Extended nonlinear analytical models of compliant parallelogram mechanisms: third-order models

This paper proposes extended nonlinear analytical models, third-order models, of compliant parallelogram mechanisms. These models are capable of capturing the accurate effects from the very large axial force within the transverse motion range of 10% of the beam length through incorporating the terms associated with the high-order (up to third-order) axial force. Firstly, the free-body diagram method is employed to derive the nonlinear analytical model for a basic compliant parallelogram mechanism based on load-displacement relations of a single beam, geometry compatibility conditions, and load-equilibrium conditions. The procedures for the forward solutions and inverse solutions are described. Nonlinear analytical models for guided compliant multi-beam parallelogram mechanisms are then obtained. A case study of the compound compliant parallelogram mechanism, composed of two basic compliant parallelogram mechanisms in symmetry, is further implemented. This work intends to estimate the internal axial force change, the transverse force change, and the transverse stiffness change with the transverse motion using the proposed third-order model in comparison with the first-order model proposed in the prior art. In addition, FEA (finite element analysis) results validate the accuracy of the third-order model for a typical example. It is shown that in the case study the slenderness ratio affects the result discrepancy between the third-order model and the first-order model significantly, and the third-order model can illustrate a non-monotonic transverse stiffness curve if the beam is thin enough.

Constraint-force-based approach of modelling compliant mechanisms: principle and application

Numerous works have been conducted on modelling basic compliant elements such as wire beams, and closed-form analytical models of most basic compliant elements have been well developed. However, the modelling of complex compliant mechanisms is still a challenging work. This paper proposes a constraint-force-based (CFB) modelling approach to model compliant mechanisms with a particular emphasis on modelling complex compliant mechanisms. The proposed CFB modelling approach can be regarded as an improved free-body- diagram (FBD) based modelling approach, and can be extended to a development of the screw-theory-based design approach. A compliant mechanism can be decomposed into rigid stages and compliant modules. A compliant module can offer elastic forces due to its deformation. Such elastic forces are regarded as variable constraint forces in the CFB modelling approach. Additionally, the CFB modelling approach defines external forces applied on a compliant mechanism as constant constraint forces. If a compliant mechanism is at static equilibrium, all the rigid stages are also at static equilibrium under the influence of the variable and constant constraint forces. Therefore, the constraint force equilibrium equations for all the rigid stages can be obtained, and the analytical model of the compliant mechanism can be derived based on the constraint force equilibrium equations. The CFB modelling approach can model a compliant mechanism linearly and nonlinearly, can obtain displacements of any points of the rigid stages, and allows external forces to be exerted on any positions of the rigid stages. Compared with the FBD based modelling approach, the CFB modelling approach does not need to identify the possible deformed configuration of a complex compliant mechanism to obtain the geometric compatibility conditions and the force equilibrium equations. Additionally, the mathematical expressions in the CFB approach have an easily understood physical meaning. Using the CFB modelling approach, the variable constraint forces of three compliant modules, a wire beam, a four-beam compliant module and an eight-beam compliant module, have been derived in this paper. Based on these variable constraint forces, the linear and non-linear models of a decoupled XYZ compliant parallel mechanism are derived, and verified by FEA simulations and experimental tests.