Machinery's Handbook, 31st Edition
168
Force Systems Algebraic Solution of Force Systems — Forces Not in Same Plane
Resolving a Single Force Into Its Three Rectangular Components:
z F z
z
The diagram shows how a force F may be resolved at any point O on its line of action into three concurrent components each of which is perpendicular to the other two. The x , y , z components F x , F y , F z of force F are determined from the accompanying relations in which θ x , θ y , θ z are the angles that the force F makes with the x , y , z axes.
O
y
y
F x
F
y
x
x
cos cos cos = = = F F F F x y z 2 2 2 = + + F F F F F F x y z x y z i i i
Finding the Resultant of Any Number of Concurrent Forces:
z
F
1
To find the resultant of any number of noncoplanar concurrent forces F 1 , F 2 , F 3 , etc., use the procedure outlined below.
F 4
O
z2
y
y2
F 3
x2
F 2
x
1) Draw a set of x , y , z axes at O , the point of concurrency of the forces. The angles each force makes measured counterclockwise from the positive x , y , and z coordinate axes must be known in addition to the magnitudes of the forces. For force F 2 , for example, the angles are θ x 2 , θ y 2 , θ z 2 as indicated on the diagram. 2) Apply the first three formulas given under the heading “Resolving a Single Force Into Its Three Rectangular Components” to each force to find its x , y , and z components. Tabulate these calculations as shown below for a system of three forces. Algebraically add the calculated components to find ∑ F x , ∑ F y , and ∑ F z which are the components of the resultant. Force Angles Components of Forces F θ x θ y θ z F x F y F z F 1 θ x 1 θ y 1 θ z 1 F 1 cos θ x 1 F 1 cos θ y 1 F 1 cos θ z 1 F 2 θ x 2 θ y 2 θ z 2 F 2 cos θ x 2 F 2 cos θ y 2 F 2 cos θ z 2 F 3 θ x 3 θ y 3 θ z 3 F 3 cos θ x 3 F 3 cos θ y 3 F 3 cos θ z 3 ∑ F x ∑ F y ∑ F z 3) Find the resultant of the system from the formula R F F F x y z 2 2 2 R R R = + + ^ ^ ^ h h h 4) Calculate the angles θ xR , θ yR , and θ zR that the resultant R makes with the respective coordinate axes:
F
R
cos cos cos
x
=
i
R
xR
F
R
y
=
i
R
yR
F
R
z
=
i
R
zR
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