Case study for robust design and tolerance analysis

Transcription

1 Subject Case study for robust design and tolerance analysis DfSS.nl A good practice in development projects is to take production variation of components into account when making design choices. The properties of components, e.g. the diameter, will vary later on in mass production. Transfer functions (an important aspect in the Design for Six Sigma methodology) are useful in describing the relation between CTQ (Critical to Quality) parameters, and more technical parameters that are perhaps linked to the components. The transfer function and the expected variation in the technical parameters can then be combined to make a prediction of the variation that we should expect in the CTQ. This is known as a tolerance study. This document describes a case study of these ideas, and even how to balance the requirements on multiple CTQs. We assume that the reader is somewhat familiar with some basic concepts of statistics, such as mean and standard deviation. The case study dates from a long time back but remains illustrative. It concerns the tape deck of a video recorder, known as Charlie deck. Somewhere in the device is a rotating head that the tape passes over at high speed. This head has an axle that rotates inside a drum. Let A be the diameter of the axle and B the diameter of the drum. Let C=B-A be the clearance. With mass production there will be some variation in A and B, for it can t be the case that throughout all the years of mass production each axle will have exactly the same diameter to the nanometer. Temperature, the composition of metals used, wear-and-tear to the machines that make the axles, the Monday morning blues of factory workers all these factors bring with them an element of variation.

2 Pagina / page 2 / 6 Let s take A and B as random variables, both normally distributed. We have still to decide on their averages for the development project, and for the spread we will use reasonable values based on experience, for example, A is normally distributed with expectation mm and a standard deviation of mm, and B is normally distributed with expectation mm and a standard deviation of mm. The axles and drums are manufactured independently of one another, so we take A and B as independent random variables. An initial simple model, with which we can answer an important question: will some shafts fail to fit into the drums? This we can answer because we can determine the probability distribution of C=B-A, and calculate the extent to which the distribution of C stays above the lower limit 0, by calculating the probability that C<0. An alternative measure of the "process capability" that we are going to use is the so-called capability index, defined by k, where LSL and USL are the lower and upper specification limits, μ is the process average and σ is the standard deviation. So in our case, this is the distance from E(C) to the nearest outer limit, relative to the standard deviation of C (times three). Entire factories work with the requirement that every feature of every process must have a Cpk>1.33. This ensures that the process average remains reasonably far away from the specification limits (i.e. at least four standard deviations). L U C pk = 2 6-sigma process C pk = 1 3-sigma process Time to look more closely at our example. The cylinder has a so-called spiral-groove bearing. This has the properties stiffness S (in N/m) and torque = torque T (in mn m), defined by: The stiffness of the design should be as high as possible, but at least 1.00 N/m, whereas the torque should be as low as possible, and at most 0.30 mnm.

3 Pagina / page 3 / 6 These physics formulae use a number of new variables, shown in the table below. We ll assume these variables are normally distributed and mutually independent (except for the parameters C, S and T, because these are a consequence of the other variables). parameter description unit average st.dev. A diameter of the axis mm B diameter of the bush mm C clearance B-A mm η rotation speed of the axis rpm ω viscosity of the grease in the bearing mpas D length of the grooved part mm Note that the transfer functions for S and T, and the values in the table, are domain knowledge. Once these are known, the quantitative analysis can start. Given the current design proposal, the product distribution for S and T can be determined; see the histograms in fig. below. The vertical lines indicate the specification limits (LSL for stiffness, USL for torque). The distribution of stiffness is pretty close to the specification limit, which in practice can create problems (Cpk=0.55 for stiffness, Cpk = 1.30 for torque). We can still tweak the design. Normally speaking, the averages of the design parameters can easily be adapted: for example, the averages of A and B can be reduced or increased, which can create a better scenario in terms of stiffness. It is often expensive to reduce spread: it might be that a spread reduction for ω can only be achieved with a more expensive electric motor, while the average adaptation could be achieved simply by adjusting the voltage. To improve the design we formulate an optimization problem. - Two objective functions: Cpk of stiffness and Cpk of Torque, both of which we want to be as large as possible (i.e. the distributions to lie far from the allowable limits).

4 Pagina / page 4 / 6 - Five adjustable variables: these are the process averages of A, B, ω, η, and D. - Pre-conditions, Cpk of C be at least 1.33, and with upper and lower limits given for the permitted process averages of A, B, ω, η, and D. But how does one deal with the fact that there are two objective functions? We could make them one, by taking the minimum of the two Cpk's as the sole objective function. One strategy that offers a good insight into the possibilities is to take the stiffness as the objective function, and ``torque Cpk >x'' as a constraint, whereby you raise your x in steps and repeatedly solve the optimization problem. The balance between good stiffness and good torque is thus mapped out, letting you then make a strategic choice. Note that we now have a mathematical description of the optimization problem and the strategies. The technical implementation is not all that simple, because the evaluation of the objective function(s) and limiting conditions is not straightforward: they contain expectations and standard deviations. This you would ideally want to express analytically, which for the distribution of C is easy. But the formulae for S and T are complex, and in practice one needs to simulate to determine the expectations and standard deviations of S and T. For example by drawing values for all ``input variables, so the outcomes for S and T then provide an estimate for the average and standard deviations. Within the optimization problem you therefore have to simulate, as well as for each step in the optimization algorithm. Suppose that the averages of the design parameters may be tweaked within the following bounds. parameter lower bound upper bound Distribution st.dev. A normal B normal eta normal 0.25 omega normal 2 D normal Maximizing the Cpk for stiffness results in the following solution. Note that the result for torque is not acceptable, since the distribution is partly above the lower spec of 0.3.

5 Pagina / page 5 / 6 Maximizing the Cpk for torque results in the following solution. Note that the result for stiffness is not acceptable. The multi-objective optimization approach results in the following graph:

6 Pagina / page 6 / 6 This graph show that there is indeed a design that has good Cpk for both torque and stiffness. Simulating that design gives the following result: In this case study, we have shown that it is possible to derive a robust design using explicit transfer functions, even when these functions are quite complex. This design is called robust because the variation in production that we will encounter will affect the quality of the Charlie deck only slightly. This is not straightforward; many design choices are non-robust for variation in production.