ESULTANTS
orce Couple Systems = eduction of a orce to an Equivalent orce and Moment (Moving a orce 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. 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=d, 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 a free vector, its location is of no concern. The combination of the force and couple is referred to as a force-couple system. M is
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. orce can be moved to a point by constructing a moment equal in magnitude and M opposite in direction. The magnitude and direction of M remains the same, but its new line of action will be d distance away from point B.
1. A force of magnitude 50 N is exerted on the automobile parkingbrake lever at the position x=50 mm. eplace the force by an equivalent force-couple system at the pivot point O.
. The device shown is part of an automobile seat-back-release mechanism. The part is subected to the 4 N force exerted at A and a 300 Nmm 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 forcecouple system at point O.
Simplification of orce Systems esultants 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 orce Systems If the resultant of all forces 1,, 3,..., n lying in a single plane such as xy is, this resultant is calculated by the vector sum of these forces.... 1 3 n x x y y tan x y 1 y x
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. M o M d d M o
380 mm C 160 mm 800 N 150 mm 800 N A 30 mm B 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. eplace 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., ). AB CD
Three Dimensional orce 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 1,, 3,..., n 1) 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, 3 1... z y x z z y y x x n
C C The resultant couple moment, The selection of point O is arbitrary, but the magnitude and direction of M M direction of r C will depend on this point; whereas, the magnitude and 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 M? O
7. epresent 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 subected to the force of 100 N and a couple of 40 N.m as shown. eplace 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 70 N. Suppose that you want to replace these two cables by a single cable E 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 E and what are the coordinates of points E and?
10. The threading dye is screwed onto the end of the fixed pipe which is bent through an angle of 0. eplace the two forces by an equivalent force at O and a couple. ind and calculate the magnitude M of the moment which tends to screw the pipe into the fixed block about its angled axis through O.
150 00 50 M o 0.15sin 0i 0.15cos 0k 0.k 0.5i 00 0.15sin 0i 0.15cos 0k 0.k 0.5i 150 M o 17i 85k eoc sin 0i cos 0k 0.34i M M M OC o 0.94k 17i 85k 0.34i 0.94k 85.68 Nm
tan 15 8 1700 N 500 N y 34 cm 3400 N 4 3 50 cm 30 cm 800 N. m tan 8 15 z 50 cm x 50 cm 11. The pulleys and the gear are subected to the loads shown. or these forces, determine the equivalent force-couple system at point A.
i i i i i i i i i 350 1040 800 1500 500 70 040 800 1500 17 8 1700 17 15 1700 500 70 040 5 4 3400 5 3 3400 3 1 k i C r M k C k i i k r k i r k i r i k i i i k r C r M A A 1139.6 1770 960 800 750 750 400 800 1500 0.5 0.5 150 500 0.3 489.6 100 1360 70 040 0.5 0.16 0.3 70 040 17 8 17 15 0.34 0.5 3 3 1 1 1 1
* 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 1. The direction cosines of robot arm AB are cos x =0.6, cos y ( y <90 ) and cos z =0 dır. or arm BC the direction cosines are, cos x =7/9, cos y =4/9 and cos z =4/9. A force of magnitude =50 N and a couple of magnitude C=7 Nm along the axis BC are applied to the end C of arm BC. Determine the moment about line AD. eplace 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 the resultant force is perpendicular to, 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 and opposite in direction M M M out. The distance d will be equal to d=m/., which is equal in magnitude, so that they will cancel each other
Wrench esultants When the resultant couple vector force, the resultant is called a wrench. is parallel to the resultant 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. M
Wrench esultants 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
// // 1 M M M M e e M M M Equivalent force-couple system at point O M is resolved into components M 1 along the direction of and M normal to. Positive wrench M d
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 forcecouple 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.
14.
Z C 1 = 30 N = 75 N 3 = 40 N C 1 = 60 Nm C = 100 Nm (in yz plane) X B 6 m 37 C 1 A 45 1 60 C 3 G y C O 53 E 30 3 C 3 = 80 Nm (in plane ABCD) y > 90 o Y O (0, 0, 0) m A (1, 0, 0) m B (in xz plane) C (1, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m 4 m D 15. eplace 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.
0.78k 0.63 0.015i 96.5 77.88 1.95 96.5k 77.88 1.95i e 96.5k 77.88 1.95i 34.64k 16 1i 40cos 30k 40cos 60sin53 40cos 60cos 53i 46.88k 46.88 35.15i 8 8 4 8)k (0 0) (8 i 18 1 75 15k 15 1.1i cos60k cos60 30 cos45 i 60 1 cos 60 cos 45 cos 14.04 3 1 o y y orce:
Moment: M o r C r1 1 r r3 3 C1 60 cos45 i C 100i BA BD 6i 8k 4 3i 4k 3i 4k C3 80 64i 3k 3 4 M M o o 10i 4 4k 1.1i 15 15k 1i 8 35.15i 46.88 46.88k 6i 10 3k 1i 16 34.64k cos 60 cos 60k 4.4i 40 65.16 65.16k 375.04i 56.56 843.84k 394.4i 43.84 4k 4.4i 30 30k 100i 64i 3k 847.86i 841.56 867k 65.16 65.16k 375.04i 56.56 843.84k 394.4i 43.84 4k 30 30k
Equivalent force-couple system at point O 1.95i 77.88 96.5k M 847.86i 841.56 867k o eduction to a wrench in yz plane M // M o e 847.86i 841.56 867k 0.015i 0.63 0.78k 133.35 M M e 133.35 0.015i 0.63 0.78k i 84.0 104.0 // // Positive wrench 1.95i 77.88 96.5k M i 84.0 104.0k // k M M o Nm z Positive wrench M // 1.95i 77.88 96.5k i 84.0 104.0k O x y
The coordinates of point P, of which the line of action of the wrench cuts through the yz plane: M Mo M // M 849.86i 95.58 76.98k y zk 1.95i 77.8 96.5k r 847.86i 841.56 867k i 84.0 104.0k 849.86i 95.58 76.98k 1.95yk 96.5yi 1.45z 77.8zi 849.86i 95.58 76.98k 1.95z 95.58 z 474.66 m k 1.95y 76.98 y 391.7 m M z 1.95i 77.88 96.5k Positive wrench r M // i 84.0 104.0k P(0;391.7;474.66) x O y