Research in EngineeringDesign(1996) 2:99-115~ 1996Springer-VerlagLondonLimited Researchin Engineering Design Functional Tolerancing: A Design for Manufacturing Methodology R. S. Srinivasan, K. L. Wood and D. A. McAdams Department of MechanicalEngineering,The Universityof Texas, Austin, USA Abstract. The problem addressed in this paper is the development of a physico-mathematical basis for mechanical tolerances. The lack of such a basis has fostered a decoupling of design (function) and manufacturing. The groundwork for a tolerancing methodology is laid by a model of profile errors, whose components are justified by physical reasoning and estimated using mathematical tools. The methodology is then presented as an evolutionary procedure that harnesses the various tools, as required, to analyze profiles in terms of a minimum set of profile parameters and to re-generate them from the parameters. This equips the designer with a rational means for estimating performance prior to manuJ~zcturing, hence integrating design and manufacturing, The utility of the functional tolerancing methodology is demonstrated with performance simulations of a lathe-head-stock design,focusing on gear transmission with synthesized errors. Keywords. Design methodology; Functional tolerancing; Lathe design; Machine precision; Wavelets 1. Perspectives on Tolerances in Design for Manufacturing Tolerancing is an important issue in the context of modern design. Its evolution is presented as a motivation to the theme of this paper. In this theme, there are two important aspects (among a myriad) of engineering: an idea emanating to satisfy some need, and the physical embodiment of the idea, an artifact. In existent terminology, the former is a component of design, and transformation to the latter includes manufacturing. The idea results from knowledge of, or intuition about, the physical laws of nature, gained by experience and honed by intellectual introspection. The artifact, on the other hand, results from knowledge of, intuition about, and craftsman skills in fabrication technologies and their applications. Correspondence and offprint requests to Prof. K. L. Wood, Departmentof MechanicalEngineering,ETC 4.132,The University of Texas, Austin,TX 78712-1063, USA. Prior to the industrial revolution, design and manufacturing activities were physically unified, in that an artisan typically designed and also made the artifacts [34]. As the artifacts became more complex, and the users more diverse, a gradual dichotomy of design and manufacturing was inevitable. Today, design and manufacturing are distinct specializations, in stark contrast to their unified genesis. An important ramification of this dichotomy is the need to re-integrate design and manufacturing through information representation and sharing. A critical component of this integration is tolerance information [26, 28, 29, 30], linking the precision of manufacturing to the performance (or function) of design concepts. 1.1. The Need for Functional Tolerancing? Tolerances have important repercussions on several areas of product development, e.g., production, metrology, assembly, and performance; there are several methods that address specific areas in the above list [301. However, there is no technique that explicitly links tolerance to function [34], i.e., to describe systematically function-driven tolerance assignments, and the means to achieve them [32]. This forms the niche of this research, i.e., the development of a methodology for functional toIerancing within the context of design for manufacturing (DFM). The importance of tolerances is reiterated in the context of design for manufacturing. The paucity of research in functionally-oriented tolerances identifies the problem areas for the paper. In a recent mechanical tolerancing workshop, Tipnis [32] indicates that '... participants could not present any documented case studies as to why specific tolerances were chosen and how these tolerances were achieved...'. Likewise, in a recent review of engineering design research, it is stated [7], 'Although tolerances are critical to both functional performance and manufacturing cost, tolerances have received very little theoretical treatment . . . Research into the effects of tolerances This model is well suited to analyze and synthesize profiles across multiple wavelengths or scales. and incorporating both surface texture and integrity in the processes of both design and manufacture. waviness. waviness is a significant factor. and waviness as relatively tow frequency variations. as opposed to being prejudiced about short or long wavelengths. the need to exercise control on these deviations requires information feedback of various error sources from manufacturing to design. the precursor to assigning appropriate tolerances. tangible. This. The Tolerance-Function Relationship The inter-relationship between tolerances. and similar practical examples. incorporating a novel fractal and wavelet representation. Relating Surface Profile Structures and Design Function The need for a tolerance that is specified by a desired device function is made clear by examining some of the shortcomings of current tolerancing techniques and error descriptions. or mesogeometric errors. The polishing process reduced the contact area further. roughness is considered as high frequency variations. Manufacturing. and the functionality of machine elements remains ill defined and poorly understood . The quantification of these criteria is expressed in terms of cut-off values. there is R. in the course of this realization. Corrective procedures were installed in the defective production line to reduce the roughness. This information is incorporated in subsequent design specifications. The methodology and tools presented in this paper provide a first step toward monitoring. including material specifications. representating. as opposed to imposing an arbitrary cut-off. 2. Although this paper does not address all of the issues necessary for a complete description of geometric tolerance. Both these factors play significant roles in determining the mechanical characteristics. an example from industry. The result was an exponential increase in the number of failures! A complete profile analysis revealed significant waviness in the journal. or material and tool properties. An automobile manufacturer traced the catastrophic failures in engines to an increase in the surface roughness amplitudes in the crankshaft journals. Geometry is the usual language used for representing such descriptions. and a precautionary polishing process was added. and (perhaps) non-unique physical descriptions. reinforce the trend towards a complete characterization. leading to the increased failures. and wavelength. Each of these error sources conceivably possesses a different natural frequency. The dividing line between these various components is ill-defined [25]. To illustrate this point. The waviness was finally attributed to a defective bearing in a machine tool. Waviness constitutes geometric errors in the tolerance scales. roughness is considered to be the significant parameter influencing the function of the surface. result in parts with non-ideal geometries and material characteristics. While these are conventional classifications. and ultimately the functionality of the product.2. 1. which reduced normal contact area between the journal and the bearing. the fractal representation provides a foundation by which a methodology for functional toterancing can be developed.' These statements reveal the need for a basic theory and methods to assess the effect of tolerance scale errors on part performance. In addition to assigning a tolerance by a device function. A new superposition approach. In addition.S. and the surface profile is the result of the extremely complex interaction between these errors. in reality. A full surface characterization would also be more logical from a manufacturing standpoint. cited in [31]. In traditional descriptions of manufactured surfaces. an understanding of the manufacturability of the tolerance is also essential for a successful design for manufacturing approach. In most cases. and waviness is relegated to a secondary status. Hence it is more rational. the machining system is subjected to a number of error sources. into quantitative. a predominantly physical domain. several significant contributions are made toward such a goal. On a qualitative level. to assess the (complete) profile impartially. However. an emerging school of thought that favors a complete surface description scheme [18]. and even ignored [18. and overall form errors. Therefore. These design descriptions are communicated to manufacturing. and the resulting profile is studied. where qualitative customer requirements are transformed through functional prescription. is used to analyze and describe a geometric form tolerance. is summarized below. inherent variabilities in the manufacturing process. Srinivasanet al. In the course of machining a part. In practice all errors with wavelengths longer than the cut-off are filtered. the errors are divided into roughness. which are usually determined by instrument or computational limitations [18]. Engineering design is primarily an informational domain. 31]. and more general.100 on functional performance is even more limited. involves the realization of the artifact. The desired output. Closely related problems are comprehension of the dynamics of current and emerging manufacturing processes. and control the errors on machine elements. Also shown is i. . methodology. f ( ) . especially in conjunction with the geometric and material choices d. of the design is represented by the performance parameters :. SurfaceRepresentation (Minimum ParameterSet) RealizedArtifact Fig. 1. General steps in a design methodology. distinct from d. Functional Tolerancing in a Design Methodology [32]. A design example is presented in this section and developed throughout the paper to illustrate explicitly the influence of manufacturing errors on design performance. yet illustrative abstraction of parametric design. The space [ is a subspace of the general design space d. The elucidation of such a relationship is a pressing need in the modern contest of advances in materials and manufacturing technology. in a cost-efficient and effect manner. understand. 2. As motivation for the theory. the designers do not have the tools to specify. For example. Figure 1 shows a simple. Design Task 1 Clarification. consider functional tolerancing in the context of modern engineering design. Tolerance and design parameters affecting performance in the design space. The black box contains a functional representation of the design. the tolerance assignments to the design. with special emphasis on the relation between errors and product function. as a consequence of more accurate aerofoil profiles in the rotor and stator blades of axial compressors. The decisions that the designer may make are represented by the design parameters d.Functional Tolerancing 101 2. the Rolls-Royce company predicts a significant increase in gear box capacity for aerospace geared powered transmissions. QFD Analysis I DesignMetrics System/Conceptual Design DesignAttributes l I EmbodiedDesign Concept Evaluation. Embodiment Manufacturing . These problems are definitive of the impediments in DFM. and are addressed in this research. Figure 2 is a -[ Fig. with a reduction in the gear tooth composite error [15]. and characterizations of their precision. because tolerance decisions are the subspace of interest in this study.including the tolerance sub-problem. or metric.ll ) Tolerance Subproblem (Profile Synthesis) d /" I Mfg. It is shown here. While the predicted improvement in performance is promising. New materials and manufacturing techniques pave the road for achieving higher precision in machine components.1. The same company also projects improved compressor efficiency. and results presented in this paper. . Stagnation pressure... efficiency.. wear. wear. heat transfer... long-life. Line uniformity Line width. Load carrying capacity Friction. The engineering requirement for accurate parts can now be cast as minimizing vibration amplitude A and maintaining the vibrations frequency oa above. Pressure. based on a mathematical representation of the design metrics. 1) provides our definition for a functional tolerance. One major factor in lathe accuracy is how much it vibrates. such as noise.. Srinivasanet al. Examples of tolerance sub-problems categorized by element interface.. noise. encompassing a wide range of critical functional issues. perhaps. the functional performance of the examples is obviously affected by a form tolerance. Translating the customer requirements into engineering requirements is.. the functional performance of a device may be used to specify geometric tolerances directly.. the functional description and sub-function solutions show the source of lathe vibrations.. Alignment.. (6) provide .. a certain range.damping. In addition to being a necessary step in the design process. uniformity.. Category Tolerance sub-problem Form tolerance of relevance Design metric Solid-solid Gears [15] Profile Piston-cylinder Cam-follower [41] Circularity Profile Press fit bearings [3 t] Ball bearings [29] Circularity Circularity Brakes Journal bearings [27.102 R. and illustrate the need to represent manufacturing directly in design through performance metrics.2. 2)? 2. the tolerance subproblem. with the tolerance subproblem shown separately for emphasis.. profile Circularity Ball point pen Profile Solid-gas Air foil Profile Solid-light Valve regulator [17] Computer disc [31] Optics [37] Roundness. or below.. Vibration Shock. A primary customer of the lathe head stock is a machinist or manufacturing firm. Vibration Deformation. The examples listed in Table 1 further provide motivation for our work. Three needs of the customer are accuracy. (5) hold and locate cutting tool. as embodied in Fig. Thus... In the context of design metrics representing functionality. This relationship between design metrics.. S. noise.. an example is used. A general set of operational functions for the lathe are: (1) provide cutting power..... and heat transfer. straightness Profile Profile Transmission error Backlash. The design task is to design a lathe. The major components of design are shown. force. The lathe is a standard manual feed lathe. the most important step ofa QFD analysis. We now generate a brief functional description of the lathe. in general. illustrating the categories of physical interfaces that tend to govern the function-tolerance interaction. 31] Flatness.. Table 1. There are many design factors that contribute to the ability of the lathe to produce accurate parts.. The basic question then becomes how do we abstractly represent manufacturing surface profiles with a minimum set of parameters so that we may incorporate these parameters directly within the design metric relationships (Fig. efficiency. an electro-mechanical system). Friction. It is clear from the table that many important tolerance subproblems exist. and quiet operation. 5. The example is the design of a lathe head stock (or. (3) convey cutting motion and power to workpiece. Solid-liquid further refinement of the design process. wear.wear. The example presented is not intended to be a complete application of a design methodology but rather to clarify the use of the methodology for functional tolerancing. (2) generate cutting motion (rotation). Flow rate.. drag. the tolerance subproblem is a critical element of the design process in many cases. A graphical description for the tolerance subproblem is further refined in Sections 3 through 6. A Motivating Example To clarify the general design methodology. While the list in Table 1 is not exhaustive. and mathematical representation (Fig.. Gloss. stiffness. A range of examples are listed in Table t. focal point.seal force. Lift... Efficiency. (4) hold and locate workpiece.. The gear pairs are used to select different speeds for the lathe spindle from the drive shaft connected to the motor by means of a belt drive. For the lathe concept shown in Fig. the design metric is now transmission error. a model for the tolerancing subproblem is needed. 3. . The head stock houses the spindle and power transmission elements of the lathe. Forces at the bearings then excite the housing. Tolerance Subproblem: Gear Transmission After applying the general design methodology to generate design concepts (Fig. The spindle provides the main cutting motion for the lathe. i. (7) house components. Two of the most important design problems confronting a gear designer are the vibration and noise of gears [9]. One of the significant factors affecting gear vibration is the geometric errors in the meshing gears. Schematicof Lathe Head Stock. In addition. relative to the expected position that the driven gear would occupy if both gears were geometrically perfect and undeformed [9]. the appropriate tolerance for the gears. The tool post mounted on the carriage carries the tool and moves the tool perpendicular to the workpiece.3. is used as the performance parameter [10]. and (8) provide support structure. In other words.de Section of HeadStod~ ~Mo~r ~ kIlhe A. The question remains: What is the appropriate form tolerance for the gears and how do we assure that it is achieved? 2. Gear noise is identified as being the result of force and displacement excitations at the gear mesh which then cause a dynamic response of the shafts and bearings of the transmission system. Relating both of these back to the customer requirements. the designer must choose.. in addition. to drill longitudinal holes. An articulative tolerance subproblem for the lathe head stock focuses on the relationship between profile errors in gear teeth and transmission error. The vibratory energy originating at each gear mesh is conducted to the housing and its supporting structure through the interconnecting structural paths. an engineering need is to reduce transmission error. Another ergonomic effect is the noise caused by the vibration. Motion parallel to the workpiece is provided by the feed rod. The necessary model is formulated below. In this example. Gears are popular machine elements used in rotary power or motion transmissions. relative motion between the tool and the workpiece in two.3. The head stock is constructed on the lathe bed. 3B is a schematic cross section of the head stock. The chuck attachment to the spindle is used to hold the workpiece. The tail stock is used to support long workpieces and. 3.~'-hem. Vibration in the gears affects the accuracy of the machine as well as causing noise in the shop environment. and subsequently used to illustrate the functional toterancing methodology. The geometric errors lead to vibration and ultimately translate into workpiece inaccuracies. Shown in Fig. Main Gomponentso~a Lathe Fig. vibration is an important performance criterion and engineering requirement. causing it to vibrate. a parameter called transmission error. Relating back to the QFD analysis.. The cutting power is transmitted from the motor to the spindle using a belt and a set of gear pairs. monitoring gear noise gives early indications of failure or fatigue [9]. the motor provides the main motive power. 2.e. This brings to focus the tolerance subproblem of design. orthogonal directions. design.ibration. Thus. The head stock houses all the above components. Each pair of meshing gears in the gear box is a source of vibratory excitation. The speeds are selected by engaging a different gear pair using the speed change lever. 2) and isolating the important design metrics. Definition of Transmission Error The transmission error is defined as the deviation of the position of the driven gear.Functional Tolerancing 103 B. and the elastic deformation of the teeth under load E14]. The two main contributions to the transmission error as inferred from the definition are manufacturing errors in the tooth geometries. a parameter of gear noise/x. In order to control the transmission error. which has strong correlations with gear noise.1. a gear pair with infinite stiffness and perfectly involute profiles would have zero transmission error [14]. A machined surface is assumed to be composed of straight line elements. Surface geometry is a complex imprint of the multitude of physical mechanisms generating the manufactured surface. a measure of the significance or 'strength' of the trend component.e..n= 1 ki(X ) O<_x<_L (1) where W is the toad transmitted by the mesh. i. n is the number of simultaneously meshing tooth pairs in the zone of contact. Hence. . If the sample length is L.e. a model is needed of manufactured et al. an intercept Yto. Such an assumption is valid and necessary in the example used in this paper as the data is sampled using digital computers. 4. a rational basis for relating component errors to performance.. error carried over from a prior process. Straightness is assumed to be the prototypical form error in the following description. within the framework of Design for Manufacturing.e. without resorting to expensive trials. and are denoted y(n). the profile is sampled at N points spaced uniformly along the x axis. with involute teeth. as shown in Fig. A novel representation of a machined profile is to resolve surface profiles into mutually uncorrelated deterministic. isolated by using linear regression techniques [24]. There are physical mechanisms associated with each mathematical component. A further practical assumption is that the error values are available in discrete form. This procedure compresses the trend information into two parameters. and investigate the surface based on a superposition model. ei(x) and ks(x) are the local composite error and the local stiffness of tooth pair i within the zone of contact respectively. the variations in the height of the profile are measured as a function of the horizontal coordinate. 0 <_ x <_ L (2) The linear regression procedure also yields a correlation coefficient. periodicities. 14]: TE(x)= W+~l=~ei(x)ki(x) ~. the slope can be interpreted as a static error. and a slope s t. i. we can define a rectangular zone of contact. and face width F. The following sections present such a basis couched in a methodology for the tolerancing problem.104 R. i. Since contact takes place across the face width. The variations inherent to manufacturing processes have a random component. x. Trend Component Physically. Contact zone for spur gear pair. then the equivalent discretized representation is (0 _< n < N . and nondeterministic components [. and by definition ei(x) -> 0. 4.. and also prescribe a model representing that component.. Consider a spur gear pair. Actual machining profiles possess a number of characteristics. i. We approximate the trend yt(x) with a straight line. Any model development must be preceded by a decomposition of the profile into mathematically tractable components. trends. The composite error is defined as the sum of the profile errors on meshing teeth at the point of contact. 3. e. is required in order to realize the benefits of improved performance. which are loosely interpreted as slopes. unleveled machine foundation. S. which cannot be predicted before a gear is manufactured [t4]. the locus of the points of contact (from the beginning to the end of tooth engagement). and expressed as y(x). The following sections examine and relate the possible causes of a specific component to the dynamics of the process and/or the error sources. and so on [-36].g. the problem is cast as follows. is defined by a straight line. The path of contact. the errors are expressed as a function of n. The popular manufacturing processes for gears are hobbing and shaping. Grinding and lapping are finishing processes.e.1). This model is needed to determine the abstract parameters for the design metrics. The absolute value of the correlation coefficient varies between 0 and 1. In order to facilitate an assessment of manufacturing errors on performance.1. error in the fixture. roughing. ( 0 _ x < L). surface profiles. 26]. with sides defined by the length of contact L. etc. which can be attributed to any one or more of the following causes: error in machine table. Srinivasan c D i P = Htch Point AD = Face Width (F) APB = Path of Contact ABCD = Zone of Contact Fig. The expression for transmission error is given by [-10... used for precision gears. Yt(x) = Yto + s t x 3. Modeling of Machining Errors As a first step in developing a functional tolerancing methodology. and is based on the surface geometry.21. 3. The central idea of fractals as applied to surface profiles is examined below.1.2. it indicates the degree of similarity between a profile. However. Recall the profile data is available in discrete form.e. is a measure of the dependence structure in the profile. These are 'ideal' geometries. the correlogram indicates the presence of a slope. heterogeneous workpiece hardness) that occur in the course of machining.i. estimated as [2]: 1 ?(h) - N-l-h N--l--h ~. the mathematical expression for the autocorrelation function is obtained in terms of the autovariance function. etc. identically distributed (i. . denoted p(h). such models can be found in Srinivasan [26]. 4. a function of the fractal dimension. and where applicable. 0 < n < N . D:. This function is model specific. in any typical manufacturing process.. The reason for adopting the fractal model stems from the complex interactions of deterministic effects (e. and the main parameter used is the fractal dimension. an effective descriptor for the complexity in a geometric entity [13]. The model used in this paper is based on the following spectral property of fractal profiles: S(~) oc ~-p(o:) (4) where ~ is the frequency.2 . Fractal dimensions are non-integer and hence permit the characterization of irregular geometries. This complexity engenders a specific structure (e. Tools/Models for Detection/Estimation of Form Error Two techniques are used to test the profile for the presence of deterministic structures. Geometric objects are traditionally described from an Euclidean viewpoint as having (integer) dimensions of 1 (line). .3. a periodic component in the profile is reflected in a periodic correlogram [2]. .. and a copy of itself. vibrations) and random effects (e. which are elegantly modeled using fractals [26]. explain the calculation of relevant model parameters.1.3..g. They are: autocorrelation function and power spectrum. long-term correlations) in the resulting errors. . and fi(D:) is the spectral exponent.Functional Tolerancing 105 3. Autocorrelation Function The autocorrelation function. the fractional part of the dimension is a measure of the deviation from the ideal geometry. In the following sections. The discrete counterpart of f~ is referred to as 'frequency number' [263. Yp0 is an offset (from an arbitrary datum of zero). p(h) is interpreted as a measure of the dependence of the profile value at a given location. ?(h). y(n). on the profile value h units downstream. A plot ofp(h) as a function of h is referred to as a correlogram [2].d. as the lag h increases [2].g.g. 24].1. i. da is the amplitude. 2.) [2] random variates. . Periodic Component When the surface profiles exhibit a repetitive pattern. 2 (plane). The following model is used for evaluating the periodic component yp(x): y p ( x ) = yvo + dasin(27rfrx/L) O< x < L (3) where. Relating this to the deterministic sub-components. by exhibiting slow decay of p(h). 4. N / 4 (5) The autocorrelation function is then estimated as: p(h) = 7(h)/y(O) (6) This normalizes the autocorrelation function with respect to unity at zero lag. A novel approach used in this research is the application of fractal concepts and parameters to model the nondeterministic component in manufactured profiles.. we develop the necessary manufacturing representation in Fig. there is an intricate interplay of random and (remnant) deterministic effects. and f~ is the frequency. Fractal Model Fractals are used for the description of irregular objects. . leading to spatial interdependence between the errors [26]. Equivalently. translated by h units (h is referred to as the lag) along the horizontal axis.=o y(n)y(n + h) h = O . 3. 1. There are several fractal models based on a variety of scaling properties [13]. in contrast. We estimate this component from the surface profile by using a nonlinear regression procedure [23. the presence of a periodic component is indicated. we describe the mathematical tools used in the identification of the presence of each component. in addition to their nondeterministic counterparts. The physical interpretations and mathematical definitions of the above quantities are given below. in that there are no errors. Nondeterministic Component The nondeterministic components are usually assumed to be independent. By so doing. wavelet theory is used as a 'mathematical profiler' to extract the fractal parameters [26]. The approximation and detail extraction steps are represented as filtering and sampling operations of the original profile. i. given a surface profile.3. The basis functions of the detail space are constructed from the dilations of translations of a wavelet function. approximation.. While the above tools suffice for identifying and quantifying the deterministic components.e.1. Wavelets share the properties of statistical self-similarity and nonstationarity with fractals [8]. 4. 1 f~ e 2~X~dx 2 S(¢) = Z y(x) . It can be interpreted as a measure of the energy per unit length (or the power) contained in the signal as a function of spatial frequency. Since the profile is usually given in terms of the height (from a specified datum) at discrete points along the workpiece. The power spectrum reveals the presence of offsets by a peak at zero frequency. and G respectively. and periodic structures by peaks at the underlying frequencies. computed by projecting the profile onto the corresponding detail space. Power Spectrum The power spectrum is the frequency domain counterpart of the autocorrelation function and is defined as the square of the Fourier Transform (amplitude) per unit length [20]. (3) in the nonlinear regression. and the number of coefficients are directly proportional to the regularity of the functions. the fractal parameters describing the nondeterministic component are estimated using the wavelet transform.y)a = n ~ h2k-. described in the next section..e. . 2m. depending on the instrument capabilities and the desired measurement scale. 4. Hence it can be used to obtain a preliminary estimate of the frequency.106 R.(A~y)d + gk_2. Srinivasanet al. The essentials of the theory are summarized below. where d indicates the discrete nature of the data.y)d = n = --o0 The reconstruction of the original profile from the approximations and details at various resolutions (coarse-to-fine) can also be computed recursively as: Reconstruction k (A. and then the square of the magnitude is computed to obtain the power spectrum. and detail D. The information lost in stepping from a finer approximation to a coarser one is called the detail. y. This is taken to be the 'base' resolution for the multiresolution analysis and is denoted by m = 0.+ly)d (9) ~00 Detail (D~. before using more rigorous computations to determine the exact parameter values. These operations are performed recursively over several scales.(Am+ 1 Y)d (8) O2k_. Wavelet-Based Method for Fractal Parameters The theory of wavelets originated from tools developed for seismic studies. The basis functions of these approximation spaces are built from the dilations and translations of the so-called scaling function. The approximations and details at resolutions m < 0 (find-to-coarse) can be obtained as shown below.3.+~y)d = 2 ~ n ~ [hk_z. as described in Srinivasan [26]. which in turn is built from the scaling function. detail extraction.e.. we indicate resolution level by m. this data sequence is indicated by (A~ y)~. Approximation (A~. This is decomposed into approximation Amy. Such an estimate of the power spectrum is called the periodogram [2].(A~. The corresponding filter coefficients are hk and gk respectively.2.(D~y)d] --oo (101 The three operations presented above. Hence.. An approximation of a profile at a certain scale is defined as the projection of the profile onto the corresponding approximation space. Implementation In the following development. 12]. by passing it through two filters/4. Profile Analysis with Wavelets Wavelet analysis yields two sets of information: approximations and details [4. Am+lY represents the signal at resolution m + 1. the discrete Fourier transform (DFT) [35] of the data is calculated. i. the value of the discrete frequency which exhibits a peak in the spectrum is used as an initial estimate for fr in Eq." (7) For calculation purposes. i. 4. These coefficients embody the characteristics of the corresponding basis functions. and the corresponding scales are dyadic.S. The output of a profiling instrument is at a specific resolution.. and reconstruction can be shown in terms of 'filters' interpretation. In addition. Performance Evaluation 10.g. Validation. In statistical parlance. Representation 4. feed) lead to well-defined components. While these three mechanisms would capture the predominant structural information in the profile. With the sub-division of the deterministic component into trend and periodicity. An empirical guideline in time series analysis suggests that the periodic component is independent if its amplitude is relatively constant [2]. Furthermore. These issues are addressed in the next section. static fixture error) and periodicity (e. In the next section. second. slope) 7. Synthesis of realistic part models 11.2. widespread in current industrial practices.. error analysis and representation. Performance evaluation Fig. a function of D:. After obtaining the details at all scales.sign problem 2. However. 5. and their adequacy for a complete description arc unclear thus far. IdcnIificationof determinsiti¢ structures 6.g. 5. De-perio~zin~. A review of published error profiles indicates a constant periodic amplitude [36]. The structure of this methodology reflects the interaction between the Problem Identification/Definition 4. Steps in proposed tolerancing methodologyfor DFM. they are also added to the model to obtain: y(x) = yt(x) + yp(x) + y:(x) + yo(x) (13) where yo(X) is called the outlier component. At this point. the superposition approach is still valid. (12). However. Hence a given surface profile can be decomposed into approximations and details at successively coarser scales. discrete details are generated as the samples of a zero-mean Gaussian process. the profile is synthesized at successively finer resolutions using Eq. sake of completeness. a concerted and cogent procedure for the effective use of these tools is also required to realize truly their potential in DFM.4. as discussed below. as shown in Fig. The first stage is problem identification. Fractal Profile Synthesis with Wavelets Fractal profiles and surfaces are characterized by power spectra S(~) of the form: s(~) oc ~-~(~':~ (11) where ~ is the frequency and fl is the spectral exponent. T h e S t e p s in the Methodology The tools introduced in previous sections for extracting structural information provide a suitable abstraction of manufactured profiles. (10). the instructions or steps to use these tools are provided.3. The first and last stages are in the design domain. 5. and for the 3. synthesis and validation. amplitude. . This overview of tools and models. Wornetl [40] presents a scaling argument in terms of the variance of the discrete detail signals (D~y)d: a2[( Dk Y)d] = 1/O2-P~DJ)" (12) where Vois the magnitude factor.Functional Tolerancing 107 4. such unrepresentative points are called 'outliers' [1].Establish precision of maa~-acturing prcoesses I l~rror A n ~ w i ~ . the fractal dimension. which can be treated as independent of the fractal component. with variance as calculated above. comprises one half of the methodology. Wavelet Analysis: [ I (f~t~ dimension. from the causal and mathematical extraction points of view. leading to a complete model of the manufactured surface. the mechanisms causing trend (e. it is possible to obtain a few points that do not conform to this overall structure. the actual composition of the various components. while the second stage is in the manufacturing domain. the various components of profile structure have been examined. (offset. Data Agquisition 5.ma~t~d~factor) ) Synthesis. De-t~ending: (intercept. The development of such a methodology would reduce the subjectivity in tolerance assignment. Comparison • Faergy • Entropy • Visual 12. A Novel Superposition Approach 1Jdentify and clarify ~.. An important assumption in this model is that these components are independent of each other. and establish a systematic approach for functional tolerancing.Isolate tolerance sab-problem The basic superposition model for representing the deterministic and stochastic components of machining errors was discussed earlier. The tolerancing methodology has three main stages. fi'equenqO 8. Since the profile information is abstracted in terms of fl(D:) and V0 in Eq. and third. these parameters are calculated from the slope and intercept of the log-log plot of the variance of the details versus the scale 2". Outlier correction 9. ydet(X). e. the detail extraction step is very sensitive to singularities [8]. However. ya~t(X) = y(x) -. This implies the use of suitable measurement methods. De-periodizing: Again the autocorrelation and the spectrum are used as indicators of periodicity on the data. (2)) are determined by linear regression. The three stages are further decomposed into several steps to present the methodology in a logical and evolutionary fashion. defects in the grinding wheel. etc. (1).. this step can involve the decomposition of the overall problem into manageable sub-problems. it is assumed that the transformations induced by the measurements are negligible. Then. e. All the different tools described hitherto are called upon to identify. which is the irregular component of the profile. wavelet analysis is at the heart of this procedure for detecting long-range correlation in surface errors. Identification of Deterministic Structures: The R.yt(x) (14) 7. Identify and clarify design problem: This generic step is mandatory to comprehend the requirements of the customer clearly [19]. depending on their significance. etc.. 1. De-trending: If the autocorrelation and spectrum indicate the presence of a significant trend component. the outliers pose serious impediments for the following reasons. e. to obtain the residual. are estimated with a nonlinear regression procedure using Eq. long-life. To obtain this knowledge. slope. that are in turn represented by mathematical models that prescribe the necessary geometric tolerance. identifying the appropriate size and geometric tolerances to be prescribed. they could be symptomatic of some very unusual mechanisms in the machining process. They detract from the quality of the data. However. frequency of underlying periodic function. these are the outliers mentioned in Section 4. and influence the subsequent calculation of the detail variances. Table 1 lists a representative range of tolerance sub-problems. 2. the periodic parameters. and establishing the equations governing the interrelationship. y(x). Consequently the presence of a few outliers will inflate the detail values. if the absolute value of a data point deviates from the mean by more than twice the standard deviation of the entire data set.g. Figures 1 and 2 provide an approach for completing this step. and quiet lathe..y p ( x ) (15) 8.g. e. but at the same time. (3). offset. ( x ) = yde~(x) . This study is restricted to form errors that can be interpreted as signals in one dimension.g. Initially. In the example presented in Section 2. and identifying conceptual solutions for each. then the value is replaced by that of its predecessor [1]. like determining performance parameters susceptible to surface errors. and frequency. Outlier points: The residual profile data are carefully studied. this phase of the methodology moves into the manufacturing domain. As motivated in previous sections. required to address the tolerance problem. Srinivasan et aL autocorretation function and the power spectrum are computed. the sub-problem is power transmission. amplitude. in the contest of surface models. given by Eq. without recourse to examples. Establish precision of manufacturing processes: At this juncture. is obtained.S. As described above. There is no attempt made to distinguish measurement errors from actual profile errors in this paper. y .108 design and manufacturing domains. extract and represent the various features. In the lathe design example the tolerance sub-problem is the profile tolerance on the gear teeth. The performance parameter (or design metric) is transmission error. these provide valuable qualitative information about the presence of deterministic structures. The raw profile data are then de-trended to obtain the de-trended profile. 4.2. periodicity. an outlier correction procedure is adopted. the slope and intercept parameters (Eq. The data are then de-periodized. Isolate tolerance sub-problem: This step involves a number of nontrivial steps. all the steps are presented. 9. The following steps comprise the tolerancing methodology in DFM. At this point a decision to include or exclude the deterministic features in the model is made. especially for the presence of a few data points which apparently do not conform to the overall structure. 6. correlation coefficient. If present. 3. In order to avoid this problem.4.g. For example.. 5. the designer must have a knowledge of the precision or the capabilities of the manufacturing processes at his or her disposal. wherein customer needs are transformed into design metrics. Wavelet analysis: The wavelet decomposition . each step is illustrated with examples. possible measurement errors. The customer need is an accurate. Data acquisition: The experimental data. The analyst can also glean some preliminary quantitative estimates for some parameters. and that the measured values are a true indication of the profile. where. Functional Tolerancing 109 (refer to Section 4. 6). i Order statistics: If the profile errors are arranged in ascending order.2. The initial phase. and a candidate manufacturing process. These decisions are important at this stage of the design. this measure is not a substitute for the other precise. The following are used as 'measures' to evaluate the two profiles. The performance parameter is transmission error. The application of the methodology is then revisited to illustrate the utility of the theory for the designer. with respect to the experimental. The following design example illustrates a design problem. given by Eq. and the workpiece is clamped 6. entropy increases with increasing disorder in the profile [6]. the true validation of the synthesis would be to evaluate the performance of the part. in the succeeding stages. even before invoking the quantitative criteria [16]. and y(. but in reverse order. (1). the issue of the 'goodness' of the synthesized profile. ll. as this is the next step after synthesis. Surface grinding is the manufacturing operation studied (Fig. The generation of the trend and periodic components is trivial. This is the last step in the error analysis phase of the methodology. Tolerances have typically been relegated to the final stages of design. • Energy: This is intended as a global measure of the size of the error.g. The next action is the evaluation of the performance using the governing equations established in Step 3. 6. The design problem considered for study is established in Section 2.3. Eq. In this quest. Synthesis: In this phase. as shown in Fig. The synthesis procedure is a replica of the analysis procedure. The Design Problem and the Tolerance Sub-Problem where m is a positive integer smaller than N/2. and also consummates the methodology. Performance: While the above criteria do offer some means of comparison. An estimate for the entropy is given by [33]: J ( y ) = g -1 ~ log Y(n+m)--Y(n-m~ (17) 12. as they follow their respective models (Eqs (2) and (3)). embodiment design. 10. manufacturing processes. and well-defined criteria. and detail design [19]. Here a grinding wheel is mounted on an horizontal spindle. e. 2. the designer has the information about the precision of manufacturing processes.. (13). However.2. then Y~0)is the first order statistic. The design problem is clarified in the next section. a part model is synthesized using the parameters obtained from the preceding steps. The methodology for functional tolerancing can theoretically fit into any one of these stages. Establish Precision: The Gringing Operation . highlighting tolerance selection. and the next step advances into the synthesis and validation stage. Comparison: While the synthesis yields an error profile. the complete profile is generated by the superposition model. The steps outlined above are explained with reference to a particular example design problem. but is only to be used in conjunction with them.3) is used on the corrected residual data to obtain the fractaI dimension and magnitude factor. to Y(N-1)the largest. Implementation of the Methodology A methodology for the overall design process comprises three stages: conceptual design. but it is potentially most useful in the conceptual and embodiment design stages. warrants some criteria for comparison. including the analysis of machined profiles through an experimental design strategy. since the designer does not possess the knowledge or the tools to represent and manipulate the manufacturing variabilities.2. from Y(0)the smallest. n=O 6. The tolerance problem is the gear transmission. since an awareness of the tolerances would help direct the choice of resources. Y~I)is the second order statistic.1. • Visual Comparison: Any radical differences can be immediately detected by a qualitative visual comparison. Hence performance evaluation serves a dual purpose: it acts as an additional criterion for comparing the experimental and synthesized profiles. and so on.) are the order statistics 1. Using these different components. using the proposed methodology. and is defined as [3]: N-1 ~(y) = ~ j y(n)l ~ (16) n=O • Entropy: This criterion is used as a relative measure of the disorder in the profile. is implemented in the following subsections. This falls into the natural framework of the methodology. comprising steps 1 through 3. The synthesis of the irregular component uses the wavelet synthesis procedure as described in Section 4. due to the random orientations of the individual grains. Data Acquisition Note: Vis peripheral grindingwheelspeed v is work table reciprocatingspeed Fig. the steps would equally apply for steel alloys or exotic gear materials. and to record the height variations. This material is used only for the purpose of illustrating the methodology. and the second 'b'. The methodology (steps 4 through 12) is applied to the profiles obtained from the grinding process. Hence. medium (hardness) grade. along with the example grinding profile. C~ V vi ~-~uc chuck . Providence. alumina wheel. each at two levels. The number of data points is N = 256. for a representative case.g. typically using a magnetic chuck [11]. 6. Surfacegrindingoperation. No coolant is used. vitrified bond. No indication of periodic structure is apparent for the grinding profiles.8 _< m < . on a reciprocating table. For example.S. where resolution levels are in terms of 2% . The work table speed ranges from 3500 fpm to 6.0005 in. These results are summarized in Table 2.1 .. In this case. Transverse motion is obtained by cross feed as shown in Fig. Depth of cut is fixed at 0. The order of experiments is randomized in each replicate. and as the table reciprocates beneath the grinding wheel. 6. Since there are several experimental profiles. the results involving graphical plots. and ranges from 6259 fpm. and the steps are presented below.3.~I~ 6. The horizontal resolution is 0. A suitable depth of cut is set. the first measurement is denoted 'a'. material is removed by abrasive action. Deterministic effects due to vibration are also present [11]. 6. The grinding experiments are carried out on a Brown and Sharpe 618 Micromaster surface grinding machine. It is desired to conduct the experimental study with multiple combinations of process parameters. 7. Based on previous studies of the grinding operation [5]. Almost all grinding profiles indicate these patterns.625 fpm. as can be inferred from the profile plots... Two different measurements are made for each test piece (errors measured along two different elements on the surface). In addition. to reduce the effects of unknown variations [22]. The grinding wheel specification is A46HV.Da implies the first measurement (a) on the fourth test piece (D) of the second replicate (II). down to 53. This makes intuitive sense.5. to yield eight levels of resolution in wavelet analysis and synthesis. Srinivasanet aL gage experimental error. Each experiment is replicated to The height variations along a linear element of the surface are recorded using the Surfanalyzer 5000 Profiling System) This system has an accurate linear drive. with each abrasive grain acting like a single point tool. Identification of Deterministic Structures The autocorrelation and power spectrum are calculated. all grinding profiles have an intercept. are not presented for all the cases. For the experiments in this study. The wheel speed is changed by using different sized grinding wheels. the detrending operation prescribed by Eq. i. two parameters are considered.094 fpm. The spectrum indicates the presence of an offset. which implies a medium grit. the geometry of the cutting tools is inconsistent. The grinding wheel can be considered as an agglomeration of multiple cutting edges. and more definitively from the power spectra.1 mm. Rhode Island. and a sensitive stylus to traverse the surface. (14) is carried 2 A product of Federal Products Corporation.e. The combination of these numerous error mechanisms produces co~nplex structural effects in the error profiles. ahead of the cutting region.110 R. autocorrelation function. These functions. power spectrum. The effects of premature deformation of the workpiece. II. etc. . e. the slow decay of the autocorrelation functions leads to the inference of a slope. as randomness prevails in ground surfaces [36]. 6. are shown in Fig. are also cited in the same reference. and a slope. However. Therefore the grinding process has a predominant random component. they are the peripheral speed of the wheel (V) and the work table (reciprocating) speed (v). chosen to accommodate the smallest diameter of the probes used for tolerance measurement [38].4. De-trending A linear regression analysis is carried out on the profile data. aluminum is chosen as the work material due to its availability and cost. 748268 -0. '. 0 50 .)°°t '''~" . Grind-II.6 2 o.. ....2 < -6 t | 1~ 1~ -10 0 50 t -0.805 0.. 1OO | i 150 200 x (e-1 1"rim) Fig.% .Ba I.. °....142413 -0.2. 250 .2 0 250 200 10 20 30 lagh x~lmm) 40 50 60 0. The few points lying outside these limits do not contribute to the overall structure.011787 1.600 0..t..Aa LAb I..009690 0.023930 -0. irrespective of the correlation coefficient. Plot of typical grinding residual.0...522975 1.Functional Tolerancing 111 10 Grind-lI... 6...276 0.0.012156 0.. No... In other words...Ba II.. nor the autocorrelation functions and spectra indicate the presence of periodic structures.278721 1.Da LDb 0.... 7. ....% .. this operation is not carried out for the grinding profiles.Db 0. (15).894023 0.0...011039 .. This is a common feature in all grinding profiles..Ca II. 8..677 0...°r~. .05 0 .713972 -0.011962 -0..Ab II..086330 .Bb II..15 OA 0.178365 0.Aa II. g...029827 0.493 -0. O.Ca LCb I...121255 0. ..Bb I. The limits corresponding to twice the standard deviation are also shown as dotted lines... and since the offset is accounted *~ ° .. 6.019312 0..0.158 for in de-trending. -2 0.779 -0.. -2 50 i .. . • t "' .Da II. autocorrelation function..012707 0... and power spectrum. The corrected residuals are used for the estimation of fractal parameters.146 0... T a b l e 2..700 .570 0.Db - - 0.. 8...399 0.. 100 150 fi'equcaey number 200 250 Fig..698 -0.842 0.002408 ... REP. ._~__..1..006601 0..Db -0. .359 .022987 -0..003322 0..Db: Profile...588932 0.019406 .._ 2 o 1 out for all grinding profiles.- ..2...8 6 .. Slope Intercept (~m) Correlation coefficient I. An example scatter plot is shown in Fig...0..752679 1. Outlier Correction A scatter plot is used in place of the normal residual plot to perceive the outliers clearly.473 0..... yp(x)= 0 for grinding.. . The residual is calculated using Eq.045278 1.~.7... T r e n d p a r a m e t e r s for g r i n d i n g ..496771 0..2 0.0.815722 . ..25 Spectrum Grind-lI.IYo Acf Gdnd-II..6. De-periodizing ! Since neither the grinding profiles.- 0 -1 ¢.385428 0.885780 0..673 II..Cb II.821609 .¢.020890 0.. and therefore the outlier correction is applied.. T a b l e 3.066 0.Db. or foundation of this machine. Synthesis and Comparison The above parameters (minimal.073059 0.982 0.794 0.179861 1.R.Aa II. It is clear that the energy and entropy values for grinding experimental and synthesized profiles are in good agreement.Db 6 6 2 2 + -2 -6 -2 -6 -lO o i i i i :50 100 150 200 x (e-lmm) -lO 250 o i i 50 1OO Fig. REP. i 150 x (e-lmm) i 200 250 . abstract set) are used to synthesize the corresponding profiles.088261 1.065793 0. The Daubechies scaling and wavelet functions with twelve coefficients [4] are used in the analysis. This step is the link to design. the possible cause is the premachining operation.158817 0. No Spectral exponent Fractal dimension D} Magnitude factor (~tm 2) Correlation coefficient I .404507 0. • There is no periodic component in any of the grinding profiles.247561 0.Aa I. 6. as exemplified by the fractal parameters.Db 0.030870 0.143456 t. 9.983 0. The example grinding profile.244190 0.464151 0.065129 0.Da I. 9.Cb II.407101 1. After an initial roughing.188157 0.999 0.10. Wavelet Analysis The wavelet analysis technique is used to calculate the fractal dimension and the magnitude factor.103456 0.286029 1.309 0.185799 0.287216 1.Ba II. where they were again milled on another milling machine. • These exists a significant trend component in all the profiles.245994 1.239665 0.035566 0.- Grind-lI.Ca II.508013 0.110229 0. The presence of trend in all the cases points to a static displacement in the vise. The energy and entropy are shown in Table 4. 6. In view of this consistency.427942 0.120276 0.995 0.057347 0.640279 0. Experimental and synthesized profiles: Grind-II.Ab II.162700 t.8. The two are visually similar.823478 0.482217 t. and the synthesized version are shown in Fig. As the spectral exponent lies in the interval (0.Ba LBb I. machine table.252747 t. and the fractal dimension is obtained from Ds = ( 3 .102393 0.508043 0.336 0.Bb II.Ab l.522 II.267925 1.130880 0.-II.Ca I.794 0.887 0.Db 0. 112 6.Db .862 0. the test pieces were moved to a different location for griding. The results are presented in Table 3.959 0.271902 t.079817 0.151776 1.418650 0.136429 0. wherein synthesized profiles map manufacturing processes to the design metric models. as explained in Srinivasan [26]. as there is no incremental feed leading to periodicities in Fractal parameters for grinding.062795 0.Cb I.958 IO lO Synthd.696449 0. This is expected. are summarized below.355 0.425596 0.448804 1./ ? ) / 2 . Discussion of Grinding Profile Parameters The characteristics of grinding profiles. 1).067165 0. fractional Gaussian noise is used as the underlying model. Srinivasan et al.S.456197 0.9.Da II.245979 0.504878 1.697 0. 465446 -4. In addition. One of the major sources of gear vibration and noise is transmission error [14]. The parameters for the gear pair used in this study are given in Houser [9].g. this component is zero. and I. where ko is a constant stiffness [10]..003304 0. the fractal model is proved valid for ground profiles in the tolerance scales. First. the lathe gear transmission problem described in Sections 2.001549 0.296089 -4. the component due to the compliance.617923 -5.001571 0.u. Grinding profiles are used in this study.001901 0. decreases with the progression of meshing [10].978600 -5.Aa LAb I.185005 II.002104 0.002105 0.004958 0. to obtain 'synthesized transmission error'.< 1..Cb II.006872 0.944433 -5.5.009132 0.Ca I. and so on. and Rtp is the tip circle radius of the pinion.Functional Tolerancing 113 Table 4.006200 0.206314 -4.781622 -4. No Entropy (e. where the feed gives rise to a periodic component [36].498378 -4. (x) = w + e~(x) ki(x) ki(x) O<_x<_L (19) The stiffness values k(x) are calculated by assuming k(x) = ko'x.Aa II.595289 -4.E.Db 0. It also implies 'persistence' [13]. Rbg is the radius of the base circle for the gear. The component due to the profile errors is called the transmission error of 'unloaded gears' [9].695406 -5.Db 0. 10. the expression for transmission error becomes: T.793002 -4. the length of contact L is calculated as [14]: L--~g--Rb~+X/~.Da and I.004091 0.419618 --5.406432 single-pass grinding.004159 0.002140 0. for the purpose of verification.5. • The fractal parameters are in the range 1 < D} < 1.008646 0. 7.Db.Da II. (Eq.558823 -5.001697 0.448349 -4. The plots of these errors for one mesh cycle (excluding initial contact) are shown in Fig.918731 -4. the total transmission .441804 -5. For a perfect gear pair.000593 0.3.126316 -4.001403 0.Ca II. (1).004893 0.002590 0.p--Rbp--tan O(Rbg+Rbp ) (18) where q5 is the pressure angle.307492 -4. The performance comparison is presented with the aid of a design example.098579 -5. verifying the synthesis approach. As the contact ratio is unity. it means that a positive value in the past will be followed by a positive value in the future. characteristic of profiles with D~.001029 0. specifically the profiles I.Ca and I.003589 0. Energy (mm 2) REP.000862 0. Note the similarity in maximum and minimum values. Equation (1) is used to calculate 'experimental transmission error'. The experimental and synthesized profile data are used in this relationship to investigate their effects on transmission error in the course of meshing.820007 -4. However.761717 -4. Using these values.Bb II.005979 0.000889 -5.Db as error on driven).Bb I.001723 -4.Ab II.769976 -4.749470 -4.) Experimental Synthesized Experimental Synthesized I.. and the relationship between profile errors and transmission is given by Eq.408412 -5.005233 0. (19)). Performance Comparison We now transition from a novel representation of manufacturing processes to incorporating the process representations in design. as opposed to the more abstract and simplified model. and then the calculation is repeated with the corresponding synthesized errors.001506 0.001061 0.Ba I.215548 -5. in contrast to a process like milling. This indicates low (but still important) irregularity when compared to the ideal dimension of unity.Cb for another.Da I.977645 -4.003775 0.000767 0. viz.e.230689 --4.Ba II. I.173342 -5. i. The direct experimental data are superimposed on the gear teeth profiles initially.003103 0. I.010212 0.Cb I.2 and 2. W/ki(x). for one gear pair. the experimental data are used as instances of error profiles on a pair of meshing gear teeth (e.394058 -4. From the definition of the transmission error.Da as error on driver. and the overall pattern of the error spectra. Energy and entropy for gringing profiles.726974 -4. or a change in profile geometry. Following a step by step procedure for identifying and characterizing each component (analysis). The above example establishes the applicability of the methodology in synthesizing realistic part models.I. The patterns in the transmission errors reflect to a certain degree. S r i n i v a s a n et al. i. Traditional and novel mathematical tools are used to detect and estimate various structural components in a machined profile.. indicating the need for more precise gear finishing methods. Any opinions. which also yield the shorter manufacturing time. higher speeds and feeds which provide a higher rate of material removal are preferable. As the magnitude factor dominates the transmission error in this case study..g. For more critical applications. and Desktop Manufacturing Inc. by The National Science Foundation. The use of fractal parameters offers a means for quantifying the structure of the errors. while being intuitively perceptible. T r a n s m i s s i o n error: E x p e r i m e n t a l a n d s y n t h e s i z e d g r i n d i n g profiles. conclusions or recommendations expressed in this publication are . S . For example. their regularities of the corresponding fractal dimensions. Acknowledgements This material is based upon work supported. Measures for determining the 'goodness' of the synthesis are also presented. for example in a concrete mixer.D . A design example is presented to illustrate the implementation of the methodology..D. 6 o~ 2 "d / 4 2 I. the designer can compare the performance of the system under various process parameter combinations. consider the performance of gear pair I. is explicitly derived using a methodological approach. t-... An extension of the experimental component to address other manufacturing processes and parameters is also part of future research. Another benefit that can be derived from this approach is the choice of process parameter combinations that are a compromise between performance and time to manufacture. 10. Conclusions and Future Work This paper presents a methodology for assigning function-driven tolerances for mechanical components within the framework of DFM. x (ram) ! 1 16 18 Fig. . other manufacturing processes and material choices represented by the superposition model may be synthesized and analyzed to determine the process that will satisfy the gear transmission function. This result.D .Da also influences the corresponding transmission error... For example.e. findings. then used in the synthesis of realistic part models. The current research only addresses one-dimensional form errors.. which can be attributed to the larger magnitude factor. the designer would prefer to choose the parameters of I.. an NSF Presidential Young Investigator Award. using data from a grinding process. could be acceptable. error for both the experimental and synthesized profiles is increasing in a mesh cycle. and research grants from Ford Motor Company. 10-13 gin.R . x (nun) . e. a hitherto unaddressed issue in tolerances. 8. The issues of assemblability also warrant attention.D. Grant No. The significant slope in I. 114 12 12 10 g 8 s .. Texas Instruments. DDM-91 t 1372. the profile is synthesized from the parameters.. like machine tool gears for the lathe head stock. ! i 2 4 1 ~ i . The former has a larger range of transmission error..C-I. Such an approach provides a rational basis for the designer to address tolerancing problems.C versus I. this error is unacceptable. I t 16 18 0 ! 2 ! 4 T ! I i 1 6 8 10 12 14 Path of Contact. a different choice of materials. provided the performance is acceptable for the application in question.. The power of the methodology lies in abstracting the error information in terms of a minimmn set of parameters. rc . The future challenges tie in extending and/or adapting this methodology to handle two-dimensional form errors and other geometric and size errors. in part.. the magnitude of the error. 1 I 6 8 10 12 14 Path of Contact. 1990. G. Srinivasan. F. C. Tabenkin. pages 731-740. 113:233-238. New York. Brecker. 1993. SAE. 40. 5. MA. Monroe. 30. 1990. Wang. Melamed. Austin. S. Wick. Dearborn. John Wiley & Sons. Cavin. R. B. ASME. L. Nonaka. Dearborn. 12. September 1992. Srivastava. Engines. NSF. Chichester. Foundations of Digital Signal Processing and Data Analysis. June 1978. MI. 14. 11. New York. 13. Srinivasan and K. Macmillan Publishing Company. Orthonormal basis of compactly supported wavelets. Phoenix. J. 36. New York. NC. 38:54-59. 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