- Original Article
- Open Access
Analytical Compliance Modeling of Serial Flexure-Based Compliant Mechanism Under Arbitrary Applied Load
- Li-Ping Wang^{1, 2},
- Yao Jiang^{3} and
- Tie-Min Li^{1, 2}Email author
https://doi.org/10.1007/s10033-017-0138-9
© The Author(s) 2017
- Received: 20 December 2015
- Accepted: 20 April 2017
- Published: 17 May 2017
Abstract
Analytical compliance model is vital to the flexure- based compliant mechanism in its mechanical design and motion control. The matrix is a common and effective approach in the compliance modeling while it is not well developed for the closed-loop serial and parallel compliant mechanisms and is not applicable to the situation when the external loads are applied on the flexure members. Concise and explicit analytical compliance models of the serial flexure-based compliant mechanisms under arbitrary loads are derived by using the matrix method. An equivalent method is proposed to deal with the situation when the external loads are applied on the flexure members. The external loads are transformed to concentrated forces applied on the rigid links, which satisfy the equations of static equilibrium and also guarantee that the deformations at the displacement output point remain unchanged. Then the matrix method can be still adopted for the compliance analysis of the compliant mechanism. Finally, several specific examples and an experimental test are given to verify the effectiveness of the compliance models and the force equivalent method. The research enriches the matrix method and provides concise analytical compliance models for the serial compliant mechanism.
Keywords
- Compliant mechanism
- Compliance modeling
- Matrix method
1 Introduction
Compared with conventional mechanisms, flexure-based compliant mechanisms can provide motions without friction, backlash, and wear [1]. Therefore, they are widely used in the applications where high positioning accuracy is required, such as the scanning tunneling microscope, precision positioning stage, X-ray lithography, and wafer alignment in microlithography [2–4].
Flexure-based compliant mechanisms transmit the motions entirely through the deformations of the flexure hinges. Therefore, a concise and accurate compliance model, which establishes the relationship between the deformations and the applied loads, is important and necessary for the optimal design, performance analysis, and accurate motion control of the compliant mechanisms [5]. Many works related to the compliance modeling of the compliant mechanism have been done and the main methods include the pseudo-rigid-body (PRB) model, the Castigliano’s theorem, the matrix method, and the finite element model(FEM). The PRB model, initially presented by L L Howell [6], treats the flexure hinge as a revolute joint with an attached torsional spring and provides a simple and intuitive way to estimate the mechanism’s compliance. However, this method shows some inaccuracies because the axial and transverse deformations of the flexure hinges are not taken into consideration. H H Pham, et al. [7], used an extended PRB model, named the PRB-D model, to derived the kinematic model of a flexure-based parallel mechanism. The experimental tests showed that the calculation error of the PRB-D model was only 1/3 compared with the PRB model. P Xu, et al. [8], used the PRB model to establish the compliance model for a variety of beam-based compliant mechanisms with large deformations. S Z Liu, et al. [9], studied the frequency characteristics and sensitivity of the large-deformation compliant mechanism based on a modified PRB model. The Castigliano’s theorem derives the compliance model of the compliant mechanism based on the strain energy. N Lobontiu, et al. [10], formulated an analytical compliance model for the planar compliant mechanisms with single-axis flexure hinges based on the Castigliano’s displacement theorem. In this research, closed-form equations were produced and a parametric study of the mechanism performance was also carried out. X Jia, et al. [11], obtained the input stiffness model of the active arm of a 3-DOF compliant parallel positioning stage on the basis of the Castigliano’s first theorem. The matrix method transforms the local compliance of the flexure members to a global frame for easily obtaining the compliance model of the entire mechanism. Y KOSEKI, et al. [12], applied the matrix method to the kinematic analysis of a translational 3-DOF micro parallel compliant mechanism. H H Pham, et al. [13], used the matrix method to present an analytical stiffness model of a flexure parallel mechanism. S L Xiao, et al. [14], conducted the compliance modeling of a novel compliant micro-parallel positioning stage by means of the matrix method. N Lobontiu [15] proposed a basic three-point compliance matrix method to model the direct and inverse quasi-static response of constrained and over-constrained planar serial compliant mechanisms. The FEM [16–19] has been widely used in the structural mechanics field and is by far the most accurate computational method in calculating the compliance of the compliant mechanism. Since the FEM is a numerical method, the intrinsic characteristic of the compliance of the compliant mechanism cannot be explicitly identified.
From the introductions above, the calculation accuracy of the PRB model is relative low, but it is simple and useful in the early stage of the prototype design and performance analysis of the compliant mechanisms. The FEM, by contrast, is accurate but only appropriate to verify the calculation accuracy of the analytical model or analyze the performance of the compliant mechanism before fabrication. Additionally, compared with the Castigliano’s theorem, the matrix method is more simple and effective for the analytical compliance modeling of the compliant mechanism. Therefore, the matrix method is adopted in this article. Though it has been widely used, the matrix method is not well developed in the compliance modeling for the closed-loop serial or parallel compliant mechanism. For example, the matrix method is not well adopted for analyzing the performance of a bridge-type amplifier, including the input stiffness and displacement amplification ratio. Instead, the elastic beam theory together with the kinematic analysis are used to solve this problem [10, 20, 21]. However, the analysis procedure is complex and there are no simple calculation expressions provided for this case. Additionally, the matrix method is not applicable for the compliance analysis when the loads are applied on the flexure members. For instance, when the compliant mechanism is inserted into the dual-stage for accuracy compensation within a long stroke [22], the deformations of the flexure members under inertial forces and gravity cannot be calculated by using the matrix method. Though the Castigliano’s theorem can deal with this issue, it involves the partial differential calculation. Therefore, it is meaningful to develop a more effectiveness method for the real-time accuracy compensation control.
This paper focuses on the analytical compliance modeling of the serial flexure-based compliant mechanisms under arbitrary external loads. According to the relative positional relationship between the applied loads and the displacement output point, concise and explicit compliance models of the open- and closed-loops serial compliant mechanisms are derived based on the matrix method. Then an equivalent method is proposed to transform the external loads applied on the flexure members to the concentrated forces applied on the rigid links. The transformation process satisfies the static equilibrium condition and also guarantees that the deformations at the displacement output point remain unchanged. Thus, the matrix method can be still used to analyze the compliance of the compliant mechanisms. Finally, several specific examples are given to illustrate the effectiveness of the proposed method.
2 External Load Applied on Rigid Links
In this section, the compliance modeling of the open- and closed-loops serial compliant mechanisms when the external loads are applied on the rigid links is discussed.
2.1 Compliance Modeling of the Open-Loop Serial Compliant Mechanism
Considering the deformations of the flexure hinges are linear and small, the deformation at a given point of the compliant mechanism can be calculated through the superposition principle. According to the relative position of the applied loads, the compliance modeling of the open-loop serial compliant mechanism is discussed in two difference conditions. Fig. 1(a) shows the loading point is closer to the fixed end than the displacement output point, and Fig. 1(b) shows the displacement output point is closer to the fixed end.
2.2 Compliance Modeling of the Closed-Loop Serial Compliant Mechanism
The closed-loop serial compliant mechanism is over- constrained because its two fixed ends produce more than three unknown reaction forces, which brings difficulty in obtaining its compliance model. Generally, the geometric relationship of the deformations and equations of static equilibrium are provided to deal with this issue. However, this method is relative complex and there are no concise and explicit expressions for the compliance calculation.
Due to the over-constrained characteristic of the closed- loop serial compliant mechanism, it is cut out at the displacement output point and the inner forces are applied at the sections, as shown in Fig. 2.
- (1)When the loading point and the displacement output point are coincident, the external load satisfies F _{ e }=F _{ iel } and F _{ jer }=0, or F _{ e }=F _{ jer } and F _{ iel }=0. The deformation at the displacement output point can be expressed as$$\varvec{\delta}= \left( {\varvec{C}_{gl} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{gr} } \right)\varvec{F}_{e} = \left( {\varvec{C}_{gr} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{gl} } \right)\varvec{F}_{e} .$$(16)Therefore, the compliance matrix of the mechanism can be written as$$\varvec{C} = \frac{{\partial\varvec{\delta}}}{{\partial \varvec{F}_{e} }} = \varvec{C}_{gl} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{gr} = \varvec{C}_{gr} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{gl} .$$(17)The stiffness matrix of the mechanism can be obtained as$$\varvec{K} = \left( {\varvec{C}_{gl} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{gr} } \right)^{ - 1} = \varvec{C}_{gl}^{ - 1} + \varvec{C}_{gr}^{ - 1} = \varvec{K}_{gl} + \varvec{K}_{gr} .$$(18)
Eq. (18) indicates that when the loading point and the displacement output point are coincident, the stiffness of the closed-loop serial compliant mechanism can be obtained by adding the stiffness of its subsystems together.
- (2)When the external load is applied on the left subsystem, it satisfies F _{ e }=F _{ iel } and F _{ jer }=0. The deformation at the displacement output point can be obtained from Eq. (15) as$$\varvec{\delta}=\varvec{\delta}_{r} = \left( {\varvec{C}_{gr} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{el} } \right)\varvec{F}_{e} .$$(19)Therefore, the compliance matrix of the mechanism can be written as$$\varvec{C} = \frac{{\partial\varvec{\delta}}}{{\partial \varvec{F}_{e} }} = \varvec{C}_{gr} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{el} .$$(20)
- (3)When the external load is applied on the right subsystem, it satisfies F _{ e }=F _{ jer } and F _{ iel }=0. The deformation at the displacement output point can be obtained from Eq. (14) as$$\varvec{\delta}=\varvec{\delta}_{l} = \left( {\varvec{C}_{gl} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{er} } \right)\varvec{F}_{e} .$$(21)Therefore, the compliance matrix of the mechanism can be given as$$\varvec{C} = \frac{{\partial\varvec{\delta}}}{{\partial \varvec{F}_{e} }} = \varvec{C}_{gl} {\kern 1pt} \varvec{B}^{ - 1} {\kern 1pt} \varvec{C}_{er} .$$(22)
According to the discussions, concise and explicit compliance models of the closed-loop serial compliant mechanism can be obtained, which will be helpful in its performance analysis and optimal design.
When multiple loads are applied on the mechanisms, the deformation at the displacement output point is calculated as
$$\varvec{\delta}_{o} = \sum {\left( {\varvec{C}_{gr} {\kern 1pt} \varvec{B}_{i}^{ - 1} {\kern 1pt} \varvec{C}_{iel} } \right)\varvec{F}_{iel} } + \sum {\left( {\varvec{C}_{gl} {\kern 1pt} \varvec{B}_{j}^{ - 1} {\kern 1pt} \varvec{C}_{jer} } \right)\varvec{F}_{jer} } .$$(23)
3 External Loads Applied on the Flexure Members
The relationship between the deformations of the flexure members and the external loads is nonlinear when the loads are applied on the flexure members. The matrix method is only appropriate to the linear problem, and therefore cannot be used to derive the compliance model of the compliant mechanisms in this case. Though the Castigliano’s theorem can deal with this issue, it involves the partial differential which leads to a complex calculation process. In this section, an effective equivalent method is proposed to handle with the external loads applied on the flexure members. This method aims at transforming the external loads from the flexure members to the rigid links. The transformation process should satisfy the equations of static equilibrium and guarantee that the deformations at the displacement output point remain unchanged.
It is complicated to solve Eqs. (24) and (25) directly. Assuming that forces F _{ i } and F _{ j } satisfy Eq. (24), and we apply forces -F _{ i } and -F _{ j } on the original flexure member at points i and j, respectively. The deformations at the two points caused by the two forces are -δ _{ i } and -δ _{ j }, respectively. They are superposed on the original deformations caused by force q results in a fixed-fixed flexure member. Therefore, forces F _{ i } and F _{ j } are of course satisfying the equations of static equilibrium represented in Eq. (25).
It can be verified that the equivalent forces satisfy Eqs. (24) and (25). Once the external loads are transformed to the concentrated forces applied on the rigid links, the matrix method can be still adopted to analyze the compliance of the compliant mechanisms by using the models derived in section 2.
4 Applications in the Analysis of Serial Flexure-Based Compliant Mechanism
Several specific examples are given in this part to illustrate the effectiveness and validity of the compliance modeling of serial flexure-based compliant mechanisms and the force equivalent method.
4.1 Compliance Analysis of an Open-Loop Serial Compliant Mechanism
The motion of this compliant mechanism is mainly provided by the deformations of all the circular flexure hinges. Therefore, an accurate compliance equation of the flexure hinge is important for obtaining a precise compliance model of the entire mechanism. An empirical compliance equation[23] for the circular flexure hinge was adopted in this paper.
Calculation errors of each compliance component %
Compliance component | Thickness t / mm | ||
---|---|---|---|
1 mm | 3 mm | 5 mm | |
C _{1x-F1x } | –3.83 | 0.04 | 2.96 |
C _{1y-F1x } | –1.64 | 2.16 | 5.10 |
C _{1x-F1y } | –1.64 | 2.16 | 5.10 |
C _{1y-F1y } | 1.74 | 5.53 | 9.15 |
Calculation errors of each compliance Component %
Compliance component | Thickness t / mm | ||
---|---|---|---|
1 mm | 3 mm | 5 mm | |
C _{1x-F1x } | –4.14 | –2.51 | –2.37 |
C _{1y-F1x } | –1.96 | –0.35 | –0.08 |
C _{1x-F1y } | –1.96 | –0.35 | –0.08 |
C _{1y-F1y } | 1.36 | 2.62 | 3.18 |
Compliance of the mechanism calculated through the FEA μm / N
Compliance component | Thickness t / mm | ||
---|---|---|---|
1 mm | 3 mm | 5 mm | |
C _{3x-F2} | –117.07 | –8.779 | –2.753 |
C _{3y-F2} | 526.60 | 38.692 | 11.905 |
Compliance of the mechanism calculated through the analytical model μm / N
Compliance component | Thickness t / mm | ||
---|---|---|---|
1 mm | 3 mm | 5 mm | |
C _{3x-F2} | –113.45 | –8.391 | –2.627 |
C _{3y-F2} | 518.65 | 37.526 | 11.519 |
The simulation analyses show that the matrix method is applicable to the compliance modeling of the open-loop serial compliant mechanism related to any given output points and external loads applied on the rigid links. Additionally, the compliance of the rigid links cannot be neglected when the flexure hinges are relative thick and the length of the open-loop serial chain is long.
4.2 Transformation of the External Loads Applied on Flexure Members
The equivalent forces F _{1} and F _{2} were calculated in the global coordinate frame as F _{1} = (0–52.05 N
–0.187 Nm)^{T} and F _{2} = (0–67.95 N –0.213 Nm)^{T}.
Similarly, external load q _{2}(x) was transformed to two equivalent forces applied at points 3 and 4 respectively as F _{3} = (0–82.34 N –0.187 Nm)^{T} and F _{2} = (0 82.34 N –0.187 Nm)^{T}.
The results show that the deformations at the given displacement output point caused by the original external loads and the equivalent forces are almost the same. Additionally, it can be verified that the equivalent forces satisfy the equations of static equilibrium. The analysis proves the effectiveness and validity of the proposed equivalent transformation method for handling the external loads applied on the flexure members.
4.3 Amplification Ratio Analysis of a Displacement Amplifier
From Fig. 16, the following conclusions can be drawn:
(1) The displacement amplification ratio decreases as geometric parameters P, C, and t increase, and increases as geometric parameter L increases.
(2) The displacement amplification ratio at first increases shapely when angle\(\theta\)increases, whereas decreases when angle\(\theta\)increases continuously.
(3) The displacement amplification ratio calculated by the analytical model without considering the compliance of the rigid links is always larger.
(4) The comparisons of the analytical and FEA results demonstrate the accuracy and effectiveness of the derived compliance model of the closed-loop compliant mechanism. Additionally, the concise expression form will be helpful to the further performance analysis and optimal design.
Therefore, small geometric parameters P, C, and t, large geometric parameter L, and optimized angle\(\theta\)can be chosen to improve the displacement amplification ratio of the amplifier. The optimization of the amplification ratio can be realized based on Eq. (38) with the consideration of other factors, such as the size dimension, input/output compliance, and frequency.
Displacement amplification ratios of the amplifiers
Geometric parameters | Amplification ratio | |||||
---|---|---|---|---|---|---|
C / mm | P / mm | L / mm | d / mm | t / mm | Analytical result | Experimental result |
30 | 25 | 35 | 1.5 | 1 | 12.46 | 11.92 |
30 | 25 | 35 | 4.5 | 1 | 7.38 | 7.14 |
5 Conclusions
- (1)
According to the relative positional relationship between the applied loads and the given displacement output point, the concise and explicit compliance models of the open- and closed-loop serial compliant mechanisms are derived based on the matrix method.
- (2)
An equivalent method is proposed to transform the external loads applied on the flexure members to the concentrated forces applied on the rigid links. Therefore, the matrix method can be still used to analyze the deformations and compliance of the compliant mechanism.
- (3)
Several specific simulation analyses and an experimental test are carried out. The results verify the effectiveness and accuracy of the derived compliance models and the force equivalent transformation method.
Notes
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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