in static equilibrium Algorithm
The static equilibrium algorithm is a computational method used to analyze and solve problems involving structures or systems that are in a state of equilibrium, meaning that there is no net force or torque acting on them. In engineering and physics, this type of analysis is essential for predicting the behavior of structures, such as bridges or buildings, under various load conditions. The algorithm works by breaking down a complex structure into simpler components and applying the fundamental principles of equilibrium, i.e., the sum of the forces and the sum of the moments acting on each component must be equal to zero.
In applying the static equilibrium algorithm, the first step is to identify all the external forces and moments acting on the structure, such as gravitational forces, tensions, and compressive forces. Next, the structure is divided into smaller, manageable components, such as beams, columns, and joints, and the equilibrium equations are written for each component. These equations are based on the principles of force balance (ΣF = 0) and moment balance (ΣM = 0) and often involve multiple unknowns, which require simultaneous equations to be solved. Once the unknown values are obtained, these can be used to determine the internal forces and moments within the structure, allowing engineers to assess the structural integrity and stability under various load conditions. This information is crucial for designing safe and efficient structures that can withstand the demands of their intended use.
"""
Checks if a system of forces is in static equilibrium.
python/black : true
flake8 : passed
mypy : passed
"""
from numpy import array, cos, sin, radians, cross # type: ignore
from typing import List
def polar_force(
magnitude: float, angle: float, radian_mode: bool = False
) -> List[float]:
"""
Resolves force along rectangular components.
(force, angle) => (force_x, force_y)
>>> polar_force(10, 45)
[7.0710678118654755, 7.071067811865475]
>>> polar_force(10, 3.14, radian_mode=True)
[-9.999987317275394, 0.01592652916486828]
"""
if radian_mode:
return [magnitude * cos(angle), magnitude * sin(angle)]
return [magnitude * cos(radians(angle)), magnitude * sin(radians(angle))]
def in_static_equilibrium(
forces: array, location: array, eps: float = 10 ** -1
) -> bool:
"""
Check if a system is in equilibrium.
It takes two numpy.array objects.
forces ==> [
[force1_x, force1_y],
[force2_x, force2_y],
....]
location ==> [
[x1, y1],
[x2, y2],
....]
>>> force = array([[1, 1], [-1, 2]])
>>> location = array([[1, 0], [10, 0]])
>>> in_static_equilibrium(force, location)
False
"""
# summation of moments is zero
moments: array = cross(location, forces)
sum_moments: float = sum(moments)
return abs(sum_moments) < eps
if __name__ == "__main__":
# Test to check if it works
forces = array(
[polar_force(718.4, 180 - 30), polar_force(879.54, 45), polar_force(100, -90)]
)
location = array([[0, 0], [0, 0], [0, 0]])
assert in_static_equilibrium(forces, location)
# Problem 1 in image_data/2D_problems.jpg
forces = array(
[
polar_force(30 * 9.81, 15),
polar_force(215, 180 - 45),
polar_force(264, 90 - 30),
]
)
location = array([[0, 0], [0, 0], [0, 0]])
assert in_static_equilibrium(forces, location)
# Problem in image_data/2D_problems_1.jpg
forces = array([[0, -2000], [0, -1200], [0, 15600], [0, -12400]])
location = array([[0, 0], [6, 0], [10, 0], [12, 0]])
assert in_static_equilibrium(forces, location)
import doctest
doctest.testmod()
