Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms
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Abstract
There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the cross-section (i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.
Keywords
Compliant mechanism Maximum torsion stress Rectangular beam Lamina emergent joint1 Introduction
A compliant mechanism achieves its mobility through the deflections of its compliant elements [1]. In most compliant mechanisms, the compliant elements are designed to produce motion through bending deflections [2, 3, 4]. In fact, torsional deflections could be another valuable source for obtaining mobility in compliant mechanisms. There have been successful designs utilizing torsional deflections, for example, a split-tube flexure based on the torsion of an open-section hollow shaft was presented [5], a revolute joint comprised of two crossed torsion plates which shows a good performance in resisting axis drift was presented [6], torsional micromirrors were proposed for optical switches and optical displays [7, 8], a torsional micro-resonator was fabricated for mass sensing [9], lamina emergent torsional (LET) joints were devised to facilitate the design of compliant mechanisms that can be fabricated from a planar material but have motion that emerges out of the fabrication plane [10, 11, 12] and were employed in precision adjustment mechanisms [13], torsion hinges were proposed as surrogate folds in origami-based engineering design [14, 15], and torsional beams were successfully used for achieving static balancing of an inverted pendulum [16].
There are several design equations available for calculating the torsional compliance and maximum torsion stress of a rectangular cross-section beam. However, before using these equations, one of the two dimensions (i.e., the thickness and the width) of the cross-section must be defined as the wider of the two dimensions [17]. This situation might be troublesome and error-prone during the design phase because we always do not know which dimension is larger in advance. This is especially true for an optimization design process considering that the two dimensions of the torsion beam(s) may change greatly during the design iteration process including the relative size of the two dimensions. In our previous work [17], general compliance equations that are symmetric with respect to the two dimensions were obtained to facilitate the design of torsional beams in compliant mechanisms. These equations had been used in characterizing parasitic motions of compliant mechanisms [18, 19] and spatial deflections modeling [20, 21, 22]. However, there still lacks an equation for predicting the maximum stress in torsional beams that is symmetric with respect to the two dimensions.
To complement previous work [17], this paper is going to address this absence. The organization of this paper is as follows: Section 2 presents a brief summary on various equations for predicting the maximum stress in torsional beams; two maximum stress equations that are symmetric with respect to the two dimensions of the cross-section are formulated in Section 3; Section 4 offers some design insights for outside lamina emergent joints using the proposed equations; and Section 5 has concluding remarks.
2 Various Equations for Calculating Maximum Shearing Stress
For narrow rectangular sections (t/w<0.1), Q approximately equals 1.
When using the expressions for \(\tau_{\text{max} }\) given in Eqs. (1), (6), (8) and (10), one of the two dimensions of the cross-section must be defined as the wider of the two dimensions because they assume that w ≥ t. That is to say, if w (width) is no larger than t (thickness), this expression for \(\tau_{\text{max} }\) needs to be changed by switching w and t. This situation might be troublesome and error-prone during design phase (we always do not know which one is bigger in advance).
3 General Equations with Symmetric Relation of t and w
A two step procedure was taken to obtain a symmetric and accurate expression for maximum torsion stress \(\tau_{\text{max} }\).
As shown in Figure 3, this equation may result in an error up to 20% (the error is defined as \(E_{c} = (Q_{c} - Q_{s} )/Q_{s}\)).
In the following section, we will demonstrate the use of this general (width-thickness independent) equation for designing lamina emergent torsional (LET) joints.
4 Outside LET Joint: Design Considerations
4.1 Torsional Segment: Stiffness vs. Stress
It is obvious that increasing the length can significantly decrease F, which is preferred if the space is allowed. If L is fixed, there is a local minimum at t/w = 1. However, there are two maxima at t/w = 1.5 and t/w = 0.67 (these geometries are feasible and preferred both for design and manufacture), which are suggested to be avoided in design.
4.2 LET Joint: Torsional Stiffness vs. Compressive/Tensile Stiffness
Ideally a LET joint would have low torsional stiffness while maintaining high stiffness in the other directions [10]. However, a LET joint is susceptible to undesired motion when compressive/tensile load is applied because the torsional segments are placed into bending, as illustrated in Figure 5(c).
4.3 Design Examples
A few LET joint designs
t (mm) | w (mm) | t/w | L (mm) | E (GPa) | G (GPa) | |
---|---|---|---|---|---|---|
Design 1 | 2 | 2 | 1 | 10 | 205 | 79.37 |
Design 2 | 1.16 | 3.45 | 0.336 | 10 | 205 | 79.37 |
Design 3 | 3.45 | 1.16 | 2.97 | 10 | 205 | 79.37 |
Design 4 | 2 .5 | 1.6 | 1.56 | 10 | 205 | 79.37 |
Design 5 | 1.6 | 2.5 | 0.64 | 10 | 205 | 79.37 |
For the purpose of comparison, we calculated \(\tau_{\text{max} }\) using Eq. (36) by assuming \(\alpha = 0.1\) rad.
Calculated results
\(K_{\alpha x}\)(N·m) | \(K_{\Delta y}\)(N/m) | \(\tau_{\text{max} }\) | |
---|---|---|---|
Design 1 | 17.8347 | 3280000 | 1 |
Design 2 | 11.2398 | 1103900 | 0.336 |
Design 3 | 11.2398 | 9764900 | 2.97 |
Design 4 | 16.3246 | 5125000 | 1.56 |
Design 5 | 16.3246 | 2099200 | 0.64 |
5 Conclusions
This work presented closed-form symmetric equations for calculating maximum torsion stress of a rectangular cross-section beam. Together with the symmetric equations in our previous work [17], these equations are independent of the relative magnitude of the two dimensions (i.e., the thickness and the width) of the cross-section, thus are more convenient and effective for designing and optimizing torsional beams in compliant mechanisms.
These equations were utilized to analyze outside lamina emergent torsional joints and some useful design insights were obtained and described.
Notes
Authors’ contributions
GC and LLH conceived the idea, GC carried out the calculation, and GC and LLH drafted the manuscript. All authors read and approved the final manuscript.
Authors' Information
Gui-Min Chen is a professor at School of Mechanical Engineering, Xi’an Jiaotong University, China. He received his PhD, MS and BS degrees from Xidian University, China. He was a visiting professor at Brigham Young University, US. His research interests include compliant mechanisms and their applications. He serves as an associate editor for the Journal of Mechanisms and Robotics.
Larry L. Howell is a professor and associate dean of the Department of Mechanical Engineering, Brigham Young University, US, where he also holds a University Professorship. Prior to joining BYU in 1994 he was a visiting professor at Purdue University, a finite element analysis consultant for Engineering Methods, and an engineer on the design of the YF-22 (the prototype for the U.S. Air Force F-22). He received his PhD and MS degrees from Purdue University and his BS from Brigham Young University. He is a licensed professional engineer and the recipient of a National Science Foundation CAREER Award, a Theodore von Kármán Fellowship, the BYU Technology Transfer Award, the Maeser Research Award, several best paper awards, and the ASME Mechanisms & Robotics Award. He is a Fellow of ASME, associate editor for the Journal of Mechanisms and Robotics, past chair of the ASME Mechanisms & Robotics Committee, past co-chair of the ASME International Design Engineering Technical Conferences, and a past Associate Editor for the Journal of Mechanical Design. Prof. Howell’s technical publications and patents focus on compliant mechanisms, including origami-inspired mechanisms, microelectromechanical systems, and medical devices. He is the author of the book Compliant Mechanisms published by John Wiley & Sons.
Acknowledgements
Supported by National Science Foundation Research of the United States (Grant No. 1663345), National Natural Science Foundation of China (Grant No. 51675396), and Fundamental Research Fund for the Central Universities (Grant No. 12K5051204021).
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
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