Jean-François Rameau

Dimensional perturbation of rigidity and mobility

Abstract Mechanisms, defined as assemblies of dimensioned rigid bodies linked by ideal joints, can be partitioned in three mobility states: the rigid state (where bodies can have only one position relative to each other), the mobile state (where bodies can move relatively to each other) and the impossible state (where bodies dimensions and specified joints cannot lead to a feasible assembly). It is also clear that although bodies dimensions can vary in a continuous way, assemblies may experience quite abrupt changes across those states. This paper proposes a new approach to this problem with the goal of being able to predict the mobility class of an assembly of arbitrary complexity, and how it can be affected by a perturbation of the dimensions of its bodies. It does so by proposing a simple and general state transition framework including the three above defined states and seven transitions describing how a dimensional perturbation can affect them. Using this framework, the mobility of a mechanism is easier to capture and predict, using only dimensional ( u ) and positional ( p ) parameters involved in an appropriate equation ( F ( u , p ) = 0 ). This is achieved by focusing on how F ( ) behaves when u and p get perturbed, and the impact of this reaction on the mobility state of the assembly. As a result of this more mathematic approach to the problem, previously used notions of iso-constraint, over-constraint and paradoxical assembly, traditionally used to describe such assemblies, can be rigorously defined and thus clarified.

Clearance vs. tolerance for rigid overconstrained assemblies

Abstract In order to manage quality, companies need to predict performance variations of products due to the manufacturing components deviations. Usually, to enable the assembly of overconstrained mechanical structure, engineers introduce clearance inside joints. We call mechanical assembly, a set of undeformable components connected together by mechanical joints. This paper presents a solution: firstly, to compute the minimum value of clearance for any given components sizes, and, secondly, to simulate variation of the minimum clearance value when the components dimensions vary between two limits. To achieve this goal, a regularized closure function G is defined. It depends on dimensional parameters, u , representing components dimensions, on positional parameters, p , representing components positions and on clearance parameters, j , representing mechanical joints clearance. A constrained optimization problem is solved to determine the minimum clearance value. An imaginative solution based on numerical integration of an ordinary differential equation is proposed to show the clearance variation. The method is designed to be used during the preliminary phase of overconstrained assemblies design. An advantage is the small number of input data unlike the tolerance analysis dedicated software.