Force - Moment Vectors PDF

Chapter one Force - Moment Vectors Previous Knowledge ** Vector presentation:-(2-D) 1) Magnitude & Direction:- Force vector defined in plane (2-D) by its magnitude F and its direction angle θ it makes with horizontal x-axis F  F  Vector...

Chapter one Force - Moment Vectors Previous Knowledge ** Vector presentation:-(2-D) 1) Magnitude & Direction:- Force vector defined in plane (2-D) by its magnitude F and its direction angle θ it makes with horizontal x-axis F  F  Vector Magnitude Direction 2) Cartesian vector notation:- The Cartesian coordinates in x-y F  Fx  Fy  F  Fx i  Fy j Fx  F cos  _, _ Fy  F sin   F  F cos  i  F sin  j F  Fx 2  Fy 2   tan 1  Fy / Fx   ** Vector Notation:-(3-D) In space vectors expressed in x,y,z coordinates in form of cosines of its direction angles with x,y,z axes  Fx  F cos  x  Fy  F cos y  Fz  F cos  z k  F  F cos x i  F cos  y j  F cosz k Where the vector magnitude calculated as F  Fx2  Fy2  Fz2 And its direction cosines. Fx Fy Fz  cos  x   cos  y   cos  z  F F F -: ‫** ملحوظة‬ cos2 x  cos2  y  cos2 z  1 In case of missing the direction angles, vector represented using the unit vector eAB between two points A,B lies on its line of action a.) rAB  (B x  A x )i  (B y  A y ) j  (B z  A z )k rAB b.) e AB  rAB c.) FAB  FAB. e AB In case of the force defined by its projection Fxy on a plane and the angle with the axis Z perpendicular to that plane. Fxy  F cos  Fz  F sin  Fx  Fxy cos   F cos  cos  Fy  Fxy sin   F cos  sin  F  Fx i  Fy j  Fz k  Moment  Moment: - It's the ability of a force to make a rotation about either a point or a line around its axis. The moment vector 3D:- Mo  r F Moment vector Force vector of force F about point o Position vector from point o to any point on the line of action of force F Where the magnitude of the moment vector is:  M o  r. F.sin  r.sin   d  Mo  F. d Magnitude of moment about point Force magnitude Perpendicular distance between point and force Remember that ***Vector Product OR Cross Product between two vectors A&B can performed as a.) A  B   A. B.sin   e n A θ i j k B b.) A  B  A x Ay Az Bx By Bz  A  B  (A y B z  A z B y )i  (A x B z  A z B x ) j  (A x B y  A y B x )k  **Moment of force about line :- can be calculated in simple steps:- 1. First find the moment of the force about any point on the line. 2. Then find its projection on the line using dot product between the moment vector and the unit vector of the line. 3. Write the obtained moment about line in vector form using the unit vector of the line M A  rAD  FCD FCD , M AB  M A. e AB  M AB  e AB. (rAD  FCD ) A B Simply, magnitude of the moment of force about line can be obtained using triple product procedure: ̅𝐴𝐵 | = 𝑒̅𝐴𝐵 ∙ 𝑟̅𝐴𝐷 × 𝐹̅𝐶𝐷 |𝑀 Remember triple product between three vectors processed by determinant as: Ax Ay Az A.(B C )  B x By Bz = scalar Cx Cy Cz Special cases: 1. In case that the force is parallel to the line, It has no moment about this line M L  0 2. In case that the force intersects the line, It has no moment about this line M L  0 **Moment of couple (3-D) :- Couples are those two forces that are equal in magnitude & opposite in direction separated with distance d Moment of MC  r F The force vector to which couple vector position vector is pointed Position vector directed from one force to the other force The moment of couple is a free vector as it has no specific point of application.  SHEET ONE 1. The shaft S exerts three force components on the die D. Find the magnitude and coordinate direction angles of the resultant force. Also find the resultant moment about x axis. 2. Specify the magnitude of F3 and its coordinate direction angles α3, β3, so that the resultant force FR = {9j} kN. Solution: 3. Knowing that the tension is 1425 N in cable AB and 2130 N in cable AC, determine the magnitude and direction of the resultant of the forces exerted at A by the two cables. Find the moment of resultant about z axis. Solution: 4. Knowing that the tension is 425 lb in cable AB and 510 lb in cable AC, determine the magnitude and direction of the resultant of the forces exerted at A by the two cables. 5. A crate is supported in equilibrium by three cables as shown. Determine the weight of the crate knowing that the tension in cable AB is 750 lb. Solution: 6. Express each of the forces in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force. also fine the moment of the resultant about z axis 7- Determine the moment of the force 80 lb about hinge B and about axis AB. 8- To loosen a frozen valve, a force F with a magnitude of 70 lb is applied to the handle of the valve. Knowing that θ = 25°, Mx = -61 lb.ft, and Mz = -43 lb.ft, determine ϕ and d. 9- The frame ACD is hinged at A and D and is supported by a cable that passes through a ring at B and is attached to hooks at G and H. Knowing that the tension in the cable is 450 N, determine the moment about the diagonal AD of the force exerted on the frame by portion BH of the cable. 10- Determine the magnitude of the horizontal force F = -F i acting on the handle of the wrench so that this force produces a component of the moment along the OA axis (z axis) of the pipe assembly of Mz = {4k} N.m. Both the wrench and the pipe assembly, OABC, lie in the y-z plane. Suggestion: Use a scalar analysis. 11- The force of acts on the bracket as shown. Determine the moment of the force about the axis of the pipe aa if α = 60°, β = 60°, and γ = 45°. 12- A twist of is applied to the handle of the screwdriver. Resolve this couple moment into a pair of couple forces F exerted on the handle and P exerted on the blade. 13- If the resultant couple of the three couples acting on the triangular block is to be zero, determine the magnitude of forces F and P. 14- Express the moment of the couple acting on the pipe assembly in Cartesian vector form. What is the magnitude of the couple moment? 15- The gears are subjected to the couple moments shown. Determine the magnitude and coordinate direction angles of the resultant couple moment. 16- Shafts A and B connect the gear box to the wheel assemblies of a tractor, and shaft C connects it to the engine. Shafts A and B lie in the vertical yz plane, while shaft C is directed along the x axis. Replace the couples applied to the shafts by a single equivalent couple, specifying its magnitude and the direction of its axis. 17- Determine the magnitudes of couple moments M1, M2, and M3 so that the resultant couple moment is zero. 18- Three couples are formed by the three pairs of equal and opposite forces. Determine the resultant M of the three couples.