Engineering Mechanics: Statics

Transcription

1 Engineering Mechanics: Statics Chapter 2: Force Systems Part A: Two Dimensional Force Systems Force Force = an action of one body on another Vector quantity External and Internal forces Mechanics of Rigid bodies: Principle of Transmissibility Specify magnitude, direction, line of action No need to specify point of application Concurrent forces Lines of action intersect at a point 1

2 2D Force Systems Rectangular components are convenient for finding the sum or resultant R v of two (or more) forces which are concurrent v v v R = F + F = ( F ˆi + F ˆj ) + ( F ˆi + F ˆj ) 1 2 1x 1y 2x 2y = ( F + F ) ˆi + ( F + F ) ˆj 1x 2x 1y 2y Actual problems do not come with reference axes. Choose the most convenient one! Moment In addition to tendency to move a body in the direction of its application, a force tends to rotate a body about an axis. The axis is any line which neither intersects nor is parallel to the line of action This rotational tendency is known as the moment M of the force Proportional to force F and the perpendicular distance from the axis to the line of action of the force d The magnitude of M is M = Fd 2

3 Moment The moment is a vector M perpendicular to the plane of the body. Sense of M is determined by the right-hand rule Direction of the thumb = arrowhead Fingers curled in the direction of the rotational tendency In a given plane (2D),we may speak of moment about a point which means moment with respect to an axis normal to the plane and passing through the point. +, - signs are used for moment directions must be consistent throughout the problem! Moment A vector approach for moment calculations is proper for 3D problems. Moment of F about point A maybe represented by the cross-product M = r x F where r = a position vector from point A to any point on the line of action of F M = Fr sin α = Fd 3

4 Example 2/5 (p. 40) Calculate the magnitude of the moment about the base point O of the 600-N force by using both scalar and vector approaches. Problem 2/50 (a) Calculate the moment of the 90-N force about point O for the condition θ = 15º. (b) Determine the value of θ for which the moment about O is (b.1) zero (b.2) a maximum 4

5 Couple Moment produced by two equal, opposite, and noncollinear forces = couple M = F(a+d) Fa = Fd Moment of a couple has the same value for all moment center Vector approach M = r A x F + r B x (-F) = (r A - r B ) x F = r x F Couple M is a free vector Couple Equivalent couples Change of values F and d Force in different directions but parallel plane Product Fd remains the same 5

6 Force-Couple Systems Replacement of a force by a force and a couple Force F is replaced by a parallel force F and a counterclockwise couple Fd Example Replace the force by an equivalent system at point O Also, reverse the problem by the replacement of a force and a couple by a single force Problem 2/76 (modified) The device shown is a 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. Find an equivalent force-couple system at point O of the 4-N force 6

7 Resultants The simplest force combination which can replace the original forces without changing the external effect on the rigid body Resultant = a force-couple system v v v v v R= F + F + F + K=ΣF x =Σ x, y =Σ y, = ( Σ x) + ( Σ y) R F R F R F F θ = tan -1 R R y x Resultants Choose a reference point (point O) and move all forces to that point Add all forces at O to form the resultant force R and add all moment to form the resultant couple M O Find the line of action of R by requiring R to have a moment of M O M =Σ M= Σ( Fd) O v v R=ΣF Rd = M O 7

8 Problem 2/87 Replace the three forces acting on the bent pipe by a single equivalent force R. Specify the distance x from point O to the point on the x-axis through which the line of action of R passes. Problem 2/76 The device shown is a 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. 8

9 Force Systems Part B: Three Dimensional Force Systems Three-Dimensional Force System Rectangular components in 3D Express in terms of unit vectors î, ĵ, ˆk v F= F ˆi + F ˆj + F kˆ x y z F = Fcos θ, F = Fcos θ, F = Fcosθ x x y y z z F= F + F + F x y z cosθ x, cosθ y, cosθ z are the direction cosines cosθ x = l, cosθ y = m, cosθ z = n v F= F( li ˆ + mj ˆ + nkˆ ) 9

10 Three-Dimensional Force System Rectangular components in 3D If the coordinates of points A and B on the line of action are known, v v v AB ( x ˆ ˆ ˆ 2 x1) i + ( y2 y1) j + ( z2 z1) k F = FnF = F = F AB ( x x ) + ( y y ) + ( z z ) If two angles θ and φ which orient the line of action of the force are known, F = Fcos φ, F = F sinφ xy F = Fcosφ cos θ, F = Fcosφ sinθ x z y Problem 2/98 The cable exerts a tension of 2 kn on the fixed bracket at A. Write the vector expression for the tension T. 10

11 Three-Dimensional Force System Dot product v v P Q= PQcosα Orthogonal projection of Fcosα of F in the direction of Q Orthogonal projection of Qcosα of Q in the direction of F We can express F x = Fcosθ x of the force F as F x = F v v i v v If the projection of F in the n-direction is F n Example Find the projection of T along the line OA 11

12 Moment and Couple Moment of force F about the axis through point O is M O = r x F r runs from O to any point on the line of action of F Point O and force F establish a plane A The vector M o is normal to the plane in the direction established by the right-hand rule Evaluating the cross product ˆi ˆj kˆ M = r r r O x y z F F F x y z Moment and Couple Moment about an arbitrary axis v M = v ( r v λ F v v n) n known as triple scalar product (see appendix C/7) The triple scalar product may be represented by the determinant rx ry rz v M = M = F F F λ λ x y z l m n where l, m, n are the direction cosines of the unit vector n 12

13 Sample Problem 2/10 A tension T of magniture 10 kn is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment M z of T about the z-axis passing through the base O. Resultants A force system can be reduced to a resultant force and a resultant couple v v v v v R= F1 + F2 + F3 L= F v v v v v v M= M1 + M2 + M3 + L= ( r F) 13

14 Problem 2/154 The motor mounted on the bracket is acted on by its 160-N weight, and its shaft resists the 120-N thrust and 25-N.m couple applied to it. Determine the resultant of the force system shown in terms of a force R at A and a couple M. Wrench Resultants Any general force systems can be represented by a wrench 14

15 Problem 2/143 Replace the two forces and single couple by an equivalent force-couple system at point A Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts Resultants Special cases Concurrent forces no moments about point of concurrency Coplanar forces 2D Parallel forces (not in the same plane) magnitude of resultant = algebraic sum of the forces Wrench resultant resultant couple M is parallel to the resultant force R Example of positive wrench = screw driver 15

16 Problem 2/151 Replace the resultant of the force system acting on the pipe assembly by a single force R at A and a couple M Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts 16