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Rotating machinery produces vibrations depending upon the structure of the mechanism involved in the process. Any faults in the machine can increase or excite the vibration signatures. Vibration behavior of the machine due to imbalance is one of the main aspects of rotating machinery which must be studied in detail and considered while designing.
If we condense the skew entries into a vector, (x,y,z), then we produce a 90° rotation around the x-axis for (1, 0, 0), around the y-axis for (0, 1, 0), and around the z-axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as x → ∞, (x, 0, 0) does approach a 180° rotation around the x axis, and similarly for ...
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
For example, Ansys software may simulate how a bridge will hold up after years of traffic, how to best process salmon in a cannery to reduce waste, or how to design a slide that uses less material without sacrificing safety. [4] Most Ansys simulations are performed using the Ansys Workbench system, [45] which is one of the company's main ...
The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, [1] is defined for incompressible (divergence-free), two-dimensional flows. The Stokes stream function , named after George Gabriel Stokes , [ 2 ] is defined for incompressible, three-dimensional flows with axisymmetry .
LS-DYNA originated from the 3D FEA program DYNA3D, developed by Dr. John O. Hallquist at Lawrence Livermore National Laboratory (LLNL) in 1976. [4] DYNA3D was created in order to simulate the impact of the Full Fuzing Option (FUFO) or "Dial-a-yield" nuclear bomb for low altitude release (impact velocity of ~ 40 m/s).
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is
Fig.1: simple rotations (black) and left and right isoclinic rotations (red and blue) Fig.2: a general rotation with angular displacements in a ratio of 1:5 Fig.3: a general rotation with angular displacements in a ratio of 5:1 All images are stereographic projections. Every rotation in 3D space has a fixed axis unchanged by rotation.