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Evaluation of flexure mechanism motion constraints during design (Part I and II) – Dr. Stuart Smith, UNC Charlotte; Dr Kumar Arumugam, NIST; and Dr. Marijn Nijenhuis, University of Twente

Part I: Review of flexure mechanisms – a mobility perspective

Part II: Interactive web application for mobility visualization using flexible multibody analysis

Flexures are found in stages and mechanisms used in precision applications including lithography, metrology instruments, mass balances, telescope mirror positioning, and medical devices. Flexure mechanisms offer frictionless, zero play, limited or zero hysteresis, and contaminant free motion in ranges of nanometers to millimeters and potentially 180° rotations. While designing these mechanisms it is necessary to consider numerous parameters including but not limited to volume, dynamics, lifetime, stresses in the flexural elements, range of travel, parasitic motion, rotation center shift, and load bearing capability. Therefore, the process of designing these mechanisms requires a vast range of design and analysis approaches to ensure its functionality.

The fundamental step during the early design process is to determine an optimal design with suitable constraints and freedoms. One simple equation that helps to understand the constraints and freedoms is ‘mobility assessment’ as discussed in Part I of these two tutorials. From the evaluation of mobility, it is possible to identify the undesired constraints that result in stresses during the motion of the mechanism. Once identified, overconstraints can be reduced by adding compliance at appropriate locations in the kinematic chain. However, overconstraints are sometimes unavoidable, for example, to improve the ability of the mechanism to withstand off-axis loading. Mobility assessment also helps to avoid undesired freedoms of the links in a mechanism, thereby improving the dynamics and load bearing capability. These improvements can be achieved by adding constraints in appropriate locations of the kinematic chain. Mobility analysis using the equation alone does not provide quantitative information about the consequences of the over and under constraints due to their directional dependence and complex geometries. To address the need to quantify these consequences, a multibody kinematic approach is used to visualize the free motions and constraining forces and moments in a given design, and this is discussed in Part II of these two tutorials.

Part I of this tutorial discusses mobility assessment of flexure mechanisms using a simple equation typically attributed as Grübler’s equation. The discussions cover the fundamentals and nuances in applying this equation, including the criteria to determine freedoms and constraints in a variety of flexural joints. Case studies and physical prototypes made out of monolithic and assembled elements will be used to develop intuition when applying this equation to assess mobility of the mechanism. As a lead-in to considerations of more complex 2D and 3D mechanisms, a quick review of conventionally used mechanisms, their mobility and their relative advantages and limitations will be provided.

Part II of this tutorial initially focuses on a flexible multibody formulation which uses kinematic equations for deriving a Jacobian matrix that relates coordinates and deformations. This helps to capture orientation dependent behavior of constraints in a mechanism. Later, the audience will be provided with an interactive web application for modeling and visualizing the mobility of flexure mechanisms. A brief introduction to a full multibody software which provides quantitative numbers indicating the effect of overconstraints on stresses and frequency modes will also be given. This can be helpful when considering design trade-offs between requirements such as stress of over-constraint competing against the stress of an off-axis load.

In summary, Part 1 establishes intuition, basic principles, and equations for a preliminary assessment of mobility in flexure mechanisms, and Part 2 introduces a web-based multibody software and the related theory to visualize the consequences of constraints in different mechanism designs. 

Dr. Stuart Smith, UNC Charlotte

Dr. Stuart Smith has been working in engineering starting in 1977 as a factory maintenance apprentice with Miles Redfern Ltd (UK). He is now a Professor of Mechanical Engineering in the Center for Precision Metrology at UNC Charlotte. Throughout the years his major focus has been the development of machines, instrumentation and sensor technologies primarily aimed towards the challenges of atomic scale discrimination and modifications. This work has resulted in the formation of four manufacturing companies, twenty patents, over a hundred journal publications, and the (co)authorship of four, soon to be five, books.

Dr. Kumar Arumugam, NIST

Kumar Arumugam is currently a postdoctoral associate at the National Institute of Standards and Technology working on ‘small’ mass and force metrology. He received his doctoral degree at the Center for Precision Metrology, UNC Charlotte in 2021. His research interests include flexure mechanisms, frequency modulated interferometry, monochromatic confocal microscopy, and stylus profilometer characterization for surface metrology. Kumar has been involved in the ASPE community since 2017. He has given talks, presented research posters, and participated (later served as a committee member) in the student challenges conducted in the ASPE conferences.

Dr. Marijn Nijenhuis, University of Twente

Marijn Nijenhuis is an assistant professor at the chair of Precision Engineering of the University of Twente in the Netherlands. He holds a doctorate degree from the same university on the topic of the nonlinear analysis of flexure mechanisms. His research focuses on the analytical and numerical modeling of mechanical systems for the purpose of design and control. Research interests include flexible multibody dynamics and (electro)mechanical metamaterials.