Introduction to Mechanical Engineering Design | Ch.1, Lecture notes of Mechanical Systems Design

Download Introduction to Mechanical Engineering Design | Ch.1 and more Lecture notes Mechanical Systems Design in PDF only on Docsity! 3 Chapter Outline 1–1 Design 4 1–2 Mechanical Engineering Design 5 1–3 Phases and Interactions of the Design Process 5 1–4 Design Tools and Resources 8 1–5 The Design Engineer’s Professional Responsibilities 10 1–6 Standards and Codes 12 1–7 Economics 12 1–8 Safety and Product Liability 15 1–9 Stress and Strength 15 1–10 Uncertainty 16 1–11 Design Factor and Factor of Safety 17 1–12 Reliability 18 1–13 Dimensions and Tolerances 19 1–14 Units 21 1–15 Calculations and Significant Figures 22 1–16 Design Topic Interdependencies 23 1–17 Power Transmission Case Study Specifications 24 1Introduction to MechanicalEngineering Design bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 3 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 4 Mechanical Engineering Design Mechanical design is a complex process, requiring many skills. Extensive relationships need to be subdivided into a series of simple tasks. The complexity of the process requires a sequence in which ideas are introduced and iterated. We first address the nature of design in general, and then mechanical engineering design in particular. Design is an iterative process with many interactive phases. Many resources exist to support the designer, including many sources of information and an abundance of computational design tools. Design engineers need not only develop com- petence in their field but they must also cultivate a strong sense of responsibility and professional work ethic. There are roles to be played by codes and standards, ever-present economics, safety, and considerations of product liability. The survival of a mechanical component is often related through stress and strength. Matters of uncertainty are ever-present in engineer- ing design and are typically addressed by the design factor and factor of safety, either in the form of a deterministic (absolute) or statistical sense. The latter, statistical approach, deals with a design’s reliability and requires good statistical data. In mechanical design, other considerations include dimensions and tolerances, units, and calculations. The book consists of four parts. Part 1, Basics, begins by explaining some differ- ences between design and analysis and introducing some fundamental notions and approaches to design. It continues with three chapters reviewing material properties, stress analysis, and stiffness and deflection analysis, which are the principles necessary for the remainder of the book. Part 2, Failure Prevention, consists of two chapters on the prevention of failure of mechanical parts. Why machine parts fail and how they can be designed to prevent fail- ure are difficult questions, and so we take two chapters to answer them, one on pre- venting failure due to static loads, and the other on preventing fatigue failure due to time-varying, cyclic loads. In Part 3, Design of Mechanical Elements, the concepts of Parts 1 and 2 are applied to the analysis, selection, and design of specific mechanical elements such as shafts, fasteners, weldments, springs, rolling contact bearings, film bearings, gears, belts, chains, and wire ropes. Part 4, Analysis Tools, provides introductions to two important methods used in mechanical design, finite element analysis and statistical analysis. This is optional study material, but some sections and examples in Parts 1 to 3 demonstrate the use of these tools. There are two appendixes at the end of the book. Appendix A contains many use- ful tables referenced throughout the book. Appendix B contains answers to selected end-of-chapter problems. 1–1 Design To design is either to formulate a plan for the satisfaction of a specified need or to solve a specific problem. If the plan results in the creation of something having a physical reality, then the product must be functional, safe, reliable, competitive, usable, manu- facturable, and marketable. Design is an innovative and highly iterative process. It is also a decision-making process. Decisions sometimes have to be made with too little information, occasion- ally with just the right amount of information, or with an excess of partially contradictory information. Decisions are sometimes made tentatively, with the right reserved to adjust as more becomes known. The point is that the engineering designer has to be personally comfortable with a decision-making, problem-solving role. bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 4 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: Introduction to Mechanical Engineering Design 7 quantified in terms of established metrics.1 As the fleshing out of the scheme progresses, analyses must be performed to assess whether the system performance is satisfactory or better, and, if satisfactory, just how well it will perform. System schemes that do not survive analysis are revised, improved, or discarded. Those with potential are optimized to determine the best performance of which the scheme is capable. Competing schemes are compared so that the path leading to the most competitive product can be chosen. Figure 1–1 shows that synthesis and analysis and optimization are intimately and iteratively related. We have noted, and we emphasize, that design is an iterative process in which we proceed through several steps, evaluate the results, and then return to an earlier phase of the procedure. Thus, we may synthesize several components of a system, analyze and optimize them, and return to synthesis to see what effect this has on the remaining parts of the system. For example, the design of a system to transmit power requires attention to the design and selection of individual components (e.g., gears, bearings, shaft). However, as is often the case in design, these components are not independent. In order to design the shaft for stress and deflection, it is necessary to know the applied forces. If the forces are transmitted through gears, it is necessary to know the gear specifica- tions in order to determine the forces that will be transmitted to the shaft. But stock gears come with certain bore sizes, requiring knowledge of the necessary shaft diame- ter. Clearly, rough estimates will need to be made in order to proceed through the process, refining and iterating until a final design is obtained that is satisfactory for each individual component as well as for the overall design specifications. Throughout the text we will elaborate on this process for the case study of a power transmission design. Both analysis and optimization require that we construct or devise abstract models of the system that will admit some form of mathematical analysis. We call these mod- els mathematical models. In creating them it is our hope that we can find one that will simulate the real physical system very well. As indicated in Fig. 1–1, evaluation is a significant phase of the total design process. Evaluation is the final proof of a success- ful design and usually involves the testing of a prototype in the laboratory. Here we wish to discover if the design really satisfies the needs. Is it reliable? Will it compete successfully with similar products? Is it economical to manufacture and to use? Is it easily maintained and adjusted? Can a profit be made from its sale or use? How likely is it to result in product-liability lawsuits? And is insurance easily and cheaply obtained? Is it likely that recalls will be needed to replace defective parts or systems? The project designer or design team will need to address a myriad of engineering and non-engineering questions. Communicating the design to others is the final, vital presentation step in the design process. Undoubtedly, many great designs, inventions, and creative works have been lost to posterity simply because the originators were unable or unwilling to properly explain their accomplishments to others. Presentation is a selling job. The engineer, when presenting a new solution to administrative, management, or supervisory persons, is attempting to sell or to prove to them that their solution is a better one. Unless this can be done successfully, the time and effort spent on obtaining the solution have been largely wasted. When designers sell a new idea, they also sell themselves. If they are repeatedly successful in selling ideas, designs, and new solutions to management, they begin to receive salary increases and promotions; in fact, this is how anyone succeeds in his or her profession. 1An excellent reference for this topic is presented by Stuart Pugh, Total Design—Integrated Methods for Successful Product Engineering, Addison-Wesley, 1991. A description of the Pugh method is also provided in Chap. 8, David G. Ullman, The Mechanical Design Process, 3rd ed., McGraw-Hill, 2003. bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 7 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 8 Mechanical Engineering Design Design Considerations Sometimes the strength required of an element in a system is an important factor in the determination of the geometry and the dimensions of the element. In such a situation we say that strength is an important design consideration. When we use the expression design consideration, we are referring to some characteristic that influences the design of the element or, perhaps, the entire system. Usually quite a number of such charac- teristics must be considered and prioritized in a given design situation. Many of the important ones are as follows (not necessarily in order of importance): 1 Functionality 14 Noise 2 Strength/stress 15 Styling 3 Distortion/deflection/stiffness 16 Shape 4 Wear 17 Size 5 Corrosion 18 Control 6 Safety 19 Thermal properties 7 Reliability 20 Surface 8 Manufacturability 21 Lubrication 9 Utility 22 Marketability 10 Cost 23 Maintenance 11 Friction 24 Volume 12 Weight 25 Liability 13 Life 26 Remanufacturing/resource recovery Some of these characteristics have to do directly with the dimensions, the material, the processing, and the joining of the elements of the system. Several characteristics may be interrelated, which affects the configuration of the total system. 1–4 Design Tools and Resources Today, the engineer has a great variety of tools and resources available to assist in the solution of design problems. Inexpensive microcomputers and robust computer soft- ware packages provide tools of immense capability for the design, analysis, and simu- lation of mechanical components. In addition to these tools, the engineer always needs technical information, either in the form of basic science/engineering behavior or the characteristics of specific off-the-shelf components. Here, the resources can range from science/engineering textbooks to manufacturers’ brochures or catalogs. Here too, the computer can play a major role in gathering information.2 Computational Tools Computer-aided design (CAD) software allows the development of three-dimensional (3-D) designs from which conventional two-dimensional orthographic views with auto- matic dimensioning can be produced. Manufacturing tool paths can be generated from the 3-D models, and in some cases, parts can be created directly from a 3-D database by using a rapid prototyping and manufacturing method (stereolithography)—paperless manufac- turing! Another advantage of a 3-D database is that it allows rapid and accurate calcula- tions of mass properties such as mass, location of the center of gravity, and mass moments of inertia. Other geometric properties such as areas and distances between points are likewise easily obtained. There are a great many CAD software packages available such 2An excellent and comprehensive discussion of the process of “gathering information” can be found in Chap. 4, George E. Dieter, Engineering Design, A Materials and Processing Approach, 3rd ed., McGraw-Hill, New York, 2000. bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 8 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: Introduction to Mechanical Engineering Design 9 as Aries, AutoCAD, CadKey, I-Deas, Unigraphics, Solid Works, and ProEngineer, to name a few. The term computer-aided engineering (CAE) generally applies to all computer- related engineering applications. With this definition, CAD can be considered as a sub- set of CAE. Some computer software packages perform specific engineering analysis and/or simulation tasks that assist the designer, but they are not considered a tool for the creation of the design that CAD is. Such software fits into two categories: engineering- based and non-engineering-specific. Some examples of engineering-based software for mechanical engineering applications—software that might also be integrated within a CAD system—include finite-element analysis (FEA) programs for analysis of stress and deflection (see Chap. 19), vibration, and heat transfer (e.g., Algor, ANSYS, and MSC/NASTRAN); computational fluid dynamics (CFD) programs for fluid-flow analy- sis and simulation (e.g., CFD++, FIDAP, and Fluent); and programs for simulation of dynamic force and motion in mechanisms (e.g., ADAMS, DADS, and Working Model). Examples of non-engineering-specific computer-aided applications include software for word processing, spreadsheet software (e.g., Excel, Lotus, and Quattro-Pro), and mathematical solvers (e.g., Maple, MathCad, MATLAB,3 Mathematica, and TKsolver). Your instructor is the best source of information about programs that may be available to you and can recommend those that are useful for specific tasks. One caution, however: Computer software is no substitute for the human thought process. You are the driver here; the computer is the vehicle to assist you on your journey to a solution. Numbers generated by a computer can be far from the truth if you entered incorrect input, if you misinterpreted the application or the output of the program, if the program contained bugs, etc. It is your responsibility to assure the validity of the results, so be careful to check the application and results carefully, perform benchmark testing by submitting problems with known solu- tions, and monitor the software company and user-group newsletters. Acquiring Technical Information We currently live in what is referred to as the information age, where information is gen- erated at an astounding pace. It is difficult, but extremely important, to keep abreast of past and current developments in one’s field of study and occupation. The reference in Footnote 2 provides an excellent description of the informational resources available and is highly recommended reading for the serious design engineer. Some sources of information are: • Libraries (community, university, and private). Engineering dictionaries and encyclo- pedias, textbooks, monographs, handbooks, indexing and abstract services, journals, translations, technical reports, patents, and business sources/brochures/catalogs. • Government sources. Departments of Defense, Commerce, Energy, and Transportation; NASA; Government Printing Office; U.S. Patent and Trademark Office; National Technical Information Service; and National Institute for Standards and Technology. • Professional societies. American Society of Mechanical Engineers, Society of Manufacturing Engineers, Society of Automotive Engineers, American Society for Testing and Materials, and American Welding Society. • Commercial vendors. Catalogs, technical literature, test data, samples, and cost information. • Internet. The computer network gateway to websites associated with most of the categories listed above.4 3MATLAB is a registered trademark of The MathWorks, Inc. 4Some helpful Web resources, to name a few, include www.globalspec.com, www.engnetglobal.com, www.efunda.com, www.thomasnet.com, and www.uspto.gov. bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 9 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 12 Mechanical Engineering Design 1–6 Standards and Codes A standard is a set of specifications for parts, materials, or processes intended to achieve uniformity, efficiency, and a specified quality. One of the important purposes of a standard is to limit the multitude of variations that can arise from the arbitrary cre- ation of a part, material, or process. A code is a set of specifications for the analysis, design, manufacture, and con- struction of something. The purpose of a code is to achieve a specified degree of safety, efficiency, and performance or quality. It is important to observe that safety codes do not imply absolute safety. In fact, absolute safety is impossible to obtain. Sometimes the unexpected event really does happen. Designing a building to withstand a 120 mi/h wind does not mean that the designers think a 140 mi/h wind is impossible; it simply means that they think it is highly improbable. All of the organizations and societies listed below have established specifications for standards and safety or design codes. The name of the organization provides a clue to the nature of the standard or code. Some of the standards and codes, as well as addresses, can be obtained in most technical libraries or on the Internet. The organiza- tions of interest to mechanical engineers are: Aluminum Association (AA) American Bearing Manufacturers Association (ABMA) American Gear Manufacturers Association (AGMA) American Institute of Steel Construction (AISC) American Iron and Steel Institute (AISI) American National Standards Institute (ANSI) American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) American Society of Mechanical Engineers (ASME) American Society of Testing and Materials (ASTM) American Welding Society (AWS) ASM International British Standards Institution (BSI) Industrial Fasteners Institute (IFI) Institute of Transportation Engineers (ITE) Institution of Mechanical Engineers (IMechE) International Bureau of Weights and Measures (BIPM) International Federation of Robotics (IFR) International Standards Organization (ISO) National Association of Power Engineers (NAPE) National Institute for Standards and Technology (NIST) Society of Automotive Engineers (SAE) 1–7 Economics The consideration of cost plays such an important role in the design decision process that we could easily spend as much time in studying the cost factor as in the study of the entire subject of design. Here we introduce only a few general concepts and sim- ple rules. First, observe that nothing can be said in an absolute sense concerning costs. Materials and labor usually show an increasing cost from year to year. But the costs bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 12 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: of processing the materials can be expected to exhibit a decreasing trend because of the use of automated machine tools and robots. The cost of manufacturing a single product will vary from city to city and from one plant to another because of over- head, labor, taxes, and freight differentials and the inevitable slight manufacturing variations. Standard Sizes The use of standard or stock sizes is a first principle of cost reduction. An engineer who specifies an AISI 1020 bar of hot-rolled steel 53 mm square has added cost to the prod- uct, provided that a bar 50 or 60 mm square, both of which are preferred sizes, would do equally well. The 53-mm size can be obtained by special order or by rolling or machining a 60-mm square, but these approaches add cost to the product. To ensure that standard or preferred sizes are specified, designers must have access to stock lists of the materials they employ. A further word of caution regarding the selection of preferred sizes is necessary. Although a great many sizes are usually listed in catalogs, they are not all readily avail- able. Some sizes are used so infrequently that they are not stocked. A rush order for such sizes may add to the expense and delay. Thus you should also have access to a list such as those in Table A–17 for preferred inch and millimeter sizes. There are many purchased parts, such as motors, pumps, bearings, and fasteners, that are specified by designers. In the case of these, too, you should make a special effort to specify parts that are readily available. Parts that are made and sold in large quantities usually cost somewhat less than the odd sizes. The cost of rolling bearings, for example, depends more on the quantity of production by the bearing manufacturer than on the size of the bearing. Large Tolerances Among the effects of design specifications on costs, tolerances are perhaps most sig- nificant. Tolerances, manufacturing processes, and surface finish are interrelated and influence the producibility of the end product in many ways. Close tolerances may necessitate additional steps in processing and inspection or even render a part com- pletely impractical to produce economically. Tolerances cover dimensional variation and surface-roughness range and also the variation in mechanical properties resulting from heat treatment and other processing operations. Since parts having large tolerances can often be produced by machines with higher production rates, costs will be significantly smaller. Also, fewer such parts will be rejected in the inspection process, and they are usually easier to assemble. A plot of cost versus tolerance/machining process is shown in Fig. 1–2, and illustrates the drastic increase in manufacturing cost as tolerance diminishes with finer machining processing. Breakeven Points Sometimes it happens that, when two or more design approaches are compared for cost, the choice between the two depends on a set of conditions such as the quantity of pro- duction, the speed of the assembly lines, or some other condition. There then occurs a point corresponding to equal cost, which is called the breakeven point. Introduction to Mechanical Engineering Design 13 bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 13 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 14 Mechanical Engineering Design As an example, consider a situation in which a certain part can be manufactured at the rate of 25 parts per hour on an automatic screw machine or 10 parts per hour on a hand screw machine. Let us suppose, too, that the setup time for the automatic is 3 h and that the labor cost for either machine is $20 per hour, including overhead. Figure 1–3 is a graph of cost versus production by the two methods. The breakeven point for this example corresponds to 50 parts. If the desired production is greater than 50 parts, the automatic machine should be used. Figure 1–2 Cost versus tolerance/ machining process. (From David G. Ullman, The Mechanical Design Process, 3rd ed., McGraw-Hill, New York, 2003.) Figure 1–3 A breakeven point. 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 Rough turn Semi- finish turn Finish turn Grind Hone Machining operations Material: steel C os ts , % Nominal tolerances (inches) Nominal tolerance (mm) 0.030 0.015 0.010 0.005 0.003 0.001 0.0005 0.00025 0.75 0.50 0.50 0.125 0.063 0.025 0.012 0.006 0 0 20 40 60 80 100 20 40 60 80 100 120 140 Breakeven point Automatic screw machine Hand screw machine Production C os t, $ bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 14 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: Introduction to Mechanical Engineering Design 17 maximum allowable parameter. Here the parameter can be load, stress, deflection, etc. Thus, the design factor nd is defined as nd = loss-of-function parametermaximum allowable parameter (1–1) If the parameter is load, then the maximum allowable load can be found from Maximum allowable load = loss-of-function load nd (1–2) EXAMPLE 1–1 Consider that the maximum load on a structure is known with an uncertainty of ±20 per- cent, and the load causing failure is known within ±15 percent. If the load causing fail- ure is nominally 2000 lbf, determine the design factor and the maximum allowable load that will offset the absolute uncertainties. Solution To account for its uncertainty, the loss-of-function load must increase to 1/0.85, whereas the maximum allowable load must decrease to 1/1.2. Thus to offset the absolute uncer- tainties the design factor, from Eq. (1–1), should be Answer nd = 1/0.851/1.2 = 1.4 From Eq. (1–2), the maximum allowable load is found to be Answer Maximum allowable load = 2000 1.4 = 1400 lbf Stochastic methods (see Chap. 20) are based on the statistical nature of the design parameters and focus on the probability of survival of the design’s function (that is, on reliability). Sections 5–13 and 6–17 demonstrate how this is accomplished. 1–11 Design Factor and Factor of Safety A general approach to the allowable load versus loss-of-function load problem is the deterministic design factor method, and sometimes called the classical method of design. The fundamental equation is Eq. (1–1) where nd is called the design factor. All loss-of-function modes must be analyzed, and the mode leading to the smallest design factor governs. After the design is completed, the actual design factor may change as a result of changes such as rounding up to a standard size for a cross section or using off-the-shelf components with higher ratings instead of employing what is calculated by using the design factor. The factor is then referred to as the factor of safety, n. The factor of safety has the same definition as the design factor, but it generally differs numerically. Since stress may not vary linearly with load (see Sec. 3–19), using load as the loss-of- function parameter may not be acceptable. It is more common then to express the design factor in terms of a stress and a relevant strength. Thus Eq. (1–1) can be rewritten as nd = loss-of-function strengthallowable stress = S σ (or τ ) (1–3) bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 17 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 18 Mechanical Engineering Design The stress and strength terms in Eq. (1–3) must be of the same type and units. Also, the stress and strength must apply to the same critical location in the part. EXAMPLE 1–2 A rod with a cross-sectional area of A and loaded in tension with an axial force of P  2000 lbf undergoes a stress of σ = P/A. Using a material strength of 24 kpsi and a design factor of 3.0, determine the minimum diameter of a solid circular rod. Using Table A–17, select a preferred fractional diameter and determine the rod’s factor of safety. Solution Since A = πd2/4, σ = P/A, and from Eq. (1–3), σ = S/nd , then σ = P A = P πd2/4 = S nd Solving for d yields Answer d = ( 4Pnd πS )1/2 = ( 4(2000)3 π(24 000) )1/2 = 0.564 in From Table A–17, the next higher preferred size is 58 in  0.625 in. Thus, when nd is replaced with n in the equation developed above, the factor of safety n is Answer n = πSd 2 4P = π(24 000)0.625 2 4(2000) = 3.68 Thus rounding the diameter has increased the actual design factor. 1–12 Reliability In these days of greatly increasing numbers of liability lawsuits and the need to conform to regulations issued by governmental agencies such as EPA and OSHA, it is very important for the designer and the manufacturer to know the reliability of their product. The reliabil- ity method of design is one in which we obtain the distribution of stresses and the distribu- tion of strengths and then relate these two in order to achieve an acceptable success rate. The statistical measure of the probability that a mechanical element will not fail in use is called the reliability of that element. The reliability R can be expressed by R = 1  pf (1–4) where pf is the probability of failure, given by the number of instances of failures per total number of possible instances. The value of R falls in the range 0  R  1. A reli- ability of R = 0.90 means that there is a 90 percent chance that the part will perform its proper function without failure. The failure of 6 parts out of every 1000 manufactured might be considered an acceptable failure rate for a certain class of products. This rep- resents a reliability of R = 1 − 6 1000 = 0.994 or 99.4 percent. In the reliability method of design, the designer’s task is to make a judicious selec- tion of materials, processes, and geometry (size) so as to achieve a specific reliability bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 18 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: goal. Thus, if the objective reliability is to be 99.4 percent, as above, what combination of materials, processing, and dimensions is needed to meet this goal? If a mechanical system fails when any one component fails, the system is said to be a series system. If the reliability of component i is Ri in a series system of n components, then the relia- bility of the system is given by R = n∑ i=1 Ri (1–5) For example, consider a shaft with two bearings having reliabilities of 95 percent and 98 percent. From Eq. (1–5), the overall reliability of the shaft system is then R = R1 R2 = 0.95 (0.98) = 0.93 or 93 percent. Analyses that lead to an assessment of reliability address uncertainties, or their estimates, in parameters that describe the situation. Stochastic variables such as stress, strength, load, or size are described in terms of their means, standard deviations, and distributions. If bearing balls are produced by a manufacturing process in which a diameter distribution is created, we can say upon choosing a ball that there is uncertainty as to size. If we wish to consider weight or moment of inertia in rolling, this size uncer- tainty can be considered to be propagated to our knowledge of weight or inertia. There are ways of estimating the statistical parameters describing weight and inertia from those describing size and density. These methods are variously called propagation of error, propagation of uncertainty, or propagation of dispersion. These methods are integral parts of analysis or synthesis tasks when probability of failure is involved. It is important to note that good statistical data and estimates are essential to per- form an acceptable reliability analysis. This requires a good deal of testing and valida- tion of the data. In many cases, this is not practical and a deterministic approach to the design must be undertaken. 1–13 Dimensions and Tolerances The following terms are used generally in dimensioning: • Nominal size. The size we use in speaking of an element. For example, we may spec- ify a 1 12 -in pipe or a 1 2 -in bolt. Either the theoretical size or the actual measured size may be quite different. The theoretical size of a 1 12 -in pipe is 1.900 in for the outside diameter. And the diameter of the 12 -in bolt, say, may actually measure 0.492 in. • Limits. The stated maximum and minimum dimensions. • Tolerance. The difference between the two limits. • Bilateral tolerance. The variation in both directions from the basic dimension. That is, the basic size is between the two limits, for example, 1.005 ± 0.002 in. The two parts of the tolerance need not be equal. • Unilateral tolerance. The basic dimension is taken as one of the limits, and variation is permitted in only one direction, for example, 1.005 +0.004−0.000 in • Clearance. A general term that refers to the mating of cylindrical parts such as a bolt and a hole. The word clearance is used only when the internal member is smaller than the external member. The diametral clearance is the measured difference in the two diameters. The radial clearance is the difference in the two radii. Introduction to Mechanical Engineering Design 19 bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 19 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 22 Mechanical Engineering Design The International System of Units (SI) is an absolute system. The base units are the meter, the kilogram (for mass), and the second. The unit of force is derived by using Newton’s second law and is called the newton. The units constituting the newton (N) are F = ML T 2 = (kilogram)(meter) (second)2 = kg · m/s2 = N (1–9) The weight of an object is the force exerted upon it by gravity. Designating the weight as W and the acceleration due to gravity as g, we have W = mg (1–10) In the fps system, standard gravity is g  32.1740 ft/s2. For most cases this is rounded off to 32.2. Thus the weight of a mass of 1 slug in the fps system is W = mg = (1 slug)(32.2 ft /s2) = 32.2 lbf In the ips system, standard gravity is 386.088 or about 386 in/s2. Thus, in this system, a unit mass weighs W = (1 lbf · s2/in)(386 in/s2) = 386 lbf With SI units, standard gravity is 9.806 or about 9.81 m/s. Thus, the weight of a 1-kg mass is W = (1 kg)(9.81 m/s2) = 9.81 N A series of names and symbols to form multiples and submultiples of SI units has been established to provide an alternative to the writing of powers of 10. Table A–1 includes these prefixes and symbols. Numbers having four or more digits are placed in groups of three and separated by a space instead of a comma. However, the space may be omitted for the special case of numbers having four digits. A period is used as a decimal point. These recommenda- tions avoid the confusion caused by certain European countries in which a comma is used as a decimal point, and by the English use of a centered period. Examples of correct and incorrect usage are as follows: 1924 or 1 924 but not 1,924 0.1924 or 0.192 4 but not 0.192,4 192 423.618 50 but not 192,423.61850 The decimal point should always be preceded by a zero for numbers less than unity. 1–15 Calculations and Significant Figures The discussion in this section applies to real numbers, not integers. The accuracy of a real number depends on the number of significant figures describing the number. Usually, but not always, three or four significant figures are necessary for engineering accuracy. Unless otherwise stated, no less than three significant figures should be used in your calculations. The number of significant figures is usually inferred by the number of figures given (except for leading zeros). For example, 706, 3.14, and 0.002 19 are assumed to be bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 22 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: numbers with three significant figures. For trailing zeros, a little more clarification is nec- essary. To display 706 to four significant figures insert a trailing zero and display either 706.0, 7.060 × 102, or 0.7060 × 103. Also, consider a number such as 91 600. Scientific notation should be used to clarify the accuracy. For three significant figures express the number as 91.6 × 103. For four significant figures express it as 91.60 × 103. Computers and calculators display calculations to many significant figures. However, you should never report a number of significant figures of a calculation any greater than the smallest number of significant figures of the numbers used for the calculation. Of course, you should use the greatest accuracy possible when performing a calculation. For example, determine the circumference of a solid shaft with a diameter of d = 0.40 in. The circumference is given by C = πd . Since d is given with two significant figures, C should be reported with only two significant figures. Now if we used only two significant figures for π our calculator would give C = 3.1 (0.40) = 1.24 in. This rounds off to two signif- icant figures as C = 1.2 in. However, using π = 3.141 592 654 as programmed in the calculator, C = 3.141 592 654 (0.40) = 1.256 637 061 in. This rounds off to C = 1.3 in, which is 8.3 percent higher than the first calculation. Note, however, since d is given with two significant figures, it is implied that the range of d is 0.40 ± 0.005. This means that the calculation of C is only accurate to within ±0.005/0.40 = ±0.0125 = ±1.25%. The calculation could also be one in a series of calculations, and rounding each calcula- tion separately may lead to an accumulation of greater inaccuracy. Thus, it is considered good engineering practice to make all calculations to the greatest accuracy possible and report the results within the accuracy of the given input. 1–16 Design Topic Interdependencies One of the characteristics of machine design problems is the interdependencies of the various elements of a given mechanical system. For example, a change from a spur gear to a helical gear on a drive shaft would add axial components of force, which would have implications on the layout and size of the shaft, and the type and size of the bear- ings. Further, even within a single component, it is necessary to consider many differ- ent facets of mechanics and failure modes, such as excessive deflection, static yielding, fatigue failure, contact stress, and material characteristics. However, in order to provide significant attention to the details of each topic, most machine design textbooks focus on these topics separately and give end-of-chapter problems that relate only to that specific topic. To help the reader see the interdependence between the various design topics, this textbook presents many ongoing and interdependent problems in the end-of-chapter problem sections. Each row of Table 1–1 shows the problem numbers that apply to the same mechanical system that is being analyzed according to the topics being presented in that particular chapter. For example, in the second row, Probs. 3–40, 5-65, and 5–66 correspond to a pin in a knuckle joint that is to be analyzed for stresses in Chap. 3 and then for static failure in Chap. 5. This is a simple example of interdependencies, but as can be seen in the table, other systems are analyzed with as many as 10 separate prob- lems. It may be beneficial to work through some of these continuing sequences as the topics are covered to increase your awareness of the various interdependencies. In addition to the problems given in Table 1–1, Sec. 1–17 describes a power trans- mission case study where various interdependent analyses are performed throughout the book, when appropriate in the presentation of the topics. The final results of the case study are then presented in Chap. 18. Introduction to Mechanical Engineering Design 23 bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 23 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 24 Mechanical Engineering Design 1–17 Power Transmission Case Study Specifications A case study incorporating the many facets of the design process for a power transmis- sion speed reducer will be considered throughout this textbook. The problem will be introduced here with the definition and specification for the product to be designed. Further details and component analysis will be presented in subsequent chapters. Chapter 18 provides an overview of the entire process, focusing on the design sequence, the interaction between the component designs, and other details pertinent to transmis- sion of power. It also contains a complete case study of the power transmission speed reducer introduced here. Many industrial applications require machinery to be powered by engines or elec- tric motors. The power source usually runs most efficiently at a narrow range of rota- tional speed. When the application requires power to be delivered at a slower speed than supplied by the motor, a speed reducer is introduced. The speed reducer should trans- mit the power from the motor to the application with as little energy loss as practical, while reducing the speed and consequently increasing the torque. For example, assume that a company wishes to provide off-the-shelf speed reducers in various capacities and speed ratios to sell to a wide variety of target applications. The marketing team has Table 1–1 Problem Numbers for Linked End-of-Chapter Problems* 3–1 4–50 4–74 3–40 5–65 5–66 3–68 4–23 4–29 4–35 5–39 6–37 7–7 11–14 3–69 4–24 4–30 4–36 5–40 6–38 7–8 11–15 3–70 4–25 4–31 4–37 5–41 6–39 7–9 11–16 3–71 4–26 4–32 4–38 5–42 6–40 7–10 11–17 3–72 4–27 4–33 4–39 5–43 6–41 7–11 7–19 7–20 7–34 11–27 11–28 13–38 14–36 3–73 4–28 4–34 4–40 5–44 6–42 7–12 7–21 7–22 7–35 11–29 11–30 13–39 14–37 3–74 5–45 6–43 7–13 11–41 13–42 3–76 5–46 6–44 7–14 11–42 13–42 3–77 5–47 6–45 7–15 11–18 13–40 14–38 3–79 5–48 6–46 7–16 11–19 13–41 14–39 3–80 4–41 4–71 5–49 6–47 3–81 5–50 6–48 3–82 5–51 6–49 3–83 5–52 6–50 3–84 4–43 4–73 5–53 5–56 6–51 3–85 5–54 6–52 3–86 5–55 6–53 3–87 5–56 *Each row corresponds to the same mechanical component repeated for a different design concept. bud29281_ch01_002-030.qxd 11/12/2009 6:40 pm Page 24 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 1–10 When one knows the true values x1 and x2 and has approximations X1 and X2 at hand, one can see where errors may arise. By viewing error as something to be added to an approximation to attain a true value, it follows that the error ei , is related to Xi , and xi as xi = Xi + ei (a) Show that the error in a sum X1 + X2 is (x1 + x2) − (X1 + X2) = e1 + e2 (b) Show that the error in a difference X1 − X2 is (x1 − x2) − (X1 − X2) = e1 − e2 (c) Show that the error in a product X1 X2 is x1x2 − X1 X2 = X1 X2 ( e1 X1 + e2 X2 ) (d ) Show that in a quotient X1/X2 the error is x1 x2 − X1 X2 = X1 X2 ( e1 X1 − e2 X2 ) 1–11 Use the true values x1 = √ 7 and x2 = √ 8 (a) Demonstrate the correctness of the error equation from Prob. 1–10 for addition if three cor- rect digits are used for X1 and X2. (b) Demonstrate the correctness of the error equation for addition using three-digit significant numbers for X1 and X2. 1–12 A solid circular rod of diameter d undergoes a bending moment M = 1000 lbf  in inducing a stress σ = 16M(πd3). Using a material strength of 25 kpsi and a design factor of 2.5, deter- mine the minimum diameter of the rod. Using Table A–17 select a preferred fractional diameter and determine the resulting factor of safety. 1–13 A mechanical system comprises three subsystems in series with reliabilities of 98, 96, and 94 percent. What is the overall reliability of the system? 1–14 Three blocks A, B, and C and a grooved block D have dimensions a, b, c, and d as follows: a = 1.500 ± 0.001 in b = 2.000 ± 0.003 in c = 3.000 ± 0.004 in d = 6.520 ± 0.010 in (a) Determine the mean gap w̄ and its tolerance. (b) Determine the mean size of d that will assure that w ≥ 0.010 in. 1–15 The volume of a rectangular parallelepiped is given by V = xyz. If x = a ±a, y = b ±b, z = c ±c, show that V V̄ =a ā +b b̄ +c c̄ d a b B C D A cw Introduction to Mechanical Engineering Design 27 Problem 1–14 bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 27 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: Use this result to determine the bilateral tolerance on the volume of a rectangular parallelepiped with dimensions a = 1.500 ± 0.002 in b = 1.875 ± 0.003 in c = 3.000 ± 0.004 in 1–16 A pivot in a linkage has a pin in the figure whose dimension a ± ta is to be established. The thickness of the link clevis is 1.500 ± 0.005 in. The designer has concluded that a gap of between 0.004 and 0.05 in will satisfactorily sustain the function of the linkage pivot. Determine the dimension a and its tolerance. 1–17 A circular cross section O ring has the dimensions shown in the figure. In particular, an AS 568A standard No. 240 O ring has an inside diameter Di and a cross-section diameter d of Di = 3.734 ± 0.028 in d = 0.139 ± 0.004 in Estimate the mean outside diameter D̄o and its bilateral tolerance. For the table given, repeat Prob. 1–17 for the following O rings, given the AS 568A standard number. Solve Problems 1–18 and 1–19 using SI units. Solve Problems 1–20 and 1–21 using ips units. Note: The solutions require research. Problem number 1–18 1–19 1–20 1–21 AS 568A No. 110 220 160 320 1–22 Convert the following to appropriate ips units: (a) A stress, σ = 150 MPa. (b) A force, F = 2 kN. (c) A moment, M = 150 N  m. (d ) An area, A = 1 500 mm2. (e) A second moment of area, I = 750 cm4. ( f ) A modulus of elasticity, E = 145 GPa. (g) A speed, v = 75 km/h. (h) A volume, V = 1 liter. 1–18 to 1–21 Di d D0 Pin a ± ta 0.042 ± 0.002 1.500 ± 0.005 Snap ring Clevis 28 Mechanical Engineering Design Problem 1–16 Dimensions in inches. Problem 1–17 bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 28 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: 1–23 Convert the following to appropriate SI units: (a) A length, l = 5 ft. (b) A stress, σ = 90 kpsi. (c) A pressure, p = 25 psi. (d ) A section modulus, Z = 12 in3. (e) A unit weight, w = 0.208 lbf/in. ( f ) A deflection,  = 0.001 89 in. (g) A velocity, v = 1 200 ft/min. (h) A unit strain,  = 0.002 15 in/in. (i) A volume, V = 1830 in3. 1–24 Generally, final design results are rounded to or fixed to three digits because the given data can- not justify a greater display. In addition, prefixes should be selected so as to limit number strings to no more than four digits to the left of the decimal point. Using these rules, as well as those for the choice of prefixes, solve the following relations: (a) σ = MZ, where M = 1770 lbf  in and Z = 0.934 in3. (b) σ = FA, where F = 9440 lbf and A = 23.8 in2. (c) y = Fl 33EI, where F = 270 lbf, l = 31.5 in, E = 30 Mpsi, and I = 0.154 in4. (d) θ = TlGJ, where T = 9 740 lbf  in, l = 9.85 in, G = 11.3 Mpsi, and d = 1.00 in. 1–25 Repeat Prob. 1–24 for the following: (a) σ = Fwt, where F = 1 kN, w = 25 mm, and t = 5 mm. (b) I = bh312, where b = 10 mm and h = 25 mm. (c) I = πd 464, where d = 25.4 mm. (d) τ = 16 Tπd 3, where T = 25 N  m, and d = 12.7 mm. 1–26 Repeat Prob. 1–24 for: (a) τ = FA, where A = πd 24, F = 2 700 lbf, and d = 0.750 in. (b) σ = 32 Faπd 3, where F = 180 lbf, a = 31.5 in, and d = 1.25 in. (c) Z = π (do4  di4)(32 do) for do = 1.50 in and di = 1.00 in. (d) k = (d 4 G)(8 D3 N), where d = 0.062 5 in, G = 11.3 Mpsi, D = 0.760 in, and N = 32 (a dimensionless number). Introduction to Mechanical Engineering Design 29 bud29281_ch01_002-030.qxd 11/11/2009 5:35 pm Page 29 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas: