Budynas-Nisbett:Shigley's I.Basics 3.Load and Stress Analysis T©The McGraw-Hill Mechanical Engineering Companies,2008 Design,Eighth Edition Load and Stress Analysis 69 Free-Body Diagrams We can greatly simplify the analysis of a very complex structure or machine by successively isolating each element and studying and analyzing it by the use of free-body diagrams. When all the members have been treated in this manner,the knowledge can be assembled to yield information concerning the behavior of the total system.Thus,free-body diagram- ming is essentially a means of breaking a complicated problem into manageable segments, analyzing these simple problems,and then,usually,putting the information together again. Using free-body diagrams for force analysis serves the following important purposes: The diagram establishes the directions of reference axes,provides a place to record the dimensions of the subsystem and the magnitudes and directions of the known forces,and helps in assuming the directions of unknown forces. The diagram simplifies your thinking because it provides a place to store one thought while proceeding to the next. The diagram provides a means of communicating your thoughts clearly and unam- biguously to other people. Careful and complete construction of the diagram clarifies fuzzy thinking by bringing out various points that are not always apparent in the statement or in the geometry of the total problem.Thus,the diagram aids in understanding all facets of the problem. The diagram helps in the planning of a logical attack on the problem and in setting up the mathematical relations. The diagram helps in recording progress in the solution and in illustrating the methods used. The diagram allows others to follow your reasoning,showing all forces. EXAMPLE 3-1 Figure 3-la shows a simplified rendition of a gear reducer where the input and output shafts AB and CD are rotating at constant speeds and respectively.The input and output torques(torsional moments)are T=240 Ibf.in and To respectively.The shafts are supported in the housing by bearings at A,B.C,and D.The pitch radii of gears G and G2 are ri=0.75 in and r2=1.5 in,respectively.Draw the free-body diagrams of each member and determine the net reaction forces and moments at all points. Solution First,we will list all simplifying assumptions. 1 Gears G and G2 are simple spur gears with a standard pressure angle=20 (see Sec.13-5). 2 The bearings are self-aligning and the shafts can be considered to be simply supported. 3 The weight of each member is negligible 4 Friction is negligible. 5 The mounting bolts at E,F,H,and are the same size. The separate free-body diagrams of the members are shown in Figs.3-1b-d.Note that Newton's third law,called the law of action and reaction,is used extensively where each member mates.The force transmitted between the spur gears is not tangential but at the pressure angle中.Thus,N=Ftan中.Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 3. Load and Stress Analysis 74 © The McGraw−Hill Companies, 2008 Load and Stress Analysis 69 Free-Body Diagrams We can greatly simplify the analysis of a very complex structure or machine by successively isolating each element and studying and analyzing it by the use of free-body diagrams. When all the members have been treated in this manner, the knowledge can be assembled to yield information concerning the behavior of the total system. Thus, free-body diagram￾ming is essentially a means of breaking a complicated problem into manageable segments, analyzing these simple problems, and then, usually, putting the information together again. Using free-body diagrams for force analysis serves the following important purposes: • The diagram establishes the directions of reference axes, provides a place to record the dimensions of the subsystem and the magnitudes and directions of the known forces, and helps in assuming the directions of unknown forces. • The diagram simplifies your thinking because it provides a place to store one thought while proceeding to the next. • The diagram provides a means of communicating your thoughts clearly and unam￾biguously to other people. • Careful and complete construction of the diagram clarifies fuzzy thinking by bringing out various points that are not always apparent in the statement or in the geometry of the total problem. Thus, the diagram aids in understanding all facets of the problem. • The diagram helps in the planning of a logical attack on the problem and in setting up the mathematical relations. • The diagram helps in recording progress in the solution and in illustrating the methods used. • The diagram allows others to follow your reasoning, showing all forces. EXAMPLE 3–1 Figure 3–1a shows a simplified rendition of a gear reducer where the input and output shafts AB and C D are rotating at constant speeds ωi and ωo, respectively. The input and output torques (torsional moments) are Ti = 240 lbf · in and To, respectively. The shafts are supported in the housing by bearings at A, B, C, and D. The pitch radii of gears G1 and G2 are r1 = 0.75 in and r2 = 1.5 in, respectively. Draw the free-body diagrams of each member and determine the net reaction forces and moments at all points. Solution First, we will list all simplifying assumptions. 1 Gears G1 and G2 are simple spur gears with a standard pressure angle φ = 20° (see Sec. 13–5). 2 The bearings are self-aligning and the shafts can be considered to be simply supported. 3 The weight of each member is negligible. 4 Friction is negligible. 5 The mounting bolts at E, F, H, and I are the same size. The separate free-body diagrams of the members are shown in Figs. 3–1b–d. Note that Newton’s third law, called the law of action and reaction, is used extensively where each member mates. The force transmitted between the spur gears is not tangential but at the pressure angle φ. Thus, N = F tan φ