機(jī)械工程導(dǎo)論(雙語(yǔ))課件四

1、mechanical engineering,An introduction to,,,1,Lecturer: Liu Ju-rong,CHAPTER 4 Forces in Structures,2,Vocabulary,,3,Mechanics 力學(xué)net effect 凈效應(yīng)lever arm 桿臂Quadrant 象限Bracket

2、 支架Protractor 量角器Eyebolt 吊環(huán)螺栓pivot point 支點(diǎn)Particle 質(zhì)點(diǎn)Rigid Body 剛體Free Body Diagram 自由體受力圖boldface notation 粗體字的標(biāo)識(shí),1 OVERVIEW,4,Mechanical engineers use mathemat

3、ics and physical laws to design hardware better and faster than would be possible otherwise.,By applying the principle of force balance, for instance, an engineer can often analyze a design to a reasonable level of accur

4、acy before any hardware is built.,Engineers reduce the time and expense associated with constructing and testing prototypes by first refining their designs on paper.,Computer-aided engineering tools further increase the

5、level of sophistication that is available for such analyses.,1 OVERVIEW--- The ability to be available in this chapter,5,Describe a force in terms of its rectangular and polar components.,Calculate the resultant of a sy

6、stem of forces by using the vector algebra and polygon methods.,Calculate the moment of a force about a point using the perpendicular lever arm and moment component methods.,Understand the requirements for equilibrium, a

7、nd calculate unknown forces.,Explain the difference between laminar and turbulent flowing fluids.,Calculate and describe the dimensionless Reynolds and Mach numbers.,Discuss the fluid forces known as buoyancy, drag, and

8、lift, and calculate them in certain application.,1 OVERVIEW,6,FINGURE 4.1 Heavy construction equipment is designed to support the large forces developed during operation.Source: Reprinted with permission of mechanical

9、Dynamics, Incorporated, and by Caterpillar Incorporated.,2 FORCES AND RESULTANTS---Rectangular and Polar Forms,7,In this textbook, we will use boldface notation-- F--to denote force vectors.,A common method to describe

10、 a force is in terms of its horizontal and vertical components. The projection of F in the horizontal direction (the x Rectangular axis) is called Fx, and the vertical projection (y axis) is called Fy. In fact, the pa

11、ir of numbers (Fx, Fy) is just the coordinates of the force vector's tip.,2 FORCES AND RESULTANTS---Rectangular and Polar Forms,8,the unit vectors i and j are used to indicate the directions in which Fx and Fy act

12、. Vector i points along the positive x direction, and j is a vector pointing in the positive y direction.,F=Fx i+Fy j,FINGURE 4.2 Representing a force vector in terms of its rectangular components (Fx, Fy), and its pola

13、r components (F,θ).,2 FORCES AND RESULTANTS---Rectangular and Polar Forms,9,The latter viewpoint is based on polar coordinates. As also shown in Figure, F acts at the angle θ, which is measured relative to the horizont

14、al axis.,The magnitude or length of the force vector is a scalar quantity, and it is denoted by F = |F|, where the |·| notation designates the vector's absolute value.,Instead of specifying Fx and Fy, we can no

15、w view Vector direction the force vector F in terms of the quantities F and θ.,Fx = F cosθ and Fy = F sinθ,2 FORCES AND RESULTANTS---Rectangular and Polar Forms,10,FINGURE 4.3 Determining the angle of action for

16、 a force that (a) lies in the first quadrant and (b) lies in the second quadrant.,2 FORCES AND RESULTANTS---Resultants,11,A force system is a collection of several forces that simultaneously act on an object.,With N in

17、dividual forces denoted by Fi (i=1, 2,…, N), they are summed according to by using the rules of vector algebra.,2 FORCES AND RESULTANTS---Resultants,12,In this technique, each force Fi is broken down into its ho

18、rizontal and vertical components, which we label as Fxi and Fyi for the it’s force.The resultant's horizontal component Rx is found byLikewise, we separately sum the vertical components through The resultant

19、 force is then expressed as R = Rxi + Ryj. If we are interested in the magnitude R and direction θof R, we apply the expressions,2 FORCES AND RESULTANTS---Resultants,13,Alternatively, the resultant of a force system c

20、an be found by sketching a polygon to represent addition of the Fi vectors. The magnitude and direction of the resultant are then determined by applying rules of trigonometry to the polygon's geometry. Referring to

21、 the mounting post of Figure 4.4, the vector polygon for those three forces is drawn by adding the individual Fi's in a chain Head-to-tail(頭尾相接 ) rule according to the head-to-tail rule.,2 FORCES AND RESULTANTS--

22、-Resultants,14,FINGURE 4.4 A mounting post and bracket that are loaded by three forces.,FINGURE 4.5 The bracket R extends from the start to the end of the chain formed by adding F1, F2 and F3 together,3 MOMENT OF A F

23、ORCE ---perpendicular lever arm,15,The term torque can also be used to describe the effect a force acting over a lever arm, but mechanical engineers generally reserve torque to describe moments that cause rotation of a s

24、haft in a motor, engine, or gearbox.,The magnitude of a moment is found from its definition Mo = Fd,and d is the perpendicular lever arm distance from the force's line of action to point O.,3 MOMENT

25、 OF A FORCE ---perpendicular lever arm,16,the unit for Mo is the product of force and distance.,In fact, F could be applied to the bracket at any point along its line of action, and the moment produced about O would rema

26、in unchanged because d would likewise not change.,3 MOMENT OF A FORCE ---moment components,17,We first choose the following sign convention: A moment that is directed clockwise is positive, and a counterclockwise momen

27、t is negative.The sign convention is just a bookkeeping tool for combining the various clockwise and counterclockwise moment components.,3 MOMENT OF A FORCE ---moment components,18,FINGURE 4.11 Calculating moments b

28、ased on components. (a) Both Fx and Fy create clockwise moments about point O. (b) Fx exerts a clockwise moment, but Fy exerts a counterclockwise moment.,3 MOMENT OF A FORCE ---moment components,19,Regardless of which

29、method you use to calculate a moment, when reporting an answer you should state (1) the numerical magnitude of the moment, (2) the units, and (3) the direction. (CW or CCW ),In the general case of the mome

30、nt components method, we write Mo = ±Fx ?y ± Fy ?x,20,the physical dimensions of the object are unimportant in calculating forces.,the length, width, and breadth of an object are important for the proble

31、m at hand.,4 EQUILIBRIUM OF FORCES AND MOMENTS-- Particles and Rigid Bodies,21,A particle is in equilibrium if the forces acting on it balance with zero resultant. Because forces combine as vectors, the resultant mus

32、t be zero in two perpendicular directions, which we label x and y:,For a rigid body to be in equilibrium, it is necessary that (1) the resultant of all forces is zero, and (2) the net moment is also zero.,4 EQUILIB

33、RIUM OF FORCES AND MOMENTS-- Particles and Rigid Bodies,∑Fxi = 0 and ∑Fyi = 0,∑Fxi = 0 and ∑Fyi = 0,∑Moi = 0,It is not possible to obtain more independent equations of equilibrium by resolving moments about an alternati

34、ve point or by summing forces in different directions.,22,Free body diagrams(abbreviated FBD) are sketches used to analyze the forces and moments that act on structures and machines, and their construction is an importan

35、t skill.The FBD is used to identify the mechanical system that is being examined and to represent all of the known and unknown forces that are present.,4 EQUILIBRIUM OF FORCES AND MOMENTS– Free body diagrams,23,the

36、physical dimensions of the object are unimportant in calculating forces.,the length, width, and breadth of an object are important for the problem at hand.,4 EQUILIBRIUM OF FORCES AND MOMENTS– Free Body Diagrams,4 E

37、QUILIBRIUM OF FORCES AND MOMENTS– Free Body Diagrams,24,Three main steps are followed when a FBD is drawn:,3. In the final step, all forces and moments are drawn and labeled.,2. The coordinate system is drawn next to in

38、dicate the positive sign conventions for forces and moments.,1. Select an object that will be analyzed by using the equilibrium equations. Imagine that a dotted line is drawn around the object, and note how the line woul

39、d cut through and expose various forces.,4 EQUILIBRIUM OF FORCES AND MOMENTS– Free Body Diagrams,25,SOLUTION(a) The free body diagram of the buckle is shown in Figure 4.15(b). The xy coordinate system is also drawn t

40、o indicate our sign convention for the positive horizontal and vertical directions. Three forces act on the buckle: the two given 300-1b forces and the unknown force in the anchor strap. For the buckle to be in equilibri

41、um, these three forces must balance. Although both the magnitude T and direction 0 of the force in strap A B are unknown, both quantities are shown on the free body diagram for completeness.,EXAMPLE During crash testin

42、g of an automobile, the lap and shoulder seat belts each become tensioned to 300 lb, as shown in Figure 4.15(a). Treating the buckle B as a particle, (a) draw a free body diagram, (b) determine the tension T in the ancho

43、r strap AB, and (c) determine the angle at which T acts.,4 EQUILIBRIUM OF FORCES AND MOMENTS– Free Body Diagrams,26,FINGURE 4.15 Equilibrium analysis of the seat belt latch in Example 4.5.,4 EQUILIBRIUM OF FORCES A

44、ND MOMENTS– Free Body Diagrams,27,(b) We sum the three forces by using the vector polygon approach, as shown in Figure 4.15(c). The polygon's start and end points are the same because the three forces acting togethe

45、r have zero resultant; that is, the distance between the polygon's start and end points is zero. The tension is determined by applying the law of cosines (equations for oblique triangles are reviewed in Appendix B) t

46、o the side-angle-side triangle in Figure 4.15(c): T2 = (300 lb)2 + (300 lb)2 - 2(300 lb)(300 lb) cos 120ofrom which we calculate T = 519.6 lb.(c) The anchor strap's angle is found from the law of sines:

47、 and θ= 30o.,SUMMARY,28,SELF-STUDY AND REVIEW,29,,,,,,,What should you review?,What are the units for force and moment?,How do you calculate the resultant of a force system by using the vector algebra and vector polygo

48、n methods?,How do you calculate a moment by using the perpendicular lever arm and force components methods?,Why is a sign convention used when calculating moments using the component method?,What are the equilibrium requ