Force Analysis of the Overconstrained Mechanisms Based on Equivalent Stiffness Considering Limb Axial Deformation

Considering the limb axial deformation, taking typical 2SS + P and 7-SS passive overconstrained mechanisms, 2SPS + P and 7-SPS active overconstrained mechanisms, and 2SPS + P and 7-SPS passive-input overconstrained mechanisms as examples, a new force analysis method based on the idea of equivalent stiffness is proposed. The equivalent stiffness matrix of passive overconstrained mechanism is derived by combining the force balance and deformation compatibility equations with consideration of axial elastic limb deformations. The relationship between the constraint wrench magnitudes and the external force, limb stiffness is established. The equivalent stiffness matrix of active overconstrained mechanism is derived by combining the force balance and displacement compatibility equations. Here, the relationship between the magnitudes of the actuated wrenches and the external force, limb stiffness is investigated. Combining with the equivalent stiffness of the passive overconstrained mechanism, an analytical relationship between the actuated forces of passive-input overconstrained mechanism and the output displacement, limb stiffness is explored. Finally, adaptability of the equivalent stiffness to overconstrained mechanisms is discussed, and the effect of the limb stiffness on overconstrained mechanisms force distribution is revealed.

Statistical tolerance analysis applied on overconstrained mechanisms with form deviations

One method for modeling geometric variations in hyperstatic (i.e. overconstrained) systems is to use sets of constraints. Different models have been developed in this way, e.g. domains, T-maps, and polytopes. In general, if the intersection of the contact constraints between two parts potentially in contact is nonempty, the parts can be assembled without interference, and their relative positions determined. In this study, the polytope method is used with a statistical approach to define the behavior of an assembly. In the first part, geometric variations including form deviations of individual parts are defined. The relations between these variations resulting from the architecture of a mechanism are then defined. In the second part, contact constraints are introduced and the general method to conform the constraints into double description polytopes is presented. The general process to simulate the compliance of the mechanism with respect to functional conditions is described. A failure rate is obtained for a simulated population of manufactured parts using the Monte Carlo method. In the third part, an application to a flange is described, an example from an industrial case study. We show how to take advantage of double description of polytopes when simulating the assembly and the misalignment of the two parts that make up the flange. Finally, we present our conclusions and prospects for future studies.

Degree of Freedom and Dynamic Analysis of the Multi-Loop Coupled Passive-Input Overconstrained Deployable Tetrahedral Mechanisms for Truss Antennas

Recently, the truss antennas with deployable tetrahedron unit mechanisms have been successfully applied in orbit, owing to the advantages of large calibers, high accuracy, and large folding ratios. As multiloop coupled mechanisms, deployable tetrahedral mechanisms have multiple different output links, whose supporting limbs connecting output links and the base are mutually coupled. These mechanisms are also called the passive-input overconstrained mechanisms because their passive torsion springs are used as drivers and because the number of the drivers contained is more than the degrees of freedom (DOFs). In this work, a method based on the equivalent concept of first link-removing and then restoring is proposed for the DOF analysis of the multiloop coupled deployable tetrahedral mechanisms. With one coupled chain removed, the equivalent serial chains between the coupled components and the base are established in the remainder of the mechanisms. Then, the coupled chain removed is restored and the equivalent of the multiloop coupled mechanisms is obtained. The Lagrange method is used to establish the dynamic equation of the passive-input overconstrained mechanisms; the influence of the stiffness and number of torsion springs on the unfolding motion is examined.

A Double-Faced 6R Single-Loop Overconstrained Spatial Mechanism

This paper deals with a 6R single-loop overconstrained spatial mechanism that has two pairs of revolute joints with intersecting axes and one pair of revolute joints with parallel axes. The 6R mechanism is first constructed from an isosceles triangle and a pair of identical circles. The kinematic analysis of the 6R mechanism is then dealt with using a dual quaternion approach. The analysis shows that the 6R mechanism usually has two solutions to the kinematic analysis for a given input and may have two circuits (closure modes or branches) with one or two pairs of full-turn revolute joints. In two configurations in each circuit of the 6R mechanism, the axes of four revolute joints are coplanar, and the axes of the other two revolute joints are perpendicular to the plane defined by the above four revolute joints. Considering that from one configuration of the 6R mechanism, one can obtain another configuration of the mechanism by simply renumbering the joints, the concept of two-faced mechanism is introduced. The formulas for the analysis of plane symmetric spatial triangle are also presented in this paper. These formulas will be useful for the design and analysis of multiloop overconstrained mechanisms involving plane symmetric spatial RRR triads.

Overconstrained Mechanisms Derived From RPRP Loops

This paper addresses the assembly strategy capable of deriving a family of overconstrained mechanisms systematically. The modular approach is proposed. It treats the topological synthesis of overconstrained mechanisms as a systematical derivation rather than a random search. The result indicates that a family of overconstrained mechanisms can be constructed by combining legitimate modules. A spatial four-bar linkage containing two revolute joints (R) and two prismatic joints (P) is selected as the source-module for the purpose of demonstration. All mechanisms discovered in this paper were modeled and animated with computer-aided design (CAD) software and their mobility were validated with input–output equations as well as computer simulations. The assembly strategy can serve as a self-contained library of overconstrained mechanisms.

Force Analysis of Spatial Single Closed-Loop Overconstrained Mechanisms

Taking the Bennett and Schatz mechanisms as examples, force analyses of spatial single closed-loop (SSCL) overconstrained mechanisms are demonstrated aiming to obtain the driving forces/torques and joint reactions of this kind of mechanisms. Firstly, regarding the SSCL overconstrained mechanisms as parallel mechanisms with two supporting limbs, the constraint wrenches and actuation wrenches imposed on the moving platform by the two limbs are discussed, and the mobility of each mechanism is analyzed based on the screw theory. Then, the compliance matrices of the limbs’ constraint wrenches are derived, which contribute to solve the statically indeterminate force problem of the mechanisms. Next, by combining the force and moment equilibrium equation of the moving platform with the deformation compatibility equation of the corresponding mechanism, the magnitudes of all constraint wrenches and actuation wrenches are solved. Furthermore, the driving forces/torques and joint reactions are derived. Finally, the numerical and simulation results of the two mechanisms are presented.