Introduction
Tolerance is the allowable variation for any given size in order to achieve a proper function, and equals the difference between lower and upper limit dimensions. Tolerances can be defined in various ways e.g nominal size, basic size, bilateral tolerance, unilateral tolerance, etc.
Parts of a mechanical product / object are designed in order to make them function properly. The moving parts have a definite relationship with each other; how precisely they work together as an entity determines the quality of that product. Tolerance plays an important role in determining the quality; the tighter the tolerance, better the quality of the object / assembly.
Why is tolerance consideration important?
Tolerance is a critical bridge between designing and manufacturing. All manufacturers would like to maintain high quality in their products. However, the cost of production grows exponentially when the manufacturer strives to attend ‘zero tolerance’. Tolerance analysis is therefore very critical for reducing manufacturing cost while maintaining desired product quality. As an example, a tolerance of 2 mm is quite acceptable while considering the overall length of an automobile. However, a tolerance (play) of 2 mm between a block and the piston of the cylinder is completely unacceptable, as it would affect the performance of the engine drastically. In general, the tighter the tolerances, the more expensive the components and the machines that are needed to produce these components. It is therefore necessary to achieve optimum tolerances so that the performance of the product is not hampered and that it becomes economically viable for the manufacturer.
Both the ANSI and ISO have devised various standards to define tolerance. In addition, there are international standards (International Tolerance Grade) that define their own grades of tolerances.
Tolerance Stack Up
Mechanical tolerances are fact of life and cannot be avoided. While manufacturing complex mechanical products, it is essential to divide them into components and sub assemblies. Some of the parts are bound to have relative motion between parts, intermeshing, material differentiation and so on. Unfortunately, once the product is divided into separate parts for the sake of manufacturing, how these parts assemble together to create the overall characteristics and features of the product become a key concern. And this is where the problem arises. Once these components are assembled back, each component brings with it its own tolerance limit. This variability in assembly dimensions is commonly referred to as tolerance stack up where tolerance refers to the allowable limits of variability assigned to each part and stack up refers to the overall variation in assembly dimension that results. In simple words, tolerance stack up is the cumulative effect of adding different tolerances. As you can visualize, tolerance stack up can result in serious functional and manufacturing problems if it is not considered and controlled.
Tolerance design is the process of designing products to guard against and avoid as far as possible the negative consequences of tolerance stack up. It involves both the development of the part decomposition and the assignment of tolerances to individual component dimensions.
Tolerance stack up analysis is the science of establishing the dimensional relationships within a part or assembly. The purpose of stack up analysis is to establish the dimensional relationships within a part or assembly. It enables tolerances in the parts to be optimized while maintaining functionality and maximum part interchangeability at a minimal manufacturing cost. One of the most important reasons for using tolerance stack analysis is that problems can be discovered and solved on paper rather than in the prototype or production, and thus evaluation and modification can be done at the early stage of design. Thanks to modern stack up analysis software, it is become simpler to make revisions at the design stage itself.
The Tolerance Stack up Models
Optimum tolerance stack up analysis is quite challenging, and there are various mathematical models that are used. It is pertinent to note that none of the models described below are perfect; each has its own set of strengths and weaknesses and are generally chosen depending on the movement / meshing of the parts.
The vector Loop Model
In this model for tolerance analysis, a graph-like schematization is adopted where any relevant linear dimension in the assembly is represented by a vector. Each vector represents either a component dimension or assembly dimension. The vectors are arranged in chains or loops representing those dimensions that stack together in determining the final functional requirements of the assembly. Three types of variations are modeled in the vector loop model: dimensional variations, kinematic variations, and geometric variations. Stack-up functions are built by combining the variations associated with vectors involved in each chain into mathematical expressions, which can then be solved with different approaches.
The Unified Jacobian Torsor Model
The unified Jacobian-Torsor model uses the Torsor model for tolerance representation and the Jacobian matrix for tolerance propagation. The matrix model is based on the positional tolerancing and the technologically and topologically related surfaces criteria, while the Torsor model comprises three translational vectors and three rotational vectors. The unified method is based on the effects that small displacements of a series of functional elements have on working requirements of the product. It is then expressed as a set of equations in matrix form which provides the desired analysis relationship.
The unified Jacobian–Torsor model has been very effective in predicting worst case computer-aided tolerancing.
The Variational Model
A variational model is a computer representation of a variational class and stands for a collection of different instances of the part or assembly modeled in CAD. It is based on parametric geometric modelling, where geometry can be modelled by mathematical equations that allow shape and size attributes to be changed and controlled through a reduced set of parameters. The variational modelling starts by reproducing small variations in an assembly part, within the ranges defined by a given tolerance. The variational model is general in that it allows performing analysis by both worst case and statistical approaches. The approach of this model aims at deriving the explicit mathematical representation of the geometry of each tolerance region, which is done through displacement matrices (matrices that describe the small displacements without violating the tolerances.)
Software Tools for Tolerances and Stack Analysis
Powerful CPUs, coupled with advances in CAD and CAE software has proved to be a game changer for tolerance and stack up analysis. What used to be a manual chore has now been simplified with sophisticated software. Tolerance and Stack up Analysis software allows design engineers to provide product development teams reliable information that help them improve product quality, accelerate product maturity and achieve optimum productivity.
The better class of tolerance analysis software offers following features:
The markets today demand shorter product lifecycles, faster time-to-market and tighter profit margins. Tolerance and Stack Analysis software facilitate manufacturers achieve this goal.