Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

Chinese Journal of Mechanical Engineering, Feb 2018

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.

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Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

Chen and Howell Chin. J. Mech. Eng. Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms Gui‑Min Chen 0 1 2 Larry L. Howell 0 1 0 Department of Mechanical Engineering, Brigham Young University , Provo, UT 84602 , USA 1 Department of Mechanical Engi‐ neering, Brigham Young University , Provo, UT 84602 , USA 2 State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University , Xi'an 710049, Shaanxi , China 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 crosssection (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. Compliant mechanism; Maximum torsion stress; Rectangular beam; Lamina emergent joint 1 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 [ 2–4 ]. 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 [ 10–12 ] 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 [ 20–22 ]. 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 torsion of rectangular sections, the stress at each corner is zero, and the maximum shearing stress τmax occurs at the middle points of the longer sides (the points most remote from the centroid of the cross section), as illustrated in Figure 1. It can be obtained using the membrane analogy by assuming w ≥ t [ 23 ]:  8 ∞ 1  τmax = Gθ t1 − π2 n=1,3,5... n2 cosh(nπw/2t) , (1) where θ is the angle of twist per unit length. The infinite series on the right side converges rapidly. Because the torque as a function of θ is given as [ 23 ] T = Gθ3wt3 1 − 1π952wt ∞ n=1,3,5... tanh(nπw/2t)  n5 . (2) Dividing Eq. (1) by Eq. (2) yields the maximum shearing stress as a function of the torque in the form 3T τmax = wt2 Q, where Q (denoted as Qs for this infinite series expression) is a constant whose value depends only on ratio t/w (t/w ≤ 1): Qs = 1 − π82 1 − 1π952 wt ∞ 1 n=1,3,5... n2 cosh(nπw/2t) ∞ . tanh(nπw/2t) n5 n=1,3,5... For narrow rectangular sections (t/w<0.1), Q approximately equals 1. A polynomial fit of Qs as a function of t/w given in Eq. (4) [ 24 ]: which leads to another expression for τmax: (6) (7) (8) (9) 3T wt2 1 1 − 0.63 wt + 0.25 wt22 . The following expression employing a linear approximation of Q [ 26 ] is often used by researchers [ 10 ]: t Ql = 1 + 0.6 , w and we have T 3T τmax = w2t2 (3w + 1.8t) = wt2 t w 1 + 0.6 which approximately equals the first two terms of Eq. (6). , because it is symmetric with respect to t and w due to |log(t/w)| = |log(w/t)|. By fitting the results of Ec using a quadratic/quadratic rational function, the following compensation function is found to reduce the maximum error to 0.4% (see Figure 4). This leads to the following expression forτmax: This compensation function reduces the maximum error to 0.07% (see Figure  4) and leads to the following expression for τmax: (21) 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 Figure 5 shows an outside LET joint fabricated from a planar sheet of material whose modulus of elasticity and modulus of rigidity are E and G, respectively. The joint consists of two parallel sets of torsional segments connected by connecting segments in bending. The geometric parameters of the torsional segments are shown in Figure 5, and I = wt3/12 represents the area moment of inertia in the sheet plane. 4.1 Torsional Segment: Stiffness vs. Stress The stiffness of each torsional segment is expressed using the symmetric equation obtained in Ref. [ 17 ] as Kt = wt22 + 2.609 wt + 1 = GL · kc, (22) where kc is the torsional stiffness constant [ 23 ] that is solely determined by the dimensions of the cross-section: kc = 2t3w3 7 t2 + w2 · As to the maximum stress, we define a stress constant as (24) which is also solely determined by the dimensions of the cross-section. If using meters as the length dimensions (e.g., t and w), the units for kc and τc are m4 and m−3, respectively. To compare the twist performance of rectangular cross-sections with different aspect ratios (i.e., t/w), we define the following non-dimensionalized term called the torsional aspect-ratio factor: Tc = kc3τc4 = 0.7738tw(t + w)4 · t2 + w2 3 expressed by the Bi-BCM (the first bending mode) [ 27 ] as: in which fo and po are the normalized axial and transverse forces, respectively, defined as and yo is the normalized displacement defined as Tc is also symmetric with respect to t and w because it is only determined by the width-thickness ratio of the cross-section. Figure  6 plots Tc as a function of t/w. For a LET joint design, we always expect kc to be small so as to lower the torsional stiffness in the desired direction for reducing actuation effort, and τc  to be small to increase the maximum allowed rotation angle. Tc reaches its maximum, 2.2747, at t/w = 0.658 and t/w = 1.52. Tc reaches its local minimum, 1.4942, at t/w = 1 (square cross-section). Tc is also smaller than 1.4942 for t/w < 0.35 and t/w > 2.86. In general, cross-sections of 0.35 <  t/w < 1 and 1 < t/w < 2.86 are suggested to be avoided for the torsional segments in LET joints. 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). When an outside LET joint is subject to a compressive/tensile load, the torsional segments can be treated as fixed-guided segments with the axial force Po = 0 (because the connecting segments is floating) as illustrated in Figure  5. The transverse force Fo, can be (26) (27) (28) (29) , (30) (31) (32) (33) (34) A well-designed LET joint should have large Rs so as to achieve low torsional stiffness but provide good constraint in the compression/extension direction. The right By substituting Eqs. (27) and (28) into Eq. (26), the parasitic compression/extension stiffness of the whole LET joint can be obtained as: K y = Fyy = 22YFoo = 12LE3I = ELw3t3 . The equivalent torsional compliance of the outside LET joint along the x-axis [ 10 ] is: Kαx = + k1k2 + k1k5 + k2k5 k3k4 + k3k6 + k4k6 k1k2k5 k3k4k6 where the stiffnesses of the torsional segments are k1 = k2 = k3 = k4 = Kt , while the connecting segments in bending can be considered stiff because they are short, i.e., k5 = k6 = ∞. Then we have Kαx = side of Eq. (34) is divided into four terms, among which the first is a constant, the second is material-related term, the third is inversely proportional to the square of the length of the torsional segments, while the last monotonically increases with t/w. For isotropic materials, the Poisson’s ratio ν  ranges from 0.2 to 0.5 [ 28 ], thus the range of the second term E/G = 2(1+ν) is from 2.4 to 3. However, one can use composite materials to obtain larger E/G (e.g., multi-layer structures). The third term indicates that Rs can be significantly increased by decreasing the length of the torsional segments. However, decreasing the length of the torsional segments will increase the torsional stiffness (as indicated by Eq. (22)) and further decrease rotational range of the joint. The fourth term is plotted in Figure  7 as a function of w/t, which shows that Rs dramatically increases as w/t increases. Considering Tc shown in Figure 6, w/t > 2.86 is suggested if parasitic motion along the compression/extension direction is required to be small. 4.3 Design Examples This section provides a few LET joint designs to demonstrate the design considerations in the previous section. The parameters of 5 designs are listed in Table 1. These Figure 7 Rs as a function of t/w For the purpose of comparison, we calculated τmax using Eq. (36) by assuming α = 0.1 rad. The results of Kαx, K y and τmax for the 5 designs are listed in Table 2. Among the 5 joints, Design 3 is the best design, with the smallest Kαx, the largest K y, and the lowest τmax. Design 2 has the same values for Kαx and τmax as Design 3, but has the lowest compression/extension stiffness. Design 1 has modest values for K y and τmax but the largest Kαx. Design 4 and Design 5 are the worst designs because they have the largest Kαx and τmax. 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. 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 profes‑ sor 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 profes‑ sional 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 Engi‑ neering 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, microelec‑ tromechanical 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. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in pub‑ lished maps and institutional affiliations. 1. L L Howell . Compliant mechanisms . New York: Wiley, 2001 . 2. N Wang , L X iang , X Zhang . Pseudo‑rigid‑body Model for Corrugated Cantilever Beam Used in Compliant Mechanisms . Chinese Journal of Mechanical Engineering , 2014 , 27 ( 1 ): 122 - 129 . 3. U Hanke , P Hampel , A Comsa , et al. Compliant mechanism synthesis by using elastic similitude . Chinese Journal of Mechanical Engineering , 2015 , 28 ( 4 ): 769 - 775 . 4. S Liu , J Dai , A Li , et al. 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Gui-Min Chen, Larry L. Howell. Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms, Chinese Journal of Mechanical Engineering, 2018, 14, DOI: 10.1186/s10033-018-0214-9