# Insights into Bending Stiffness Modeling of Elementary Flexure Hinges
## 1 Introduction  
In traditional mechanics, mechanisms usually consist of rigid links connected by joints based on the principle of conjugate surfaces, typically lubricated and in sliding contact. Recent research efforts have been directed toward designing mechanisms and mechanical systems at different scales, from macro to nano. However, they typically have some drawbacks, such as assembly, friction, and lubrication problems. Among the technical solutions proposed to overcome them, compliant mechanisms (CMs) have become increasingly popular in mechanical design, especially for micro and nano manipulators in the measurement industry. CMs are mechanisms in which conventional kinematic pairs are replaced by flexure hinges [1]. They have been extensively studied in the scientific literature, with examples in many fields of application and with different objectives: mechanisms featuring constant force [2] and multistable equilibrium [3], micro-electromechanical systems (MEMS) [4], positioning stages and precision grippers [5; 6; 7], micro/nano manipulators [8], fast servo tools for precision machining [9], servo valves [10] and other devices in the fields of energy harvesting [11], micro-vibration suppression [12], alignment of optics and robotic actuation [13], and many others. In general, planar CMs have received more attention than their spatial counterparts. In fact, the former are more popular because they can be easily manufactured with EDM (Electrical Discharge Machining) technology. Moreover, their position kinematics can be more easily modeled, allowing for easier implementation in small-scale mechatronic systems that make use of flexures. Many are the geometries of flexure hinges that have been studied: circular [14], elliptical [15], corner-filtered [16], parabolic and hyperbolic [17], cycloidal [18], filledted Vshaped [19], conical with a generalized model [20; 21] and various hybrid flexure hinges [22; 23]. Some of these geometries are shown in the chart of Figure 1.  
Many techniques and methods have been proposed to study various flexure hinges and compliant mechanisms, especially in the linear field. Some of them are based on elastic beam theory [24; 25; 26], some on stiffness and compliance matrices [27; 28], some on elliptic integrals [29; 30; 31] and Castigliano's theorem [17; 23]. [PERSON] et al. [32] proposed a classification of the various methods based on the objectives of the analysis at hand, such as the need for static or dynamic models of CMs [33; 34; 35]. The search for analytical models that describe the behavior of flexure hinges is of great use. In controllers of compliant mechanisms, easy-to-implement control logic based on kinematic models that are simple and quick to compute is required. Although finite element models typically provide more accurate solutions than analytical models, the former are usually implemented in dedicated simulation environments and require more computation time. In closed kinematic chain systems where there are many joints, it becomes crucial to have kinematic models of the whole mechanism that quickly and accurately provide the relation between the actuated joints and the end-effector operating in the mechanism workspace. Therefore, the simpler the model of the joint turns out to be, the simpler the model associated with the mechanism will be. In the field literature, much more attention has been given to analyses of flexure hinges in small deformations. In contrast, only a small fraction of the papers have proposed studies on flexure hinges in large deformation [36; 37; 38; 39; 40]. In the present work, the bending of a flexure joint with a variable profile is modeled in large displacements by means of the Euler-Bernoulli beam Elastica Theory [41]. Some simplifications are applied to the nonlinear model to arrive at a model that can be solved in analytical form. The simplification process is described in detail to understand what is neglected in the process of obtaining an analytical solution and what it entails.