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Unformatted text preview: FORCE SYSTEM RESULTANTS
MOMENT OF A COUPLE LEARNING OBJECTIVES Be able to define a couple Be able to determine the moment of a couple PREREQUISITE KNOWLEDGE Units of measurements Trigonometry concepts Vector concepts Rectangular components concepts Moment concept Cross product and dot product DEFINITION OF COUPLE
A couple is defined as two parallel forces of the same magnitude acting in opposite direction and separated by a constant distance. F F Resultant force of a couple = 0 A moment of a couple (also called free vector) or a couple moment has a constant direction and magnitude regardless of the location of the positions of the forces to the reference point MOMENT OF A COUPLE
Scalar formulation Magnitude of couple moment : M = ( F )(d ) Where d is the perpendicular distance between the couple forces The direction of the couple moment follows the righthand rule EXAMPLE  (Scalar Formulation)
Determine the magnitude and direction of the couple moment Magnitude of couple moment : d = 6 cos 30 = 5.20 in M = ( F )(d ) = (25)(5.20) = 130 lb  in MOMENT OF A COUPLE
Vector formulation M = rF
Where r is any position vector projecting from one couple force to another As couple moment is a free vector Also, M = rA (  F ) + rB F EXAMPLE 1 Determine the couple moment in vector format if F = (25k) N SOLUTION 1 The couple force vector at B: F = ( 25k )N
Position vector from A to B:
r rAB = {350i  200 j}mm
The couple moment vector :
j  200 0 k 0 25 i M = rAB F =  350 0 = (  200 25  0 ) i  (  350 25  0 ) j = {5000i + 8750 j}N  mm EXAMPLE 2 Determine the distance d so that the resultant couple moment has a magnitude of 20 Nm SOLUTION 2
The couple force vectors at B and C :
rAB rAC FB = (35k ) N FC = (  50i ) N The position vectors from A to B and A to C are: rAB = {d cos 30j + d sin 30k} mm
rAC = {250i  d cos 30 j + d sin 30k} mm SOLUTION 2Contd.
Resultant moment couple :
M = rAB FB + rAC FC i =0 0 j 0.866d 0 k i 0.5d + 250 35  50 j  0.866d 0 k 0 .5 d 0 = (  0.866d 35  0 ) i  ( 0  (  50 ) 0.5d ) j + ( 0  (  50 ) (  0.866d ) ) k = d {  30.31i  25 j  43.30k } N  mm SOLUTION 2Contd.
Resultant of couple moments : M = (d / 1000){  30.31i  25 j  43.30k} N  m The magnitude of the resultant of couple moments : M = ( d / 1000) (  30.31) 2 + (  25) + (  43.30 )
2 2 = (0.05847)(d ) N  m For M = 20 Nm, d can be calculated as: d = 20/0.05847 = 342 mm ...
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This note was uploaded on 02/03/2012 for the course CE 221 taught by Professor Buch during the Summer '08 term at Michigan State University.
 Summer '08
 Buch

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