acting on a body has two effects:

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2 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.

3 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.

4 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 =d, which is counterclockwise for this case.

5 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. is

6 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 opposite in direction. The magnitude and direction of remains the same, but its new line of action will be d distance away from point B.

7 1. The tie-rod AB exerts the 250 N force on the steering knuckle AO as shown. eplace this force by an equivalent force-couple system at O.

8 If two force systems are creating the same external effect on the rigid body they are exerted on, they are said to be. 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.

9 Coplanar orce Systems If the resultant of all forces 1, 2, 3,..., n lying in a single plane such as xy is, this resultant is calculated by the vector sum of these forces n x x y y tan 2 x y 1 y x 2

10 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. o d d o

11 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 Nmm restoring moment exerted by a hidden torsional spring. Determine the y-intercept of the line of action of the single equivalent force.

12 3. 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

13 Three Dimensional orce Systems The same principles can be enlarged to three dimensional force systems. The resultant of forces,,..., acting on a 1, 2 3 body can be obtained by moving them to a desired point. In this way, the given force system will be converted to n 1) Three dimensional, concurrent forces comprising the same magnitudes and directions as the original forces, 2) Three dimensional couples.

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

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

16 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 out. The distance d will be equal to d=s/., which is equal in magnitude, so that they will cancel each other

17 4. The special-purpose milling cutter is subjected to the force of 1200 N and a couple of 240 N.m as shown. eplace the given force system with an equivalent force-couple system at O.

18 5. epresent the resultant of the force system acting on the pipe assembly by a single force at A and a couple.

19 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 if they point in opposite directions.

20 Wrench esultants A common 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.

21 // // n n 2 1 Equivalent force-couple system at point O is resolved into components 1 along the direction of and 2 normal to. Positive wrench d 2

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23 6. 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.

24 7. The force-couple system acting at O is equivalent to the wrench acting at A. If 600i 1400 j 700k N and // 1200 N m, determine o.

25 8. a) educe the general three-dimensional force system to a force-couple system at O. b) eplace the force-couple system obtained by a wrench and determine the coordinates of the point in the yz plane through which the line of the wrench passes. Dimensions in meters.

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27 25 N. m 115 N 100 N 9. Two forces and a couple (applied to portion AB) are applied to the car s exhaust system as shown. eplace the given force system with an equivalent force-couple system at D.

28 * Line N 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. 10. The direction cosines of robot arm AB are cos x =0.6, cos y ( y <90 ) and cos z =0. or arm BC the direction cosines are, cos x =7/9, cos y =4/9 and cos z =4/9. A force of magnitude =250 N and a couple of magnitude C=27 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.

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

30 S 150 j 200 j 50 j S o 0. 15sin 20i 0. 15cos 20k 0. 2k 0. 25i 200 j 0. 15sin 20i 0. 15cos 20k 0. 2k 0. 25i 150 j So 17i 85k noc sin 20i cos 20k S OC o 0. 34i 0. 94k 17i 85k 0. 34i 0. 94k Nm

31 tan N 500 N y 34 cm 3400 N cm 30 cm 800 N. m tan 8 15 z 50 cm x 50 cm 12. The pulleys and the gear are subjected to the loads shown. or these forces, determine the equivalent force-couple system at point A.

32 j i j i i j i j i j i i j i j i S k j i C r k C k j i j i k j r k i j r k j i r j i k j i j i j i k r C r A A S S S S

33 Z C 2 1 = 30 N 2 = 75 N 3 = 40 N C 1 = 60 Nm C 2 = 100 Nm (in yz plane) X B 6 m 37 C 1 A C 3 2 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 (12, 0, 0) m B (in xz plane) C (12, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m 4 m D 13. 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.

34 orce: Z C 2 1 = 30 N 2 = 75 N 3 = 40 N C 1 = 60 Nm C 2 = 100 Nm (in yz plane) X 6 m 37 C 1 A C 3 2 G y C O 53 E 30 3 C 3 = 80 Nm (in plane ABCD) y > 90 o Y B 4 m D 2 cos cos 2 60 cos 2 y 1 cos45i cos60 j cos60k i 15 j 15k y 60 1 o

35 orce: Z C 2 1 = 30 N 2 = 75 N 3 = 40 N C 1 = 60 Nm C 2 = 100 Nm (in yz plane) X 6 m 37 C 1 A C 3 2 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 (12, 0, 0) m B (in xz plane) C (12, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m B 4 m D 12 18i ( 8 0 ) j ( 0 8)k i j k

36 orce: Z C 2 1 = 30 N 2 = 75 N 3 = 40 N C 1 = 60 Nm C 2 = 100 Nm (in yz plane) X 6 m 37 C 1 A C 3 2 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 (12, 0, 0) m B (in xz plane) C (12, 8, 0) m E (6, 10, -3) m G (10, 4, 4) m B 4 m D 3 40cos 60cos 53i 40cos 60 sin 53 j 40cos 30k 12i 16 j k S 1. 95i j k 1. 95i j k n i j 0. 78k

37 r r r oment: o Sr SC 10i 4 j 4k i 15 j 15k j k 12i j. i. j. k i j 6i j k i j. k i j 24k Z k X 6 m 37 C 1 A C 3 2 G y C O 53 E C Y 1 = 30 N 2 = 75 N 3 = 40 N C 1 = 60 Nm C 2 = 100 Nm (in yz plane) C 3 = 80 Nm (in plane ABCD) y > 90 o 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 B 4 m D

38 C1 60 cos45i cos 60 j cos 60k i C2 100i BA BD 6i k 8 4 j 32i 24k 32i 24k C i 32k 2 2 Z j 30k C 2 1 = 30 N 2 = 75 N 3 = 40 N C 1 = 60 Nm C 2 = 100 Nm (in yz plane) C 3 = 80 Nm (in plane ABCD) y > 90 o 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 X B 4 m 6 m 37 D C 1 A C 3 2 G y C O 53 E 30 3 Y

39 oment: o o j k i j k i j 24k 4. i j k i i 32k i j 867k Equivalent force-couple system at point O 1.95i j 96.52k i j 867k o

40 Equivalent force-couple system at point O 1.95i j 96.52k i j 867k o eduction to a wrench in yz plane // // o // n n Positive wrench i j 867k i j 0. 78k.. i. j. k i j k i j 96.52k 2i j k // o z Positive wrench // 2i j k Nm 1.95i j 96.52k O x y

41 The coordinates of point P, of which the line of action of the wrench cuts through the yz plane: o // i j k yj zk 1.95i 77.8 j 96.52k r i j 867k 2i j k i j k 1.95yk 96.52yi 1.45zj 77.8zi i j k j 1.95z z m k 1.95y y m z 1.95i j 96.52k Positive wrench r // 2i j k P(0;391.27;474.66) x O y