Predicting Nonlinear Stiffness, Motion Range, and Load-Bearing Capability of Leaf-Type Isosceles-Trapezoidal Flexural Pivot Using Comprehensive Elliptic Integral Solution

A leaf-type isosceles-trapezoidal flexural (LITF) pivot consists of two leaf springs that are situated in the same plane and intersect at a virtual center of motion outside the pivot. )e LITF pivot offers many advantages, including large rotation range and monolithic structure. Each leaf spring of a LITF pivot subject to end loads is deflected into an S-shaped configuration carrying one or two inflection points, which is quite difficult to model. )e kinetostatic characteristics of the LITF pivot are precisely modeled using the comprehensive elliptic integral solution for the large-deflection problem derived in our previous work, and the strengthchecking method is further presented. Two cases are employed to verify the accuracy of the model. )e deflected shapes and nonlinear stiffness characteristics within the range of the yield strength are discussed. )e load-bearing capability and motion range of the pivot are proposed.)e nonlinear finite element results validate the effectiveness and accuracy of the proposed model for LITF pivots.


Introduction
A leaf-type isosceles-trapezoidal flexural (LITF) pivot consists of two leaf springs located in the same plane [1]. e two leaf springs are arranged symmetrically and intersect at a virtual center of motion outside the pivot, as shown in Figure 1. e parallelogram flexure is a special type in which the two leaf springs intersect at infinity. LITF pivots have been utilized in many accurate mechanisms [2][3][4] due to their obvious advantages such as low cost, monolithic manufacturing, reduced weight, and smooth motion [5][6][7].
When delivering movement, the leaf springs of a LITF pivot undergo nonlinear large deflection that may carry one or two inflection points (where the resultant moment is equal to zero [8]), which complicates the accurate modeling of LITF pivots. e remote center location and stiffness of a LITF pivot are presented by the model of screw theory based on the small-deflection assumption [9], which limits the application range of the model. e two pseudorigid-body models with small-deflection assumption [10], i.e., a four-bar model and a pin-joint model, were proposed for the analysis of the moment-angle characteristics of LITF pivots subject to horizontal force and moment. However, the influence of vertical force to LITF pivots was neglected. e analytic models for stiffness and center shift were presented by using the beam constraint model (BCM) method [11], which can be used to solve the nonlinear characteristics of LITF pivots, i.e., the rotation angle is in the range of ± 15 ∘ . e efficiency of uniform-strength composite leaf springs under various loading conditions [12] was analyzed. erefore, the accurate nonlinear analysis and load-bearing capability solution within the entire stress range are indispensable for the application of LITF pivots.
Because the leaf spring is so thin and flexible that the effects on axial elongation and shear are negligible, the elliptic integral solution is often considered to be the most accurate model for modeling this kind of large deflection beams. Howell [13] presented the elliptic integral solutions for the large deflection beam with no inflection point. An elliptic integral solution for the beam with an inflection point was derived by Kimball and Tsai [14]. In our previous research [15], we developed the comprehensive elliptic integral solution to solve the large deflections of beams with multiple inflection points and subject to any kinds of load cases. Because each of the deflected leaf spring carry one or two inflection points, LITF pivots can be modeled by the comprehensive elliptic integral solution. e model can be used to solve the exact deflected shapes and nonlinear stiffness of LITF pivots subject to different loads. rough the stress analysis for deflected leaf springs, the maximum motion range and allowable loads of LITF pivots are solved. e rest of this paper is organized as follows. In Section 2, the accurate kinetostatic model and stress check for LITF pivots are proposed. In Section 3, two examples are calculated to demonstrate the accuracy of the model for LITF pivots. e nonlinear stiffness and workspace evaluation of the two examples are then discussed. In Section 4, concluding remarks are presented. Figure 2, two springs (O 1 A and O 2 A, length L) of a LITF pivot intersect at point O ′ and the angle between two leaf springs is 2β. e lengths of O 1 O 2 and AB are w 1 and w 2 , respectively. Letting N � w 2 /w 1 , when N ≠ 1, we then have

LITF Pivot. As shown in
When N � 1, the LITF pivot becomes a parallelogram flexure, as shown in Figure 2(c), β � 0 ∘ and w 1 � w 2 . e global coordinate system OXY is established for the LITF pivot with the X axis oriented along O 2 O 1 and the origin located at the midpoint of O 2 O 1 , as shown in Figure 2. e initial angle between leaf spring O 1 A and the X axis is θ 1 , and the angle between leaf spring O 2 B and the X axis is θ 2 .
e local coordinate systems O 1 X 1 Y 1 and O 2 X 2 Y 2 for leaf springs O 1 A and O 2 B are established with the origins placed at the fixed end and the X 1 and X 2 axes oriented along the leaf springs, respectively. e deflected end coordinates and the end angle of spring O 1 A with respect to the local coordinate O 1 X 1 Y 1 are a 1 , b 1 , and θ 1o , respectively. Similarly, the corresponding end coordinates and angle of O 2 B with respect to the local coordinate O 2 X 2 Y 2 are denoted a 2 , b 2 , and θ 2o , respectively. e horizontal displacement ΔX, vertical displacement ΔY, and rotation angle Δθ of the freedom for the LITF pivot in the global coordinate system are expressed as e loop closure equations are given as (3) Figure 3 shows the free-body diagrams for link AB and leaf springs O 1 A and O 2 B. When the pivot is subject to horizontal force F x , vertical force F y , and moment M at the midpoint C of link AB, the horizontal and vertical components of the end force and moment loaded at the O 1 A and O 2 B are F 1x , F 1y , M 1o and F 2x , F 2y , M 2o , respectively. Applying the static equilibrium for link AB yields Similarly, for O 2 B,

Mathematical Problems in Engineering
e deflection of each leaf spring can be modeled using the comprehensive elliptic integral solution summarized in the following section. e comprehensive elliptic integral solution for each leaf spring, together with the loop closure equation equation (3) and the static equilibrium equation (4), constitute the kinetostatic model for LITF pivots.

Comprehensive Elliptic Integral Solution.
Each of the two springs of a LITF pivot can be viewed as a cantilever beam subject to an end vertical force P, an end horizontal force nP, and an end moment M o , as shown in Figure 4. e tip coordinates and tip angle of the deflected beam are denoted as a, b, and θ o , respectively. Each deflected spring may carry m inflection points (m � 1 or 2). e comprehensive solution [15] for the beam with inflection points, as summarized in the following, formulates the load parameters (κ, n, and α) and deflection parameters (a, b, and θ o ) by introducing m as the shape parameter:  where α is defined as the force index (EI is the flexural rigidity of the beam), κ is the load ratio, and S r is the sign of the resulting moment at the fixed end of the beam, Moreover, are the incomplete elliptic integrals of the first and second kinds [16], respectively, in which γ is called the amplitude and t(− 1 ≤ t ≤ 1) is the modulus. When c � π/2, they become the complete elliptic integrals of the first and second kinds and are denoted as F(t) and E(t), respectively. ϕ represents the angle of the force applied at the free end (as marked in Figure 4): e coordinates (x, y) of an arbitrary point A on the beam (shown in Figure 4) can be written as where θ is the deflected angle at point A and e value m(θ) is equal to the number of the inflection points between the fixed end and point A.
Most deflected springs mainly loaded by the horizontal force F x and the moment M have one inflection point (m � 1). When the vertical force dominates in the applied loads, the deflected spring may contain two inflection points (m � 2). e kinetostatic model applied for LITF pivots can be calculated by the built-in function "fsolve" in MatLab (MathWorks, USA). e deflected shapes and the nonlinear stiffness characteristics of LITF pivots are further shown in Section 3. However, the stress of deflected leaf springs should be less than the maximum yield stress, which determines the motion range and load-bearing capacity of LITF pivots.

Stress Analysis.
e maximum bending stress, the important index of checking the LITF pivot, decides the maximum rotation angle and the maximum allowable loads. For deflected LITF pivots, the maximum bending stress σ max of each spring should be less than the yield strength S y of the material used; thus, for the rectangular cross section I � wh 3 /12 (w is the width and h is the height), we have [17] where |M| max is the maximum resultant moment distributing in leaf springs. Substituting curvature K � M/EI into equation (15) yields where |K| max is the maximum curvature occurring in the two deflected leaf springs. e motion range and load-bearing capability of LITF pivots subject to different loads, as two of the most important criteria with which to compare the compliant joints, can be obtained by the kinetostatic model and stresschecking equation (16). e flowchart of the solution to the deflected shape and workspace evaluation of LITF pivots is shown in Figure 5.

Case Studies
In this section, a LITF pivot and a parallelogram flexure are employed as two cases to demonstrate the effectiveness of the comprehensive elliptic integral model. e parameters of the two pivots are given in Table 1, and the materials are polypropylene in which E � 1.4 GPa and S y � 34 MPa [13].

Solution for LITF Pivot.
e parameters of the LITF pivot are shown in Table 1. e lengths of link O 1 O 2 and AB solved by equation (1) are w 1 � 0.1167 m and w 2 � 0.0467 m, respectively. e deflected shapes of the pivot subject to different loads, the load-bearing capacity, and the corresponding motion range of the pivot will be discussed here.

Deflected Shapes under Different Loads.
e deflected results of the pivot subject to different loads are obtained separately by using the comprehensive elliptic integral solution and a nonlinear finite element analysis (NFEA) model, as shown in Figures 6-10. For the NFEA model built with the ANSYS software, springs O 1 A and O 2 B are meshed into 100 elements with BEAM188, respectively, and the large displacement analysis option is turned on. BEAM188 is suitable for analyzing slender to moderately stubby beam structures. is element is based on the Timoshenko beam theory. Shear deformation is included. e results of the comprehensive elliptic integral solution agree well with NFEA.
For the LITF pivot subject to pure moment loaded at point C, the relationship between the rotation angle Δθ and the moment M is shown in Figure 6. e LITF pivot reveals fine linearity for Δθ less than 10 ∘ (the dashed line in Figure 6 expresses the linear approximation of the LITF pivot with small deformation). However, when Δθ is larger than 10 ∘ , the nonlinearity of the stiffness for this kind of pivot becomes remarkable. For M � 0.49 N·m, the maximum curvature of the pivot occurring at point A is equal to 48.5519·m − 1 , which is substituted into equation (16) to obtain the maximum stress, σ max � 33.986 MPa, close to the yield strength. Meanwhile, Δθ attains 27.04 ∘ , for which the deflected shape of the pivot is shown in Figure 7.
Using equation (9), draw the deflected shape and solve the maximum curvature Equation (16) is satisfied?
Loads meet the strees need Loads exceed the bearing capacity    ′ are equal to zero, so that the pivot returns to the original position. e corresponding deflected shapes of the pivot for F x � − 5 N, F y � 0 N, and M � 0 ∼ 0.5 N·m, as shown in Figure 11, incline to the left and then to the right.

Workspace Evaluation.
e stress of the deflected pivot solved by the kinetostatic model is checked by equation (16), and then the load-bearing capacity in different load cases and the motion range of the pivot are obtained.
(1) Horizontal Force and Moment. Figure 12 shows that the pivot subject to different M and F x can bear a range of horizontal force. e arrows drawn in Figure 12 roughly mark the descending direction of the stress, and the covered area is the safe working region. e pivot only subjected to horizontal force, i.e., M � 0, can bear the maximum horizontal force reaching F x � ± 13.6 N, for which the corresponding rotation angles are Δθ � ± 16.9 ∘ , as shown in Figure 13. With the incremental moment, the maximum positive horizontal force of the pivot gradually decreases and the anticlockwise rotation angle of link AB shows the increasing tendency.
For M � 0.49 N·m, the maximum stress of the pivot without horizontal force reaches the yield strength. When negative horizontal force and moment act on the pivot, the negative allowable horizontal force increases gradually and the corresponding angle decreases slightly with the increasing moment, as shown in Figures 12 and 13. For M � 0.5 N·m, the allowable negative force is F x � − 24.12 N and Δθ � − 13.06 ∘ , as shown in Figure 14, and the maximum curvature also happens at point A. e relative errors of the rotation angles between the comprehensive solution (Δθ CS ) and the nonlinear finite element (Δθ FEA ) results are expressed as e errors of the positive rotation angles depicted in Figure 13 between the comprehensive solution and the nonlinear finite element results are less than 1.5%, which is shown in Figure 15.
(2) Vertical Force and Moment. For the pivot subject to M and F y , Figure 16 draws the maximum vertical force that the pivot subject to different moments can bear. Similarly, the declining direction of the stress is masked roughly by the arrows in Figure 16. Positive vertical force can counteract the rotation angle of the pivot caused by the moment. It should be noted that the tensile stress might lead to the failure of the pivot when positive vertical force reaches a certain value because the Bernoulli-Euler beam theory neglects the effect of axial elongation and the maximum positive vertical force cannot be predicted, the discussion of which is outside the scope of this paper.  For M � 0 N·m, the pivot only subject to vertical force and the buckling of the spring may take place. For the buckled LITF pivot, the maximum bending stress σ max may be less than the yield strength of the material used, but the LITF pivot has been invalidated, so the maximum negative vertical force for M � 0 N · m is equal to the critical buckling force. e buckled springs can be seemed as the fixed-guided beams with two inflection points (m � 2) that perhaps have two deformed shapes (I) and (II), as shown in Figure 19. e vertical displacement of the freedom is δ, and the end slope of the buckled spring remains constant, i.e., θ o ≡ 0.

Mathematical Problems in Engineering
For the buckled springs, the coordinates of the free end are given as e vertical force F y can be solved as Substituting m � 2, θ o ≡ 0, and equation (8) into equation (9) yields From equations (18), (21), and (22), n is

Substituting equations (20) and (23) into equation (19) yields
When F y reaches the critical buckling force, we have n ⟶ ∞ and (a/L) ⟶ 0, and then equation (21) reduces to (F(t)/E(t)) ⟶ 1 and has t � 0. (25) We have F(t) � π/2; then, the critical buckling force F cr from equation (24) is us, for M � 0, the maximum negative vertical force of the pivot F y is determined by the critical buckling force solved by equation (26) and equal to − 41.0272 N, as shown in Figure 16. When the pivot is loaded only by the vertical force, the leaf with two inflection points includes two deflection paths, which are shown in the left-hand leaf and right-hand leaf of Figure 19. e choice of the two solutions is decided by the processing factor of the leaf.

Parallelogram Flexure.
A parallelogram flexure is a onedegree-of-freedom device that obtains accurate motion by the bending of the springs. Many authors have contributed to this problem; for example, Awtar et al. [18] proposed a beam constraint model and Dibiasio et al. [19] presented a pseudorigid-body model to simplify the derivation and    Mathematical Problems in Engineering calculation. In the paper, the kinetostatic model is also suitable to analyze the parallelogram flexure. For the parameters of the mechanism given in Table 1, the leaf springs O 1 A and O 2 B with one inflection point guide the motion of link AB with minimal rotation. When F x is applied at point C, the horizontal displacement ΔX is obtained to arrive at a static equilibrium state, as shown in Figure 20. With increasing horizontal force, the nonlinear characteristics of the curve are gradually obvious and the rotation angle Δθ is slowly increasing, as shown in Figure 21.
When F x � 8.5 N is loaded at point C, ΔX � 0.032 m and Δθ � − 0.145 ∘ , for which the maximum curvature occurring at point O 2 is K max � 46.9396 m − 1 and the maximum bending stress σ max solved by equation (16) is slightly less than S y . e rotation angle Δθ of link AB is a parasitic error motion that is undesirable in response to the horizontal force F x , which may be eliminated by an appropriate combination of moment M or vertical force F y [18]. When F x and M are loaded simultaneously at point C to ensure Δθ ≡ 0, we have, from equations (4)- (6), For a parallelogram flexure because each deflected leaf spring carries one inflection point, where the resultant moment is equal to zero and the rotation angles at the fixed and free ends of each deflected leaf spring are both equal to zero, the inflection point occurs at the middle of the deflected spring, i.e., x � a 1 /2. e moment at the inflection point is Substituting equation (28) into equation (27) yields For F y � 0, the ratios in equation (29) during the intermediate stage are approximatively constant, which agree well with the results of Ref. [18] equal to 0.5L, as shown in Figures 22 and 23. en, the ratios between M and F x are less than 0.5L with increasing horizontal force F x and the corresponding transverse stiffness gradually increases for protecting Δθ ≡ 0. As listed in Table 2, the corresponding moments and displacements for F x � 2 ∼ 10 N solved by the elliptic integral solution agrees well with the load-deflection relationship expressed in equation (29) and the deflected shapes of the pivot subject to F x and M are shown in Figure 24.
When F x and F y are loaded at point C simultaneously, M is needed to ensure that Δθ ≡ 0. In this case, the inflection points also appear in the middle of the deflected leaf springs. However, if F y is a tensile force, the ratios between M and F x are less than a 1 /2 because of n being less than zero. On the contrary, for F y as a pressure, the ratios between M and F x are greater than a 1 /2. e more obvious nonlinearity of the Figure 19: Buckling deformation of the LITF pivot. pivot appears with increasing pressure F y , as shown in Figure 23. Until the pressure reaches the critical buckling force calculated by equation (26) (F y � − 47.3741 N), the buckling of the pivot leads directly to failure.

Conclusions
e comprehensive elliptic integral solution was used for building the generalized model of LITF pivots and solving nonlinear deflection problems. For the LITF pivot, the accurate deflected shapes are described subject to different horizontal forces, vertical forces, and moments. Furthermore, based on the strength check and the analysis of the critical buckling force, motion range and load-bearing capability for the pivot are evaluated. For the parallelogram flexure, two cases for free rotation angle and constant rotation angle are discussed. e more accurate ratio between horizontal force and moment is proposed to ensure that the rotation angle remains constant. e analytical results for the maximum rotation angle of the LITF pivot subject to horizontal force and moment solved by the comprehensive elliptic integral solution are within 1.5 percent error compared to the finite element analysis results.

Data Availability
e calculation data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.