The Beam Constraint Model (BCM) was developed for the purpose of accurately and analytically modeling nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections (10 % of the beam length). The BCM is expressed in the form of Taylor's expansion associated with the axial force. It has been found that the BCM may yield large predicting errors (> 5 %) when the applied axial force goes beyond a certain boundary, even the deflection is still in the intermediate range. However, this boundary has not been clearly identified so far. In this work, we mathematically determine the non-dimensional boundary of the axial force by the condition that the strain energy expression of the BCM is a positive definite quadratic form, and by the buckling condition relate to compressing axial force. Several examples are analyzed to demonstrate the effects of the axial force on the modeling errors of the BCM. When using the BCM for modeling, it is always suggested to check if the axial force is within this boundary to avoid large modeling errors. If the axial force is beyond the boundary, the Chained Beam Constraint Model (CBCM) can be used instead.

Introduction

The Beam Constraint Model (BCM), developed by a decade ago, offers a parametric and closed-form model for accurately and analytically capturing nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections (typically when transverse motion is less than 10 % of the beam length). The BCM is represented in the form of Taylor's expansion associated with the axial force. The commonly-used BCM keeps the first-order terms of the axial force , termed as the first-order BCM. The study objective of using the BCM is the slender beam (Euler beam) where beam length is 10 times longer than beam thickness in the bending direction. Due to its simplicity and analytical form, the BCM has been successfully used in characterizing the performances of tilted-beam compliant mechanisms , XY flexure mechanisms , cross-spring flexural pivots , fully compliant bistable mechanisms , and compound multibeam parallelogram mechanisms . The BCM can also capture the first fixed-free beam buckling mode where the buckling load pcrit=-2.5, which is less than 1.3 % deviation from the classical beam buckling prediction of pcrit=-π2/4.

It was demonstrated that the maximum error of the first-order BCM is less than 5 % for the non-dimensional transverse displacements (i.e., rotation θ and translation y=ΔY/L, as illustrated in Fig. ) within ±0.1, intermediate deflection range, and the normalized axial force (p=PL2/(EI)) within ±10 . The non-dimensional transverse displacements are confined within ±0.1 because BCM was derived based on the linearized beam-curvature assumptions . Moreover, the range of the normalized axial force p was “empirically” restricted to [-10,10] to maintain less than 2 % truncation error , which may limit the application of the BCM.

For compliant mechanisms that use flexible elements experiencing axial-force dominant loads, compressive force or tensile force , there is a necessity to check if the maximum axial forces carried by the flexible elements are within the BCM's capability before using it for modeling. Moeen researched the nonlinear static load-displacement relationships of beam-based flexure modules with an intermediate semi-rigid element  and considered the load-stiffening effects caused by axial loads using the BCM . showed that the results of the first-order BCM for compound multi-beam parallelogram mechanisms significantly deviate from the results of the BCM including third-order terms of the axial force (termed as the third-order BCM) when the axial force exceeds a certain value. Although the third-order BCM is more accurate than the first-order BCM and extends the use of the BCM, what is the allowable axial force range of the third-order BCM remains unknown.

The determination of the range of allowable axial force includes finding the lower axial force boundary pl (i.e. the maximum compressive axial force) and the upper boundary pu (i.e. the maximum tensile axial force). Therefore, in this paper, by using the conditions: (1) the positive definite quadratic condition of the strain energy expression of the BCM; (2) the characteristic that the BCM can only capture the first buckling mode (the deflected beam carrying no more than one inflection point), we mathematically derive the upper and lower bounds of the allowable axial force for the third-order BCM, corresponding to the maximum tensile and compressive forces that the BCM can take, respectively. The third-order model is considered in this work because showed that the BCM with the third-order terms accurately captures the relevant nonlinearities.

The rest of this paper is organized as follows. Section  briefly reviews the BCM equations and other related wor. The range of allowable axial force for BCM is mathematically obtained in Sects.  and . Three examples are utilised in Sects.  and to demonstrate how the axial force influences the modeling errors of the third BCM. Finally, Sect.  draws remarks of this study.

Literature review Revisiting the Beam Constraint Model (BCM)

A simple beam flexure subject to combined force and moment loads.

Beam characteristic coefficients of the BCM matrices .

k11(0) 12 k12(0) -6 k22(0) 4 k11(1) 6/5 k12(1) -1/10 k22(1) 2/15 k11(2) -1/700 k12(2) 1/1400 k22(2) -11/6300 k11(3) 1/63 000 k12(3) -1/126 000 k22(3) 1/27 000 g11(3) -37/97 020 000 g12(3) 37/194 040 000 g22(3) -509/291 060 000

Figure  illustrates a uniform cross-section beam flexure imposed by a transverse force F, an axial force P and a moment M at its free end, resulting in transverse and axial deflections ΔX, ΔY and a tip slope θ. The parameters of the beam include: the length L, the in-plane thickness T, the out-of-plane thickness H, and the Young's modulus of the material E. A=HT and I=HT3/12 are the area and the second moment of inertia of the beam's cross-section, respectively. The BCM formulates the nonlinear load-deflection relations of the beam using a set of parametric and closed-form equations . Following are the BCM equations with the third-order terms of the axial force included : fm7=k11(0)k12(0)k12(0)k22(0)yθ+pk11(1)k12(1)k12(1)k22(1)yθ+p2k11(2)k12(2)k12(2)k22(2)yθ+p3k11(3)k12(3)k12(3)k22(3)yθ x=t2p12L2-12yθk11(1)k12(1)k12(1)k22(1)yθ-pyθk11(2)k12(2)k12(2)k22(2)yθ-32p2yθk11(3)k12(3)k12(3)k22(3)yθ-p3yθg11(3)g12(3)g12(3)g22(3)yθ where the non-dimensional coefficients k's and g's in the matrices are entirely independent of the beam shape and are referred to as the beam characteristic coefficients, as listed in Table . t is the normalized thickness given as t=T/L. The variables, m, f and p, are the normalized load parameters and y and x are the non-dimensionalized (normalized) deflection parameters, which are, respectively, given as: m=MLEI,f=FL2EI,p=PL2EI,y=ΔYL,x=ΔXL In addition, the third-order strain energy stored in the deflected beam is formulated as  v=12t2p212L2+12yθk11(0)k12(0)k12(0)k22(0)yθ-12p2yθk11(2)k12(2)k12(2)k22(2)yθ-p3yθk11(3)k12(3)k12(3)k22(3)yθ in which v is the normalized strain energy with respect to the beam parameters given as v=VLEI where V is the actual strain energy.

Note that the values of coefficients in Eqs. ()–() are listed in Table .

Other related work

proposed an analytical method for modeling spatial deflection of a slender beam in its intermediate deflection range, which combines two orthogonal beam constraint models (BCMs) with a torsional deflection model in the deformed configuration. Based on the Timoshenko beam theory, the Timoshenko Beam Constraint Model (TBCM) was developed for the purpose of including shear effects in the model . , presented a spatial BCM (SBCM) for the purpose of accurately predicting the nonlinear spatial constraint characteristics of thin bisymmetric beams in their intermediate deflection range (a bisymmetric beam refers to a beam whose cross section has equal moments of area and zero product of inertia). proposed a Chained BCM (CBCM), which extends the BCM for modeling large planer deflections of beams in compliant mechanisms by utilizing a discretization strategy. The CBCM also alleviates the limitation of the maximum axial load of the BCM through discretization. further developed a Chained SBCM (CSBCM), which extends the SBCM for modeling large spatial deflections of bisymmetric flexible beams.

Upper bound of axial force Determining upper bound

The strain energy stored in the deflected beam (Eq. ) can be rewritten as: v=VLEI=12t2p212L2+12yθKe11(p)Ke12(p)Ke21(p)Ke22(p)yθ where the first term on the right hand side of Eq. () represents the strain energy due to stretching or compressing and the following second term represents the strain energy due to bending. The Ke(p)'s in the strain energy formulation are Ke11(p)=k11(0)-k11(2)p2-2k11(3)p3Ke12(p)=k12(0)-k12(2)p2-2k12(3)p3Ke21(p)=k12(0)-k12(2)p2-2k12(3)p3Ke22(p)=k22(0)-k22(2)p2-2k22(3)p3 Because the strain energy expression of the third-order BCM is in a positive definite quadratic form for the case with a tensile axial force. The following conditions should be satisfied : Ke11(p)>0Ke11(p)Ke22(p)-Ke12(p)Ke21(p)>0 Combining Eqs. () and () and substituting the beam characteristic coefficients in Table  into the result yields the following inequalities: -p331500+p2700+12>0p6476280000-11p579380000+p4504000-13p315750+19p21050+12>0 By solving the above inequalities, we have: p<34.05 That is to say, pu=34.05 is the upper bound of axial force when using the third-order BCM. It should also be noted that a Maximum truncation error of 2.4 % is occurred with the neglecting of the item with the fourth power of p in the polynomial.

Case study

A compliant parallelogram mechanism, as shown in Fig. , is widely used in precision positioning as compliant translational joints . The geometric nonlinearities associated with deflections of the flexible beams and the resulting nonlinear behaviors always complicate the modeling. For example, when applying a vertical force Fy at the motion stage of the compliant parallelogram mechanism, the internal tensile axial force along the flexible beam can become very large while the transverse displacement ΔY restrict to within 10 % of the beam length. The BCM is a reasonable choice to model the flexible beams in compliant parallelogram mechanisms. used the third-order BCM for modeling compound compliant parallelogram mechanisms. But when the transverse displacement ΔY increases to a certain value, the difference between FEA and the BCM becomes large so that the third-order BCM may fail to predict the correct force-displacement relationship. The reason is that when the non-dimensional axial force p exceeds the upper boundary axial force pu given in Eq. (), the third-order BCM begins to predict the inaccurate results, in other words, the third-order BCM should be used within the allowable upper boundary of the axial force. More details are followed below.

A fixed-guided beam in compliant parallelogram mechanism.

Geometric parameters of the compliant parallelogram mechanism. H is the out-of-plane thickness of the beam.

Parameter H L T E Value 20 mm 45 mm 1 mm 69 GPa

A beam flexure in compliant parallelogram mechanism.

Since the compliant parallelogram mechanism has a symmetric configuration, only a single beam of the mechanism is modeled. As shown in Fig. , the beam parameters listed in Table  are identical to those used in . The coordinate XOY is established with its origin at the fixed end attached to the base. The normalized end displacement can be written as: x=0,y=ΔYL,θ=0 Let ΔY increase from the initial position to 0.1L and substitute Eq. () into Eqs. () and (), the relationships of the reaction force Fy and the normalized internal axial force p along the beam against the transverse displacement ΔY can be illustrated in Figs.  and , respectively.

A finite element model of the compliant parallelogram mechanism was built with the ANSYS software where the beam was meshed into 200 elements with Beam 188. The results are also plotted in Fig. . It can be seen from Fig.  that the results obtained by the third-order BCM clearly deviate from FEA results after p exceeds pu. The maximum error of the BCM comparing with FEA is 20 % when the non-dimensional end displacement is 0.1. The axial tensile loads are also given in Fig.  which shows that the maximum internal axial tensile force of the third-order BCM is larger than 100, which is much larger the upper boundary of the allowable force pu for the BCM.

Comparion of the load-displacement relationship of the compliant parallelogram mechanism.

In order to show the effect of the axial tensile force on the modeling error of the third-order BCM, we compare its results to those obtained using CBCM. Considering the effectiveness and efficiency, the CBCM results were obtained by dividing the beam into 3 elements . In such a way, the element axial tensile loads by normalization is reduced to pe=P(L/3)2/EI=p/9 which is ensured to be less than pu. The maximum error of the CBCM is reduced to 1 %, and the curves of the CBCM error and the normalized internal axial force during the deflection are also plotted in Fig. . Although not discussed in this work, it should be noted that the maximum stress in the beams may exceed the yield stress of the material when the mechanism translates 0.1L along the transverse direction, inducing failure of the mechanism.

Comparision of normalised axial force and error of the compliant parallelogram mechanism.

Lower bound of axial force Determining lower bound

When the compressive force applied at the end of a flexible beam exceeds a certain value p, the beam can buckle due to the instability of the beam. Actually, the buckling corresponds to a singularity of the stiffness matrix in the force-displacement formula. This buckling load is also defined as the critical load fcr. As the third-order BCM is derived based on a fixed-free beam shown in Fig. , the compressive force related to the buckling can be calculated by setting the determinant of the nonlinear bending stiffness matrix to be zero. Rewrite Eq. () as: fm=Ksyθ=Ks11(p)Ks12(p)Ks21(p)Ks22(p)yθ where Ks is the nonlinear stiffness matrix that relates the tip loads and the tip displacements, in which each Ks11(p)=k11(0)+k11(1)p+k11(2)p2+k11(3)p3Ks12(p)=k12(0)+k12(1)p+k12(2)p2+k12(3)p3Ks21(p)=k12(0)+k12(1)p+k12(2)p2+k12(3)p3Ks22(p)=k22(0)+k22(1)p+k22(2)p2+k22(3)p3 By letting the determinant of Ks be zero and using the parameters given in Table , the following equation can be obtained: p61905120000-11p5158760000+71p41512000-109p363000+277p22100+26p5+12=0 Equation () is solved to obtain the following roots: p=47.16+83.74i47.16+83.74i31.82+61.26i31.82-61.26i-23.50-2.47 The first four roots are complex numbers with no physical meaning, which thus are neglected. The fifth and sixth roots are real solutions associated with two buckling modes. However, when the normalized compressive axial force increases to the lower one, p=-2.47, the stiffness matrix Ks will exhibit a singularity, which the beam buckles, so the larger one (fifth root) is not considered. The axial force fa corresponding to the first buckling mode of fixed-free beam is given as: pa=p=-2.47 It is indicated in the knowledge of mechanics of materials that the critical load fcr corresponding to buckling is closely related to the boundary condition applying at the end of the flexible beam. In order to find the lower boundary of the axial force that defines the third-order BCM formula, we consider others boundary conditions: fixed-pivoted and fixed-fixed, as illustrated in Fig. . The critical loads corresponding to the fixed-free, fixed-pivoted and fixed-fixed beam buckling modes (Fig. ) can be written, respectively, as: pa=fcrpb=40049fcrpc=16fcr As the BCM can only capture the first buckling mode, according to the results in Eq. (), the lower boundary of the axial force, pl, for different boundary conditions can be expressed as: pl=-2.47Fixed-free beam-20.16Fixed-pivoted beam-39.52Fixed-fixed beam

(a) fixed-free beam buckling; (b) fixed-pivot beam buckling; (c) fixed-fixed beam buckling.

Case study

The bistable mechanism shown in Fig.  has many potential applications such as switches, static balancing mechanisms. The negative stiffness and the bistable behaviors have always complicated the modeling. Furthermore, the compressive axial force along the flexible beam can become very large while the transverse displacement ΔY (perpendicular to the beam) is still limited within 10 % of the beam length. Among all the methods, the BCM is a prior choice to model the behaviors of bistable mechanisms. But, when the compressive axial force exceeds a certain value, the BCM results may show an error. In this section, two different bistable mechanism examples were used to demonstrate the effectiveness of the range of the axial force allowable for the BCM. The first example is chosen to show the performance of the BCM when the compressive axial force is larger than the allowable lower boundary. The second example, on the contrary, is a case for the compressive axial force within the allowable lower boundary.

It should be demonstrated that for both examples the non-dimensional transverse displacements perpendicular to the beam are strictly limited within ±0.1 in all the deflected cases in order to guarantee the effectiveness of the linearized approximate expression for curvature. Details are elaborated as follows.

A bistable mechanism.

Considering that the geometry and loading of the bistable mechanism are symmetrical, a single fixed-guided limb is chosen to analyze. Figure  illustrates the schematic of the fixed-guided beam. Establish the coordinate system along the beam with its origin placed at the fixed end and denote the initial angle of the fixed-guided beam as β. The end displacement can be written as: x=ΔYcos⁡βL,y=ΔYsin⁡βL,θ=0 Where x is the normalized displacement along the beam, y is the normalized displacement perpendicular to the beam, and θ is the tip rotation of the beam, which stays unchanged during the deflection. The parameters of Example I are given in Table . Let ΔY increase from the initial position to 0.1L with the help of submitting Eq. () into Eqs. () and (), the relationships of the reaction force Fy and the normalized internal axial force along the beam against the transverse displacement ΔY can be obtained as shown in Figs. 9 and 10, respectively.

Schematic of the bistable mechanism.

Parameters of the examples. H is the out-of-plane thickness of the beam.

Parameter Example I Example II H 10 mm 10 mm L 80 mm 60 mm θ 3.5 2 t 1 mm 1 mm E 1.379 GPa 1.379 GPa

Comparison of the load-displacement relationship of Example I.

Comparision of normalized axial force and error of Example I.

Modelling the deflected beam of Example I using an FEA model built with the ANSYS software (the beam was meshed into 200 elements with Beam 188), the FEA force-displacement relationship is also plotted in Fig. . It can be observed that the results achieved by the BCM are consistent well with those obtained by FEA at the beginning of the force-deflection curve. When the tip displacement ΔY getting larger, the third-order BCM fails to predict the force-displacement relationship. The error between the results obtained by the BCM and FEA is up to 60 %, as shown in Fig. . The normalized internal axial force along the beam during the deflection is also illustrated in Fig.  (using the other y axis for comparison in the figure). It can be shown that when the normalized compressing axial force magnitude p is larger than 39.52, which exceeds the lower boundary of the axial force pl, the third-order BCM fails to deal with this problem.

The problem in Example I can be easily solved and the axial force at each displacement step was obtained by numerically solving the CBCM equations , The curve obtained by the CBCM is given in Fig. . The error of the force displacement relationship obtained by comparing the CBCM and FEA results is also plotted in Fig.  and the maximum error is less than 1 %.

As for Example II, we also model it using both the third-order BCM and FEA methods. The force-displacement relationship can be obtained as shown in Fig. . It can be seen that the results obtained by the BCM and FEA match well and the maximum error is 2.5 % as shown in Fig. . The normalized internal axial loads along the beam during the deflection are also displayed in Fig.  for comparison. The maximum compressing axial force magnitude p=21 is lower than |pl| and the tip displacement ΔY is kept in the intermediate deflection range, so the third-order BCM can correctly capture the force-displacement of the mechanism.

Comparion of the load-displacement relationship of Example II between FEA and the BCM.

Comparision of normalized axial force and error of Example II.

Conclusions

In this work, we determine the range of the allowable axial force of the third-order BCM. The upper bound pu are equal to 34.05, and the lower bound pl is set to different values with respect to different boundary conditions: pl=-2.47 for fixed-free buckling, pl=-20.16 for fixed-pivoted buckling, and pl=-39.52 for fixed-fixed buckling. Generally speaking, a complaint mechanism is analyzed in a series of incremental displacement steps or a series of incremental load steps. At each step, the proposed bounds should be used to check the axial force p to guarantee the effectiveness of the BCM equations.

Three examples are analyzed to demonstrate the effects of the axial force on the modeling errors of the third-order BCM, and the CBCM with 3 elements were used in this work to guarantee the accuracy of the results so that they can be used as the exact solutions for the purpose of comparison. Firstly, the case that the axial tensile force is beyond the upper boundary of the allowable force is given showing the effectiveness of pu. Secondly, the cases that the axial compressing forces arewithin and beyond the lower boundary of the allowable force are presented, the results have verified the effectiveness of the value of pl. When using the third-order BCM for modeling, it is always necessary to check if the axial force is in this allowable range to avoid (in case) large modeling errors.

For the non-dimensional transverse displacements within ±0.1 and the normalized axial force (p=PL2/(EI)) within [pl,pu], the majority of the modeling errors of the third-order BCM comes from the truncation of the Taylor's expansion.

It should be acknowledged that the CBCM expands the range of the allowable axial force for the BCM through discretizing the beam . For example, if a beam is discretized into two elements with equal length (the length of each element is L/2), then the corresponding normalized compressive axial force of the model for each element becomes 1/4 times of the original normalised one according to Eq. (). For a beam that carries an axial force beyond the derived bounds in this paper, the chained Beam- Constraint-Model (CBCM) and the comprehensive elliptic integral solution can be used to model it instead despite they much less analytical than the third-order BCM.

All the data used in this manuscript can be obtained on request from the corresponding author.

The authors declare that they have no conflict of interest.

Acknowledgements

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 51675396, and the Fundamental Research Funds for the Central Universities under No. JB170403/K5051204021. Edited by: Xianwen Kong Reviewed by: two anonymous referees

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Eng., 35, 369–382, 2011. </app></app-group></back> <!--<article-title-html>Determining the range of allowable axial force for the third-order Beam Constraint Model</article-title-html> <abstract-html><p class="p">The Beam Constraint Model (BCM) was developed for the purpose of accurately and analytically modeling nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections (10 % of the beam length). The BCM is expressed in the form of Taylor's expansion associated with the axial force. It has been found that the BCM may yield large predicting errors ( >  5 %) when the applied axial force goes beyond a certain boundary, even the deflection is still in the intermediate range. However, this boundary has not been clearly identified so far. In this work, we mathematically determine the non-dimensional boundary of the axial force by the condition that the strain energy expression of the BCM is a positive definite quadratic form, and by the buckling condition relate to compressing axial force. Several examples are analyzed to demonstrate the effects of the axial force on the modeling errors of the BCM. When using the BCM for modeling, it is always suggested to check if the axial force is within this boundary to avoid large modeling errors. If the axial force is beyond the boundary, the Chained Beam Constraint Model (CBCM) can be used instead.</p></abstract-html> <ref-html id="bib1.bib1"><label>Awtar and Sen(2010a)</label><mixed-citation> Awtar, S. and Sen, S.: A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Load-Displacement Formulation, ASME J. Mechan. 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