# Compliant Systems: Mechanics of Elastically Deformable Mechanisms, Actuators and Sensors

## Book Preface

Compliance is no longer considered a disadvantage of mechanical systems. Instead, the re-sulting advantages, such as the utilization of elastic restoring forces, possibilities for energy storage, structure coherence etc., are used specifically to provide a system with new qualita-tive properties. In many application areas, such as in medical technology and human-machine interaction, specific and differentiated compliance is even a requirement for tech-nical systems ([7], [125]). Conventional rigid-body mechanisms are being successfully sup-plemented or replaced more and more with compliant systems even in classical branches of mechanical engineering, often to take over motion or force transmission tasks, for example in gripping devices (Fig. 1.1). This tendency is being facilitated by the development of new materials and corresponding production technologies. Highly-elastic materials allow for actuators that can be designed and integrated into a compliant system in such a way that such systems take on inherent actuator properties. In combination with the use of functional mate-rials, a system can be given inherent sensing properties, while allowing for a compact design and higher multi-functionality of the system as a whole.

By first investigating thoroughly its complex behavior, both in deformation and motion, a focused application of the actuators and functional materials can be guaranteed, while also ensuring a sensitive design for the compliant system as a whole. The most suitable method is by using model-based investigations, especially if an analytical model can be constructed to this end. The complex relationships between different parameters can thus be written in a transparent way, and dependencies between them can be revealed. Even if a solution cannot be achieved entirely through analysis, the resulting relationships between the mechanical parameters of a system can help in understanding it and comprehending its behavior. Com-pliant systems are considered under the effects of different loads, which are mostly caused by moments and forces or distributed moments and forces designated as line loads or area loads. These loads as well as other vector parameters, such as radius vectors or displacements are characterized by both a direction and a value. The terms for the vectors are given in bold text, e.g. F for a force vector, while the values of the vectors or scalar parameters are given in italics, such as the value of a force: F. In order to differentiate them from vectors, matrices are written in bold and underlined: T.

In the following chapters, classifications of compliant systems are first presented, which have been collected and systematized in the field of compliant mechanisms and actuators. Compliant systems with linear elastic properties are the focus of the modeling. Linear theory, which leads to the linear differential equations, is used to describe the actual behaviors of elastic deflections in a system undergoing small deflections. For large deflections, non-linear theory should be used when constructing a usable model for given deflections. Both methods of modeling are based on classical methods and have been both expanded upon and methodi-cally generalized, in order to account for special cases, while also providing a formalism for simplifying the modeling process.

Before choosing an analytical method for describing the deformation of a compliant sys-tem, the character of this system, mirrored in its geometry and material, should first be estab-lished. In this step, a decision is made about which parts of the system must be modeled as compliant bodies and which as rigid bodies, thus excluding the compliant system parts to be modeled from the mechanical system as a whole. Subsequently, a decision must be made about which modeling methods are suitable for a given situation, based on the behavior and the uses of the system. The behavior of a system is understood here as both the deformation behavior of a compliant system part, as well as its motion behavior. The expected defor-mations, regarded as either large or small deflections, should be modeled accordingly using either a non-linear or linear theory. The intended application area of the system and corre-sponding objectives should also be taken into account. The next important step before formu-lating a mathematical model is determining the boundary conditions. Decisions are made here about where in the system loads will be applied, and which loads can be modeled as idealized concentrated forces and moments, or as distributed forces and moments. This also includes decisions about the bearing of the compliant system parts, which determines the boundary conditions for forces and moments.

After completing the modeling process, the calculated results of the model and the mod-el-based simulation should be evaluated, based on the assumptions made and the starting requirements. If, for example, a compliant element with a cross-section measuring a third of its length is described using the theory for thin rods, it can be assumed that the results will correlate less accurately, if at all, than real deformations in an actual thin rod. The success of a model-based investigation depends on a keen examination of modeling method theory and a critical consideration of the results.

Based on existing research, a simple, usable method, based on the rigid-body replacement approach, of synthesizing compliant mechanisms with concentrated compliance is presented in the final chapter. The numerous possibilities of realizing a flexure hinge are first listed and evaluated based on their uses in precision engineering technology. A synthesis method based on the rotation angles of all hinges is especially suitable for mechanisms with different opti-mized flexure hinge contours, and is explained using the example of a four-bar mechanism.

The derivations of the modeling methods and their solutions to the examples presented are kept both clear and concise, in order to minimize the time and effort required to become acquainted with the material and to learn the methods. The calculations of mathematical equations are carried out using the software Mathematica® and MATLAB®. Finite elements method (FEM) simulations are performed with ANSYS Workbench®. Where possible, ana-lytical solutions have been preferred over numerical solutions.

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November 22, 2021 |