Brouwer, Dannis

Naves, Mark

Nijenhuis, Marijn

English

Leafsprings are commonly used in flexure mechanisms to enable repeatable motion and provide support stiffness. In various applications, the torsional stiffness needs to be increased. This paper presents the concept of using warping constraints distributed along the leafspring to increase the torsional stiffness relative to the stiffness in other directions. A beam-based model of nonuniform torsion is used to estimate the relationship between the geometrical design parameters and the expected increase in torsional stiffness. The model predicts that an order of magnitude increase in stiffness is feasible. Compared to other solutions, the distributed warping constraints have a less obtrusive shape that offer more design freedom.</p

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              euspen’s 22nd International Conference & Exhibition, Geneva, CH, May/June 2022 www.euspen.eu  High torsion stiffness leafspring flexure element using distributed warping constraints  Marijn Nijenhuis, Mark Naves, Dannis Brouwer   Precision Engineering, Faculty of Engineering Technology, University of Twente, Enschede, The Netherlands      m.nijenhuis@utwente.nl        Abstract Leafsprings are commonly used in flexure mechanisms to enable repeatable motion and provide support stiffness. In various applications, the torsional stiffness needs to be increased. This paper presents the concept of using warping constraints distributed along the leafspring to increase the torsional stiffness relative to the stiffness in other directions. A beam-based model of nonuniform torsion is used to estimate the relationship between the geometrical design parameters and the expected increase in torsional stiffness. The model predicts that an order of magnitude increase in stiffness is feasible. Compared to other solutions, the distributed warping constraints have a less obtrusive shape that offer more design freedom.   Flexure mechanism, leaf spring, warping, torsion stiffness, twisting, constraints 1. Introduction Leafsprings are common components of flexure mechanisms [1-5]. Their slender shape enables repeatable motion in the three directions of low stiffness, while constraining motion in the three directions of high stiffness. As the torsional stiffness is relatively low, twisting motion is generally considered one of the three unconstrained degrees of freedom (DOFs). In some applications, it may be desirable to increase the torsional stiffness of the leafspring relative to the stiffness in the two other DOFs.  In this paper, we present the use of distributed warping constraints to significantly increase the torsional stiffness relative to the stiffness in the other directions, essentially yielding a torsionally stiffened leafspring with only two DOFs remaining. These warping constraints may eliminate the need for additional flexures that would otherwise serve as torsional constraints. Other applications are flexure mechanisms with a large range of motion, in which the support stiffness tends to deteriorate with deflection due to unwanted torsional compliance. Existing solutions for increasing the torsional stiffness include reinforced midsections [5], the infinity flexure [6-7] and a cell-based metamaterial [8]. Compared to these solutions, the distributed warping constraints of the present work have a less obstructive shape, offering more design freedom and application-specific tailoring. This paper establishes the design concept and provides a theoretical model for the mode of action and the design parameters that are involved. 2. Warping      A leafspring can be considered as a slender structure with rectangular cross-sections. When it is subjected to a torsional moment, the cross-sections rotate and warp [9-10]. The warping of the cross-section is of interest, because it affects the torsional stiffness of the leafspring and it can be influenced by design choices. If all cross-sections are free to warp, the torsion angle changes linearly over the length of the beam. However, if a cross-section is constrained from warping, additional normal and shear stresses develop in the vicinity of the constraint. These stresses require additional torque for the same torsion angle and essentially increase the torsional stiffness.  Warping can be constrained by a section with a large warping stiffness, such as the clamps or fillets commonly present at the ends of a leafspring. Since the additional stresses in the leafspring vanish at a distance from the warping constraint, the effect on the torsional stiffness is nontrivial and dependent on geometrical parameters. 3. Beam-based stiffness model When the additional stresses due to warping constraints are significant, the torsion angle no longer changes linearly over the length of the beam and the St. Venant uniform torsion model loses validity. Instead, the torsion angle 𝜙𝑥(𝑠) can be modeled by the ordinary differential equation [9-11] 𝐺𝐽𝑑2𝜙𝑥𝑑𝑠2− 𝐸𝐼𝑤𝑑4𝜙𝑥𝑑𝑠4= 0 (1) where 𝐸 is Young’s modulus, 𝐺 the shear modulus and 𝑠 the coordinate along the beam’s neutral line ranging from 0 to 𝐿. The St. Venant torsion constant for a wide rectangular beam with width 𝑤 and thickness 𝑡 can be approximated by 𝐽 ≈13𝑤𝑡3 (2) and the warping constant by 𝐼𝑤 ≈1144𝑤3𝑡3 (3) The torsional moment at a position 𝑠 is given by 𝑀𝑥(𝑠) = 𝐺𝐽𝑑𝜙𝑥𝑑𝑠− 𝐸𝐼𝑤𝑑3𝜙𝑥𝑑𝑠3(4) The quantity 𝐵(𝑠) = 𝐸𝐼𝑤𝑑2𝜙𝑥𝑑𝑠2(5)    is referred to as the bimoment and represents the stress resultant associated with the additional stresses due to warping constraints. If warping is free, then 𝐵 = 0. If warping is perfectly constrained, i.e. with infinite warping stiffness, then 𝜙𝑥′ = 0, where a prime denotes differentiation with respect to 𝑠.  To obtain the stiffness matrix, the ODE can be solved for the torsion angle by supplying the four boundary conditions 𝜙𝑥(0) = Φ0 𝜙𝑥′ (0) = 𝑊0 𝜙𝑥(𝐿) = ΦL 𝜙𝑥′ (𝐿) = 𝑊𝐿  The stiffness matrix can be found by substituting the solution for 𝜙𝑥(𝑠) in equations 4 and 5, and evaluating the torsional moment and bimoment at both ends 𝑠 = 0 and 𝑠 = 𝐿. This yields [𝑀0𝑀𝐿𝐵0𝐵𝐿] = 𝑲 [Φ0Φ𝐿𝑊0𝑊𝐿] (6) The full expression of stiffness matrix 𝑲 is given in the appendix. Each matrix component depends on 𝐺, 𝐽, 𝐿 and 𝜆. Here, the dimensionless parameter 𝜆 has been introduced. It is given by 𝜆 = 𝐿√𝐺𝐽𝐸𝐼𝑤≈𝐿𝑤√241 + 𝜈(7) and relates the warping stiffness 𝐸𝐼𝑤  to the uniform torsion stiffness 𝐺𝐽. It can be interpreted as a spatial decay rate for the warping stresses. For the present discussion, we isolate component 2,2 (𝑑𝑀𝐿/𝑑Φ𝐿) from 𝑲 and consider it to be the torsional stiffness in the presence of perfect warping constraints at both ends (i.e. 𝑊0 = 𝑊𝐿 = 0). It is given by 𝑘𝑥 =𝐺𝐽𝐿(𝜆𝜆 − 2 tanh (𝜆2)) (8) Since the torsional stiffness in the case of free warping is 𝐺𝐽/𝐿, we can see that the warping constraints lead to a stiffening factor of 𝛾 =𝜆𝜆 − 2 tanh (𝜆2)(9) as shown in figure 1 for 𝐿/𝑤 ratios from 1/4 to 15 and 𝜈 = 0.3. This range corresponds with 𝜆 from 1 to 64. It can be seen that the stiffness remains unaffected for slender beams with a large 𝐿/𝑤 ratio (i.e. a large decay rate 𝜆), but strongly increases for shorter beams.   Figure 1. Increase in torsional stiffness due to constrained warping, based on a beam model for nonuniform torsion in leafsprings.  4. Distributed warping constraints This paper introduces the idea of using a number of warping constraints distributed along the leafspring’s length dimension, in order to increase the total torsional stiffness. Each warping constraint represents a local increase in warping stiffness that hinders torsional deformation. This reinforced leafspring can be regarded as a series of very short clamped-clamped leafspring segments with warping constrained at both ends. Each segment has a small spatial decay rate 𝜆 (i.e. a small length-to-width ratio), such that the warping stresses cannot significantly diminish over the length of the segment. Figure 2 shows a 3D-printed prototype of a 150 x 50 x 1 mm leafspring with 14 local warping constraints, effected by 14 segments of 20 x 2 mm distributed equally along the length.  Figure 2. Photograph of a leafspring with distributed warping constraints.   Figure 3. Schematic of the leafspring with distributed warping constraints.   4.1. Perfect-constraint model    The theoretical stiffness increase, an upper bound based on the perfect-constraint model of the previous section, can be obtained by considering the reinforced leafspring as a series connection of 𝑁 individual torsion compliances 1/𝑘𝑥, see figure 3. The length of each segment is 𝐿/𝑁. Each segment is comprised of the warping constraint with length 𝛼𝐿/𝑁 and a flexible part with length (1 − 𝛼)𝐿/𝑁. With this definition, 𝛼 represents the fraction of the total leafspring length 𝐿 that is used by the warping constraints. The total stiffness of the reinforced leafspring is given by the reciprocal sum 𝑘𝑥,reinf = (1𝑘𝑥,1+ ⋯+1𝑘𝑥,𝑁)−1(10) which, for identical segments, reduces to  𝑘𝑥,reinf =𝐺𝐽𝐿(1 − 𝛼)(𝜆𝑒𝜆𝑒 − 2 tanh (𝜆𝑒2)) (11) This expression takes the same form as the one for the regular leafspring (equation 8). Parameter 𝐿 still represents the total length of the full leafspring, but now the stiffening factor (figure 1) for the entire reinforced leafspring is dependent on 𝜆𝑒, which is the decay rate of a single segment. It is given by 𝜆𝑒 =𝐿(1 − 𝛼)𝑁𝑤√241 + 𝜈(12) Equations 11 and 12 show how the design parameters affect the stiffness increase that can be obtained with the reinforced leafspring. Comparing with equation 7, the ratio 𝐿/𝑤 plays the same role as for the regular leafspring, but we also see that increasing the number of constraints 𝑁 drastically reduces the spatial decay rate and hence, as per figure 1, increases the torsional stiffness. 𝑘𝑥 𝑘𝑥 𝑘𝑥 𝑘𝑥 𝐿 𝛼𝐿/𝑁 (1 − 𝛼)𝐿/𝑁     Figure 4. Torsional stiffening factor due to warping constraints, equally distributed over the leafspring.  Figure 4 shows the stiffening factor, expressed as 𝑘𝑥,reinf  normalised with 𝐺𝐽/𝐿 (the stiffness in the free-warping case), for various numbers of constraints 𝑁.   Figure 5. Profile of torsion angle 𝜙𝑥(𝑠) as a function of the length coordinate for various numbers of warping constraints 𝑁. The black dots indicate the location of the warping constraint, where locally 𝜙𝑥′ = 0.  Figure 5 shows the torsion angle 𝜙𝑥(𝑠) as a function of the length coordinate 𝑠, based on the solution of equation 1 (see appendix). The dimensions and material parameters can be found in the appendix. Graphs for various numbers of constraints have been plotted; in all cases the same torsional moment has been applied. The warping constraint length 𝛼 has been set to 0. In the free-warping case, the torsion angle varies linearly with 𝑠 and achieves the largest value of all cases at the end at 𝑠 = 𝐿. When perfect warping constraints are present, the torsion angle has zero slope locally, retarding the progression of twist, leading to smaller torsion angles at the end and to a larger stiffness.  4.2. Finite-length perfect-constraint model    Parameter 𝛼 represents the length of the warping constraint segments relative to the flexible length. An ideal warping constraint would provide infinite stiffness and have zero length (𝛼 = 0), but in reality both will be finite. When 𝛼 > 0, the effective flexible length decreases and the torsional stiffness increases (also see figure 4). While that is a desirable property, it will also lead to a larger stiffness in the two intended DOFs of the leafspring. Bending stiffness 𝐸𝐼/𝐿 becomes 𝐸𝐼/(𝐿(1 − 𝛼)), which may either be acceptable or should be compensated for by a smaller thickness 𝑡 =2𝜎max𝐿(1 − 𝛼)𝜙𝑧,max𝐸(13) to ensure the same bending stress levels 𝜎max given a certain prescribed bending rotation angle 𝜙𝑧,max. Similar considerations can be had for the compliant translational direction.   5. Discussion Realistic warping constraints will have a finite length and a finite warping stiffness. As the beam-based model of section 4 assumed an infinite warping stiffness, it serves as an upper bound for the achievable stiffness increase. Some error is incurred here, since the individual segments have very small length-to-width ratios, which are pushing the limits of validity of beam theory. It is conceivable that plate-like effects, such as the nonlinear Wagner torque for larger torsion angles and constrained anticlastic bending [12-14], will play a role as well. Stress considerations will limit the number and the length of the warping constraints that can be implemented, representing another bound on the maximally achievable stiffness increase. The optimal shape of the warping constraints themselves has not been discussed in this paper. It will be an application-specific trade-off, since adding too much mass may hinder dynamic performance, using up too much volume may cause self-collision depending on the intended range of motion in the DOFs, and intricate shapes may be hard to manufacture, even with additive manufacturing.  Depending on the load case of the leafspring, equally spaced warping constraints may not be ideal. For instance, if the applied load is not a pure torsional moment, but instead a lateral force that causes a varying internal torsional moment when the leafspring is bent, warping constraints could be concentrated around where the torsional moment is largest. This could mitigate some of the support stiffness decrease observed in large range of motion flexures, which can be due to undesired torsional compliance [13,15]. Experiments to validate this model and design considerations for the shape of the warping constraints are topics for future work. 6. Conclusion This paper has presented the concept of using warping constraints distributed over a leafspring’s length to increase the torsional stiffness relative to the stiffness in other directions. A theoretical foundation has been established by means of a beam-based stiffness model. The model predicts that a stiffening factor of an order of magnitude is feasible. Reinforcing leafsprings this way can be useful since the warping constraints offer a lot of design freedom that can be leveraged for the specifics of an application. It has practical relevance since there are numerous flexure mechanism cases where torsional compliance is unwanted.  References   [1] Eastman F S 1935 Flexure pivots to replace knife edges and ball bearings Engineering Experiment Station Bulletin 86 [2] Jones R V 1956 A parallel-spring cross-movement for an optical bench J. Scientific Instruments 33 279–280 [3] Slocum A H 1992 Precision Machine Design (Prentice Hall) [4] Smith S T 2000 Flexures: Elements of Elastic Mechanisms (CRC Press)  [5] Soemers H 2011 Design Principles for precision mechanisms (T-pointpress)  [6] Wiersma D H, Boer S E, Aarts R G K M, Brouwer D M 2013 Design and Performance Optimization of Large Stroke Spatial Flexures J. Computational and Nonlinear Dynamics 9  [7] Naves M, Brouwer D M, Aarts R G K M 2017 Building Block-Based Spatial Topology Synthesis Method for Large-Stroke Flexure Hinges J. Mechanisms and Robotics 9 041006–9  [8] Andree D J, Nijenhuis M, Brouwer D M 2018 Design of a meta-material for the torsional reinforcement of a leaf spring Proceedings of The Thirty-Third Annual Meeting of the American Society for Precision Engineering 563-568    [9] Timoshenko S, Goodier J N 1969 Theory of Elasticity (McGraw-Hill)  [10] Trahair N S 1993 Flexural-Torsional Buckling of Structures (CRC Press) [11] Van Eijk J 1985 On the design of plate-spring mechanisms (Delft University of Technology)  [12] Nijenhuis M, Meijaard J P, Brouwer D M 2020 A spatial closed-form nonlinear stiffness model for sheet flexures based on a mixed variational principle including third-order effects Precision Engineering 66 429-444 [13] Brouwer D M, Meijaard J P, Jonker J B 2013 Large deflection stiffness analysis of parallel prismatic leaf-spring flexures Precision Engineering 37 505–521 [14] Meijaard J P 2011 Refinements of Classical Beam Theory for Beams with a Large Aspect Ratio of Their Cross-Sections IUTAM Symposium on Dynamics Modeling and Interaction Control in Virtual and Real Environments 285–292 [15] Nijenhuis M, Meijaard J P, Mariappan D, Herder J L, Brouwer D M, Awtar S 2017 An Analytical Formulation for the Lateral Support Stiffness of a Spatial Flexure Strip J. Mechanical Design 139 Appendix Torsion angle solution The solution of the ordinary differential equation with the specified boundary conditions is given by  𝜙𝑥(𝑠) = 𝑐1𝐿2𝜆2ⅇ𝑥𝜆𝐿 + 𝑐2𝐿2𝜆2ⅇ−𝑥𝜆𝐿 + 𝑐3 + 𝑐4𝑥  where 𝑐1 =𝜆 (𝐿(−1 + ⅇ𝜆 − 𝜆)𝑊0 + 𝐿 (1 + ⅇ𝜆(−1 + 𝜆))𝑊𝐿 + (−1 + ⅇ𝜆)𝜆(𝛷0 − 𝛷𝐿))(−1 + ⅇ𝜆)𝐿2(2 + ⅇ𝜆(−2 + 𝜆) + 𝜆)  𝑐2 = −ⅇ𝜆𝜆 (𝐿 (1 + ⅇ𝜆(−1 + 𝜆))𝑊0 + 𝐿(−1 + ⅇ𝜆 − 𝜆)𝑊𝐿 + (−1 + ⅇ𝜆)𝜆(𝛷0 − 𝛷𝐿))(−1 + ⅇ𝜆)𝐿2(2 + ⅇ𝜆(−2 + 𝜆) + 𝜆)  𝑐3 =𝐿(1 + ⅇ2𝜆(−1 + 𝜆) + 𝜆)𝑊0 + 𝐿(−1 + ⅇ2𝜆 − 2ⅇ𝜆𝜆)𝑊𝐿 + (−1 + ⅇ𝜆)𝜆 ((1 + ⅇ𝜆(−1 + 𝜆) + 𝜆)𝛷0 − (−1 + ⅇ𝜆)𝛷𝐿)(−1 + ⅇ𝜆)𝜆(2 + ⅇ𝜆(−2 + 𝜆) + 𝜆)  𝑐4 = −(−1 + ⅇ𝜆)𝐿𝑊0 + (−1 + ⅇ𝜆)𝐿𝑊𝐿 + (1 + ⅇ𝜆)𝜆(𝛷0 − 𝛷𝐿)𝐿(2 + ⅇ𝜆(−2 + 𝜆) + 𝜆)     Stiffness matrix Stiffness matrix 𝑲 is given by  [            −𝐺𝐽𝜆𝐿𝜆 − 2𝐿tanh[𝜆2]𝐺𝐽𝜆𝐿𝜆 − 2𝐿tanh[𝜆2]𝐺𝐽2 − 𝜆coth[𝜆2]𝐺𝐽2 − 𝜆coth[𝜆2]−𝐺𝐽𝜆𝐿𝜆 − 2𝐿tanh[𝜆2]𝐺𝐽𝜆𝐿𝜆 − 2𝐿tanh[𝜆2]𝐺𝐽2 − 𝜆coth[𝜆2]𝐺𝐽2 − 𝜆coth[𝜆2]𝐺𝐽2 − 𝜆coth[𝜆2]𝐺𝐽−2 + 𝜆coth[𝜆2]𝐺𝐽𝐿csch[𝜆2]2(−𝜆cosh[𝜆] + sinh[𝜆])2𝜆(−2 + 𝜆coth[𝜆2])𝐺𝐽𝐿csch[𝜆2]2(𝜆 − sinh[𝜆])2𝜆(−2 + 𝜆coth[𝜆2])𝐺𝐽−2 + 𝜆coth[𝜆2]𝐺𝐽2 − 𝜆coth[𝜆2]𝐺𝐽𝐿csch[𝜆2]2(−𝜆 + sinh[𝜆])2𝜆(−2 + 𝜆coth[𝜆2])𝐺𝐽𝐿csch[𝜆2]2(𝜆cosh[𝜆] − sinh[𝜆])2𝜆(−2 + 𝜆coth[𝜆2]) ]              where csch represents the hyperbolic cosecant.  Figure 5 details   Leafspring dimensions are 100 x 30 x 1 mm. Young’s modulus is 210 GPa. Shear modulus is 70 GPa.  The applied torsional moment is 0.1 Nm.      

High torsion stiffness leafspring flexure element using distributed warping constraints

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High torsion stiffness leafspring flexure element using distributed warping constraints

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