Mech. Mach. Theory Vol. 25, No. 3, pp. 305-324, 1990 0094-114X/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1990 Pergamon Press plc MODELING IMPRECISION AND UNCERTAINTY IN PRELIMINARY ENGINEERING DESIGN KRISTIN L. WOOD and ERIK K. ANTONSSON Division of Engineering and Applied Science, California Institute of Technology, Mail Code 10 A. ~, Pasadena, CA 91125, U.S.A. Abstract--Each stage of the engineering design process, and particularly the preliminary phase, includes both imprecision and uncertainty. This paper presents a method by which the imprecision and uncertainty in the description of the design's components can be represented. Fuzzy set theory provides the foundation of the approach. Extended hybrid numbers are introduced to handle the two separate representations of imprecision and uncertainty. These representations include the designer's judgements. An application of the theory to machine design is presented, emphasizing the imprecision usually found in the preliminary design of machine elements. Results show the performance of each design alternative compared to its functional requirements, as well as the coupling betwen the design parameters and the resulting performance parameters. NOMENCLATURE dp= pulley (sheave) diameter, belt configuration Ks = size factor, fatigue strength d, = shaft diameter K,¢ = stress concentration factor, shaft strength m = module K~r= service factor mG= speed ratio K~ = application shock factor nf = factor of safety, fatigue strength Kt = temperature factor, fatigue strength n~ = speed, rpm KA = velocity factor nR = rated speed, bearing life KL = life factor ns = factor of safety, surface durability Kj = geometry factor w r = face width K~ = reliability factor, surface durability y, = acceptable shaft deflection KT = temperature factor, surface durability C = bearing load rating L~ = expected belt life Cc --- belt force conversion factor L, = shaft length Cp = belt force conversion factor L v = bearing design life E = modulus of elasticity L R = rated bearing life Fp = peak belt force N = number of teeth He = Brinell hardness ArT= total belt passes for the belt sheaves K, = surface finish factor, fatigue strength P = power Kb = belt bending force factor R = Reliability Kc = belt centrifugal force factor Sr = failure stress Kd ----- depth factor, V-belt S t = tensile strength Kdf = design factor, shaft strength Tr = tension ratio K~ = misc. effects factor q~ = pressure angle, spur gear Ke~as = elastic coefficient 0t = transverse pressure angle, helical gear Kf = surface finish and environment factor 0, = normal pressure angle, helical gear Kk = service factor, shaft strength ~b = helix angle KI = load-distribution factor K0 = overload factor Ko, = oscillation factor Note: superscripts s and h denote spur gear and helical Kp= preloading factor gear configuration, respectively: Superscript r denotes K, = reliability factor, fatigue strength "requirement." 1. INTRODUCTION The engineering design process can be characterized as a collection of phases by which a description of a need is transformed into a physical artifact, proceeding from a highly imprecise preliminary stage to a final configuration in the form of a precise, physical design description. Many tools exist for the later design stages (e.g. solid modeling and mechanism analysis) by which designers may analyze precisely described configurations in order to verify configuration performance or to make final configuration choices. Because of the imprecision and uncertainty inherent in the preliminary design phase, computational tools developed for this phase must be able to manipulate both imprecise and uncertain representations of design alternatives, while directly incorporating the 305