Machinery's Handbook, 31st Edition
Force Systems
169
Finding the Resultant of Parallel Forces Not in the Same Plane:
z
– x
F 1
y 1
In the diagram, forces F 1 , F 2 , etc. represent a system of noncoplanar parallel forces. To find the resultant of such systems, use the procedure shown below.
x 1
O
– y
y
F 3
F 2
x
1) Draw a set of x , y , and z coordinate axes through any point O in such a way that one of these axes, say the z axis, is parallel to the lines of action of the forces. The x and y axes then will be perpendicular to the forces. 2) Set the distances of each force from the x and y axes in a table as shown below. For example, x 1 and y 1 are the x and y distances for F 1 shown in the diagram. 3) Calculate the moment of each force about the x and y axes and set the results in the table as shown for a system consisting of three forces. The algebraic sums of the moments ∑ M x and ∑ M y are then obtained. (In taking moments about the x and y axes, assign counterclockwise moments a plus ( + ) sign and clockwise moments a minus ( − ) sign. In deciding whether a moment is counterclockwise or clockwise, look from the positive side of the axis in question toward the negative side.) Force Coordinates of Force F Moments M x and M y due to F F x y M x M y F 1 x 1 y 1 F 1 y 1 F 1 x 1 F 2 x 2 y 2 F 2 y 2 F 2 x 2 F 3 x 3 y 3 F 3 y 3 F 3 x 3 ∑ F ∑ M x ∑ M y 4) Find the algebraic sum ∑ F of all the forces; this will be the resultant R of the system. R F F F 1 2 f R = = + + 5) Calculate x R and y R , the moment arms of the resultant: x M R y M R R y R x = = These moment arms are measured in a direction along the x and y axes as will give the resultant a moment of the same direction of rotation as ∑ M x and ∑ M y . ' ' R R Note Concerning Interpretation of Results: If ∑ M x and ∑ M y are both 0, then the resultant is a single force R along the z axis; if R is also 0, then the system is in equilibrium. If R is 0 but ∑ M x and ∑ M y are not both 0, then the resultant is a couple M M M R x y 2 2 R R = + ^ h ^ h that lies in a plane parallel to the z axis and making an angle θ R measured in a counterclockwise direction from the positive x axis and calculated from the following formula: sin M M R R x i R =
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