Effects of dimensional errors on compliant mechanisms performance by using dynamic splines

Highlights

• Use of modified dynamic spline formulation for the simulation of compliant assemblies.

• Influence of dimensional errors on the performance of compliant assemblies.

• Test of the proposed methodology on a constant-force compliant slider crank mechanism.

Abstract

The paper deals with the description of a methodology for investigating the effects of the dimensional errors on the performance of compliant assemblies in order to perform an optimal allocation of tolerances. The proposed approach is based on the modeling of flexible elements using the dynamic spline formulation which is able to take into account the nonlinear compliance of the system using a small set of parameters. Starting from the description of the elasto-kinematic behavior of the system written in terms of Lagrange equations, the deduction of sensitivity coefficients is discussed and an algorithmic description of the tolerance allocation scheme is presented. In order to give an example of application, the methodology has been successfully applied to the optimization of a constant-force compliant mechanism.

Introduction

A compliant mechanism is a device which gains mobility thanks to the deformation of its links and flexible components rather than through the use of specific kinematic pairs as in standard mechanisms with all rigid bodies [1]. From this point of view, a compliant mechanism does not have any degree of freedom in the strict sense, but however preserves the mobility thanks to the specific shape of its parts which produce controlled deformation of the entire structure. In particular, the displacement of the structure is based on the presence of flexible elements that deform and store energy during movement. This energy can be used for driving the relative motion back to the original undeformed configuration or to produce a controlled reaction. The compliant mechanisms are easy to manufacture and easy to assemble. They can be produced by means of common molding of plastic material or by cutting of metal sheets. Moreover, they allow high performances in terms of accuracy, reliability, low wear, low weight, low noise and low maintenance. Thanks to all these characteristics, the compliant mechanisms are used in many simple and complex applications.

The main challenge in the design of compliant mechanisms concerns the careful definition of morphological and structural properties of components in order to achieve the desired articulation and required functionality [2]. In general, the proper design of a compliant mechanism is more complex than that of a standard one because kinematics and elastic performance are strictly related and the dimensions of the deforming links affect both of them together with mechanical resistance.

From this point of view, the presence of unpredictable dimensional errors on the manufacturing of a compliant mechanism may influence both kinematic and elastic performances. For this reason, it is very important to have a deep understanding of the sensitivity of these errors on the global performance of the device. This knowledge may be useful for a comprehensive allocation of tolerances in the design phases, for choosing the manufacturing technology and for assessing the actual precision.

According to the scientific literature, the design methodologies of compliant mechanisms are often addressed using the pseudo-rigid-body approach [3], [4], [5]. According to this methodology, the compliant mechanism is associated to a system with all rigid bodies, kinematic pairs and concentrated elastic elements that reproduce the global behavior of the device in a satisfactory manner. The pseudo-rigid-body model allows the application of the common design techniques for rigid-body mechanisms for both synthesis and analysis purposes.

The definition of a pseudo-rigid-body mechanism is a good compromise between simplicity in modeling and accuracy in the estimation of functional behavior. On the other hand, it has three main drawbacks. First of all, in many cases, it is not suitable for very large deformation configurations in the whole range of movement, since the approximation using concentrated articulations may lead to unacceptable errors due to linear stiffness approximations [6]. Secondly, the approach is suitable for static and quasi-static assessment, but may give very inaccurate results in dynamic simulations in case of a bad approximation of the inertia distribution of the moving parts. On the other hand, in the last year, some authors reported successful implementation of dynamic behavior using pseudo-rigid models [7]. Moreover, the 3D structure with complex shapes may be too difficult to be modeled using simple rigid bodies.

Another possible approach in the design of this type of mechanisms makes use of finite element modeling ([8] as a recent example of application). This methodology allows a precise and accurate description of the shape and the elastic behavior of each link of the device but requires considerable computational resources. This limitation is mainly due to the need for complex formulations in order to take into account the highly nonlinear effects of large displacements. Finite elements can be suitable for refinement analyses but are often too computationally onerous for preliminary design and optimization and for interactive assessment and simulation.

In a recent paper [9], another methodology has been introduced and tested on a special class of compliant mechanism. This novel approach makes use of the dynamic spline theory [10] for approaching the design and simulation of compliant assemblies with a computational efficient formulation (faster than finite elements), but able to achieve more precise results with respect to the pseudo-rigid-body model in the whole range of displacement, taking into account nonlinear behavior. The methodology is suitable for conceptual design, preliminary reviews and simulation of compliant systems or similar devices based on the use of compliant assemblies.

Starting from this brief overview, the purpose of the paper is to introduce a methodology for addressing the computation of the influence of dimensional errors on the global performance of a compliant mechanism. The problem of the estimation of the effects of these errors is very important in the design of precise mechanisms [11]. The methodology makes use of the dynamic spline approach [10] which has been improved by the authors for the specific application, by introducing a more efficient mathematical formulation based on Hermite interpolants. The mathematical description has been integrated into a constraint-based approach for the deduction of the equations describing the kinematics and a sensitivity analysis of the geometrical variations has been performed.

The paper is organized as follows. In the first part, the basics of the dynamic spline theory are presented focusing on the original approach introducing Hermite interpolants. In the second part, the generic approach for deducing the elasto-kinematic equations of a compliant mechanism is addressed. In the third part, the computation of the sensitivity coefficients which relate the dimensional errors to performance is presented. In the last part, an example of application for a constant-force compliant slider mechanism is presented.