RESULTANTS

Force – Couple Systems = Reduction of a Force to an Equivalent Force and Moment (Moving a Force to Another Point) The force acting on a body has two effects: the first one is the tendency to push or pull the body in the direction of the force, and the second one is to rotate the body about any fixed axis which does not intersect nor is parallel to the line of the force. This dual effect can more easily be represented by replacing the given force by an equal parallel force and a couple to compensate for the change in the moment of the force.

Let us consider for acting at point A in a rigid body Let us consider for acting at point A in a rigid body. It is possible to slide force along its line of action, but it is not possible to directly move it to point B without changing the external effect on the rigid body.

In order to do this, two equal and opposite forces and are added to point B without introducing any net external effects on the body. It is seen that, the original force at A and and the equal and opposite one at B constitute the couple M=Fd, which is counterclockwise for this case.

Therefore, we have replaced the original force at A by the same force acting at a different point B and a couple, without altering the external effects of the original force on the body. Since is a free vector, its location is of no concern. The combination of the force and couple is referred to as a force-couple system.

By reversing this process, we can combine a given couple and a force which lies in the plane of the couple (normal to the couple vector) to produce a single, equivalent force. Force can be moved to a point by constructing a moment equal in magnitude and opposite in direction . The magnitude and direction of remains the same, but its new line of action will be distance away from point B.

1. A force F of magnitude 50 N is exerted on the automobile parking-brake lever at the position x=250 mm. Replace the force by an equivalent force-couple system at the pivot point O.

2. The device shown is part of an automobile seat-back-release mechanism. The part is subjected to the 4 N force exerted at A and a 300 Nmm restoring moment exerted by a hidden torsional spring. Determine the y-intercept of the line of action of the single equivalent force.

3. A 50 N horizontal force is applied to the handle of the industrial water valve as shown. The force is perpendicular to the vertical plane containing line OA of the handle. Determine the equivalent force-couple system at point O.

Simplification of Force Systems – Resultants If two force systems are creating the same external effect on the rigid body they are exerted on, they are said to be “equivalent”. The resultant of a force system is the simplest combination that they can be reduced without altering the external effects they produce on the body.

Coplanar Force Systems If the resultant of all forces lying in a single plane such as xy is , this resultant is calculated by the vector sum of these forces.

The location of the line of action of the resultant force to an arbitrary point (such as point O is the origin of the xy coordinate system) can be determined by using the Varignon’s theorem. The moment of about point O will be equal the sum of the couple moments constructed by moving its components to point O.

160 mm 380 mm 150 mm 320 mm C B A 800 N 4. The forces acting on the belts on a pulley system are equal with a magnitude of 800 N. The pulley is secured to the steel column by means of two bolts at A and B. Replace the two forces with a force-couple system, in which the equivalent force will be at the midpoint of the bolts. Then, determine the force each bolt will sustain by distributing this force and couple to forces acting at points A and B.

5. Under nonuniform and slippery road conditions, the four forces shown are exerted on the four drive wheels of the all-wheel-drive vehicle. Determine the resultant of this system and the x- and y-intercepts of its line of action. Note that the front and rear tracks are equal (i. e., ).

Three Dimensional Force Systems The same principles can be enlarged to three dimensional force systems. The resultant of forces acting on a body can be obtained by moving them to a desired point. In this way, the given force system will be converted to Three dimensional, concurrent forces comprising the same magnitudes and directions as the original forces, Three dimensional couples.

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

The resultant couple moment, The selection of point O is arbitrary, but the magnitude and direction of will depend on this point; whereas, the magnitude and direction of are the same no matter which point is selected.

6. Determine the force-couple system at O which is equivalent to the two forces applied to the shaft AOB. Is perpendicular to ?

7. Represent the resultant of the force system acting on the pipe assembly by a single force at A and a couple .

8. The special purpose milling-cutter is subjected to the force of 1200 N and a couple of 240 N.m as shown. Replace this loading system by an equivalent force-couple system at O.

9. The tension in cable AB is 450 N and the tension in cable CD is 270 N. Suppose that you want to replace these two cables by a single cable EF so that the force exerted on the wall at E is equivalent to the two forces exerted by cables AB and CD on the walls at A and C. What is the tension in cable EF and what are the coordinates of points E and F?

10. The threading dye is screwed onto the end of the fixed pipe which is bent through an angle of 20°. Replace the two forces by an equivalent force at O and a couple . Find and calculate the magnitude M of the moment which tends to screw the pipe into the fixed block about its angled axis through O.

y x 500 N 1700 N 3400 N 3 4 q tan q= 30 cm 50 cm 34 cm z 800 N.m 11. The pulleys and the gear are subjected to the loads shown. For these forces, determine the equivalent force-couple system at point A.

* Line MN lies in a plane parallel to the horizontal plane * Line AD lies in the xz plane and makes a 37° angle with the x axis 12. The direction cosines of robot arm AB are cos qx=0.6, cos qy (qy <90°) and cos qz=0’dır. For arm BC the direction cosines are, cos qx=7/9, cos qy=4/9 and cos qz=-4/9. A force of magnitude F=250 N and a couple of magnitude C=27 Nm along the axis BC are applied to the end C of arm BC. Determine the moment about line AD. Replace the force and couple acting on the robot assembly with an equivalent force-couple at point A.

As a special case, if the resultant couple is perpendicular to the resultant force , these two vectors can further be simplified to obtain a single resultant force . The force can be slided a distance d to form a moment - , which is equal in magnitude and opposite in direction , so that they will cancel each other out. The distance d will be equal to d=SM/R.

Wrench Resultants When the resultant couple vector is parallel to the resultant force , the resultant is called a “wrench”. The wrench is the simplest form in which the resultant of a general force system may be expressed. By definition, a wrench is positive if the couple and force vectors point in the same direction, and negative it they point in opposite directions.

Wrench Resultants A common example example of a wrench is found with the application of a screw driver. All force systems can be reduced to a wrench acting at a particular line of action. M R

Equivalent force-couple system at point O M is resolved into components M1 along the direction of R and M2 normal to R. Positive wrench

Positive Wrench Negative Wrench

13. In tightening a bolt whose center is at point O, a person exerts a 180 N force on the ratchet handle with his right hand. In addition, with his left hand he exerts a 90-N force as shown in order to secure the socket onto the bolt head. Determine the equivalent force-couple system at O. The find the point in the x-y plane through which the line of action of the resultant force of the wrench passes.

13.

A B C O D E G 37° X Y Z 45° 60° qy 53° 30° 6 m 4 m C3 = 80 Nm (in plane ABCD) qy < 90o F1 = 30 N F2 = 75 N F3 = 40 N C1 = 60 Nm C2 = 100 Nm (in yz plane) O (0, 0, 0) m A (12, 0, 0) m B (in xz plane) C (12, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m 14. Replace the system comprising two forces, two couples and a positive wrench with an equivalent force-couple acting at point O. Then, reduce the system further into a wrench and determine the coordinates of point P, of which the line of action of the wrench cuts through the yz plane.

Force:

Moment:

Equivalent force-couple system at point O Reduction to a wrench in yz plane O x y z Positive wrench Positive wrench

The coordinates of point P, of which the line of action of the wrench cuts through the yz plane: x y z Positive wrench P(0;391.27;474.66)