Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies

Abstract

International audienceThis paper presents a generic approach, to generate symbolic dependency relations between the variations on the dimensional parameters for a family of over-constrained structure. We call structure a set of rigid parts interconnected together with mechanical linkages. A structure is over-constrained when the size of parts are not independent of each other.To achieve our goal, we propose the following method. Firstly, parameters are divided into two categories: dimensional parameters and configuration parameters. Dimensional parameters represent the size of the parts and configuration parameters represent the relative position between the parts. Symbolic closed-loop equations model the geometric problem. They represent the dependency between two types of parameters. To generate symbolic equations under dimensional parameters, we use a generic method of elimination, based on Gröbner basis computation. These symbolic relations are called “compatibility equations”, which guarantee the existence of the studied mechanical assembly. Generally, there are many more dimensional parameters than compatibility equations.In this paper, compatibility equations are regarded as implicit functions. We apply the implicit function theorem to generate symbolic differential equations which govern the variations of the clearance between the components. For that, the set of dimensional parameters is separated into two subsets: independent and dependent dimensional parameters. Maximum and minimum clearances are calculated by simulating harmonic variations of the rigid parts size. The solution of the differential equations allows a fast simulation.The presented generic approach is applied on a 2D over-constrained assembly. Symbolic and numerical results show the feasibility of this generic approach

Clearance vs. tolerance for rigid overconstrained assemblies

“…The first step is to check that (u 0 , 0, p 0 , 0) is a solution of (13). Indeed, thanks to (2), (12) and to definition 1…”

“…The second step is to investigate the inversion of the partial derivative H (j,p,λ) at (u 0 , 0, p 0 , 0). By derivating (12)…”

“…This section makes use of theorem 1 to compute numerical variations of clearance j according to the tolerance, which is the numerical variations of dimension u. The guide line stated by [12] is to deal with an ordinary differential equation, as it is particularly adapted to numerical integration.…”

“…Symbol ′ in (23) is the derivation with respect to real variable t. The partial derivative H (j,p,λ) can be detailed in terms of function G by using (14). The partial derivative H t can be detailed in terms of function G by using (12) and the chain rule. Figure 6: Case study number 1.…”

“…For numerical investigation purpose, a simple and efficient choice introduced by [12] is to design the curve t → u(t) as a harmonic function oscillating around the nominal value u 0 . In addition, a homotopy effect is used to start from the nominal solution by using the mapping…”