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Optimising tolerance allocation for mechanical components correlated by selective assembly

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Abstract

Selective assembly can enlarge the tolerances of mechanical components for easier manufacturing. However, the non-independent dimensions of correlated components make it difficult to optimise tolerance allocation for an assembly. This paper proposes a solution for this constrained optimisation problem consisting of tolerances and non-independent dimensions as design variables. The approach is to develop a simplified algorithm applying a Lagrange multiplier method to evaluate the optimal tolerances efficiently. The solution is shown to be a global optimum at the given correlation coefficients. The correlation coefficients are key elements in determining the optimal solution, which is demonstrated in the given examples. The results are helpful in designing tolerances for selective assembly.

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Abbreviations

A j :

coefficient matrix off j

B i :

coefficient of cost function

C :

total manufacturing cost function

C i :

manufacturing cost function forx i

F j :

thejth dimensional constraint function

f j :

thejth quadratic constraint function

f :

quadratic constraint vector

H j :

thejth Hessian matrix

J kj :

element ofn×m Jacobian matrix

L :

Lagrangian

m :

number of assembly dimensions

n :

number of component dimensions

p :

number of equality dimensional constraints

T :

tolerance vector of component dimensions [mm] or [°]

\(T_{x_i } \) :

tolerance ofx i [mm] or [°]

\(T_{Z_j } \) :

tolerance ofZ j [mm] or [°]

x :

component dimension vector

x :

midpoint vector

x i :

component dimension [mm] or [°]

x i :

midpoint ofx i [mm] or [°]

Z j :

assembly dimension [mm] or [°]

β j :

confidence coefficient forZ j

γ i :

confidence coefficient forx i>

δ j :

given design value ofZ j [mm] or [°]

λ :

Lagrange multiplier vector

λ j :

thejth Lagrange multiplier

λ*:

Lagrange multiplier vector at the optimum solution

\(\rho _{x_i x_k } \) :

correlation coefficient forx i andx k

σ x :

standard deviation vector

σ * x :

standard deviation vector at the optimum solution

σ 0 x :

candidate point satisfying the constraintsf(σ * x )=0

\(\sigma _{x_i } \) :

standard deviation ofx i

References

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Authors and Affiliations

  1. Department of Die and Mold Engineering, National Kaohsiung Institute of Technology, 415 Chien-Kung Road, 80782, Kaohsiung, Taiwan, ROC

    Ming-Shang Chen

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  1. Ming-Shang Chen
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Chen, MS. Optimising tolerance allocation for mechanical components correlated by selective assembly. Int J Adv Manuf Technol 12, 349–355 (1996). https://doi.org/10.1007/BF01179810

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  • DOI: https://doi.org/10.1007/BF01179810

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