Machinery's Handbook, 31st Edition
Force Systems
165
Finding a Force and a Couple which Together Are Equivalent to a Single Force: d F
To resolve a single force F into a couple of moment M and a force P passing through any chosen point O at a distance d from the original force F , use the relations P F M F d # = = The moment M must, of course, tend to produce rotation about O in the same direction as the original force. Thus, as seen in the diagram, F tends to produce clockwise rotation; hence M is shown clockwise.
O
P
O
M
Finding the Resultant of a Single Force and a Couple:
The resultant of a single force F and a couple M is a single force R equal in magnitude and direction to F and parallel to it at a distance d to the left or right of F . R F d M R ' = = Resultant R is placed to the left or right of point of application O of the original force F depending on which position will give R the same direction of moment about O as the original couple M .
F
M
O
R
d
Finding the Resultant of a System of Parallel Forces:
F 3
F 2
d 2
d 3
To find the resultant of a system of coplanar parallel forces, proceed as indicated below.
O
F 1
F 4
d 3
d 4
1) Select any convenient point O from which perpendicular distances d 1 , d 2 , d 3 , etc. to parallel forces F 1 , F 2 , F 3 , etc. can be specified or calculated. 2) Find the algebraic sum of all the forces; this will give the magnitude of the resultant of the system. R F F F F 1 2 3 f R = = + + + 3) Find the algebraic sum of the moments of the forces about O ; clockwise moments may be taken as negative and counterclockwise moments as positive: M F d F d O 1 1 2 2 f R = + + 4) Calculate the distance d from O to the line of action of resultant R:
Σ M O R = ------
d
This distance is measured to the left or right from O depending on which position will give the mo- ment of R the same direction of rotation about O as the couple ∑ M O , that is, if ∑ M O is negative, then d is left or right of O depending on which direction will make R 3 d negative. Note Concerning Interpretation of Results: If R = 0, then the resultant of the system is a couple ∑ M O ; if ∑ M O = 0 then the resultant is a single force R ; if both R and ∑ M O = 0, then the system is in equilibrium.
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