# Chapter 4: Resultants of Non-coplanar Force Systems¶

## Example 4.4-1, Page no: 48¶

In [2]:
import math
import numpy as np
#Initilization Of Variables

F1=20 #lb
F2=15 #lb
F3=30 #lb
F4=50 #lb
#Co-ordinates of Forces
C1=np.array([2,1,6])
C2=np.array([4,-2,5])
C3=np.array([-3,-2,1])
C4=np.array([5,1,-2])

#Calculations

A=(C1[0]**2+C1[1]**2+C1[2]**2)**0.5
B=(C2[0]**2+C2[1]**2+C2[2]**2)**0.5
C=(C3[0]**2+C3[1]**2+C3[2]**2)**0.5
D=(C4[0]**2+C4[1]**2+C4[2]**2)**0.5
#Calculations for cos(thetax),cos(thetay) and cos(thetaz)
theta1=(A**-1)*np.array([C1[0],C1[1],C1[2]])
theta2=(B**-1)*np.array([C2[0],C2[1],C2[2]])
theta3=(C**-1)*np.array([C3[0],C3[1],C3[2]])
theta4=(D**-1)*np.array([C4[0],C4[1],C4[2]])
#Calculations for forces (in form of force vectors)
Fa=F1*np.array([theta1[0],theta1[1],theta1[2]]) #lb
Fb=F2*np.array([theta2[0],theta2[1],theta2[2]]) #lb
Fc=F3*np.array([theta3[0],theta3[1],theta3[2]]) #lb
Fd=F4*np.array([theta4[0],theta4[1],theta4[2]]) #lb
Fx=Fa[0]+Fb[0]+Fc[0]+Fd[0] #lb
Fy=Fa[1]+Fb[1]+Fc[1]+Fd[1] #lb
Fz=Fa[2]+Fb[2]+Fc[2]+Fd[2] #lb
R=(Fx**2+Fy**2+Fz**2)**0.5 #lb
thetax=arccos(Fx*R**-1)*(180/pi) #degrees
thetay=180-((180*arccos(Fy*R**-1))/pi) #degrees
thetaz=(180*arccos(Fz*R**-1))/pi #degrees

#Result

print'The resultant of the force system is',round(R,1),"lb"
print'The angle of the resultant with respect to x axis is',round(thetax,1),"degree"
print'The angle of the resultant with respect to y axis is',round(thetay),"degree"
print'The angle of the resultant with respect to a axis is',round(thetaz,1),"degree"

# Thetax is off by 0.1 degrees

The resultant of the force system is 42.5 lb
The angle of the resultant with respect to x axis is 30.1 degree
The angle of the resultant with respect to y axis is 79.0 degree
The angle of the resultant with respect to a axis is 62.4 degree


## Example 4.4-2, Page no: 49¶

In [3]:
import math

#Initilization of variables

F=[20,-10,30] #N
#co-ordinates in meters
a=2 #m
b=4 #m
c=7 #m
d=3 #m
e=2 #m
f=4 #m

#Calculations

R=F[0]+F[1]+F[2] #N
M_o=F[0]*a+F[1]*b+F[2]*c #N.m
x=M_o*R**-1 #m
M_x=-F[2]*f-F[0]*d-F[1]*e #N.m
z=-M_x/R #m

#Result

print'The resultant is +',round(R),"N"
print'The moment about point O is +',round(M_o),"N.m"
print'The position of R is at',round(x,2),"m (from origin)"
print'The moment is',round(M_x),"N.m"
print'The z co-ordinate is +',round(z),"m"

# Here the value of R & M_o is correct which should yeild the vaue of x (M_o/R) correctly. However dueto some error in the software the correct value is not being printed.

The resultant is + 40.0 N
The moment about point O is + 210.0 N.m
The position of R is at 5.25 m (from origin)
The moment is -160.0 N.m
The z co-ordinate is + 4.0 m


## Example 4.4-3, Page No: 49¶

In [4]:
import math

#Initilization of variables

F=[100,50,-150] #Force vector N
a=2 #m
b=2 #m
c=3 #m
d=2 #m
e=4 #m
f=8 #m

#Calculations

R=F[0]+F[1]+F[2] #N
M_x=-F[0]*a+F[1]*b-F[2]*c #N.m
M_z=F[0]*d+F[1]*e+F[2]*f #N.m
C=sqrt(M_x**2+M_z**2) #N-m
thetax=arctan(M_x*-M_z**-1)*(180/pi) #degrees

#Result

print'The resultant is',round(R),"N"
print'The moment about x axis is +',round(M_x),"N.m"
print'The moment about z axis is',round(M_z),"N.m"
print'The couple acting is',round(C),"N.m"
print'The trace makes an angle with x axis of',round(thetax,1),"degrees"

# The answer for C waries by 1 N.m as compared to the book.

The resultant is 0.0 N
The moment about x axis is + 350.0 N.m
The moment about z axis is -800.0 N.m
The couple acting is 873.0 N.m
The trace makes an angle with x axis of 23.6 degrees


## Example 4.4-4, Page No: 50¶

In [22]:
import math
import numpy as np

#Initilization of variables
x1=-2
y1=2
z1=-2
x2=3
y2=0
z2=-4
x3=3
y3=2
z3=2
F1=40 #lb
F2=30 #lb
F3=20 #lb
Mxm=np.array([-92.4,-48,-19.4])
Mym=np.array([-46.2,72,9.8])
Mzm=np.array([46.2,-36,19.4])

#Calculations
mag1=(x1**2+y1**2+z1**2)**0.5
mag2=(x2**2+y2**2+z2**2)**0.5
mag3=(x3**2+y3**2+z3**2)**0.5
thetax1=(x1*mag1**-1) #degrees
thetay1=(y1*mag1**-1) #degrees
thetaz1=(z1*mag1**-1) #degrees
thetax2=(x2*mag2**-1) #degrees
thetay2=(y2*mag2**-1) #degrees
thetaz2=(z2*mag2**-1) #degrees
thetax3=(x3*mag3**-1) #degrees
thetay3=(y3*mag3**-1) #degrees
thetaz3=(z3*mag3**-1) #degrees
#Now we will define all the components in terms of matrices for simplicity of computation
F=np.array([F1,F2,F3]) #lb
Fx1=F[0]*thetax1
Fy1=F[0]*thetay1
Fz1=F[0]*thetaz1
Fx2=F[1]*thetax2
Fy2=F[1]*thetay2
Fz2=F[1]*thetaz2
Fx3=F[2]*thetax3
Fy3=F[2]*thetay3
Fz3=F[2]*thetaz3
Fx=Fx1+Fx2+Fx3 #lb
Fy=Fy1+Fy2+Fy3 #lb
Fz=Fz1+Fz2+Fz3 #lb
R=(Fx**2+Fy**2+Fz**2)**0.5 #lb
thetax=arccos(Fx*R**-1)*(180/pi) #degrees
thetay=arccos(Fy*R**-1)*(180/pi) #degrees
thetaz=arccos(Fz*R**-1)*(180/pi) #degrees
#Moment calculations
Mx=Mxm[0]+Mxm[1]+Mxm[2] #lb-ft
My=Mym[0]+Mym[1]+Mym[2] #lb-ft
Mz=Mzm[0]+Mzm[1]+Mzm[2] #lb-ft
C=(Mx**2+My**2+Mz**2)**0.5 #lb-ft
#Direction cosines
PHIx=arccos(Mx*C**-1)*(180/pi) #degrees
PHIy=arccos(My*C**-1)*(180/pi) #degrees
PHIz=arccos(Mz*C**-1)*(180/pi) #degrees

#Result
print'The result of the force is',round(R,1),"lb"
print'The angles with respect to X-Axis,Y-Axis and Z-axis are:',round(thetax,1),"degrees",',',round(thetay,1),"degrees",'and',round(thetaz,1),"degrees respectively."
print'The magnitude of resultant couple is',round(C),"lb-ft"
print'The angles are as follows: Cosphix=',round(PHIx,1),"degrees",',Cosphiy=',round(PHIy,1),"degrees",'and Cosphiz=',round(PHIz,1),"degrees"

# The answers may wary due to decimal point descrepancy

The result of the force is 50.6 lb
The angles with respect to X-Axis,Y-Axis and Z-axis are: 79.2 degrees , 49.6 degrees and 137.6 degrees respectively.
The magnitude of resultant couple is 166.0 lb-ft
The angles are as follows: Cosphix= 163.8 degrees ,Cosphiy= 77.6 degrees and Cosphiz= 79.8 degrees


## Example 4.4-5, Page no 52¶

In [48]:
import math
import numpy as np

#Initilization of variables
F=np.array([150,90,160]) #lb force vector kind of decleration
# Co-ordinates defined as [x,y,z] all the co-ordinates are in feet
C_1=np.array([2,0,0])
C_2=np.array([0,0,1])
C_3=np.array([0,-2,-1])
C_4=np.array([-1,0,-1])

#Calculations
A=C_2-C_1
B=C_4-C_3
F1=(F[0]*A)/(A[0]**2+A[1]**2+A[2]**2)**0.5
F2=(F[1]*B)/(B[0]**2+B[1]**2+B[2]**2)**0.5
R=F1+F2
# Determine C1 & C2
# Calculating the cross products of C1 & C2 as,
C1=np.array([[C_1[1]*F1[2]-C_1[2]*F1[1]],[-(C_1[0]*F1[2]-C_1[2]*F1[0])],[C_1[0]*F1[1]-C_1[1]*F1[0]]]) # lb-ft
C2=np.array([[C_3[1]*F2[2]-C_3[2]*F2[1]],[-(C_3[0]*F2[2]-C_3[2]*F2[0])],[C_3[0]*F2[1]-C_3[1]*F2[0]]]) # lb-ft
C3=np.array([[0],[0],[160]]) # lb-ft
C=C1+C2+C3

#Result
print'The resultant force couple is',round(C[0],1),"i",round(C[1],1),"j +",round(C[2],1),"k lb-ft"

The resultant force couple is 80.5 i -93.9 j + 79.5 k lb-ft