Machinery's Handbook, 31st Edition
Force Systems
171
4) Calculate the moments M x , M y , M z about x , y , and z axes, respectively, due to the F x , F y , and F z components of each force and set them in tabular form. The formulas to use are given in the accompanying table. In interpreting moments about the x , y , and z axes, consider 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 Moments of Components of F ( F x , F y , F z ) about x , y , z axes F M x = yF z − zF y M y = zF x − xF z M z = xF y − yF x F 1 M x 1 = y 1 F z 1 − z 1 F y 1 M y 1 = z 1 F x 1 − x 1 F z 1 M z 1 = x 1 F y 1 − y 1 F x 1 F 2 M x 2 = y 2 F z 2 − z 2 F y 2 M y 2 = z 2 F x 2 − x 2 F z 2 M z 2 = x 2 F y 2 − y 2 F x 2 F 3 M x 3 = y 3 F z 3 − z 3 F y 3 M y 3 = z 3 F x 3 − x 3 F z 3 M z 3 = x 3 F y 3 − y 3 F x 3 ∑ M x ∑ M y ∑ M z 5) Add the component moments algebraically to get ∑ M x , ∑ M y and ∑ M z , which are the components of the resultant couple, M , given by the formula, M M M M x y z 2 2 2 R R R = + + ^ ^ ^ h h h The resultant couple M will tend to produce rotation about an axis making angles of β x , β y , and β z with the x , y , z axes, respectively. These angles are determined from the formulas, cos cos cos M M M M M M x x y y z z β β β R R R = = = From the origin, point O , of a set of coordinate axes x , y , z , lay off on the x axis a length A representing the algebraic sum ∑ F x of the x components of all the forces. From the end of line A lay off a line B representing ∑ F y , the algebraic sum of the y components; this line B is drawn in a direction parallel to the y axis. From the end of line B lay off a line C representing ∑ F z . Finally, draw a line R from O to the end of C ; R will be the resultant of the system. Notice that R is in the three-dimensional space bounded by the xy , xz , and yz planes. General Method of Locating Resultant When Its Components Are Known: To determine the position of the resultant force of a system of forces, proceed as follows:
z
R
O
y
A
C
B
x
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