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Translational envelopes: Moving forward and backward on the X-axis. (Surge) Moving left and right on the Y-axis. (Sway) Moving up and down on the Z-axis. (Heave) Rotational envelopes: Tilting side to side on the X-axis. Tilting forward and backward on the Y-axis. Turning left and right on the Z-axis.
The position of an n-dimensional rigid body is defined by the rigid transformation, [T] = [A, d], where d is an n-dimensional translation and A is an n × n rotation matrix, which has n translational degrees of freedom and n(n − 1)/2 rotational degrees of freedom.
In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body. [1] [2] [3] [4] [5]Traditionally the ...
The "rotational temperature" to activate the rotational degrees of freedom is less than 100 K for many gases. For N 2 and O 2, it is less than 3 K. [1] The "vibrational temperature" necessary for substantial vibration is between 10 3 K and 10 4 K, 3521 K for N 2 and 2156 K for O 2. [1]
The main difference between rotational diffusion and translational diffusion is that rotational diffusion has a periodicity of + =, meaning that these two angles are identical. This is because a circle can rotate entirely once before being at the same angle as it was in the beginning, meaning that all the possible orientations can be mapped ...
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10 13 Hz to approximately 10 14 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm −1 and wavelengths of approximately 30 to 3 μm.
2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. A rotocenter is the fixed, or invariant, point of a rotation. [3] There are two rotocenters per primitive cell. Together with double translational symmetry the rotation groups are the following wallpaper groups, with axes per primitive cell: