Force Couple Systems = Replacement of a Force with an Equivalent Force and Moment (Moving a Force to Another Point)

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2 orce Couple Sstems = eplacement of a orce with an Equivalent orce and oment (oving a orce to Another Point) The force acting on a bod has two effects: The first one is the tendenc to push or pull the bod in the direction of the force, and the second one is to rotate the bod about an fied ais which does not intersect nor is parallel to the line of action of the force. This dual effect can more easil be represented b replacing the given force b an equal parallel force and a couple to compensate for the change in the moment of the force.

3 Let us consider force possible to move force acting at point A in a rigid bod. It is along its line of action, but it is not possible to directl move it to point B without altering the eternal effect it creates on the rigid bod. If it is desired to move force to a point such as B, the force must be moved b maintaining the same eternal effect it creates on the rigid bod.

4 In order to do this, two equal and opposite forces and are added to point B without introducing an net eternal effects on the bod. It is now seen that, the original force at A and the equal and opposite one at B constitute the couple = d, which is counterclockwise for this case. Since the force effect of a couple is ero and a couple onl creates a tendenc of rotation on the bod it acts, it will be possible to represent this couple b its moment effect.

5 Therefore, the original force at A is replaced b the same force acting at a different point B and a couple, without altering the eternal effects of the original force on the bod. Since is a free vector, the location of the couple is of no concern. But the force has to be eerted at point B. If it is to be moved to a third point, a new couple has to be determined.

6 Since a force can be replaced b a force and a couple, the reverse will also be possible. A given couple and a force, which lies in the plane of the couple (normal to the couple vector) can be combined to produce a single, equivalent force.

7 orce can be moved to a point b constructing a moment equal in magnitude and opposite in direction to. The magnitude and direction of remains the same, but its new line of action will be d = / distance awa from point B.

8 Simplification of orce Sstems = esultants If two force sstems are creating the same eternal effect on the rigid bod the are eerted on, the are said to be equivalent. The resultant of a force sstem is the simplest combination that forces can be reduced without altering the eternal effect the produce on the bod. or some sstems the resultant will onl be a force, for others it will onl be a couple, but in general the resultant will comprise both a force and a couple.

9 Coplanar orce Sstems If the resultant of all forces, 1 3,,..., n such as the plane is ling in a single plane, then, this 3 1 resultant is calculated b the vector sum of these forces n n n n, i j

10 The location of the line of action of the resultant force to an arbitrar point (such as point O, which is the origin of the coordinate sstem) can be determined b using the Varignon theorem. The moment of about point O will be equal to the sum of the couple moments constructed b moving its components to point O. d 1 1 d d d 1 n n d O, d O n d 3 d 1 d n O O d d

11 This calculation can be defined b an equation as: O d If the resultant force of coplanar forces is ero, but their couple moment is different from ero, the resultant will be a couple moment perpendicular to the plane containing the forces. 3 1, 1 3,,..., n n d 3 d 1 d n O O d d

12 Three Dimensional orce Sstems The same principles can be applied to three dimensional force sstems. The resultant of forces,,..., acting on a bod can be obtained b, 1 3 n moving them to a desired point. In this wa, the given force sstem will be converted to 1) Three dimensional, concurrent forces comprising the same magnitudes and directions as the original forces, ) Three dimensional couples.

13 B calculating the resultants of these forces and couples, a single resultant force and a single resultant couple can be obtained.

14 The resultant force, k j i n n n,,

15 The resultant couple moment, k j i n r r r n n,,

16 The selection of point O is arbitrar, but the magnitude and direction of will depend on this point; whereas, the magnitude and direction of will remain the same irrespective of the point it is moved to.

17 As a special case, if the resultant couple perpendicular to the resultant force is, these two vectors can further be simplified to obtain a single resultant force.

18 The force moment in direction to can be shifted a distance d to form a, which is equal in magnitude and opposite, so that the will cancel each other out. The distance d will be equal to d= /.

19 Wrench esultants When the resultant couple vector the resultant force is parallel to, the resultant is called a wrench. The wrench is the simplest form in which the resultant of a general force sstem ma be epressed.

20 B definition, a wrench is positive if the couple and force vectors point in the same direction, and negative if the point in opposite directions. A common eample of a positive wrench is found with the application of a screwdriver. All force sstems can be reduced to a wrench acting at a particular line of action.

21 Positive wrench Negative wrench

22 Positive wrench

23 Negative wrench