Compliant mechanisms are flexible structures that transmit motions or forces through elastic deformations with the advantages of low-cost and high-performance by effectively eliminating the impacts of frictions, wears and backlashes (). Thanks to the capability to deliver high precision motions, compliant mechanisms have been widely explored in advanced applications of precision engineering including micro/nano-manipulating systems, biological cell manipulations and precision instruments (). Meanwhile significant research efforts have been devoted to the design and analysis of compliant mechanisms ().

In particular, the fixed-guided beam flexure is a representative type of flexible segments in compliant mechanisms receiving increasing attentions in research literatures due to their potential to achieve large translational motions such as bistable mechanisms (), compliant parallelogram mechanisms (), and compound compliant parallelogram mechanisms (). With one end remaining a constant angle, the deflected configuration of the fixed-guided beam carries at least one inflection point, where axial deflections and axial forces have significant impacts on its mechanical properties. In particular, the equivalent stiffness will change and the rotational center will shift with respect to the connected links in the presence of load applications and the subsequent deflections. Consequently, complaint mechanisms with fixed-guided beams inevitably suffer from performance tradeoffs in terms of travel range, static stiffness and motion precision, see , , and the references therein.

Various methods have been developed for the analysis and design of complaint mechanisms with beam flexures, such as the finite element analysis, elliptical integrals, beam constraint model, as well as topological synthesis. Considering the complicated calculation of these approach, proposed the pseudo-rigid-body (PRB) modeling method to provide a simplified approach to analyze the deflection of beam flexures, which consists of two rigid links joined at a pin joint with a torsion spring. Based on the concept of PRB, presented a viable method of analyzing a fixed-guided compliant beam with an inflection point for various boundary conditions. To further approximate tip deflection of cantilever beams subject to combined end forces and moments, the PRB 2R (revolute) () and 3R () have been explored to improve the analysis accuracy. However these PRB models can only describe bending deformation, without capturing the axial deformation of the beams. To this end, the PRB PR (prismatic-revolute) (), PRR () and 3-Spring model () have been developed by adopting a prismatic pair with a linear spring to describe the axial deformations. Note that the PRB parameters are usually optimized over a large range of deflections. These models demonstrate significant modeling error for the small deflection beams adopted in the nano-manipulating systems, especially when the nonlinear effect caused by axial forces and deformations is addressed.

In this paper, we focus on the pseudo-rigid-body modeling of the fixed-guided beams employed in the nano-manipulating systems, where the transverse displacements is an order of magnitude less than the beam length but generally greater than the beam nominal thickness. Based on the beam constraint model (BCM) (), a modified pseudo-rigid-body (MPRB) modeling approach is proposed for the fixed-guided beam flexures by taking the impacts of nonlinear center-shifting and load-stiffening into account, where extension springs representing the axial stiffness and torsion springs accounting for the bending stiffness are adopted. Different from the existing PRB models, the PRB parameters of the proposed MPRB model are determined by the general tip loads. The model is successfully applied to the analysis of the compound parallelogram mechanism, the bistable mechanism and the 1-DOF translational mechanism, where the analytical results and FEA results demonstrate the effectiveness and accuracy of the proposed MPRB method.

In the rest of the paper, the BCM is first recalled in Sect. . In Sect. , the MPRB modeling method for a fixed-free beam flexure is developed. Section  establishes the MPRB model for the fixed-guided beam flexure, where the FEA analysis is also deployed to verify the effectiveness of the proposed method. In Sect.  the proposed MPRB modeling approach is applied to three common compliant mechanisms as case studies, where comparisons with existing methods are also provided. Finally, some concluding remarks are summarized in Sect. .

We start with a fixed-free beam with generalized end forces as depicted in Fig. . Note that we would like to focus on the deformed configuration that the transverse displacements are an order of magnitude less than the beam length but generally greater than the beam thickness, which agrees with many actual applications of nano-manipulating systems (). Therefore, the beam curvature can be linearized by assuming small slopes. Based on the Euler-Bernoulli equation, we have M(x)=EIdθds=EId2ydx2,M(x)=M+F(l+Δx-x)-P(Δy-y), where M(x) is the equivalent bending moment applied at the arbitrary cross section, F, P and M are the applied transverse force, axial force and bending moment, x is the distance along the undeflected beam axis, y is the transverse deflection, θ is the angular deflection, dθ/ds is the change rate of the angular deflection along the beam, E is the Young's modulus, I is the inertia moment, l is the initial length of the beam, Δx and Δy are the axial and transverse deformations of the tip point, respectively.

The above equations can be solved by applying the boundary conditions that y=0, dydx=0 at x=0 and d2ydx2=M/EI, d3ydx3=(Pdy/dx-F)/EI at x=l+Δx. The results can be further approximated by recalling  as δy=Δyl≈5f3(5+2p)+12m2(12+5p),δθ≈12f2(12+5p)+48(10+p)m5p2+208p+480,δx=Δxl=xel-xkl,δxe=xel=t2p12,δxk=xkl≈420-p700δy2-70-p700δyδθ+420-11p6300δθ2, where f=Fl2EI, p=Pl2EI and m=MlEI are the normalized forces and moment applied at the tip point, t=Tl is the dimensionless thickness, δx, δy and δθ are the normalized deflection parameters, respectively. It is obvious that the axial displacement Δx is comprised of a purely elastic component xe resulting from the elastic stretch and a kinematic component xk from the conservation of beam arc-length, as depicted in Fig. .

Accordingly, we can safely assume that the deformation of the beam can be divided into the following two steps:

The beam flexure stretches to l+xe under the action of axial force.

In this section, we consider a simple case for a fixed-free beam flexure with the length of l subject to a combined force [F,P], as shown in Fig. 2a. Based on the BCM model, the deformations of the free end D can be easily derived by inserting m=0 into Eqs. ()–().

Inspired by the PRB models proposed in , , a modified PRB model subject to a combined end force is proposed as depicted in Fig. 2b, which is composed of a rigid link OD of length γl and an extension spring joined by a pin joint O with a torsion spring. The undeflected length of the extension spring is (1-γ)l. We assume that the extension spring is not capable to bend to represent the axial stiffness, and the torsion spring represents the bending stiffness.

As illustrated in Fig. 2b, the extension spring is stretched by xe with the axial force P, i.e., the rigid link OD shifts to O′D′′. Thus the equivalent stiffness of the extension spring can be derived as ke=pδxe=12t2,Ke=keEIl3, where ke is defined as the normalized stiffness of the extension spring.

Then the free end D′′ rotates to D′ around the pin joint O′ under the combined end force. From Fig. 2b, we have the following equations: tan⁡Θ2=δxkδy,γsin⁡Θ2=δxk2+δy22, where Θ is denoted as the PRB angle and γ is the characteristic radius factor.

It follows by Eqs. ()–() and m=0 that γ=Bff212600(5p+12)2(2p+5)2+8750(5p+12)2Bf,Θ=fBf5250(5p+12)2(2p+5), where Bf=25(420-p)(5p+12)2+90(p-70)(2p+5)(5p+12)+36(2p+5)2(420-11p).

The equivalent normalized torque Tf applied at the torsion spring can be expressed as Tf=fγcos⁡Θ-pγsin⁡Θ≈γf1-Θ22-pΘ, where sin⁡Θ and cos⁡Θ are approximated by Θ and 1-Θ22 respectively because Θ is small enough.

Accordingly, the equivalent stiffness of the torsion spring can be calculated as kθ=TfΘ≈γ5250(5p+12)2(2p+5)Bf-p,KΘ=kθEIl, where kθ is defined as the normalized stiffness of the torsion spring.

Meanwhile the location of the torsion spring (i.e., pin point O) can be derived as xO=(1-γ)l,xO′=(1-γ)l+xe=(1-γ+t2p12)l.

It is straightforward that the location of the torsion spring is not only determined by the characteristic radius factor γ, but also the axial force P. The PRB parameters (characteristic radius factor γ and the equivalent torsional stiffness KΘ) are no longer constant values, but determined by the combined force. With this, the nonlinear characteristics of center-shifting and load-stiffening are incorporated in the modified PRB model. Note that the MPRB model proposed in this paper can capture the nonlinear effect of the axial deformations and forces on the equivalent torsional stiffness with some simplifications compared with the BCM. More importantly, the proposed MPRB model inherits the advantage of the PRB models, which makes a wealth of existing rigid-body mechanism analysis and synthesis knowledge available to the treatment of compliant mechanisms and is convenient for the mechanism design, kinematic synthesis, as well as the structure parameter optimization. It is also worth pointing out that due to the limitation of the BCM, the proposed MPRB model can be only applied to the modeling of beam flexures with intermediate deflections where the transverse displacements are with an order of magnitude less than the beam length.

In this section, we consider the modeling of the fixed-guided beam flexures, as illustrated in Fig. . Because the slope of the beam is equivalent at the fixed and guided ends, there must be at least one point of inflection in the deformation curve. At every point of inflection, the internal moment vanishes. Without loss of generality, the initially-straight fixed-guided beam with only one inflection point is considered in this work. Note that the methodology can be extended to the cases with more inflection points, which promises the capability of predicting the second mode bending of fixed-guided beams.

In this section, the proposed MPRB model is further applied to the analysis of three compliant mechanisms with fixed-guided beams, including a compound parallelogram mechanism, a bistable mechanism and a 1-DOF translational mechanism, where the impacts of the axial deformations and the axial forces cannot be ignored.

In this paper, we took the nonlinear effect (center-shifting and load-stiffening) caused by axial deformations and forces into consideration and developed a modified pseudo-rigid-body modeling method for compliant mechanisms with fixed-guided beam flexures. Based on the BCM, the fixed-free beam flexure was modeled as a rigid link and an extension spring joined by a pin joints with a torsion spring, where the PRB parameters including the characteristic radius factor and the equivalent torsional stiffness were shown to be determined by the general tip loads, instead of a constant value. Accordingly, a fixed-guided beam flexure was modeled as a pair of two fixed-free beam flexures jointed at the inflection point, where each fixed-free beam flexure was modeled by the proposed MPRB method. FEA simulations and three case studies, including a compound parallelogram mechanism, a fully compliant bistable mechanism and a 1-DOF translational mechanism, demonstrated significant improvement over existing results to predict performance characteristics of the compliant mechanisms with fixed-guided beams. It is worth pointing out that the proposed MPRB can be extended to the cases with more inflection points, which promises the capability of predicting the second mode bending of fixed-guided beams. Also note that the proposed MPRB modeling method is only suitable for capturing the deflection behavior of fixed-guided beam flexures with known inflection points and intermediate deformations, due to the limitation of the BCM. Future extensions along this line of research include the pseudo-rigid-body modeling for the case of larger deformations and more generalized flexure hinges (both notched hinges and beam flexures) by considering the influence of the axial force.