Machinery's Handbook, 31st Edition
170 Force Systems Finding the Resultant of Nonparallel Forces Not Meeting at a Common Point:
z
The diagram shows a system of noncoplanar, nonparallel, noncon current forces F 1 , F 2 , etc. for which the resultant is to be determined. Generally speaking, the resultant will be a noncoplanar force and a couple, which may be further com bined, if desired, into two forces that are skewed. This is the most general force system that can be devised, so each of the other systems so far described represents a special, simpler case of this general force system. The method of solution described below for a system of three forces applies for any number of forces.
F 2
F 1
O
y
x 2 z 2
y 2
F 3
x
1) Select a set of coordinate x , y , and z axes at any desired point O in the body as shown in the diagram. 2) Determine the x , y , and z coordinates of any convenient point on the line of action of each force as shown for F 2 . Also determine the angles, θ x , θ y , θ z that each force makes with each coordinate axis. These angles are measured counterclockwise from the positive direction of the x , y , and z axes. The data is tabulated, as shown in the table accompanying Step 3, for convenient use in subsequent calculations. 3) Calculate the x , y , and z components of each force using the formulas given in the accompanying table. Add these components algebraically to get ∑ F x , ∑ F y and ∑ F z , which are the components of the resultant R given by the formula, R F F F x y z 2 2 2 R R R = + + ^ ^ ^ h h h
Force
Coordinates of Force F
Components of F
F x y
z
F x
F y
F z
θ x
θ y
θ z
F 1 cos θ x 1 F 2 cos θ x 2 F 3 cos θ x 3
F 1 cos θ y 1 F 2 cos θ y 2 F 3 cos θ y 3
F 1 cos θ z 1 F 2 cos θ z 2 F 3 cos θ z 3
F 1 F 2 F 3
x 1 x 2 x 3
y 1 y 2 y 3
z 1 z 2 z 3
θ x 1
θ y 1
θ z 1
θ x 2
θ y 2
θ z 2
θ x 3
θ y 3
θ z 3
∑ F x
∑ F y
∑ F z
The resultant force R makes angles of θ xR , θ yR , and θ zR with the x , y , and z axes, respectively, and passes through the selected point O . These angles are determined from the formulas,
cos cos cos
F R F R F R x y z ' ' '
= = =
i i i
R R R
xR yR zR
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