is neglected. In other words, the [[distance]] between any two given [[Point (geometry)|point]]s of a rigid body remains constant in time regardless of external [[force]]s exerted on it. In classical mechanics a rigid body is usually considered as a continuous mass distribution, while in [[quantum mechanics]] a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics [[molecules]] (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see [[Rotational_spectroscopy#Classification_of_molecules_based_on_rotational_behavior|classification of molecules as rigid rotors]]). ==Kinematics== === Position === The position of a rigid body can be described by a combination of a [[translation (geometry)|translation]] and a [[rotation]] from a given reference position. For this purpose a [[reference frame]] is chosen that is rigidly connected to the body (see also below). This is typically referred to as a ''"local"'' reference frame (''L''). The position of its origin and the [[orientation (geometry)|orientation]] of its axes with respect to a given ''"global"'' or ''"world"'' reference frame (''G'') represent the position of the body. The position of ''G'' not necessarily coincides with the initial position of ''L''. Thus, the position of a rigid body has two components: '''linear''' and '''angular''', respectively. Each can be represented by a vector. The angular position is also called [[orientation (geometry)|orientation]]. There are several methods to describe numerically the orientation of a rigid body (see [[orientation (geometry)|orientation]]). In general, if the rigid body moves, both its linear and angular position vary with time. In the kinematic sense, these changes are referred to as [[translation]] and [[rotation]], respectively. All the points of the body change their position during a rotation about a fixed axis, except for those lying on the rotation axis. If the rigid body has any [[rotational symmetry]], not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. In two dimensions the situation is similar. In one dimension a "rigid body" can not move (continuously change) from one orientation to the other. === Other quantities === If ''C'' is the origin of the local reference frame ''L'', *the (linear or translational) '''velocity''' of a rigid body is defined as the [[velocity]] of ''C''; *the (linear or translational) '''acceleration''' of a rigid body is defined as the [[acceleration]] of ''C'' (sometimes referred at material acceleration); *the '''angular''' (or rotational) '''velocity''' of a rigid body is defined as the time derivative of its angular position (see [[angular velocity#angular velocity of a rigid body|angular velocity of a rigid body]]); *the '''angular''' (or rotational) '''acceleration''' of a rigid body is defined as the time derivative of its angular velocity (see [[angular acceleration|angular acceleration of a rigid body]]); *the '''spatial''' or '''[[screw theory#twist|twist]]''' '''acceleration''' of a rigid body is defined as the [[spatial acceleration]] of ''C'' (as opposed to material acceleration above); For any point/particle of a moving rigid body we have :<math> \mathbf{r}(t,\mathbf{r}_0) = \mathbf{r}_c(t) + A(t) \mathbf{r}_0 </math> :<math> \mathbf{v}(t,\mathbf{r}_0) = \mathbf{v}_c(t) + \boldsymbol\omega(t) \times (\mathbf{r}(t,\mathbf{r}_0) - \mathbf{r}_c(t)) = \mathbf{v}_c(t) + \boldsymbol\omega(t) \times A(t) \mathbf{r}_0 </math> :<math> \mathbf{a}(t,\mathbf{r}_0) = \mathbf{a}_c(t) + \boldsymbol\alpha(t) \times A(t) \mathbf{r}_0 + \boldsymbol\omega(t) \times \boldsymbol\omega(t) \times A(t) \mathbf{r}_0</math> :<math> \boldsymbol\psi(t,\mathbf{r}_0) = \mathbf{a}(t,\mathbf{r}_0) - \boldsymbol\omega(t) \times \mathbf{v}(t,\mathbf{r}_0) = \boldsymbol\psi_c(t) + \boldsymbol\alpha(t) \times A(t) \mathbf{r}_0</math> where *<math> \mathbf{r}_0 </math> represents the position of the point/particle with respect to the reference point of the body in terms of the local frame ''L'' (the rigidity of the body means that this does not depend on time) *<math> \mathbf{r}(t,\mathbf{r}_0) </math> represents the position of the point/particle at time <math>t\,</math> *<math> \mathbf{r}_c(t) </math> represents the position of the reference point of the body (the origin of local frame ''L'') at time <math>t\,</math> * <math>A(t)\, </math> is the [[orientation (rigid body)|orientation]] matrix, an [[orthogonal matrix]] with determinant 1, representing the [[orientation (rigid body)|orientation]] (angular position) of the local frame ''L'', with respect to the arbitrary reference orientation of frame ''G''. Think of this matrix as three orthogonal unit vectors, one in each column, which define the orientation of the axes of frame ''L'' with respect to ''G''. *<math>\boldsymbol\omega(t)</math> represents the [[angular velocity]] of the rigid body *<math>\mathbf{v}(t,\mathbf{r}_0)</math> represents the total velocity of the point/particle *<math>\mathbf{v}_c(t)</math> represents the translational velocity (i.e. the velocity of the origin of frame ''L'') *<math>\mathbf{a}(t,\mathbf{r}_0)</math> represents the total acceleration of the point/particle *<math>\mathbf{a}_c(t)</math> represents the translational acceleration (i.e. the acceleration of the origin of frame ''L'') *<math>\boldsymbol\alpha(t)</math> represents the [[angular acceleration]] of the rigid body *<math>\boldsymbol\psi(t,\mathbf{r}_0)</math> represents the [[spatial acceleration]] of the point/particle *<math>\boldsymbol\psi_c(t)</math> represents the [[spatial acceleration]] of the rigid body (i.e. the spatial acceleration of the origin of frame ''L'') In 2D the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the ''xy''-plane by an angle which is the integral of the angular velocity over time. [[Vehicle]]s, walking people, etc. usually rotate according to changes in the direction of the velocity: they move forward with respect to their own orientation. Then, if the body follows a closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is completed once, is an integer times 360°. This integer is the [[winding number]] with respect to the origin of the velocity. Compare the [[Polygon#Angles|amount of rotation associated with the vertices of a polygon]]. ==Kinetics== {{Main|Rigid body dynamics}} Any point that is rigidly connected to the body can be used as reference point (origin of frame ''L'') to describe the linear motion of the body (the linear position, velocity and acceleration vectors depend on the choice). However, depending on the application, a convenient choice may be: *the [[center of mass]] of the whole system; *a point such that the translational motion is zero or simplified, e.g on an [[axle]] or [[hinge]], at the center of a [[ball and socket joint]], etc. When the center of mass is used as reference point: *The (linear) [[momentum]] is independent of the rotational motion. At any time it is equal to the total mass of the rigid body times the translational velocity. *The [[angular momentum]] with respect to the center of mass is the same as without translation: at any time it is equal to the [[moment of inertia|inertia tensor]] times the angular velocity. When the angular velocity is expressed with respect to the [[Moment_of_inertia#Inertia_tensor|principal axes frame]] of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; the [[torque]] is the inertia tensor times the [[angular acceleration]]. *Possible motions in the absence of external forces are translation with constant velocity, steady rotation about a fixed principal axis, and also torque-free [[precession]]. *The net external force on the rigid body is always equal to the total mass times the translational acceleration (i.e., [[Newton's law|Newton's second law]] holds for the translational motion, even when the net external torque is nonnull, and/or the body rotates). *The total [[kinetic energy]] is simply the sum of translational and [[rotational energy]]. ==Geometry== Two rigid bodies are said to be [[equality (objects)|different]] (not copies) if there is no [[proper rotation]] from one to the other. A rigid body is called [[Chirality (mathematics)|chiral]] if its [[mirror image]] is different in that sense, i.e., if it has either no [[symmetry]] or its [[symmetry group]] contains only proper rotations. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version. The latter applies for [[Point groups in three dimensions|''S<sub>2n''</sub>]], of which the case ''n'' = 1 is inversion symmetry. For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down. We can distinguish two cases: *the sheet surface with the image is not symmetric - in this case the two sides are different, but the mirror image of the object is the same, after a rotation by 180° about the axis perpendicular to the mirror plane. *the sheet surface with the image has a symmetry axis - in this case the two sides are the same, and the mirror image of the object is also the same, again after a rotation by 180° about the axis perpendicular to the mirror plane. A sheet with a [[through and through]] image is achiral. We can distinguish again two cases: *the sheet surface with the image has no symmetry axis - the two sides are different *the sheet surface with the image has a symmetry axis - the two sides are the same ==Configuration space== The [[configuration space]] of a rigid body with one point fixed (i.e., a body with zero translational motion) is given by the underlying [[manifold]] of the [[rotation group]] [[SO(3)]]. The configuration space of a nonfixed (with non-zero translational motion) rigid body is ''E''<sup>+</sup>(3), the subgroup of direct isometries of the [[Euclidean group]] in three dimensions (combinations of [[translation (geometry)|translations]] and [[rotation]]s). ==See also== *[[angular velocity]] *[[Rigid body dynamics]] *[[Skew-symmetric#Infinitesimal_rotations|infinitesimal rotations]] *[[Euler's equations]] *[[Born rigidity]] *[[Rigid rotor]] ==References== *{{cite book|title=Robot Dynamics Algorithms|author=Roy Featherstone|publisher=Springer|year=1987|isbn=0898382300}}. This reference effectively combines [[screw theory]] with rigid body [[dynamics]] for robotic applications. The author also chooses to use [[spatial acceleration]]s extensively in place of material accelerations as they simplify the equations and allows for compact notation. *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]). [[Category:Rigid bodies| ]] [[Category:Introductory physics]] [[Category:Rotational symmetry]] [[be:Абсалютна цвёрдае цела]] [[bg:Абсолютно твърдо тяло]] [[cs:Tuhé těleso]] [[de:Starrer Körper]] [[es:Mecánica del sólido rígido]] [[ko:강체]] [[id:Rigid body]] [[it:Corpo rigido]] [[he:מכניקה של גוף קשיח]] [[hu:Merev test]] [[ja:剛体の力学]] [[pl:Bryła sztywna]] [[pt:Corpo rígido]] [[ru:Абсолютно твёрдое тело]] [[sk:Tuhé teleso]] [[sl:Togo telo]] [[sv:Stel kropp]] [[tr:Rijit cisim]] [[uk:Абсолютно тверде тіло]] [[zh:刚体]]