more rigid

موضوع: more rigid الجمعة ماي 16, 2008 5:29 pm

Kinematics

[edit] Position

The position of a rigid body can be described by a combination of a 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 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. There are several methods to describe numerically the orientation of a rigid body (see 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.

[edit] 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 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 of a rigid body);
the spatial or 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

where

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)
represents the position of the point/particle at time
represents the position of the reference point of the body (the origin of local frame L) at time
is the orientation matrix, an orthogonal matrix with determinant 1, representing the 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.
represents the angular velocity of the rigid body
represents the total velocity of the point/particle
represents the translational velocity (i.e. the velocity of the origin of frame L)
represents the total acceleration of the point/particle
represents the translational acceleration (i.e. the acceleration of the origin of frame L)
represents the angular acceleration of the rigid body
represents the spatial acceleration of the point/particle
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.
Vehicles, 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 amount of rotation associated with the vertices of a polygon.

flower

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موضوع: رد: more rigid الجمعة ماي 16, 2008 5:29 pm

and waettt moooooore Very Happy

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واخر دعوانا أن الحمد لله رب العالمين

موضوع: رد: more rigid السبت ماي 17, 2008 12:06 am

iam waiting with more details and more pictures
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موضوع: رد: more rigid السبت ماي 17, 2008 4:32 pm

we are waiting you with more details and more pictures in all aplied math
thank's alot shady

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