RV-8F

N772RW

Building, testing and operating an experimental aircraft

3D Statics Problem

08 Dec 2015

The following three-dimensional statics problem is taken from the book Engineering Mechanics: Statics, 8th edition, by J. L. Meriam, et al. The solution algorithm was created using the Jupyter Notebook and the Python programming language.

Problem 3/076

The light right-angle boom which supports the 410-kg cylinder is supported by three cables and a ball-and-socket joint at O attached to the vertical x-y surface. Determine the reactions at O and the cable tensions.

3D Boom Schematic
Figure 1. Right-Angle Boom and Cylinder
import numpy as np

# Set output precision
%precision 3
# Mass on a string (kg)
m = 410

# Position vectors of each point
#  (Point G is where the mass string is tied to the boom)
A = np.array([ 0.00,  0.0, 2.7])
B = np.array([ 1.10,  0.0, 2.7])
C = np.array([-0.80,  0.9, 0.0])
D = np.array([ 1.10,  1.8, 0.0])
E = np.array([ 1.10,  0.0, 0.0])
G = np.array([ 0.55,  0.0, 2.7])
# Weight force vector
W = np.array([0, -m*9.81, 0])

# Radius vectors
OA = A
OG = G
OB = B

# Tension vectors
AC = C - A
BD = D - B
BE = E - B
# Tension unit vectors
uAC = AC / np.linalg.norm(AC)
uBD = BD / np.linalg.norm(BD)
uBE = BE / np.linalg.norm(BE)
# Create a 3x3 matrix
M = np.zeros(shape=(3,3))
# Populate cols of matrix with moment vectors
M[:,0] = np.cross(OA, uAC)
M[:,1] = np.cross(OB, uBD)
M[:,2] = np.cross(OB, uBE)
print(M)
[[-0.822 -1.498 -0.   ]
 [-0.731  0.915  1.1  ]
 [ 0.     0.61   0.   ]]
# The three tension moments must cancel the weight moment
w = -np.cross(OG, W)
print(w)
[-10859.67      -0.      2212.155]
# Solve the matrix for the tension magnitudes
T = np.linalg.solve(M,w)
T_AC = T[0]
T_BD = T[1]
T_BE = T[2]
# Reaction forces at O
Ox = -T_AC * uAC[0]
Oy = -T_AC * uAC[1] - T_BD * uBD[1] - W[1]
Oz = -T_AC * uAC[2] - T_BD * uBD[2] - T_BE * uBE[2]
print("T_AC = {} N\nT_BD = {} N\nT_BE = {} N\n".format(T_AC,T_BD,T_BE))
print("Ox   = {} N\nOy   = {} N\nOz   = {} N".format(Ox,Oy,Oz))
T_AC =  6606.0 N
T_BD =  3625.5 N
T_BE =  1371.2 N

Ox   =  1787.6 N
Oy   =     0.0 N
Oz   = 10420.9 N
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