Compliant mechanisms design of flexure hinges
This book is dedicated to flexure hinges, which are the main constituents of compliant mechanisms. The flexure hinge, alternatively called flexural pivot, consists of a flexible, slender region between two rigid parts that must undergo limited relative rotation in a mechanism (which will be called compliant ) due to the presence of at least one flexure hinge. Under the combined action of external loading and actuation, the flexure hinge bends and thus produces the relative rotation between the adjacent members. Being monolithic with the rest of the mechanism for the vast majority of applications, a flexure hinge offers several advantages over classical rotation joints, such as no friction losses, no need for lubrication, no hysteresis, compactness, capacity to be utilized in small-scale applications, ease of fabrication, virtually no assembly, and no required maintenance.
The flexure hinges are incorporated in a large number of applications, both civil and military, including translation micropositioning stages, piezoelectric actuators and motors, high-accuracy alignment devices for optical fibers, missile-control devices, displacement and force amplifiers/deamplifiers, orthotic prostheses, antennas, valves, scanning tunneling microscopes, accelerometers, gyroscopes, high-precision cameras, nanolithography, robotic microdisplacement mechanisms, nanoscale bioengineering, small-scale insectlike walking robots, actuation devices for unmanned micro aerial vehicles, or nano-imprint technology. One of the rapidly growing areas where flexure hinges are massively applied is the microelectromechanical system (MEMS) sector, where the very nature of the microscale structure of such mechanisms, together with their fabrication technology, demands almost exclusive utilization of flexure hinges as connecting joints between quasi-rigid members.
The book is primarily intended for industrial practitioners, researchers, and academics involved in designing and developing flexure-based compliant mechanisms in such areas as mechanical engineering, aerospace engineering, robotics, MEMS, and biomedical engineering. It can also serve as a supplemental text for graduate students in universities where compliant mechanisms are a curriculum subject.
The book originated from the perceived need for an information source dedicated to current problems in the design of flexure hinges and flexurebased compliant mechanisms in both macroscale and MEMS applications in a manner that would resonate with the reality of a multitude of practical engineering cases. Two main directions have been taken in this book. The
first targets the design pool of flexure hinges through a systematic approach and the introduction of several new flexure configurations in the hope that the interested designer might opt for a specific flexure solution if presented vwith an ample variety of choices. This aspect is all but trivial, as minor changes in the geometry of a flexure hinge might result in substantial modifications at the output port of a compliant mechanism. The second direction of the book addresses the modeling tier by recognizing that in most cases the flexure hinges will operate under small displacements. The reader acquainted with flexure modeling and design would probably agree that, except for the finite-element analysis available through commercially available software (which is predominantly the variant of choice in a large number of applications), the accepted modeling paradigm currently in operation is based on two main hypotheses: (1) the flexure hinges are flexible constant cross-section members, and (2) they are subject to large deformations
. Consequently, the modeling procedure substitutes a flexure hinge with a purely rotational joint equipped with a torsional-spring stiffness. The resulting model of the flexure hinge is subsequently incorporated into a classical rigidlink mechanism model of the specific device being studied, and further static and dynamic calculations are performed according to standard procedures.
The reality is that only in a few occasions are the two fundamental premises mentioned above (constant cross-section and large deformations) concomitantly met. In the vast majority of practical situations, the flexure hinges are actually and deliberately designed to function within a small-displacement environment. The supporting rationale is twofold, as either the application
itself requires this type of condition (for instance, in precision mechanisms where the output displacements are inherently small) or the physical dimensions of the flexure hinge do not permit large deformations that would automatically generate stresses over the allowable limits. Moreover, the flexure hinge geometry is seldom of constant cross-section because the fabrication
technology currently in use might not allow this particular geometry to be produced in either macroscale or MEMS monolithic applications. The radius of the wire tool in electrodischarge machining, for instance, has a finite, nonzero value and, as a result, the corner of a flexure hinge fabricated by this procedure will always be filleted. Similarly, the design process itself attempts to avoid the type of geometry that would induce undesirable stress concentration in the corner areas.
Previous work supports the approach followed in this book. A very solid paper written by Paros and Weisbord in the 1960s convincingly demonstrated that a circular flexure hinge is a complex spring that not only produces the desired relative rotation between two adjacent rigid links but is also deforming axially and out of plane. The vision expressed by the authors in this fundamental
paper is compelling, and present-day flexure-based applications reveal that all deformational facets of a flexure hinge have to be accounted for if an accurate assessment of the performance of a compliant mechanism is to be achieved. The path followed by Paros and Weisbord has been revisited only recently, and the present work is a modest addition to this work.
This book attempts to provide practical answers to the problems of efficiently modeling, analyzing, deciding on, and designing devices that include flexure hinges. It contains many ready-to-use plots and simple equations describing several flexure types for professionals who need speedy solutions to their current applications. For the researcher who would prefer to find specific answers to a particular design configuration (which is probably not covered here), the book contains self-contained, easy-to-apply mathematical tools that provide guidance for real-time problem solving of further applications. Several original features are included in this book:
• The book introduces new types of single-axis flexure hinge configurations (e.g., parabolic, hyperbolic, inverse-parabolic, secant) that supplement such classic geometries as circular, corner-filleted, and elliptical and are designed for planar compliant mechanism applications.
• The same configurations are addressed for multiple-axis (revolute) flexure hinges for spatial compliant mechanism applications.
• Newly introduced are the two-axis flexure hinges that are capable of displaying a selective response over two different compliance ranges in spatial applications.
• For all the single-axis flexure hinges, both longitudinally symmetric and nonsymmetric configurations are analyzed, and their corresponding spring rates are explicitly given.
• Short flexures and the associated shearing effects are also modeled for all flexure configurations in terms of their compliance.
• Flexure hinges are studied from a performance-oriented viewpoint in a unitary manner: flexibility, precision of rotation, stress limitations, and energy consumption are factors defined and analyzed by means of closed-form compliance equations.
• Inertia and damping properties are also derived, consistent with the compliance formulation, so that a flexure hinge can be fully characterized and included in the dynamic model of a flexurebased compliant mechanism that can be solved to evaluate its free or forced response. The inertia and damping properties of a flexure hinge are modeled by following either the long (Euler–Bernoulli) or short (Timoshenko) member hypotheses.
• An original finite-element approach is developed whereby the flexure hinges are assimilated to line elements; this approach reduces the problem dimensionality and allows us to perform static and modal/time-history analyses in a simple fashion.
• Also treated are more advanced topics related to flexure hinges, such as shape optimization, buckling, torsion of noncircular variable cross-section members, nonhomogeneous flexures, thermal effects, and large deformations.
• The book includes a list of novel industrial applications, both macroand microscale (MEMS), for which flexure hinges are instrumental.
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