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from sympy import sympify |
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from sympy.physics.vector import Point, Dyadic, ReferenceFrame, outer |
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from collections import namedtuple |
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__all__ = ['inertia', 'inertia_of_point_mass', 'Inertia'] |
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def inertia(frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0): |
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"""Simple way to create inertia Dyadic object. |
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Explanation |
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=========== |
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Creates an inertia Dyadic based on the given tensor values and a body-fixed |
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reference frame. |
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Parameters |
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========== |
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frame : ReferenceFrame |
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The frame the inertia is defined in. |
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ixx : Sympifyable |
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The xx element in the inertia dyadic. |
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iyy : Sympifyable |
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The yy element in the inertia dyadic. |
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izz : Sympifyable |
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The zz element in the inertia dyadic. |
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ixy : Sympifyable |
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The xy element in the inertia dyadic. |
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iyz : Sympifyable |
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The yz element in the inertia dyadic. |
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izx : Sympifyable |
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The zx element in the inertia dyadic. |
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Examples |
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======== |
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>>> from sympy.physics.mechanics import ReferenceFrame, inertia |
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>>> N = ReferenceFrame('N') |
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>>> inertia(N, 1, 2, 3) |
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(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z) |
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""" |
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if not isinstance(frame, ReferenceFrame): |
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raise TypeError('Need to define the inertia in a frame') |
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ixx, iyy, izz = sympify(ixx), sympify(iyy), sympify(izz) |
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ixy, iyz, izx = sympify(ixy), sympify(iyz), sympify(izx) |
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return (ixx*outer(frame.x, frame.x) + ixy*outer(frame.x, frame.y) + |
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izx*outer(frame.x, frame.z) + ixy*outer(frame.y, frame.x) + |
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iyy*outer(frame.y, frame.y) + iyz*outer(frame.y, frame.z) + |
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izx*outer(frame.z, frame.x) + iyz*outer(frame.z, frame.y) + |
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izz*outer(frame.z, frame.z)) |
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def inertia_of_point_mass(mass, pos_vec, frame): |
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"""Inertia dyadic of a point mass relative to point O. |
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Parameters |
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========== |
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mass : Sympifyable |
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Mass of the point mass |
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pos_vec : Vector |
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Position from point O to point mass |
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frame : ReferenceFrame |
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Reference frame to express the dyadic in |
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Examples |
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======== |
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>>> from sympy import symbols |
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>>> from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass |
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>>> N = ReferenceFrame('N') |
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>>> r, m = symbols('r m') |
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>>> px = r * N.x |
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>>> inertia_of_point_mass(m, px, N) |
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m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z) |
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""" |
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return mass*( |
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(outer(frame.x, frame.x) + |
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outer(frame.y, frame.y) + |
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outer(frame.z, frame.z)) * |
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(pos_vec.dot(pos_vec)) - outer(pos_vec, pos_vec)) |
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class Inertia(namedtuple('Inertia', ['dyadic', 'point'])): |
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"""Inertia object consisting of a Dyadic and a Point of reference. |
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Explanation |
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=========== |
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This is a simple class to store the Point and Dyadic, belonging to an |
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inertia. |
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Attributes |
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========== |
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dyadic : Dyadic |
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The dyadic of the inertia. |
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point : Point |
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The reference point of the inertia. |
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Examples |
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======== |
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>>> from sympy.physics.mechanics import ReferenceFrame, Point, Inertia |
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>>> N = ReferenceFrame('N') |
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>>> Po = Point('Po') |
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>>> Inertia(N.x.outer(N.x) + N.y.outer(N.y) + N.z.outer(N.z), Po) |
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((N.x|N.x) + (N.y|N.y) + (N.z|N.z), Po) |
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In the example above the Dyadic was created manually, one can however also |
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use the ``inertia`` function for this or the class method ``from_tensor`` as |
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shown below. |
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>>> Inertia.from_inertia_scalars(Po, N, 1, 1, 1) |
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((N.x|N.x) + (N.y|N.y) + (N.z|N.z), Po) |
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""" |
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__slots__ = () |
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def __new__(cls, dyadic, point): |
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if isinstance(dyadic, Point) and isinstance(point, Dyadic): |
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point, dyadic = dyadic, point |
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if not isinstance(point, Point): |
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raise TypeError('Reference point should be of type Point') |
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if not isinstance(dyadic, Dyadic): |
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raise TypeError('Inertia value should be expressed as a Dyadic') |
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return super().__new__(cls, dyadic, point) |
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@classmethod |
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def from_inertia_scalars(cls, point, frame, ixx, iyy, izz, ixy=0, iyz=0, |
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izx=0): |
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"""Simple way to create an Inertia object based on the tensor values. |
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Explanation |
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=========== |
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This class method uses the :func`~.inertia` to create the Dyadic based |
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on the tensor values. |
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Parameters |
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========== |
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point : Point |
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The reference point of the inertia. |
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frame : ReferenceFrame |
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The frame the inertia is defined in. |
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ixx : Sympifyable |
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The xx element in the inertia dyadic. |
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iyy : Sympifyable |
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The yy element in the inertia dyadic. |
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izz : Sympifyable |
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The zz element in the inertia dyadic. |
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ixy : Sympifyable |
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The xy element in the inertia dyadic. |
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iyz : Sympifyable |
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The yz element in the inertia dyadic. |
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izx : Sympifyable |
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The zx element in the inertia dyadic. |
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Examples |
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======== |
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>>> from sympy import symbols |
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>>> from sympy.physics.mechanics import ReferenceFrame, Point, Inertia |
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>>> ixx, iyy, izz, ixy, iyz, izx = symbols('ixx iyy izz ixy iyz izx') |
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>>> N = ReferenceFrame('N') |
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>>> P = Point('P') |
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>>> I = Inertia.from_inertia_scalars(P, N, ixx, iyy, izz, ixy, iyz, izx) |
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The tensor values can easily be seen when converting the dyadic to a |
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matrix. |
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>>> I.dyadic.to_matrix(N) |
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Matrix([ |
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[ixx, ixy, izx], |
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[ixy, iyy, iyz], |
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[izx, iyz, izz]]) |
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""" |
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return cls(inertia(frame, ixx, iyy, izz, ixy, iyz, izx), point) |
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def __add__(self, other): |
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raise TypeError(f"unsupported operand type(s) for +: " |
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f"'{self.__class__.__name__}' and " |
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f"'{other.__class__.__name__}'") |
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def __mul__(self, other): |
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raise TypeError(f"unsupported operand type(s) for *: " |
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f"'{self.__class__.__name__}' and " |
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f"'{other.__class__.__name__}'") |
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__radd__ = __add__ |
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__rmul__ = __mul__ |
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