The driving force behind the present study has been the development of the NURBS-based element which enables an elegant framework of in-plane vibrations of arbitrarily curved Bernoulli-Euler beams, being a function only of the global Cartesian coordinates. Due to the fact that no additional ...
If a point(x,y)(x,y)on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angleθθfrom the positive x-axis, then the coordinates of the point with respect to the new axes are(x′,y′).(x′,y′).We can use the fo...
Plane rotation (4.1a)x=|r→|cosα (4.1b)y=|r→|sinα In the Cartesian plane (x´, y´), the vector r→ can be written as, (4.2a)x′=|r→|cosα−θ=|r→|cosαcosθ+sinαsinθ (4.2b)y′=|r→|sinα−θ=|r→|sinαcosθ+cosαsinθ Substituting equations (4.1) ...
rotationResultis a 1-by-3 vector whenquatis a scalarquaternionobject andcartesianPointsis a 1-by-3 vector representing a single point. Otherwise,rotationResultis anM-by-3 matrix, whereMis the maximum oflength(quat)andsize(cartesianPoints,1). ...
There is a 1:1 equivalence (a morphism) between the rotation of a 3D rigid body and the movement of a shape on the surface of a sphere. Note that this is a shape rather than a point because we need to be able to specify a rotation θ in the spherical plane in addition to Latitude...
rotation. these solutions are axisymmetric, of sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. to establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, ...
The direction of ω→ is normal to the plane of rotation (or the angular displacement Δθ). To ensure the relationship v=ωr holds, we define: v→=ω→×r→ Taking the derivative with respect to t on both sides of the equation, we derived: a→=dω→dt×r→+ω→×dr→dt a→ta...
The Cartesian plane is divided into four quadrants. The quadrants are numbered based on the counter-clockwise rotation of the angle. Quadrant 1 is between 0∘and 90∘. Quadrant II is between 90∘and 180∘. The third quadrant QIII is between 180∘to 270∘. Lastly, the fourth quadr...
def get_rotation(self, t): if (t< self.Tf): return np.eye(3,3) else: plane_to_center = np.matrix([1,0,0]).T Pitch_vector = plane_to_center - self.vector_project(plane_to_center, np.matrix(-self.base_e[2])) Pitch_vector = Pitch_vector / LA.norm(Pitch_vector) Roll_vect...
The development of plate (and shell) ÿnite elements was initially based on the so called thin plate theory following Kirchho 's main assumption of preserving orthogonality of the normals to the mid-plane [1; 4]. Indeed, most plates and shells can be classed as 'thin' structures and ...