Chapter10 - Case studies: multiple constraints and compound...

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10.1 Introduction and synopsis These case studies illustrate how the techniques described in the previous chapter really work. Two 'were sketched out there: the light, stijJ; strong beam, and the light, cheap, stiff beam. Here we develop four more. The first pair illustrate multiple constraints; here the active constraint method is used. The second pair illustrate compound objectives; here a value function containing an exchange constant. £$, is formulated. The examples are deliberately simplified to avoid clouding the illustra- tion with unnecessary detail. The simplification is not nearly as critical as it may at first appear: the choice of material is determined primarily by the physical principles of the problem, not by details of geometry .The principles remain the same when much of the detail is removed so that the selection is largely independent of these. Further case studies can be found in the sources listed under Further reading. con-rods for 10.2 Multiple constraints - high-performance engines A connecting rod in a high perfonnance engine, compressor or pump is a critical component: if it fails, catastrophe follows. Yet -to minimize inertial forces and bearing loads -it must weigh as little as possible, implying the use of light, strong materials, stressed near their limits. When cost, not perfonnance, is the design goal, con-rods are frequently made of cast iron, because it is so cheap. But what are the best materials for con-rods when performance is the objective? The model Table 10.1 sultlmarizes the design requirements for a connecting rod of minimum weight with two constraints: that it must carry a peak load F without failing either by fatigue or by buckling elastically. For simplicity, we assume that the shaft has a rectangular section A = bw (Figure 10.1). The objective function is an equation for the mass which we approximate as m = fJALp (10.1) where L is the length of the con-rod and p the density of the material of which it is made, A the cross-section of the shaft and .8 a constant multiplier to allow for the mass of the bearing housings. Case studies: multiple constraints and compound objectives 10.1 Introduction and synopsis
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Case studies: multiple constraints and compound objectives 229 Table 10.1 The design requirements: connecting rods Function Objective Minimize mass Constraints Connecting rod for reciprocating engine or pump (a) Must not fail by high-cycle fatigue, or (b) Must not fail by elastic buckling (c) Stroke, and thus con-rod length L, specified Fig. 10.1 A connecting rod. The rod must not buckle, fail by fatigue or by fast fracture (an example of multiple constraints). The objective is to minimize mass. The fatigue constraint requires that F A- - < 0, (10.2) where CT~ is the endurance limit of the material of which the con-rod is made. (Here, and elsewhere, we omit the safety factor which would normally enter an equation of this sort, since it does not influence the selection.) Using equation (10.2) to eliminate A in equation (10.1) gives the mass of a con-rod which will just meet the fatigue constraint: ml = BFL (:) (10.3) pi MI = - (10.4)
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Chapter10 - Case studies: multiple constraints and compound...

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