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1 rigid body
A rigid body is an object whose shape and size do not change during motion and after being acted upon by force, and the relative positions of its internal points do not change. Absolute rigid bodies actually do not exist, but are only an ideal model, because any object will deform to a greater or lesser extent after being subjected to force. If the degree of deformation is extremely small relative to the geometric size of the object itself, when studying the object Deformation during movement is negligible.
According to the theory of relativity, objects such as rigid bodies cannot actually exist, but objects can usually be assumed to be perfect rigid bodies, provided that the condition of moving at a speed much smaller than the speed of light must be met.
By treating many solid bodies as rigid bodies, the results obtained are generally accurate enough for engineering purposes. However, to study stress and strain, deformation must be considered. Since the deformation is generally always small, the object can be treated as a rigid body first, and the unknown forces applied to it can be obtained using the method of theoretical mechanics, and then the deformation body mechanics can be used, including material mechanics, elasticity mechanics, plasticity mechanics, etc. Theory and methods for research.
In general, a rigid body is a system of particles in which the distance between particles remains constant. The spatial position of the rigid body is determined by any three non-collinear points that are firmly connected to the rigid body .
2 General motion of rigid bodies
Rigid body motion (rigid motion) is the rotation and translation motion of a geometric object in three-dimensional space, which is called rigid body transformation.
The motion of a rigid body is usually described by two main parts: translational motion and rotational motion.
The position of a rigid body in space must be determined based on the spatial position of any point in the rigid body and the position of the rigid body when it rotates around that point (see general motion of rigid bodies), so the rigid body has six degrees of freedom in space .
1 Translational motion
Translational motion: This part is very intuitive, that is, a rigid body moves a distance in a certain direction. This is usually represented by a vector, with the direction of the vector indicating the direction of movement and the length of the vector indicating the distance of movement.
2 rotational motion
Rotational motion: This part is complicated because rotation can be about any axis, and different rotation sequences can produce different results (this is called the gimbal lock problem). Rotations are usually represented by a rotation matrix or quaternion. The rotation matrix is a 3x3 matrix that transforms a vector from one coordinate system to another. A quaternion is a four-element number that allows for more efficient representation and calculation of rotations.
The motion of a rigid body can also be represented in other ways, such as Euler angles or axial angle representations, but these are usually more mathematically complex.
Translation and rotation can be represented simultaneously by a 4x4 homogeneous transformation matrix. The upper left corner of this matrix is a 3x3 rotation matrix, the upper right corner is a 3x1 translation vector, the lower left corner is a 1x3 zero vector, and the lower right corner is 1. By multiplying this matrix, you can transform a rigid body from one position and orientation to another.
In practical applications, a series of translations and rotations are usually required to describe complex movements, such as the movement of a robot arm or the transformation of three-dimensional graphics.