1 Introduction

Flexure hinges are frequently designed to transmit the relative motion between adjacent rigid links through elastic deformation [1]. Unlike conventional revolute joints, flexure hinges are free of friction and lubrication, capable of smooth motion and infinite resolution, and moreover, almost insusceptible to temperature [2]. Thus, flexure hinges are commonly used in many applications including micro-position platform [3, 4], micro-gripper [5], and micro-electro-mechanical systems (MEMS) [6]. Among various flexure hinges, cycloidal and filleted V-shaped flexure hinge were proved to possess higher rotational precision than RCFH [7, 8]. However, RCFH has a broader range of applications where high positioning accuracy and resolution are required [9, 10]. This is mainly due to its convenience and readily available machining processes. When RCFH is integrated in a mechanism, the static and dynamic characteristics of the system mainly depend on RCFH’s behaviors [11]. As one of the most important performance indexes, compliance of RCFH needs to be accurately modeled to reduce the modeling errors.

Analytical modeling and FEA are the two most commonly used approaches in RCFH’s compliance modeling. The compliances of RCFH obtained by FEA are always more accurate with respect to experimental data [12, 13]. However, FEA lacks of physical meaning and cannot construct the analytical relationship between the applied loads and resulting deformations. Moreover, the fine meshes achieving high precision usually result in massive computations [14]. In contrast, analytical modeling can provide a set of optimized design parameters in less time. Therefore, analytical modeling is preferred for the dimension synthesis during the design and optimization stage.

Starting with Paros and Weisbord’s seminal work as early as 1965 [15], various analytical solutions for RCFH’s compliance have been provided. With some exceptions [16, 17], most of them are based on the assumption of small deformation due to RCFH’s relatively low compliance and wide applications in small displacement-oriented systems [18]. Paros and Weisbord [15] presented both exact and tractable simplified closed-form equations for RCFH. The angular and linear compliances for bending and axial loads were developed based on the theory of mechanics of materials. Wu and Zhou [19] utilized the integration of linear differential equations and provided a set of more concise closed-form equations. Their equations were proved to have the same percentage errors as Paros and Weisbord’s exact equations (neglecting signs). Lobontiu [1] developed the in-plane and out-of-plane compliances of RCFH based on the Castigliano’s second theorem. During modeling RCFH’s torsional compliance, the thick-to-width ratio of the cross-section was supposed to be equal to or larger than 1 (or equal to or less than 1) for the whole notch region [20]. Tseytlin [21] adopted the inverse conformal mapping method to predict the rotational compliance of RCFH. The predictions were likely to be much closer to FEA results and experimental data. Afterwards, Yong et al. [22] compared the previously reviewed models with FEA results. A guideline for selecting the most suitable and accurate compliance equations for RCFH was presented. Chen and Howell [23] also summarized preexisting equations for RCFH’s torsional compliance and analyzed their computational accuracy. Two new equations were proposed, which were independent to the relative magnitude of the cross-sectional thickness and width. However, the equation that Chen and Howell adopted for the torsional moment of inertia was in terms of the rectangular beam with constant cross-section [24], which will result in inevitable errors. Xu et al. [25] divided RCFH into a series of rectangular section unit beams and developed an analytical model for RCFH based on the theory of elasticity and infinitesimal method. Larger number of unit beams usually corresponds to higher accuracy and worse computational efficiency.

The thorough review of RCFH’s analytical compliance models has indicated that there are no analytical compliance models for RCFH with the stress concentration in consideration. However, the stress concentration has been proved to have significant influence on RCFH’s compliance calculation [22, 26]. In addition, preexisting equations calculating torsional compliance for RCFH with any dimensions were not available. Thus, this paper presents an analytical compliance model for RCFH, which takes the stress concentration into account. The established model is expected to be helpful in automating the design of RCFH and better understanding its behavior.

The remainder of this paper is organized as follows. In Sect. 2, RCFH’s geometric as well as generic compliance model are presented firstly. Then, using FEA, the effect of stress concentration on RCFH is analyzed. Finally, modeling method with the stress concentration in consideration are presented. The in-plane and out-of-plane compliances of RCFH are calculated as a series of closed-form equations related with geometric dimensions and employed material in Sect. 3. Comparisons with other literatures and FEA are conducted in Sect. 4 to verify the established model. Finally, the paper ends with a conclusion in Sect. 5.

2 Compliance Model

2.1 Generic Compliance Model

RCFH is usually fabricated by removing two symmetric round cutouts from the rectangular beam. Thus, it is composed of two parts, the circle notch region and rigid beams. The boundary conditions of RCFH are acceptable to be fixed-free, which has been the case with all analytical approaches to RCFH ever since the work of Paros and Weisbord [15]. Figure 1 illustrates a fixed-free generic RCFH subjected to three-dimensional loads at its free end. The three-dimensional loads are defined as the loading vector \(\left\{ L \right\} = \left[ {F_{x} , F_{y} , F_{z} , M_{x} , M_{y} , M_{z} } \right]^{T}\), including two bending moments \(M_{y}\), \(M_{z}\), two shearing forces \(F_{y}\), \(F_{z}\), one axial force \(F_{x}\), and one torsional moment \(M_{x}\). The resulting deformations are described as the deformation vector \(\left\{ \delta \right\} = \left[ {u_{x} , u_{y} , u_{z} , \theta_{x} , \theta_{y} , \theta_{z} } \right]^{T}\), including three translations \(u_{x} , u_{y} , u_{z}\), and three rotations \(\theta_{x} , \theta_{y} , \theta_{z}\). Geometric definitions of RCFH are illustrated in Fig. 1 as well, mainly including the cutting radius \(r\), the minimum thickness \(t\), and the depth \(w\). The Cartesian coordinate system is utilized, where the origin is located at the midpoint of the free end, and the \(x\)-, \(y\)- and \(z\)-axes are the longitudinal, altitudinal and depth directions, respectively.

Fig. 1
figure 1

Boundary conditions, geometric parameters, spatial loads and resulting deformations of a generic RCFH

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Due to RCFH’s relatively low compliance and wide applications in small displacement-oriented systems, this work mainly focuses on RCFH whose deformations are small (infinitesimal). Thus, the small deformation theory can be applied and the geometrically nonlinear effect can be neglected [27]. In addition, the parasitic motion is ignored in this work because it could be reduced to be negligible with some precautions [28, 29]. Then, the deformation vector \(\left\{ \delta \right\}\), loading vector \(\left\{ L \right\}\) and the compliance matrix \(\left[ {\text{C}} \right]\) of RCFH can be connected by Hooke’s law, namely \(\left\{ \delta \right\} = \left[ {\text{C}} \right]\left\{ L \right\}\) with

$$\left[ {\text{C}} \right] = \left[ {\begin{array}{*{20}c} {{\text{C}}_{{u_{x} - F_{x} }} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {{\text{C}}_{{u_{y} - F_{y} }} } & 0 & 0 & 0 & {{\text{C}}_{{u_{y} - M_{z} }} } \\ 0 & 0 & {{\text{C}}_{{u_{z} - F_{z} }} } & 0 & {{\text{C}}_{{u_{z} - M_{y} }} } & 0 \\ 0 & 0 & 0 & {{\text{C}}_{{\theta_{x} - M_{x} }} } & 0 & 0 \\ 0 & 0 & {{\text{C}}_{{\theta_{y} - F_{z} }} } & 0 & {{\text{C}}_{{\theta_{y} - M_{y} }} } & 0 \\ 0 & {{\text{C}}_{{\theta_{z} - F_{y} }} } & 0 & 0 & 0 & {{\text{C}}_{{\theta_{z} - M_{z} }} } \\ \end{array} } \right]$$
(1)

The subscript pair of any compliance in Eq. (1) corresponds to the deformation at the free end and the particular load producing that deformation. According to Maxwell’s reciprocity principle, following equations can be yielded

$${\text{C}}_{{u_{z} - M_{y} }} = {\text{C}}_{{\theta_{y} - F_{z} }} , {\text{C}}_{{u_{y} - M_{z} }} = {\text{C}}_{{\theta_{z} - F_{y} }}$$
(2)

which indicates that \(\left[ {\text{C}} \right]\) is a symmetric square matrix.

2.2 Influence of Stress Concentration

Due to its high accuracy, FEA is usually regarded as a reliable method to estimate RCFH’s behaviors. The stress and strain obtained by FEA are always accurate with respect to experimental data. Therefore, FEA is conducted here to simulate RCFH’s stress and strain. During the numerical calculations, geometric parameters of RCFH are given as \(r = 5\,{\text{mm}}\), \(t = 1\,{\text{mm}}\), and \(w = 5\,{\text{mm}}\). Details about FEA settings are demonstrated in Sect. 4.2.

It can be observed from Fig. 2 that the iso-strain and iso-stress plane are not perpendicular to the \(x\)-axis either parallel to each other when RCFH is subjected to a unit axial load \(F_{x}\). This is not consistent with theoretical assumptions and mainly because of the stress concentration happening at RCFH’s weakest point.

Fig. 2
figure 2

FEA results of RCFH’s stress and strain distributions; a Iso-strain plane, b iso-stress plane

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In order to obtain a clear view, FEA results as well as theoretical deformations along a selected path are compared in Fig. 3. The path origins from the center of the fixed end and ends at the center of the free end, as depicted in Fig. 3a. According to the results plotted in Fig. 3b and c, it is clear that RCFH’s deformations obtained by FEA are slightly larger than the theoretical values either for a unit axial force or a bending moment. This phenomenon indicates that the stress concentration has significant influence on RCFH’s compliance calculation. Therefore, more attentions must be directed towards the stress concentration effect within RCFH as well as RCFH-based mechanisms designs to reduce modeling errors.

Fig. 3
figure 3

Comparisons between FEA results and theoretical values; a definition of the selected path, b deformations for a unit axial force, c deformations for a unit bending moment

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2.3 Modeling Method

From the qualitative analyses above and Saint–Venant’s principle, it is noted that the stress concentration only affects the region near the circle notch. For the sake of generality, RCFH with two relatively long beam connections is considered in this work, which can be regarded as a flexure serial chain that is formed with five flexure members, as depicted in Fig. 4.

Fig. 4
figure 4

RCFH with two relatively long beam connections

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It is expected that the stress concentration only affects the second and forth segments and has little effect on the first and fifth segments. Compared to RCFH illustrated in Fig. 1, the configuration of Fig. 4 adds another three design parameters, the length of beam connections near to the circle notch \(l_{1}\), and the length of beam connections far away from the circle notch on both sides \(l_{2}\), \(l_{3}\). Based on the virtual work theory and the superposition relationship of the deformation, the overall compliance matrix of RCFH \(\left[ {\text{C}} \right]\) can be expressed as

$$\left[ {\text{C}} \right] = \mathop \sum \limits_{i = 1}^{5} \left[ {J_{i} } \right] \cdot \left[ {{\text{C}}_{i} } \right] + \left[ {{\text{C}}_{s} } \right]$$
(3)

where \(\left[ {J_{i} } \right]\) means the transformation matrix, \(\left[ {{\text{C}}_{i} } \right]\) stands for the compliance matrix of the segment \(i\), \(\left[ {{\text{C}}_{s} } \right]\) represents for the additional compliance caused by the stress concentration. The transformation matrix comes in the following form

$$\left[ {J_{i} } \right] = \left[ {\begin{array}{*{20}c} {\left[ {R_{i} } \right]} & {0_{4 \times 2} } \\ {0_{2 \times 4} } & {N_{i} \cdot \left[ {S_{i} } \right]} \\ \end{array} } \right]$$
(4)

where \(\left[ {R_{i} } \right]\), \(\left[ {S_{i} } \right]\) are 4 × 4, 2 × 2 unit square matrix, respectively. \(N_{i}\) is the mapping coefficient and there are

$$\left\{ {\begin{array}{*{20}l} {N_{1} = l_{1} /l} \hfill \\ {N_{2} = \left( {l_{1} + l_{2} } \right)/l} \hfill \\ {N_{3} = \left( {l_{1} + l_{2} + 2r} \right)/l} \hfill \\ {N_{4} = \left( {2l_{1} + l_{2} + 2r} \right)/l} \hfill \\ {N_{5} = 1} \hfill \\ \end{array} } \right.$$
(5)

where \(l\) is the total length of the specimen and equal to \(2l_{1} + l_{2} + l_{3} + 2r\). According to the results of Li et al. [26], \(\left[ {{\text{C}}_{s} } \right]\) can be equivalent to the compliance of a rectangular beam that locates at the same position with the circle notch. The length, width and height of the rectangular beam are \(2r\), \(2r + t\), and \(w\), respectively. Then, Eq. (3) can be reformulated as

$$\left[ {\text{C}} \right] = \left[ {{\text{C}}_{r} } \right] + \left[ {J_{3} } \right] \cdot \left[ {{\text{C}}_{3} } \right]$$
(6)

where \(\left[ {{\text{C}}_{r} } \right]\), \(\left[ {{\text{C}}_{3} } \right]\) are the compliance matrices of the rectangular beam over the whole RCFH and the circle notch region, respectively. Without loss of generality, two new variables \(h_{1}\) and \(h_{2}\) are introduced to substitute for \(l_{1} + l_{2}\) and \(l_{1} + l_{3}\), respectively, which will make Eq. (6) universal to RCFH with short beam connections.

3 Analytical Compliance Model

According to the analysis in Sect. 2.3, the main requirement for RCFH’s compliance model is to solve the compliance matrixes of the rectangular beam over the whole RCFH and the circle notch region. In this section, both of them are analytically calculated.

3.1 Compliance Matrix of the Rectangular Beam

The rectangular beam with constant cross-section has been overwhelmingly investigated. Meanwhile, the established compliance models have been proved to be precise for most practical applications. Therefore, the compliance matrix of the rectangular beam over the whole RCFH can be directly given as

$$\left[ {{\text{C}}_{r} } \right] = \left[ {\begin{array}{*{20}c} {\frac{l}{Ews}} & 0 & 0 & 0 & 0 & 0 \\ 0 & {\frac{\alpha l}{Gws} + \frac{{4l^{3} }}{{Ews^{3} }}} & 0 & 0 & 0 & {\frac{{6l^{2} }}{{Ews^{3} }}} \\ 0 & 0 & {\frac{\alpha l}{Gws} + \frac{{4l^{3} }}{{Ew^{3} s}}} & 0 & { - \frac{{6l^{2} }}{{Ew^{3} s}}} & 0 \\ 0 & 0 & 0 & {\frac{l}{{GI_{t} }}} & 0 & 0 \\ 0 & 0 & { - \frac{{6l^{2} }}{{Ew^{3} s}}} & 0 & {\frac{12l}{{Ew^{3} s}}} & 0 \\ 0 & {\frac{{6l^{2} }}{{Ews^{3} }}} & 0 & 0 & 0 & {\frac{12l}{{Ews^{3} }}} \\ \end{array} } \right]$$
(7)

where \(s\) is an intermediate variable and equal to \(2r + t\). \(E\) and \(G\) are the Young’s modulus and the shear modulus, respectively. \(\alpha\) is the shear coefficient and commonly equal to 6/5 for a rectangular cross-section. \({\text{I}}_{\text{t}}\) is the torsional moment of inertia. For a rectangular cross-section with the length of \(h\) and width of \(b\), \(I_{t}\) can be expressed as \(\beta hb^{3}\), where \(\beta\) is the torsional shape coefficient. The representative corresponding values of the length–width ratio \(k\) and \(\beta\) are available in Table 1.

Table 1 Representative corresponding values of \(k\) and \(\beta\)
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However, it is worth mentioning that these discrete values are often inadequate for engineering design. In order to satisfy the practical applications, approximate continuous equations for the torsional shape coefficient were in demand, which has been provided by Hearn [24] and Young [30]. Their equations are given as

$$\beta_{\text{H}} = \frac{{k^{2} }}{{3.5k^{2} + 3.5}} ,\quad \beta_{\text{Y}} = \frac{1}{3} - \frac{0.21}{k}\left( {1 - \frac{1}{{12k^{4} }}} \right)$$
(8)

where \(\beta_{\text{H}}\) indicates Hearn’s equation and \(\beta_{\text{Y}}\) indicates Young’s equation. In order to know which equation is more accurate, \(\beta_{\text{H}}\), \(\beta_{\text{Y}}\), as well as representative corresponding values of \(\beta\) are compared together in Fig. 5. It is apparent that \(\beta_{\text{Y}}\) is more consistent with those discrete values listed in Table 1. Therefore, \(\beta_{\text{Y}}\) is adopted here, and two cases for calculating \(I_{t}\) are summarized in Table 2.

Fig. 5
figure 5

Descriptions of representative values of \(\beta\), as well as continuous equations \(\beta_{\text{H}}\) and \(\beta_{\text{Y}}\)

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Table 2 Two cases for calculating \(I_{t}\)
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3.2 Compliance Matrix of the Circle Notch Region

In order to derive closed-form compliance equations for the circle notch region, the Castigliano’s second theorem is adopted and written as

$$\Delta = \frac{\partial U}{\partial P}$$
(9)

where \(\Delta\) is the deformation due to the applied load, \(U\) is the strain energy stored, and \(P\) is the applied load. The strain energy is determined by the applied loads and expressed as

$$U = \mathop \int \limits_{{h_{1} }}^{{h_{1} + 2r}} \frac{{F_{x}^{2} }}{2EA} + \frac{{\alpha F_{y}^{2} }}{2GA} + \frac{{\alpha F_{z}^{2} }}{2GA} + \frac{{M_{x}^{2} }}{{2GJ_{x} }} + \frac{{\left( {M_{y} + xF_{z} } \right)^{2} }}{{2EI_{y} }} + \frac{{\left( {M_{z} + xF_{y} } \right)^{2} }}{{2EI_{z} }} dx$$
(10)

where \(I_{y}\), \(I_{z}\) are cross-sectional moments of the area in the \(y\) and \(z\) direction and equal to \(w^{3} t_{x} /12\) and \(wt_{x}^{3} /12\), respectively. \(A\) is the cross-sectional area. \(J_{x}\) is the torsional moment of inertia for the infinitesimal strip \(dx\) at position \(x\). The variational thickness \(t_{x}\) can be expressed as

$$t_{x} = t + 2\left[ {r - \sqrt {2r\left( {x - h_{1} } \right) - \left( {x - h_{1} } \right)^{2} } } \right]$$
(11)

Then, the linear and angular compliances except for torsional compliance can be easily derived. Concrete expressions of these compliances and \(f_{i}\) (\(i\) = 1–4) involved are expressed as follows:

$$\left[ {{\text{C}}_{3} } \right] = \left[ {\begin{array}{*{20}c} {\frac{1}{Ew}f_{1} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {\frac{\alpha }{Gw}f_{1} + \frac{12}{Ew}f_{2} } & 0 & 0 & 0 & {\frac{{12\left( {r + h_{1} } \right)}}{Ew}f_{4} } \\ 0 & 0 & {\frac{\alpha }{Gw}f_{1} + \frac{12}{{Ew^{3} }}f_{3} } & 0 & { - \frac{{12\left( {r + h_{1} } \right)}}{{Ew^{3} }}f_{1} } & 0 \\ 0 & 0 & 0 & {\frac{1}{G}\mathop \smallint \limits_{{h_{1} }}^{{h_{1} + 2r}} \frac{1}{{J_{x} }}dx} & 0 & 0 \\ 0 & 0 & { - \frac{{12\left( {r + h_{1} } \right)}}{{Ew^{3} }}f_{1} } & 0 & {\frac{12}{{Ew^{3} }}f_{1} } & 0 \\ 0 & {\frac{{12\left( {r + h_{1} } \right)}}{Ew}f_{4} } & 0 & 0 & 0 & {\frac{12}{Ew}f_{4} } \\ \end{array} } \right]$$
(12)
$$f_{1} = - \frac{\pi }{2} + \frac{{2\left( {2r + t} \right)}}{{\sqrt {t\left( {4r + t} \right)} }}\arctan \sqrt {\frac{4r + t}{t}}$$
(13)
$$\begin{aligned} f_{2} & = \frac{{96r^{3} \left( {h_{1} + r} \right)^{2} + 32r^{2} \left( {2h_{1}^{2} + 4rh_{1} + 3r^{2} } \right)t + 8r\left( {2h_{1}^{2} + 4rh_{1} + \left( {11 + 4\pi } \right)r^{2} } \right)t^{2} + 32\left( {1 + \pi } \right)r^{2} t^{3} }}{{8t^{2} \left( {2r + t} \right)\left( {4r + t} \right)^{2} }} \\ & \quad + \frac{{2\left( {2 + 5\pi } \right)rt^{4} + \pi t^{5} }}{{8t^{2} \left( {2r + t} \right)\left( {4r + t} \right)^{2} }} + \frac{{4\left( {2r + t} \right)\left( {24r^{2} \left( {h_{1} + r} \right)^{2} + 8r^{3} t - 14r^{2} t^{2} - 8rt^{3} - t^{4} } \right)arctan\sqrt {\frac{4r + t}{t}} }}{{8t^{5/2} \left( {4r + t} \right)^{5/2} }} \\ \end{aligned}$$
(14)
$$f_{3} = - \frac{1}{2}\pi h_{1}^{2} - \pi rh_{1} - \frac{1}{4}\left( { - 4 + \pi } \right)r^{2} + \frac{1}{2}\left( {1 + \pi } \right)rt + \frac{1}{8}\pi t^{2} + \frac{{4\left( {2r + t} \right)\left( {4\left( {h_{1} + r} \right)^{2} - 4rt - t^{2} } \right)arctan\sqrt {\frac{4r + t}{t}} }}{{8\sqrt {t\left( {4r + t} \right)} }}$$
(15)
$$f_{4} = \frac{{2r\left( {t\left( {4r + t} \right)\left( {6r^{2} + 4rt + t^{2} } \right) + 6r\left( {2r + t} \right)^{2} \sqrt {t\left( {4r + t} \right)} arctan\sqrt {\frac{4r + t}{t}} } \right)}}{{t^{3} \left( {2r + t} \right)\left( {4r + t} \right)^{3} }}$$
(16)

According to Sect. 3.1, \(\beta_{\text{Y}}\) is considered to be more accurate than \(\beta_{\text{H}}\), therefore, there should be

$$J_{x} = \left\{ {\begin{array}{*{20}l} {\beta_{\text{Y}} w^{3} t_{x} \quad t_{x} \ge w } \hfill \\ {\beta_{\text{Y}} wt_{x}^{3} \quad t_{x} < w } \hfill \\ \end{array} } \right.$$
(17)

However, the high-order term existing in \(\beta_{\text{Y}}\) makes Eq. (17) difficult to integral. In terms of that, a more simplified equation \(\beta_{\text{L}} = 0.333 - 0.21/k\) proposed by Lobontiu et al. [20] is adopted. The Root Mean Square Error (RMSE) of \(\beta_{\text{Y}}\), \(\beta_{\text{L}}\) and \(\beta_{\text{H}}\) are calculated, which are 0.006, 0.0063 and 0.015, respectively. This indicates that \(\beta_{\text{L}}\) is more suitable for applications where high computational accuracy and efficiency need to be achieved simultaneously. Then, three cases for calculating \({\text{C}}_{{3,\theta_{x} - M_{x} }}\) are listed in Table 3 where \(\beta_{\text{L}}\) is adopted.

Table 3 Three cases for calculating \({\text{C}}_{{3,\theta_{x} - M_{x} }}\)
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4 Comparisons and FEA Validation

4.1 Comparisons

It should be pointed out that there are many definite integrals in the compliance equations of the circle notch region. In order to verify these mathematic relationships, comparisons are conducted between the model established in Sect. 3.2 (donated by Tuo) and other preexisting ones, mainly including Paros and Weisbord’s (both full and simplified, donated by Paros (full) and Paros (Simp.), respectively), Wu and Zhou’s (donated by Wu), Lobontiu’s (donated by Lo.), Tseytlin’s, as well as Chen’s. For all comparisons, Titanium alloy (Tc4) is chosen as the linearly elastic material. The properties of Tc4 are: Young’s modulus \(E = 110\,{\text{GPa}}\), Poisson’s ratio \(\nu = 0.34\), shear modulus \(G = 43\,{\text{GPa}}\), and density \(\rho = 4500\,{\text{kg}}/{\text{m}}^{3}\). The width \(w\) and thickness \(t\) of all compared right circle flexure hinges (RCFHs) are chosen as 10 mm and 1 mm, respectively. The results are depicted in Fig. 6, where \(r\) varies from 0.5 to 10.

Fig. 6
figure 6

Comparisons with other analytical compliance models for the circle notch region; a\({\text{C}}_{{3,u_{x} - F_{x} }}\), b\({\text{C}}_{{3,u_{y} - F_{y} }}\), c\({\text{C}}_{{3,u_{z} - F_{z} }}\), d\({\text{C}}_{{3,\theta_{x} - M_{x} }}\), e\({\text{C}}_{{3,\theta_{y} - M_{y} }}\), f\({\text{C}}_{{3,\theta_{y} - F_{z} }}\), g\({\text{C}}_{{3,\theta_{z} - M_{z} }}\), h\({\text{C}}_{{3,\theta_{z} - F_{y} }}\)

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From Fig. 6, following points can be easily drawn:

  • Results of Wu and Tuo are identical when considering the shear effect. Due to its high accuracy and wide applications, Wu’s model is usually regarded as a reliable solution for RCFH’s compliance. Thus, the mathematic relationships in Sect. 3.2 can be considered correct. However, the equation calculating RCFH’s torsional compliance was not mentioned in Wu’s work.

  • Torsional compliance calculated by Tuo’s model has the same trend with Chen’s results. However, the difference tends to be obvious as \(r\) increases. This is mainly because RCFH was regarded as a rectangular beam with the cross-sectional length of \(w\) and width of \(t\).

  • Models of Lobontiu and Tuo were derived using the same method (Castigliano’s second theorem). Thus, they should have the same results. However, results of the linear and angular compliances for \(F_{z}\) are still very different. Given that Lobontiu’s equations were relatively simple and the specific derivation progress was not provided, it is reasonable to guess that some approximations might have been made in his derivation.

4.2 FEA Validation

FEA is utilized as a benchmark in this section to examine the established model. FEA is conducted in ANSYS Workbench, where 3D static structure analysis method is adopted. Due to the relatively simple construction of RCFH, Design Modeler integrated in Workbench is used to model the geometry. In order to improve the computational accuracy, multi-zone sweep instead of a “smart” mesh is utilized. The material is chosen as Tc4, whose properties have been listed before. In each static analysis, the boundary conditions are chosen as: one end is fixed for all degrees of freedom, and the three-dimensional loads are applied at the midpoint of the opposite end. In addition, the loads are supposed to be ramped unit loads for the purpose of reducing the amount of calculation. The numerical simulations are carried out based on RCFHs with the width of 10 mm and thickness of 1 mm. Moreover, two new dimensionless variables \(h_{1} /r\) and \(h_{2} /r\) are utilized to quantify the relative magnitude between \(h_{1}\), \(h_{2}\), and \(r\), respectively.

The percentage relative error between FEA results and analytical model’s predictions is defined as \(\frac{{{\text{C}}_{\text{A}} - {\text{C}}_{\text{FEA}} }}{{{\text{C}}_{\text{FEA}} }} \times 100\%\), where \({\text{C}}_{\text{A}}\) and \({\text{C}}_{\text{FEA}}\) donate for the analytical model’s predictions and FEA results, respectively. The complete set of the results is plotted in Figs. 7, 8, and 9, where the cutting radius are 1, 3, and 5 mm respectively. In each case, \(h_{1} /r\) and \(h_{2} /r\) vary from 1 to 6.

Fig. 7
figure 7

Percentage relative errors between the model’s predictions and FEA results for the case of \(r = 5\); a\({\text{C}}_{{u_{x} - F_{x} }}\), b\({\text{C}}_{{{\text{u}}_{\text{y}} {\text{ - F}}_{\text{y}} }}\), c\({\text{C}}_{{{\text{u}}_{\text{z}} {\text{ - F}}_{\text{z}} }}\), d\({\text{C}}_{{\theta_{x} - M_{x} }}\), e\({\text{C}}_{{\theta_{y} - M_{y} }}\), \({\text{C}}_{{u_{z} - M_{y} }}\) as well as \({\text{C}}_{{\theta_{y} - F_{z} }}\)f\({\text{C}}_{{\theta_{z} - M_{z} }}\), \({\text{C}}_{{u_{y} - M_{z} }}\), as well as \({\text{C}}_{{\theta_{z} - F_{y} }}\)

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Fig. 8
figure 8

Percentage relative errors between the model’s predictions and FEA results for the case of \(r = 3\); a\({\text{C}}_{{u_{x} - F_{x} }}\), b\({\text{C}}_{{{\text{u}}_{\text{y}} {\text{ - F}}_{\text{y}} }}\), c\({\text{C}}_{{{\text{u}}_{\text{z}} {\text{ - F}}_{\text{z}} }}\), d\({\text{C}}_{{\theta_{x} - M_{x} }}\), e\({\text{C}}_{{\theta_{y} - M_{y} }}\), \({\text{C}}_{{u_{z} - M_{y} }}\) as well as \({\text{C}}_{{\theta_{y} - F_{z} }}\)f\({\text{C}}_{{\theta_{z} - M_{z} }}\), \({\text{C}}_{{u_{y} - M_{z} }}\), as well as \({\text{C}}_{{\theta_{z} - F_{y} }}\)

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Fig. 9
figure 9

Percentage relative errors between the model’s predictions and FEA results for the case of \(r = 1\); a\({\text{C}}_{{u_{x} - F_{x} }}\), b\({\text{C}}_{{{\text{u}}_{\text{y}} {\text{ - F}}_{\text{y}} }}\), c\({\text{C}}_{{{\text{u}}_{\text{z}} {\text{ - F}}_{\text{z}} }}\), d\({\text{C}}_{{\theta_{x} - M_{x} }}\), e\({\text{C}}_{{\theta_{y} - M_{y} }}\), \({\text{C}}_{{u_{z} - M_{y} }}\) as well as \({\text{C}}_{{\theta_{y} - F_{z} }}\), f\({\text{C}}_{{\theta_{z} - M_{z} }}\), \({\text{C}}_{{u_{y} - M_{z} }}\), as well as \({\text{C}}_{{\theta_{z} - F_{y} }}\)

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It is noted that the percentage relative errors decrease with the increasing ratio of \(r/t\). In addition, it is apparent that the results between FEA and established model are in good agreement with the maximum deviation being less than 20%. This indicates the established model is accurate enough and can be used to predict RCFH’s compliance during the design and optimization stage.

5 Conclusions

An analytical compliance model for RCFH has been presented in this paper, which takes the stress concentration effect into account. The stress concentration caused by changes in RCFH’s cross-section is analyzed using FEA method. Relevant results show that the stress concentration will result in additional deformations and bring calculation errors on RCFH’s compliance. To overcome compliance calculation errors, the stress concentration effect has been quantified during modeling RCFH’s compliance. A series of closed-form equations related with geometric parameters and employed material are derived on the basis of the virtual work theory, superposition relationship of the deformation, as well as Castigliano’s second theorem. Comparisons with other literatures, including Paros and Weisbord’s, Wu and Zhou’s, Lobontiu’s, Tseytlin’s, as well as Chen’s, indicate the correctness of complicated definite integrals existing in these closed-form equations. FEA results and analytical predictions are found in good agreement, with maximum errors of less than 20% for a wide range of geometric dimensions. The developed model can be utilized as guidelines for RCFH as well as RCFH-based mechanism design. In addition, in order to further verify the established model, the related experiments will be conducted in our follow-up works.