【python】【材料力学】计算任意界面的质心坐标及惯性矩 |
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背景
etabs自定义截面 公式This code uses the following formulas: Area of the polygon: A = 1 2 ∑ i = 1 n ( x i y i + 1 − x i + 1 y i ) A = \frac{1}{2} \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) A=21i=1∑n(xiyi+1−xi+1yi) Centroid coordinates: C x = 1 6 A ∑ i = 1 n ( x i + x i + 1 ) ( x i y i + 1 − x i + 1 y i ) C_x = \frac{1}{6A} \sum_{i=1}^{n} (x_i + x_{i+1})(x_i y_{i+1} - x_{i+1} y_i) Cx=6A1i=1∑n(xi+xi+1)(xiyi+1−xi+1yi) C y = 1 6 A ∑ i = 1 n ( y i + y i + 1 ) ( x i y i + 1 − x i + 1 y i ) C_y = \frac{1}{6A} \sum_{i=1}^{n} (y_i + y_{i+1})(x_i y_{i+1} - x_{i+1} y_i) Cy=6A1i=1∑n(yi+yi+1)(xiyi+1−xi+1yi) Moment of inertia: I x = 1 12 ∑ i = 0 n − 1 ( y i 2 + y i y i + 1 + y i + 1 2 ) ( x i y i + 1 − x i + 1 y i ) Ix = \frac{1}{12}\sum{i=0}^{n-1} (yi^2 + yiy{i+1} + y{i+1}^2)(xi y{i+1} - x{i+1} yi) Ix=121∑i=0n−1(yi2+yiyi+1+yi+12)(xiyi+1−xi+1yi) I y = 1 12 ∑ i = 0 n − 1 ( x i 2 + x i x i + 1 + x i + 1 2 ) ( x i y i + 1 − x i + 1 y i ) Iy = \frac{1}{12}\sum{i=0}^{n-1} (xi^2 + xi x{i+1} + x{i+1}^2)(xi y{i+1} - x{i+1} yi) Iy=121∑i=0n−1(xi2+xixi+1+xi+12)(xiyi+1−xi+1yi) I x y = 1 24 ∑ i = 0 n − 1 ( 2 x i y i + x i + 1 y i + 2 x i + 1 y i + 1 + x i y i + 1 ) ( x i y i + 1 − x i + 1 y i ) I{xy} = \frac{1}{24}\sum{i=0}^{n-1} (2xi yi + x{i+1} yi + 2x{i+1} y{i+1} + xi y{i+1})(xi y{i+1} - x{i+1} yi) Ixy=241∑i=0n−1(2xiyi+xi+1yi+2xi+1yi+1+xiyi+1)(xiyi+1−xi+1yi) 其中， n n n 是多边形的顶点数， ( x i , y i ) (xi,yi) (xi,yi) 表示多边形中第 i i i 个顶点的坐标， ( x i + 1 , y i + 1 ) (x{i+1},y{i+1}) (xi+1,yi+1) 表示多边形中第 i + 1 i+1 i+1 个顶点的坐标， I x Ix Ix 代表多边形相对于 x x x 轴的惯性矩， I y Iy Iy 代表多边形相对于 y y y 轴的惯性矩， I x y I_{xy} Ixy 代表多边形相对于 x x x 和 y y y 轴的二次惯性矩。 代码 import matplotlib.pyplot as plt def polygon_area(vertices): n = len(vertices) area = 0 for i in range(n): area += (vertices[i][0] * vertices[(i + 1) % n][1] - vertices[(i + 1) % n][0] * vertices[i][1]) return 0.5 * abs(area) def centroid(vertices): n = len(vertices) cx, cy = 0, 0 area = polygon_area(vertices) for i in range(n): factor = (vertices[i][0] * vertices[(i + 1) % n][1] - vertices[(i + 1) % n][0] * vertices[i][1]) cx += (vertices[i][0] + vertices[(i + 1) % n][0]) * factor cy += (vertices[i][1] + vertices[(i + 1) % n][1]) * factor return (1 / (6 * area)) * cx, (1 / (6 * area)) * cy def moment_of_inertia(vertices): n = len(vertices) Ix, Iy, Ixy = 0, 0, 0 area = polygon_area(vertices) for i in range(n): factor = (vertices[i][0] * vertices[(i + 1) % n][1] - vertices[(i + 1) % n][0] * vertices[i][1]) Ix += (vertices[i][1]**2 + vertices[i][1]*vertices[(i + 1) % n][1] + vertices[(i + 1) % n][1]**2) * factor Iy += (vertices[i][0]**2 + vertices[i][0]*vertices[(i + 1) % n][0] + vertices[(i + 1) % n][0]**2) * factor Ixy += (vertices[i][0]*vertices[(i + 1) % n][1] + 2*vertices[i][0]*vertices[i][1] + 2*vertices[(i + 1) % n][0]*vertices[(i + 1) % n][1] + vertices[(i + 1) % n][0]*vertices[i][1]) * factor return (1 / 12) * abs(Ix), (1 / 12) * abs(Iy), (1 / 24) * abs(Ixy) def plot_polygon(vertices, centroid): polygon = plt.Polygon(vertices, fill=None, edgecolor='r', linewidth=2) plt.gca().add_patch(polygon) plt.scatter(*centroid, color='blue', marker='o', s=100, label='Centroid') plt.axis('equal') plt.legend() plt.show() # 可增加，从cad中读取 vertices = [(0, 0), (30, 0), (30, 30), (0, 30), (-20, 20)] area = polygon_area(vertices) centroid_coordinates = centroid(vertices) Ix, Iy, Ixy = moment_of_inertia(vertices) print("Area:", area) print("Centroid:", centroid_coordinates) print("Moment of Inertia: Ix =", Ix, "Iy =", Iy, "Ixy =", Ixy) plot_polygon(vertices, centroid_coordinates) 总结常用的markdown编辑器可以渲染latex格式代码 常用公式可以编程 |
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