Traditional tolerancing considers the conformity of a batch when the batch satisfies the specifications. The characteristic is considered for itself and not according to its incidence in the assembly. Inertial tolerancing proposes another alternative of tolerancing in order to guarantee the final assembly characteristic. The inertia I2 = \sqrt{\delta^2 + \sigma^2} is not toleranced by a tolerance interval but by a scalar representing the maximum inertia that the characteristic should not exceed. We detail how to calculate the inertial tolerances according to two cases, one aims to guarantee an inertia of the assembly characteristic the other a tolerance interval on the assembly characteristic by a Cpk capability index, in the particular but common case of uniform tolerances or more general with non uniform tolerances. An example will be detailed to show the results of the different tolerancing methods.
Deep Dive into Inertial tolerancing and capability indices in an assembly production.
Traditional tolerancing considers the conformity of a batch when the batch satisfies the specifications. The characteristic is considered for itself and not according to its incidence in the assembly. Inertial tolerancing proposes another alternative of tolerancing in order to guarantee the final assembly characteristic. The inertia I2 = \sqrt{\delta^2 + \sigma^2} is not toleranced by a tolerance interval but by a scalar representing the maximum inertia that the characteristic should not exceed. We detail how to calculate the inertial tolerances according to two cases, one aims to guarantee an inertia of the assembly characteristic the other a tolerance interval on the assembly characteristic by a Cpk capability index, in the particular but common case of uniform tolerances or more general with non uniform tolerances. An example will be detailed to show the results of the different tolerancing methods.
The traditional tolerancing considers the conformity of a batch when the batch satisfies the specifications. The characteristic is considered for itself and not regarding its incidence on the final assembly resultant. It has been showed that inertial tolerancing (I = σ 2 + δ 2 which is no more based on a [Min Max] interval but on the Taguchi loss function) proposes another tolerancing method to guarantee the final assembly while allowing larger variability in the case of centered production.
This paper proposes a method to calculate the inertial tolerances of the components of a 1D mechanical assembly chain in two cases:
-we want to guarantee an inertial tolerance on the final assembly, -we want to guarantee a minimum of the Cpk index on the tolerance interval [Min; Max] of the assembly characteristic.
Different cases are considered, even the general case where the components tolerances are not uniformly distributed. A comparison with the traditional tolerancing will show the difference on the allowed variability of components. The two considered cases have different hypothesis of application. We will discuss on the choice of using the first or the second approach.
An industrial case of application will be treated as an example.
The aim of tolerancing is to determine an acceptation criterion on the components characteristics x i to guarantee the quality of the assembly resultant Y. In the case of a good design, when the x characteristic is produced on the target, the quality is optimal. As x gets an offset from the target, the function of the assembly will be more sensitive to the conditions of use and the environment, and can lead to a non-satisfaction of the customer. Taguchi demonstrated that the financial loss associated to an offset from the target is proportional to the square of this offcentering L = K(T-X)² . (Pillet et al, 2001) shows that in the case of a batch, the financial loss associated is L = K(σ² + δ²). Then he defines
This function is called Inertia by analogy to the mechanical inertia. Here I x represents the inertia of the x characteristic, σ x is the standard deviation of the batch distribution and δ x is the offset of the mean to the target.
To qualify the capability of a process with the inertial tolerancing, two capability indices have been defined:
which indicates the capability of a centered process.
which indicates the capability considering the process off-centering.
Compared to the traditional tolerancing, the proposed approach of the inertial tolerancing is quite different. The aim is no more to guarantee a rate of parts out of tolerance, but to guarantee the centering of components around the target in order to guarantee the quality of the assembly. The reflection is no more based on the proportions out of tolerances but on the inertias of the components, the normality of the batch distribution is no more a necessary criterion.
For the tolerancing of assembly systems, the problem consists of finding the elementary characteristics x i of the components in order to obtain a final characteristic Y satisfying the functional condition of the assembled product for the customers needs. As a general rule, it is possible to approximate the system behavior around the target by a linearization at the first order. The final characteristic behavior can be expressed by the following relation:
Where α i is the influence coefficient of the component i on the assembly resultant Y, α 0 is the target value of Y and n is the number of components in the assembly. For the computation of the components tolerances, the general case will be considered where tolerances are not uniformly distributed with the use a difficulty coefficient β i ≥ 1, also called feasibility coefficient. The simplified but common case will also be presented, where the tolerances are uniformly distributed β i = 1, and the incidence coefficients are all equal α i = 1 .
Before the presentation of the inertial tolerancing, here is a brief reminds of the traditional tolerancing methods of assembly systems. Three commonly used traditional methods are presented.
In this case, one considers that the final characteristic of the assembly will be respected in any cases of assembly. In the general case where tolerances are nonuniformly distributed, the β i coefficients will be used for the components. The tolerance expression of a components is R xi = β i . R x , the assembly resultant is then:
Where R Y represents the tolerance interval of the functional condition of the assembly, and R xi is the tolerance interval of the components. Tolerances can be distributed regarding different methods by changing the β i coefficients (Graves, 2001):
-Uniform distribution of the tolerance, -Considering the tolerances of standard components, or conception rules, -Proportional to the square of the nominal length, -Considering the process capabilities, In the case of uniform distribution of the tolerance β i = 1 , and same incidences o
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