Machinery's Handbook, 31st Edition
166 Force Systems Finding the Resultant of Forces Not Intersecting at a Common Point:
y
F 2
F 3
F 4
To determine the resultant of a coplanar, nonconcurrent, nonparallel force system as shown in the diagram, proceed as shown below.
4
– x
y 4
x
O
x 4
– y
F 1
1) Draw a set of x and y coordinate axes through any convenient point O in the plane of the forces as shown in the diagram. 2) Determine the x and y coordinates of any convenient point on the line of action of each force and the angle θ , measured in a counterclockwise direction, that each line of action makes with the positive x axis. For example, in the diagram, coordinates x 4 , y 4 , and θ 4 are shown for F 4 . Similar data should be known for each of the forces of the system. 3) Calculate the x and y components ( F x , F y ) of each force and the moment of each component about O . Counterclockwise moments are considered positive and clockwise moments are negative. Tabulate all results in a manner similar to that shown below for a system of three forces and find ∑ F x , ∑ F y , ∑ M O by algebraic addition.
Components of F
Moment of F about O
Force
Coordinates of F
M O = xF y − yF x
F y
F
x
y
F x
θ
F 1 cos θ 1 F 2 cos θ 2 F 3 cos θ 3
F 1 sin θ 1 F 2 sin θ 2 F 3 sin θ 3
x 1 F 1 sin θ 1 − y 1 F 1 cos θ 1 x 2 F 2 sin θ 2 − y 2 F 2 cos θ 2 x 3 F 3 sin θ 3 − y 3 F 3 cos θ 3
F 1 F 2 F 3
x 1 x 2 x 3
y 1 y 2 y 3
θ 1
θ 2
θ 3
∑ F x
∑ F y
∑ M O
4) Compute the resultant of the system and the angle θ R it makes with the x axis by using the formulas:
F h ^ h y 2 +
2
R F x = ^
R R
cos
or
tan
F R x '
F F y x '
=
=
i
R
i
R R
R
R
5) Calculate the distance d from O to the line of action of the resultant R : d M R O ' R = Distance d is in such direction from O as will make the moment of R about O have the same sign as ∑ M O . Note Concerning Interpretation of Results: If R = 0, then the resultant is a couple ∑ M O ; if ∑ M O = 0, then R passes through O ; if both R = 0 and ∑ M O = 0, then the system is in equilibrium.
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