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Fatigue analysis and experiment of leaf-spring pivots for high precision flexural static balancing instruments
Accepted Manuscript Fatigue analysis and experiment of leaf-spring pivots for high precision flexural static balancing instruments Shusheng Bi, Yongzhen Li, Hongzhe Zhao PII:
S0141-6359(18)30364-7
DOI:
https://doi.org/10.1016/j.precisioneng.2018.10.009
Reference:
PRE 6862
To appear in:
Precision Engineering
Received Date: 7 June 2018 Revised Date:
1 October 2018
Accepted Date: 17 October 2018
Please cite this article as: Bi S, Li Y, Zhao H, Fatigue analysis and experiment of leaf-spring pivots for high precision flexural static balancing instruments, Precision Engineering (2018), doi: https:// doi.org/10.1016/j.precisioneng.2018.10.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Fatigue analysis and experiment of leaf-spring pivots for high precision flexural static balancing instruments Shusheng Bi, Yongzhen Li, Hongzhe Zhao* Robotics Institute, Beihang University, XueYuan Road No. 37, HaiDian District, Beijing, 100191, China
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Abstract For distributed-compliance flexure mechanisms, their service life is affected by fluctuating stress which may result in fatigue failure, but only static failure was taken into consideration in most studies. As one of the best-known types of distributed-compliance flexural pivots, the cross-spring pivot applied to the flexural static balancing instruments is subjected to large enough vertical loads and fluctuating stresses. In this paper, the cross-spring pivot is taken to explore the methods of
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prolonging its fatigue life, and the methods can be extended to the other pivots based on leaf-springs. Meanwhile, another important performance of the pivot-the rotational stiffness remains unchanged. Firstly, in order to develop a stress-life model, the equivalent fatigue stresses of the cross-spring pivots are derived in which the stress concentration factors are obtained by
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fitting the finite element results. Then, two methods for prolonging the fatigue life of the pivot are proposed, i.e., increasing the ratio of length to thickness of leaf-springs and reducing stress concentration. To verify the theoretical analysis, the fatigue life of five cross-spring pivots with the same rotational stiffness is tested based on a developed fatigue test bench. Experimental results show that the fatigue life of the cross-spring pivot can be prolonged, which verifies the validity of the two methods. Keywords
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Introduction
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Static balancing instrument, Cross-spring pivot, Fatigue stress; Fatigue life, Stress concentration factor
Static balancing instrument, as a precision instrument for measuring the unbalanced moment of objects, plays an important role in achieving the static balance of objects. Compared with the traditional static balancing instrument, flexural
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static balancing instrument has attracted the attentions of majority of scholars due to its high measurement accuracy and wide measurement range. The typical cases are that two flexural static balancing instruments using flexural pivots were designed by Yan et al and Boynton et al, respectively [1,2]. However, the flexural static balancing instrument is more prone to fatigue
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failure due to the use of the flexural pivot as a rotation support unit. Currently, the researches on flexural static balancing instruments mainly focus on the design of new mechanisms and the improvement of measurement sensitivity, but few studies on the method of prolonging their service life. For compliant mechanisms, fatigue failure is a main failure mode that affects their service life significantly, especially under dynamic loading, which leads to a premature fatigue failure of the flexural static balancing instruments. Therefore, it is quite necessary to maximize the fatigue life of the pivots in engineering applications.
The existing flexural pivots are mainly divided into two categories: lumped-compliance and distributed-compliance flexural pivot. The main studies on the fatigue property of flexural pivots have focused on lumped-compliance pivots. Dirksen et al [3] incorporated the fatigue life cycle criteria of three notch flexural pivots into the topology optimization design of the compliant mechanism. Henein et al [4] tested the fatigue life of right circular pivots to inquire into the __________ * Corresponding author. Tel.: 86-010-82338019. E-mail addresses: [email protected]
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ACCEPTED MANUSCRIPT exportability of the fatigue measurements made on standard test-specimen by Wire cut Electrical Discharge Machining (WEDM). Schoenen et al [5] studied the influences of the WEDM method on fatigue life of notch type flexural pivots. The fatigue properties of the notch type flexural pivots used in a planar micro-positioning platform and a high-precision micro manipulator were studied by Wang [6] and Ivanov [7], respectively. As one of the best-known types of distributedcompliance flexural pivots, cross-spring pivot is employed in an increasing number of precision machines, due to its significant advantages such as maintenance-free, no backlash, diminished friction and infinitesimal resolution. Currently,
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some research groups focus on the stiffness, accuracy, rotational range and other static characteristics of the cross-spring pivot [8-11]. However, there are very few researches on the fatigue properties of the cross-spring pivot at present. Gómez et al [12,13] proposed a method of optimizing the shape of leaf-springs to reduce stress in cross-spring pivots, thus extending their fatigue life. It is relatively complicated, however, to determine the shape parameters of leaf-springs. Currently, there are limited ways to prolong the fatigue life of the cross-spring pivot. According to the similarity in the processing method,
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surface roughness and rotational frequency between the leaf-spring pivots and the notch pivots, the influences of them on the fatigue characteristics of pivot are no longer considered. Therefore, this paper will focus on researching the influences of shape parameters and stress concentration on the fatigue characteristics of the cross-spring pivot.
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As a precision measuring instrument, it is highly necessary to possess high measuring sensitivity and long service life simultaneously. In our previous work [14], a high precision flexural static balancing instrument, which used the cross-spring pivot with quasi-constant rotational stiffness as a rotational unit, was developed, as shown in Fig. 1. It was revealed that the sensitivity of the instrument is closely related to the rotational stiffness of the cross-spring pivot. For this reason, methods of prolonging the fatigue life of the cross-spring pivot with its rotational stiffness keeping constant should be proposed. The objective of this paper is to study the influences of the shape parameters (length L and thickness T) and stress concentration on the fatigue life of the cross-spring pivot. The corresponding methods are proposed to prolong the fatigue life
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of the cross-spring pivot without changing rotational stiffness. Consequently, the remainder of this paper is organized as follows. The characteristics of the cross-spring pivots to be studied are introduced in Section 2. The equivalent fatigue stress model for the cross-spring pivot is developed, in which the equations of theoretical stress concentration factors are obtained on the basis of the Finite Element Analysis (FEA) results. On this basis, the corresponding methods are proposed to reduce
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the equivalent fatigue stress of the cross-spring pivot without changing rotational stiffness in Section 3. Finally, in order to verify the correctness of the proposed methods, experimental investigations on the fatigue life of five cross-spring pivots with
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the same rotational stiffness are conducted in Section 4.
Fig. 1. (a) flexural Static balancing instrument ( spirit level,
mechanical limit,
cross-spring pivot,
counterweights,
working platform,
electronic force rebalance transducer) and (b) cross-spring pivot. 2
payloads,
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Characteristics of the cross-spring pivots for flexural static balancing instrument For precision measuring instruments, measurement sensitivity is one of the key performances of the flexural static
balancing instrument. The sensitivity of the instrument should not be changed while studying the fatigue properties of the cross-spring pivot. Since the sensitivity of the static balancing instrument is closely related to the rotational stiffness of the cross-spring pivot, so all the cross-spring pivots to be studied have the same rotational stiffness. According to the Ref. [14],
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to simplify the design variables, the non-dimensional rotational stiffness of the cross-spring pivots can be represented as Eq. (1).
Km =
2(9λ 2 − 9λ + 1) λ f cos α + m = 8 ( 3λ 2 − 3λ + 1) − + λ cos α p − ph θ 15cos α
(1)
where m=ML/(EI), p=PL2/(EI), f=FL2/(EI), I=WT3/12, h=H/L, the lower case letters are non-dimensional parameters, M, P
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and F denote bending moment, vertical load and horizontal load, respectively; L, W and T are the length, width, and thickness of the leaf-spring, respectively; H denotes the height of the action point of load P from the platform; I is the cross section moment of inertia about the neutral axis; E is the Young’s modulus of the material; λL is the distance from moving stage to
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the intersection; 2α is the angle of intersection, as shown in Fig. 1(b).
As can been seen from Eq. (1) that the rotational stiffness of the cross-spring pivot varies with the load conditions (load p and height h). However, load p and height h change with different measured objects, and therefore it is difficult to evaluate the rotational stiffness of the cross-spring pivot. This paper focuses on the influences of structural parameters on the fatigue life of cross-spring pivot under the same load conditions, which means that ph is a constant. Furthermore, all the cross-spring pivots to be studied have the approximately identical rotational stiffness as the structural parameters vary. If the coefficient of vertical load p is zero, the rotational stiffness of the cross-spring pivot will not be affected, so the following relationship
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between λ and α should be satisfied.
2(9λ 2 − 9λ + 1) + 15λ cos 2 α = 0
(2)
Accordingly, when λ is fixed, the rotational stiffness Km is an approximately constant, and α also can be determined. Furthermore, if λ=1/3 and α=50.77°, the center shift of the pivot takes the minimum value [14]. Therefore, the cross-spring
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pivots to be studied satisfy the two conditions of approximately constant rotational stiffness and minimum center shift. The above analysis result of rotational stiffness is non-dimensional, and then it is converted into a dimensional parameter. Suppose the width W of leaf-spring is fixed, hence the shape parameters L and T of the cross-spring pivot with constant
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rotational stiffness should be satisfy the Eq. (3). The relationship between L and T is shown in Fig. 2.
L / T 3 = constant
3
(3)
Fig. 2. Relationship between L and T.
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When cross-spring pivot acts as a rotational unit, the leaf-springs are treated as Euler-Bernoulli beams ignoring the shearing effect. The bending stress along the length of leaf-spring can be written as [11]
3Emi ( x)
, (i = 1,2)
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σ bi =
d
(4)
where d=12(L/T)2; i denotes one of the two leaf-springs of the cross-spring pivot; mi(x) is the internal bending moment in the two leaf-springs.
The expressions of the internal bending moment along the length of the leaf-spring are given by
m1 ( x) = ( A3 x3 − A2 x2 + A1 x + A0 )θ − ( B3 x3 − B2 x2 + B1 x + B0 )θ 2 3 2 3 2 2 m2 ( x) = ( A3 x − A2 x + A1 x + A0 )θ + ( B3 x − B2 x + B1 x + B0 )θ
(5)
Ref. [11].
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where A1, A2, A3, B1, B2 and B3 are functions of vertical load p and horizontal load f. Their specific value can be referred to If the axial stress along the length of the leaf-spring can be regarded as constant, then it can be written in the form [11]
Epi , (i = 1, 2) d
(6)
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σ ti =
where pi is the non-dimensional parameter.
Therefore, when the total bending moment and vertical load p are applied to the cross-spring pivot simultaneously, the
3
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corresponding bending and axial nominal stresses can be calculated by Eqs. (4) and (6), respectively. Fatigue analysis of the cross-spring pivots for static balancing instrument Fatigue failure caused by cyclic stress is the major failure mode of the cross-spring pivot under vertical load and bending moment. The stress-life (or S-N) curve describes the number of cycles at failure at a given stress, and the stress-life model is the most direct and common fatigue failure prediction model. Even if it is difficult to accurately predict fatigue failures for a variety of reasons, it is highly helpful to understand the prediction of fatigue failure in the design of compliant mechanisms. For most common materials, the S-N curve is usually divided into low cycle fatigue and high cycle fatigue. The S-N curve can be approximated by means of three points on the S-N curve, namely, Su (the ultimate strength), Se (the fatigue limit), and SL (the fatigue strength at a fatigue life of 103), as shown in Fig. 3 [6].
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Fig. 3. Approximate S-N diagram.
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In high cycle fatigue stage, for aluminum alloy materials with no endurance limit, the curve is started with fatigue failure
cycle region can be written in the following form
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at more than 103 cycles. For an actual aluminum alloy component, the approximation formula of the S-N diagram in the high-
S f = af N
log a f =
bf
log N2 log SL − log N1 log S f 2
bf =
log N2 − log N1
S 1 log L Sf 2 log N1 − log N2
(7) (8)
(9)
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where Sf is the fatigue strength; N is the number of cycles; af and bf are two unknowns; For aluminum without fatigue limit, N1 = 1×103, N2 = 5×108; Sf2 is the conditioned fatigue strength at 5×108 cycles for aluminum. The value of SL at fatigue life cycles of 103 can be approximated as
SL = η Su
0.9 0.75
η=
bending axial loading
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where
(10)
According to Eqs. (7)-(9), in order to obtain the relationship between stress and fatigue life (for high-cycle fatigue) of a pivot at specified fatigue strength, it is first necessary to obtain conditioned fatigue strength and the fatigue stress of the actual component. These will be discussed in Sections 3.1 and 3.2, respectively.
3.1 Conditioned fatigue strength of the cross-spring pivot The fatigue properties of materials or components are measured by fatigue strength which is reflected through fatigue limit. The fatigue limit is defined as the amplitude of cyclic stress in which the material or component can withstand the infinite stress cycle without fatigue failure. For aluminum alloy materials with no fatigue limit, their fatigue life can be estimated by the fatigue strength of a certain number of cycles. The conditioned fatigue strength of the cross-spring pivot can be obtained by modifying that of standard specimens. The conditioned fatigue strength modification factors suggested by Marin [8,15-18] are expressed as 5
ACCEPTED MANUSCRIPT S f 2 ≈ ka kb kc kd ke km S f '
(11)
where ka, kb, kc, kd, ke and km are the surface condition modification factor, the size modification factor, the load modification factor, the temperature modification factor, the reliability factor and other miscellaneous modification factors, respectively. Next, the uncorrected fatigue strength Sf’ of standard specimen needs to be obtained from fatigue testing. However, the
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process is time-consuming and laborious. The simple method of getting uncorrected fatigue strength is estimated by the ultimate tensile strength of the material. For aluminum alloys, the relations between uncorrected fatigue strength Sf’ and the ultimate tensile strength Su are depicted as [8]
3.2 Equivalent fatigue stresses of the cross-spring pivot
(12)
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Su < 330MPa 0.4Su Sf ' ≈ 130MPa Su ≥ 330MPa
The cross-spring pivots for static balancing instrument are subjected to a bending nominal stress and an axial nominal stress. The internal stresses of the cross-spring pivot contain nonzero mean stress, so the equivalent fatigue stresses can be
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used to represent the fatigue stress of the cross-spring pivot. Classical fatigue failure criteria that match experimental data quite well has been proposed by Goodman [19]. In this work, equivalent fatigue stresses Seqv of the cross-spring pivot is calculated by the modified Goodman relation.
Seqv =
σa 1 − σ m / Su
(13)
where σa and σm are equivalent stress amplitude and equivalent mean stress, respectively. The equivalent stress amplitude and
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equivalent mean stress are represented in Ref. [19]
σ a = K fbσ ba + σ m = K fbσ bm +
K ftσ ta 0.85 K ftσ tm
(14)
0.85
where σbm and σba are mean stress and stress amplitude corresponding to the bending stress; σtm and σta are mean stress and
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stress amplitude corresponding to the axial stress; Kfb and Kft are fatigue stress concentration factors for bending and axial stress, respectively. The fatigue stress concentration factor Kfi is defined as the ratio of fatigue strength of standard specimen
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to that of the specified component. The stress concentration factors will be discussed in the Section 3.3. Mean stress and stress amplitude are defined as
σ bm = (σ b max + σ b min ) / 2, σ ba = (σ b max − σ b min ) / 2 σ tm = (σ t max + σ t min ) / 2,
σ ta = (σ t max − σ t min ) / 2
(15)
where σbmax and σbmin are the maximum and minimum bending stress; σtmax and σtmin are the maximum and minimum axial stress. They can be solved by Eqs. (4) and (6). To sum up, the equivalent fatigue stress of the cross-spring pivot can be reduced by reducing the nominal stresses and the fatigue stress concentration factors, thus extending its fatigue life. Accordingly, the study will focus on exploring the methods of reducing the nominal stresses and stress concentration. According to Eq. (5), the internal bending moment of a leaf-spring is determined by the structure configuration of the cross-spring pivot. Next, the influences of the length L and thickness T of leaf-spring on the bending stress and axial stress 6
ACCEPTED MANUSCRIPT can be obtained by Eqs. (4) and (6). The relationship between internal maximum stress σmax along the length of leaf-spring versus the ratio of length to thickness of leaf-spring is shown in Fig. 4 when Eq. (3) is satisfied. It can be seen that the bending stress of the leaf-spring is much larger than axial stress, but the axial stress is not neglected to improve accuracy in this paper (the axial stress is neglected in Ref. [11]). Moreover, as the ratio of length to thickness of leaf-spring increases, the internal total stress of leaf-spring decreases at first and then increases, but the increase is not obvious. On these grounds, the maximum nominal stress of the cross-spring pivot can be reduced by increasing the ratio of length to thickness of leaf-springs.
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pivot under the condition that the rotational stiffness of pivot is constant.
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Therefore, it is of great significance to study the effects of shape parameters (L and T) on the fatigue life of the cross-spring
Fig. 4. Relationship between stress and the ratio of length to thickness.
3.3 Theoretical stress concentration factors
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It can be seen from Eq. (14) that fatigue stress concentration factors have important influences on equivalent fatigue stresses. To simplify the solution process, provided that the theoretical stress concentration factor Kti is equal to the fatigue stress concentration factor Kfi. The problem of obtaining the fatigue stress concentration factor of the cross-spring pivot is transformed into solving its theoretical stress concentration factor. For some typical notch type components, the correlation diagrams or calculation formulas of the theoretical stress
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concentration factors have been given in Ref. [20]. However, as a distributed-compliance flexural pivot, studies on theoretical stress concentration factor of the cross-spring pivot have not been found in existing literatures. Generally, stress
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concentration factors can be obtained through photo-elasticity experimentation, analytical methods (e.g., the theory of elasticity) or numerical methods (e.g., FEA). To this end, we try to obtain the bending and axial stress concentration factors of the cross-spring pivot through fitting simulation results, because it is an effective and inexpensive method. The theoretical stress concentration factor Kti is defined as
Kti =
σ i max , (i = b, t ) σ inom
(16)
where σbnom is the bending nominal stress, and can be obtained from Eq. (4). If an axial force Fx is applied along the length of leaf-spring, σtnom is the axial nominal stress which can be solved according to Eq. (6). σimax is the maximum stress of a component, and can be obtained by finite element method. The most fatigue failures of components occur first at the sites of local stress concentration, and the stress concentration in the cross-spring pivot appears at the end of the leaf-spring. The most common way to reduce stress concentration is to 7
ACCEPTED MANUSCRIPT design fillet R. Besides, Refs. [3,21] indicate that theoretical stress concentration factors of notch type pivots strongly depend on the minimum thickness and the corresponding radius of curvature. Provided that this conclusion also applies to the crossspring pivots, equations for predicting the stress concentration factors of the cross-spring pivots can be formulated based on the finite element results by defining γ=R/T as the non-dimensional stress-concentrating parameter in the following subsections. Besides, as is well known, acute angle should be avoided as much as possible in engineering design, while it exists in most of the cross-spring pivots used in practical engineering applications. Therefore, the effects of parameter γ and
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acute angle φ on stress concentration factor are studied through finite element method. The parameter definitions of the
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improved cross-spring pivot are shown in Fig. 5.
Fig. 5. The parameter definitions of the cross-spring pivot. The ranges of the shape parameters of the cross-spring pivots are: 0
R
3.5mm, 0.3mm
T
0.6mm, ignoring the
impact of other parameters. For all cross-spring pivots designed, the linear elastic material properties are: the Young’s
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modulus E=7.2×1010 Pa, Poisson’s ratio υ=0.33, and the density ρ=2700 kg/m3. The FEA model of each cross-spring pivot is meshed with SOLID186 elements in ANSYS, which is a twenty-node three-dimensional hexahedral element with three degrees of freedom at each node. A typical mesh model is shown in Fig. 6. To ensure the validity of the FEA results, the mesh of each model is refined at the end of leaf-spring so that the size of mesh does not exceed 0.2mm at the fillet position. The bottom rigid body of leaf-spring is fixed, and loaded with unit load (i.e., bending moment and axial load) on the free end
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of leaf-spring to simplify the calculation. So the corresponding maximum stress σmax is recorded, which is then used to
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calculate theoretical stress concentration factor Ktb (or Ktt) using Eq. (16).
Fig. 6. Finite element mesh model. These results leave no doubt that the maximum stress of every cross-spring pivot model appears at the root of the leaf8
ACCEPTED MANUSCRIPT spring. When the fillets R becomes larger, the maximum stress in the cross-spring pivot model gradually decreases and tends to be stable. When the angle φ changes from an acute angle to a right angle, the maximum axial stress also decreases, while the maximum bending stress is almost constant. Therefore, the methods of reducing stress concentration are obtained. The FEA results for both Ktb and Ktt confirm the previous assumption that the stress concentration factors depend on the nondimensional parameter γ, but is almost independent of the other shape parameters (e.g., L and W). When the thickness T is fixed, the stress concentration factors can be reduced within a certain range by increasing the fillet R. In addition, when R
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increases to a certain value, the stress concentration factor tends to be a constant. As a result, a rational function of γ with first-order polynomials in both the numerator and denominator could fit the FEA results for Ktb and Ktt by using MATLAB program. Ultimately, the following equations are obtained, respectively:
Ktt =
1.018γ + 0.7319 γ + 0.433 1.269γ + 1.137 γ + 0.7288
(17)
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Ktb =
(18)
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Figs. 7 and 8 depict the fitting equations of stress concentration factors Ktb and Ktt and FEA results of leaf-spring with four thicknesses for the purpose of comparison. Compared with FEA results, the maximum errors of fitting equations for Ktb and Ktt are 2.25% and 2.1 %, respectively. Hence, Eqs. (17) and (18) are applicable to a wide range of the cross-spring pivots with acceptable error.
1.6
T=0.3mm T=0.4mm T=0.5mm T=0.6mm Fitting Equation
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γ
Fig. 7. Bending stress concentration factor versus γ.
Ktt
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Ktb
1.5
1.4
1.3
1.2 0
2
4
6
8
10
γ Fig. 8. Axial stress concentration factor versus γ.
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In order to study the relationships between the stress concentration factors and the angle φ (Fig. 5) in the cross-spring pivot, the maximum stresses of the cross-spring pivots with multiple φ values are obtained by virtue of FEA, and are used to calculate Ktb (or Ktt). Figs. 9 and 10 plot Ktb and Ktt calculated from the finite element results. It can be found in Figs. 9 and 10 that, as the angle φ increases, the bending stress concentration factors of the cross-spring pivots with different φ are almost the same, while the axial stress concentration factors decrease continuously. In addition, there is an important conclusion that the value of axial stress concentration factor reaches minimum when φ = 90°.
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=30° =39.23° =45° =60° =90°
1.6 1.5
Ktt
Ktb
1.4 1.3 1.2
1
0
2
4
6
8
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1.1 10
γ
γ Fig. 9. Bending stress concentration factor versus γ.
Fig. 10. Axial stress concentration factor versus γ.
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Finally, the equivalent stress amplitude and equivalent mean stress can be obtained by substituting Eqs. (15), (17) and (18) into Eq. (14). Furthermore, the equivalent fatigue stress of the cross-spring pivot can be solved according to Eq. (13), which
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can be used to predict the fatigue life of cross-spring pivots. Experimental evaluation
4.1 Stress-life model of the cross-spring pivots
Obviously, according to the above analysis, the nominal stress of the cross-spring pivot can be reduced by increasing the ratio of length to thickness of the leaf-spring. Moreover, stress concentration has great influences on the equivalent fatigue stress of the cross-spring pivot. Next, the geometric and shape parameters of the cross-spring pivots to be tested are determined to verify the correctness of the above theoretical analysis.
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According to the analysis in Section. 2, the sensitivity of the instrument should not be changed while investigating the ways to prolong the fatigue life of the cross-spring pivot. All the cross-spring pivots have the same rotational stiffness, that is, the relationship between the length L and thickness T of the leaf-spring satisfies Eq. (3). Additionally, the cross-spring pivots satisfy the two conditions: approximately constant rotational stiffness and minimum center shift. All the cross-spring pivots
Table 1. Table 1
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have the same geometric parameters, i.e., λ=1/3 and α=50.77°. The shape parameters of five cross-spring pivots are listed in
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The parameters and equivalent fatigue stresses of five cross-spring pivots.
Pivot L(mm) W(mm) T(mm) R(mm) φ(°) Ktb Ktt Seqv(MPa) a 45 10 0.4 0 39.23 1.82 1.49 57.26 b 45 10 0.4 0.5 39.23 1.19 1.38 40.40 c 45 10 0.4 1 39.23 1.12 1.33 38.25 d 45 10 0.4 1 90 1.09 1.10 35.81 e 64.07 10 0.45 0 39.23 1.82 1.49 49.90 The fatigue failures of most components occur in the stress concentration zones, and the stress concentration of the crossspring pivot locates at the end of the leaf-springs. To reduce the effects of the stress concentration on fatigue life, the method of designing fillets is the most common. Compared with pivot a, the pivots b and c have fillets with radius of 0.5mm and 1mm at the end of the leaf-springs, respectively, while the effective length L of pivots b and c is the same as that of pivot a. In addition, the rest of the geometric and shape parameters of pivots b and c are identical to that of pivot a. 10
ACCEPTED MANUSCRIPT The fatigue crack always appears at the end of the leaf-spring and is close to the moving stage. Based on the FEA results in Section. 3.3, the axial stress concentration factor is minimum when the φ is equal to 90°. To reduce the stress concentration, the φ is set to be 90° for pivot d. In addition, pivot d also has fillets with radius 1mm at the end of the leaf-springs. The rest of the shape parameters of the pivot d are the same as that of pivot a. Moreover, to investigate the influences of the ratio of length to thickness of the leaf-spring on the fatigue life, pivot e with a larger ratio of length to thickness is tested with reference to the pivot a. However, it also ensures that the relationship
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between length L and thickness T satisfies Eq. (3).
For the flexural static balancing instrument, the maximum rotational angle of the cross-spring pivot is 2°, and vertical load is P=30N. The ultimate tensile strength of aluminum alloy 7075-T6 is Su=538MPa [22]. The theoretical stress concentration factors of five pivots can be obtained by Eqs. (17) and (18). Ultimately, the equivalent fatigue stresses of these pivots can be solved. The results are tabulated in Table 1. Compared with pivot a, the equivalent fatigue stress of the pivots b-e are
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relatively small. Hence, the effectiveness of the two methods of reducing equivalent fatigue stress is verified.
These cross-spring pivots are machined by WEDM. The prototypes of five cross-spring pivots are shown in Fig. 11. The surface roughness Ra of the leaf-spring is measured by the surface roughness measuring instrument, and the result is
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Ra≈3.05um. Then, the surface reduction factor can be obtained, i.e., ka=0.75 [3]. The remaining Marin correction factors of the specific cross-spring pivot can also be calculated. The theoretical stress concentration factors Kt of the pivots a-e are obtained by the finite element simulation. So miscellaneous-effects factor km can be calculated. According to Eq. (12), the fatigue strengths of these cross-spring pivots are estimated. According to Eq. (7), S-N curves of the high-cycle fatigue life for these cross-spring pivots are estimated and expressed as Eqs. (19)-(23).
(19)
Sb = 9003.8 Nb −0.4231
(20)
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Sa = 8098.6 N a −0.4078
Sc = 8677.6 N c −0.4178
(21)
S d = 7689.6 N d −0.4003
(22)
Se = 8098.6 N e −0.4078
(23)
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where the subscript a-e represent the pivots a-e.
Fig. 11. Five cross-spring pivots used in the instrument.
4.2 Measurement set-up 11
ACCEPTED MANUSCRIPT In order to verify the validity of the above theoretical analysis, a fatigue test bench applying to the cross-spring pivot is implemented. There are three working principles that can be used to swing the cross-spring pivot back and forth, as shown in Fig. 12. Schoenen et al [5] and Ivanov et al [7] developed a platform of the fatigue life testing for notch type flexural pivot based on crank-slider mechanism and crank-rocker mechanism, respectively. Nevertheless, the crank-slider mechanism, which consists of too many components, is a relatively complicated system. In the crank-rocker mechanism, additional forces have an effect on the fatigue life of pivots. The crank-guide mechanism is adopted, because it consists of relatively few
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components. Accordingly, it is easy to assemble and maintain the mechanical system. Furthermore, there is no additional force on the fatigue life of pivot as in the crank-rocker mechanism.
ωA
slider
ω
B
D
C guide bar specimen
O
(a)
B crank ω
A
guide bar
C
link
crank
H
link
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A
B
θ
rocker
O
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crank
specimen
(b)
specimen
O
(c)
Fig. 12. The schematic diagram of fatigue test bench: (a) crank-slider mechanism, (b) crank-rocker mechanism and (c) crankguide mechanism.
The existing literatures studied the fatigue life of notch type pivot being subjected to bending stress only, ignoring the effects of vertical loads on the fatigue life of pivot [5,7]. This work explores how to prolong the fatigue life of the cross-
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spring pivot for a specific application. Since the load on the pivot have important influences on its fatigue life, the influences of the external forces on the fatigue test results should be avoided as much as possible. While testing the fatigue life of the cross-spring pivot, there are two challenges: test instruments are not damaged by loads, and test results are not affected by external forces. Additionally, in order to adapt to the cross-spring pivot with different shape parameters, the fatigue test bench with adjustable kinematic parameters must be designed.
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Referring to the working principle of crank-guide mechanism, a fatigue test bench is designed to swing the cross-spring pivot continuously at a predetermined angle. The prototype of the fatigue test bench for the cross-spring pivot is demonstrated in Fig. 13. A rod is fixed to the crank and deviates from the center of the crank. Meanwhile, the other end of the
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rod is located in the groove of the guide bar. The fixed stage of the cross-spring pivot is fixed to the base, and the moving stage is fixed to the guide bar with a groove. The swing of the guide bar is achieved by the rotation of the crank which is actuated by a stepper motor. The swing angle of the guide bar is equal to the rotational angle of the cross-spring pivot. As illustrated in Fig. 12(c), assuming that the rotational axis of the cross-spring pivot is fixed at O, the maximum value of the rotational angle of the cross-spring pivot can be expressed as
θmax = arcsin
l1 H
(24)
where l1 is the length of the crank; H is the distance between the center of crank and the rotational axis of the cross-spring pivot. The maximum rotational angle θmax of the pivot can be changed by adjusting the eccentric distance l1. Herein, the angle θmax is set to be 2°. 12
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Fig. 13. The photo of the fatigue test bench: (a) global graph and (b) partial view
There are some measures adopted to reduce test errors. In order to reduce the friction between the rod and the guide bar, there is a roller assembled at the end of the rod and rolling in the groove of the guide bar. In this way, the effects of friction
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on the results of fatigue test can be reduced.
Before starting the test, each pivot is kept in a static state by the mechanical limits which prevent the pivot from being damaged by excessive rotational angle. During the test, the pivot swings back and forth until the fracture occurs. To prevent payloads falling and protect other equipment from being damaged, thin wire ropes are used to protect the loads. The starting and stopping of the fatigue test are determined by a control unit. One of the photoelectric switches is fixed at the end of the guide bar and provides the motor with stopping signal when the pivot fractures in fatigue. The other photoelectric switch is used to record the number of the cross-spring pivot rotations, and the number is displayed in real time
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by a counter. At the same time, a camera monitors the whole process of fatigue test and the number of the counter. The control flow chart of the fatigue test is shown in Fig. 14.
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start
counter reset control unit motor turning
counter no photoelectric switch
end
yes
failure
motor stop
Fig. 14. The control flow charts of fatigue test
4.3 Experiments and results 13
ACCEPTED MANUSCRIPT In order to ensure machining accuracy of the pivots, five AL7075-T6 prototypes of the cross-spring pivot are processed by using Charmilles Robofil 380 WEDM machine. Furthermore, for the sake of reducing assembling error, two leaf-springs of the cross-spring pivot are cut by one process at the same time. Then, one of them is turned over and they are assembled by pins. Most of the cross-spring pivots for the flexural static balancing instrument have been subjected to the vertical loads p during rotation. Consequently, the fatigue life tests of the cross-spring pivots are carried out under the action of constant
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vertical load. The vertical load of 30N is applied on moving platform of each cross-spring pivot in the experiments. Besides, the maximum rotational angle of the cross-spring pivots is set to be 2°. The rotation of the pivot is generated by the horizontal force acting on the pivot by the test bench, and the moment created by the horizontal force is equal to the moment in the theoretical analysis. In addition, the rotational frequency of all the cross-spring pivots is set to be 2Hz according to the actual application.
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The fatigue testing results of five cross-spring pivots are listed in Table 2. Taking pivot a as the reference, the life cycles of the rest of the cross-spring pivots are relatively long, which confirms the effectiveness of the improved measures. Compared with pivot b, the fatigue life cycles of pivots c and d are improved, which supports the analysis about the stress
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concentration factors (Figs. 7-10), namely, the stress concentration factor (Ktb or Ktt) decreases with increasing γ, and the value of axial stress concentration factor reaches minimum when φ = 90°. By comparing the fatigue life cycles of pivots a and e, it can be proved that the larger ratio of length to thickness of the leaf-spring can prolong the fatigue life cycles of the cross-spring pivot. Based on what has been discussed above, the fatigue life of the cross-spring pivot can be maximized when φ is set to be 90° and fillets are designed. That is, stress concentration effects can be decreased by avoiding acute angle appearing at the root of the leaf-springs and designing fillet.
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Table 2 The fatigue life cycles of five cross-spring pivots.
Pivot
Experimental results
Error (%)
Improvement(%)
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187770 225106 19.88 354140 310573 12.30 435130 377052 13.35 669080 461698 31.00 263080 334281 27.06 Nevertheless, there are a certain amount of errors between the
37.97 67.50 105.10 48.50 fatigue test results and the actual values. The potential
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a b c d e
Analytical results
reasons are analyzed in the following:
a. On account of the existence of machining errors, the dimensions of the prototype are not exactly equal to that of the theoretical model. In particular, the thickness error of leaf-spring has much influences on fatigue life based on the above analysis.
b. For the sake of simplification, this paper assumes that the theoretical stress concentration factor is equal to the fatigue stress concentration factor. The former is actually less than the latter. Therefore, this introduces certain errors to the calculation of the equivalent fatigue stress and fatigue life cycles of the pivots. c. The effects of rotational frequency of the cross-spring pivot on the fatigue life are not considered in the theoretical analysis, but that cannot be ignored in practice. Mechanical limits are employed to prevent the pivots from being damaged by excessive rotational angle, but due to the presence of inertial forces, there are collisions when the cross-spring pivot reaches 14
ACCEPTED MANUSCRIPT the maximum angle. Furthermore, the higher the rotational frequency, the stronger the collisions. So the fatigue life of the cross-spring pivots may decrease to some extent, which will be studied in future work. d. Since the fatigue test process is time consuming and laborious, only one pivot sample of each type is tested in the experiments. As a result, there is some limitation in the test results. 5
Conclusion
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In order to establish the stress-life model of the cross-spring pivot, the fatigue strength of actual component is corrected using the Marin correction factors. There are two proposed methods to prolong the service life of the cross-spring pivot used in flexural static balancing instrument, i.e. increasing the ratio of length to thickness of leaf-springs and reducing stress concentration. The equations for theoretical stress concentration factors Kti of the cross-spring pivot which are applicable to the cross-spring pivots with acceptable errors, are obtained by fitting the FEA results. On these grounds, the equivalent
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fatigue stresses of five cross-spring pivots are solved to confirm the effectiveness of two methods. The experimental studies of pivots a-e with the same rotational stiffness are carried out to further verify the validity of the two methods. It is proved that the increase of the ratio of length to thickness of leaf-springs is beneficial to prolong the fatigue life of the cross-spring
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pivot without changing its rotational stiffness. In addition, designing fillet and avoiding acute angle appearing at the end of the leaf-springs can reduce the stress concentration effects and prolong the fatigue life of the cross-spring pivot. These conclusions provide an amount of theoretical supports for the preliminary design of the cross-spring pivot. The influences of two important factors (i.e., the ratio of length to thickness of leaf-springs and stress concentration) on the fatigue life of the cross-spring pivot are considered. Future studies will focus on other factors such as rotational frequency
Acknowledgments
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of the cross-spring pivot and surface roughness of leaf-springs.
The authors gratefully acknowledge the financial supports of National Natural Science Foundation of China (Grant Nos. 51675015, 91748205). References
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[1] Yan WX, Zhan ST, Qian ZY, Fu Z, Zhao YZ. Design of a measurement system for use in static balancing a two-axis gimbaled antenna. Journal of Aerospace Engineering 2014; 0: 1–12. [2] Boynton R, Wiener K, Kennedy P. Static balancing a device with two or more degrees of freedom (the key to
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obtaining high performance on gimbaled missile seekers). In: the 62nd Annual Conference of Society of Allied Weight Engineers, Inc. 2003. p. 24–8. Paper No. 3320. [3] Dirksen F, Anselmann M, Zohdi T I, et al. Incorporation of flexural hinge fatigue-life cycle criteria into the topological design of compliant small-scale devices. Precision Engineering, 2013, 37(3):531-541. [4] Henein S, Aymon C, Bottinelli S, et al. Fatigue failure of thin wire-electrodischarge-machined flexure flexures. // Photonics East. International Society for Optics and Photonics, 1999:110-121. [5] David Schoenen, Sascha Lersch Mathias Hüsing, et al. Development, design and application of a fatigue test bench for high precision flexure flexures. Proceedings of the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference.2016.p. 21-24. [6] Wang Q, Zhang X. Fatigue reliability based optimal design of planar compliant micro positioning stages. Review of Scientific Instruments, 2015, 86(10):105117. 15
ACCEPTED MANUSCRIPT [7] I. Ivanov, B. Corves. Fatigue testing of flexure flexures for the purpose of the development of a high-precision micro manipulator. Mech. Sci., 5, 59–66, 2014. [8] L.L. Howell, Compliant Mechanisms, Wiley, New York, 2001. [9] W.E. Young, An investigation of the cross-spring pivot, Journal of Applied Mechanics 11 (1944) 113–120. [10] S.T. Smith, Flexures: Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, New York, NY, 2000.
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[11] Zhao HZ, Bi SS. Stiffness and stress characteristics of the generalized cross-spring pivot. Mechanism & Machine Theory, 2010, 45(3):378-391.
[12] Gómez J F, Booker J D, Mellor P H. 2D shape optimization of leaf-type crossed flexure pivot springs for minimum stress. Precision Engineering, 2015, 42:6-21.
[13] Griffin S F. Stress optimization of leaf-spring crossed flexure pivots for an active Gurney flap mechanism// Society
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of Photo-optical Instrumentation Engineers Conference Series. International Society for Optics and Photonics, 2015. [14] Bi SS, Zhang SQ, Zhao HZ. Quasi-constant rotational stiffness characteristic for cross-spring pivots in high precision measurement of unbalance moment. Precision Engineering, 2016, 43:328-334.
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[15] Marin, J., Mechanical Behavior of Engineering Materials, Prentice Hall, Upper Saddle River, NJ, 1962. [16] Shigley J. E., Mitchell L. D. Mechanical Engineering Design, 1983.
[17] Howell L L, Rao S S, Midha A. Reliability-Based Optimal Design of a Bistable Compliant Mechanism. Journal of Mechanical Design, 1994, 116(4):1115-1121.
[18] Mischke C R. Prediction of Stochastic Endurance Strength. Journal of Vibration & Acoustics, 1987, 109(1):113. [19] J. Shigley, C. Mischke, T. Brown. Standard Handbook of Machine Design, McGraw-Hill Standard Handbooks (McGraw-Hill Education, 2004).
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[20] Pilkey D. Peterson’s stress concentration factors. New Jersey: Wiley; 2008. [21] Chen G. Generalized Equations for Estimating Stress Concentration Factors of Various Notch Flexure Hinges. Journal of Mechanical Design, 2014, 136(3):252-261.
[22] Zhao T, Zhang J, Jiang Y. A study of fatigue crack growth of 7075-T651 aluminum alloy. International Journal of
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Fatigue, 2008, 30(7):1169-1180.
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ACCEPTED MANUSCRIPT Highlights: This paper proposes two methods of prolonging the fatigue life of the cross-spring pivots (CSP) without changing another important performance, rotational stiffness. In order to establish the stress-life model, the equivalent fatigue stresses of the CSPs are derived in
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which the stress concentration factors are obtained by fitting the finite element results.
Reference to the crank-guide mechanism consisting of few components, a fatigue test bench applying to the CSP is implemented, in which there is no additional force on the fatigue life of pivot as
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in the crank-rocker mechanism.
S0141-6359(18)30364-7
DOI:
https://doi.org/10.1016/j.precisioneng.2018.10.009
Reference:
PRE 6862
To appear in:
Precision Engineering
Received Date: 7 June 2018 Revised Date:
1 October 2018
Accepted Date: 17 October 2018
Please cite this article as: Bi S, Li Y, Zhao H, Fatigue analysis and experiment of leaf-spring pivots for high precision flexural static balancing instruments, Precision Engineering (2018), doi: https:// doi.org/10.1016/j.precisioneng.2018.10.009. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Fatigue analysis and experiment of leaf-spring pivots for high precision flexural static balancing instruments Shusheng Bi, Yongzhen Li, Hongzhe Zhao* Robotics Institute, Beihang University, XueYuan Road No. 37, HaiDian District, Beijing, 100191, China
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Abstract For distributed-compliance flexure mechanisms, their service life is affected by fluctuating stress which may result in fatigue failure, but only static failure was taken into consideration in most studies. As one of the best-known types of distributed-compliance flexural pivots, the cross-spring pivot applied to the flexural static balancing instruments is subjected to large enough vertical loads and fluctuating stresses. In this paper, the cross-spring pivot is taken to explore the methods of
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prolonging its fatigue life, and the methods can be extended to the other pivots based on leaf-springs. Meanwhile, another important performance of the pivot-the rotational stiffness remains unchanged. Firstly, in order to develop a stress-life model, the equivalent fatigue stresses of the cross-spring pivots are derived in which the stress concentration factors are obtained by
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fitting the finite element results. Then, two methods for prolonging the fatigue life of the pivot are proposed, i.e., increasing the ratio of length to thickness of leaf-springs and reducing stress concentration. To verify the theoretical analysis, the fatigue life of five cross-spring pivots with the same rotational stiffness is tested based on a developed fatigue test bench. Experimental results show that the fatigue life of the cross-spring pivot can be prolonged, which verifies the validity of the two methods. Keywords
1
Introduction
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Static balancing instrument, Cross-spring pivot, Fatigue stress; Fatigue life, Stress concentration factor
Static balancing instrument, as a precision instrument for measuring the unbalanced moment of objects, plays an important role in achieving the static balance of objects. Compared with the traditional static balancing instrument, flexural
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static balancing instrument has attracted the attentions of majority of scholars due to its high measurement accuracy and wide measurement range. The typical cases are that two flexural static balancing instruments using flexural pivots were designed by Yan et al and Boynton et al, respectively [1,2]. However, the flexural static balancing instrument is more prone to fatigue
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failure due to the use of the flexural pivot as a rotation support unit. Currently, the researches on flexural static balancing instruments mainly focus on the design of new mechanisms and the improvement of measurement sensitivity, but few studies on the method of prolonging their service life. For compliant mechanisms, fatigue failure is a main failure mode that affects their service life significantly, especially under dynamic loading, which leads to a premature fatigue failure of the flexural static balancing instruments. Therefore, it is quite necessary to maximize the fatigue life of the pivots in engineering applications.
The existing flexural pivots are mainly divided into two categories: lumped-compliance and distributed-compliance flexural pivot. The main studies on the fatigue property of flexural pivots have focused on lumped-compliance pivots. Dirksen et al [3] incorporated the fatigue life cycle criteria of three notch flexural pivots into the topology optimization design of the compliant mechanism. Henein et al [4] tested the fatigue life of right circular pivots to inquire into the __________ * Corresponding author. Tel.: 86-010-82338019. E-mail addresses: [email protected]
1
ACCEPTED MANUSCRIPT exportability of the fatigue measurements made on standard test-specimen by Wire cut Electrical Discharge Machining (WEDM). Schoenen et al [5] studied the influences of the WEDM method on fatigue life of notch type flexural pivots. The fatigue properties of the notch type flexural pivots used in a planar micro-positioning platform and a high-precision micro manipulator were studied by Wang [6] and Ivanov [7], respectively. As one of the best-known types of distributedcompliance flexural pivots, cross-spring pivot is employed in an increasing number of precision machines, due to its significant advantages such as maintenance-free, no backlash, diminished friction and infinitesimal resolution. Currently,
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some research groups focus on the stiffness, accuracy, rotational range and other static characteristics of the cross-spring pivot [8-11]. However, there are very few researches on the fatigue properties of the cross-spring pivot at present. Gómez et al [12,13] proposed a method of optimizing the shape of leaf-springs to reduce stress in cross-spring pivots, thus extending their fatigue life. It is relatively complicated, however, to determine the shape parameters of leaf-springs. Currently, there are limited ways to prolong the fatigue life of the cross-spring pivot. According to the similarity in the processing method,
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surface roughness and rotational frequency between the leaf-spring pivots and the notch pivots, the influences of them on the fatigue characteristics of pivot are no longer considered. Therefore, this paper will focus on researching the influences of shape parameters and stress concentration on the fatigue characteristics of the cross-spring pivot.
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As a precision measuring instrument, it is highly necessary to possess high measuring sensitivity and long service life simultaneously. In our previous work [14], a high precision flexural static balancing instrument, which used the cross-spring pivot with quasi-constant rotational stiffness as a rotational unit, was developed, as shown in Fig. 1. It was revealed that the sensitivity of the instrument is closely related to the rotational stiffness of the cross-spring pivot. For this reason, methods of prolonging the fatigue life of the cross-spring pivot with its rotational stiffness keeping constant should be proposed. The objective of this paper is to study the influences of the shape parameters (length L and thickness T) and stress concentration on the fatigue life of the cross-spring pivot. The corresponding methods are proposed to prolong the fatigue life
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of the cross-spring pivot without changing rotational stiffness. Consequently, the remainder of this paper is organized as follows. The characteristics of the cross-spring pivots to be studied are introduced in Section 2. The equivalent fatigue stress model for the cross-spring pivot is developed, in which the equations of theoretical stress concentration factors are obtained on the basis of the Finite Element Analysis (FEA) results. On this basis, the corresponding methods are proposed to reduce
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the equivalent fatigue stress of the cross-spring pivot without changing rotational stiffness in Section 3. Finally, in order to verify the correctness of the proposed methods, experimental investigations on the fatigue life of five cross-spring pivots with
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the same rotational stiffness are conducted in Section 4.
Fig. 1. (a) flexural Static balancing instrument ( spirit level,
mechanical limit,
cross-spring pivot,
counterweights,
working platform,
electronic force rebalance transducer) and (b) cross-spring pivot. 2
payloads,
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Characteristics of the cross-spring pivots for flexural static balancing instrument For precision measuring instruments, measurement sensitivity is one of the key performances of the flexural static
balancing instrument. The sensitivity of the instrument should not be changed while studying the fatigue properties of the cross-spring pivot. Since the sensitivity of the static balancing instrument is closely related to the rotational stiffness of the cross-spring pivot, so all the cross-spring pivots to be studied have the same rotational stiffness. According to the Ref. [14],
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to simplify the design variables, the non-dimensional rotational stiffness of the cross-spring pivots can be represented as Eq. (1).
Km =
2(9λ 2 − 9λ + 1) λ f cos α + m = 8 ( 3λ 2 − 3λ + 1) − + λ cos α p − ph θ 15cos α
(1)
where m=ML/(EI), p=PL2/(EI), f=FL2/(EI), I=WT3/12, h=H/L, the lower case letters are non-dimensional parameters, M, P
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and F denote bending moment, vertical load and horizontal load, respectively; L, W and T are the length, width, and thickness of the leaf-spring, respectively; H denotes the height of the action point of load P from the platform; I is the cross section moment of inertia about the neutral axis; E is the Young’s modulus of the material; λL is the distance from moving stage to
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the intersection; 2α is the angle of intersection, as shown in Fig. 1(b).
As can been seen from Eq. (1) that the rotational stiffness of the cross-spring pivot varies with the load conditions (load p and height h). However, load p and height h change with different measured objects, and therefore it is difficult to evaluate the rotational stiffness of the cross-spring pivot. This paper focuses on the influences of structural parameters on the fatigue life of cross-spring pivot under the same load conditions, which means that ph is a constant. Furthermore, all the cross-spring pivots to be studied have the approximately identical rotational stiffness as the structural parameters vary. If the coefficient of vertical load p is zero, the rotational stiffness of the cross-spring pivot will not be affected, so the following relationship
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between λ and α should be satisfied.
2(9λ 2 − 9λ + 1) + 15λ cos 2 α = 0
(2)
Accordingly, when λ is fixed, the rotational stiffness Km is an approximately constant, and α also can be determined. Furthermore, if λ=1/3 and α=50.77°, the center shift of the pivot takes the minimum value [14]. Therefore, the cross-spring
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pivots to be studied satisfy the two conditions of approximately constant rotational stiffness and minimum center shift. The above analysis result of rotational stiffness is non-dimensional, and then it is converted into a dimensional parameter. Suppose the width W of leaf-spring is fixed, hence the shape parameters L and T of the cross-spring pivot with constant
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rotational stiffness should be satisfy the Eq. (3). The relationship between L and T is shown in Fig. 2.
L / T 3 = constant
3
(3)
Fig. 2. Relationship between L and T.
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When cross-spring pivot acts as a rotational unit, the leaf-springs are treated as Euler-Bernoulli beams ignoring the shearing effect. The bending stress along the length of leaf-spring can be written as [11]
3Emi ( x)
, (i = 1,2)
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σ bi =
d
(4)
where d=12(L/T)2; i denotes one of the two leaf-springs of the cross-spring pivot; mi(x) is the internal bending moment in the two leaf-springs.
The expressions of the internal bending moment along the length of the leaf-spring are given by
m1 ( x) = ( A3 x3 − A2 x2 + A1 x + A0 )θ − ( B3 x3 − B2 x2 + B1 x + B0 )θ 2 3 2 3 2 2 m2 ( x) = ( A3 x − A2 x + A1 x + A0 )θ + ( B3 x − B2 x + B1 x + B0 )θ
(5)
Ref. [11].
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where A1, A2, A3, B1, B2 and B3 are functions of vertical load p and horizontal load f. Their specific value can be referred to If the axial stress along the length of the leaf-spring can be regarded as constant, then it can be written in the form [11]
Epi , (i = 1, 2) d
(6)
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σ ti =
where pi is the non-dimensional parameter.
Therefore, when the total bending moment and vertical load p are applied to the cross-spring pivot simultaneously, the
3
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corresponding bending and axial nominal stresses can be calculated by Eqs. (4) and (6), respectively. Fatigue analysis of the cross-spring pivots for static balancing instrument Fatigue failure caused by cyclic stress is the major failure mode of the cross-spring pivot under vertical load and bending moment. The stress-life (or S-N) curve describes the number of cycles at failure at a given stress, and the stress-life model is the most direct and common fatigue failure prediction model. Even if it is difficult to accurately predict fatigue failures for a variety of reasons, it is highly helpful to understand the prediction of fatigue failure in the design of compliant mechanisms. For most common materials, the S-N curve is usually divided into low cycle fatigue and high cycle fatigue. The S-N curve can be approximated by means of three points on the S-N curve, namely, Su (the ultimate strength), Se (the fatigue limit), and SL (the fatigue strength at a fatigue life of 103), as shown in Fig. 3 [6].
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Fig. 3. Approximate S-N diagram.
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In high cycle fatigue stage, for aluminum alloy materials with no endurance limit, the curve is started with fatigue failure
cycle region can be written in the following form
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at more than 103 cycles. For an actual aluminum alloy component, the approximation formula of the S-N diagram in the high-
S f = af N
log a f =
bf
log N2 log SL − log N1 log S f 2
bf =
log N2 − log N1
S 1 log L Sf 2 log N1 − log N2
(7) (8)
(9)
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where Sf is the fatigue strength; N is the number of cycles; af and bf are two unknowns; For aluminum without fatigue limit, N1 = 1×103, N2 = 5×108; Sf2 is the conditioned fatigue strength at 5×108 cycles for aluminum. The value of SL at fatigue life cycles of 103 can be approximated as
SL = η Su
0.9 0.75
η=
bending axial loading
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where
(10)
According to Eqs. (7)-(9), in order to obtain the relationship between stress and fatigue life (for high-cycle fatigue) of a pivot at specified fatigue strength, it is first necessary to obtain conditioned fatigue strength and the fatigue stress of the actual component. These will be discussed in Sections 3.1 and 3.2, respectively.
3.1 Conditioned fatigue strength of the cross-spring pivot The fatigue properties of materials or components are measured by fatigue strength which is reflected through fatigue limit. The fatigue limit is defined as the amplitude of cyclic stress in which the material or component can withstand the infinite stress cycle without fatigue failure. For aluminum alloy materials with no fatigue limit, their fatigue life can be estimated by the fatigue strength of a certain number of cycles. The conditioned fatigue strength of the cross-spring pivot can be obtained by modifying that of standard specimens. The conditioned fatigue strength modification factors suggested by Marin [8,15-18] are expressed as 5
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(11)
where ka, kb, kc, kd, ke and km are the surface condition modification factor, the size modification factor, the load modification factor, the temperature modification factor, the reliability factor and other miscellaneous modification factors, respectively. Next, the uncorrected fatigue strength Sf’ of standard specimen needs to be obtained from fatigue testing. However, the
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process is time-consuming and laborious. The simple method of getting uncorrected fatigue strength is estimated by the ultimate tensile strength of the material. For aluminum alloys, the relations between uncorrected fatigue strength Sf’ and the ultimate tensile strength Su are depicted as [8]
3.2 Equivalent fatigue stresses of the cross-spring pivot
(12)
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Su < 330MPa 0.4Su Sf ' ≈ 130MPa Su ≥ 330MPa
The cross-spring pivots for static balancing instrument are subjected to a bending nominal stress and an axial nominal stress. The internal stresses of the cross-spring pivot contain nonzero mean stress, so the equivalent fatigue stresses can be
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used to represent the fatigue stress of the cross-spring pivot. Classical fatigue failure criteria that match experimental data quite well has been proposed by Goodman [19]. In this work, equivalent fatigue stresses Seqv of the cross-spring pivot is calculated by the modified Goodman relation.
Seqv =
σa 1 − σ m / Su
(13)
where σa and σm are equivalent stress amplitude and equivalent mean stress, respectively. The equivalent stress amplitude and
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equivalent mean stress are represented in Ref. [19]
σ a = K fbσ ba + σ m = K fbσ bm +
K ftσ ta 0.85 K ftσ tm
(14)
0.85
where σbm and σba are mean stress and stress amplitude corresponding to the bending stress; σtm and σta are mean stress and
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stress amplitude corresponding to the axial stress; Kfb and Kft are fatigue stress concentration factors for bending and axial stress, respectively. The fatigue stress concentration factor Kfi is defined as the ratio of fatigue strength of standard specimen
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to that of the specified component. The stress concentration factors will be discussed in the Section 3.3. Mean stress and stress amplitude are defined as
σ bm = (σ b max + σ b min ) / 2, σ ba = (σ b max − σ b min ) / 2 σ tm = (σ t max + σ t min ) / 2,
σ ta = (σ t max − σ t min ) / 2
(15)
where σbmax and σbmin are the maximum and minimum bending stress; σtmax and σtmin are the maximum and minimum axial stress. They can be solved by Eqs. (4) and (6). To sum up, the equivalent fatigue stress of the cross-spring pivot can be reduced by reducing the nominal stresses and the fatigue stress concentration factors, thus extending its fatigue life. Accordingly, the study will focus on exploring the methods of reducing the nominal stresses and stress concentration. According to Eq. (5), the internal bending moment of a leaf-spring is determined by the structure configuration of the cross-spring pivot. Next, the influences of the length L and thickness T of leaf-spring on the bending stress and axial stress 6
ACCEPTED MANUSCRIPT can be obtained by Eqs. (4) and (6). The relationship between internal maximum stress σmax along the length of leaf-spring versus the ratio of length to thickness of leaf-spring is shown in Fig. 4 when Eq. (3) is satisfied. It can be seen that the bending stress of the leaf-spring is much larger than axial stress, but the axial stress is not neglected to improve accuracy in this paper (the axial stress is neglected in Ref. [11]). Moreover, as the ratio of length to thickness of leaf-spring increases, the internal total stress of leaf-spring decreases at first and then increases, but the increase is not obvious. On these grounds, the maximum nominal stress of the cross-spring pivot can be reduced by increasing the ratio of length to thickness of leaf-springs.
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pivot under the condition that the rotational stiffness of pivot is constant.
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Therefore, it is of great significance to study the effects of shape parameters (L and T) on the fatigue life of the cross-spring
Fig. 4. Relationship between stress and the ratio of length to thickness.
3.3 Theoretical stress concentration factors
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It can be seen from Eq. (14) that fatigue stress concentration factors have important influences on equivalent fatigue stresses. To simplify the solution process, provided that the theoretical stress concentration factor Kti is equal to the fatigue stress concentration factor Kfi. The problem of obtaining the fatigue stress concentration factor of the cross-spring pivot is transformed into solving its theoretical stress concentration factor. For some typical notch type components, the correlation diagrams or calculation formulas of the theoretical stress
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concentration factors have been given in Ref. [20]. However, as a distributed-compliance flexural pivot, studies on theoretical stress concentration factor of the cross-spring pivot have not been found in existing literatures. Generally, stress
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concentration factors can be obtained through photo-elasticity experimentation, analytical methods (e.g., the theory of elasticity) or numerical methods (e.g., FEA). To this end, we try to obtain the bending and axial stress concentration factors of the cross-spring pivot through fitting simulation results, because it is an effective and inexpensive method. The theoretical stress concentration factor Kti is defined as
Kti =
σ i max , (i = b, t ) σ inom
(16)
where σbnom is the bending nominal stress, and can be obtained from Eq. (4). If an axial force Fx is applied along the length of leaf-spring, σtnom is the axial nominal stress which can be solved according to Eq. (6). σimax is the maximum stress of a component, and can be obtained by finite element method. The most fatigue failures of components occur first at the sites of local stress concentration, and the stress concentration in the cross-spring pivot appears at the end of the leaf-spring. The most common way to reduce stress concentration is to 7
ACCEPTED MANUSCRIPT design fillet R. Besides, Refs. [3,21] indicate that theoretical stress concentration factors of notch type pivots strongly depend on the minimum thickness and the corresponding radius of curvature. Provided that this conclusion also applies to the crossspring pivots, equations for predicting the stress concentration factors of the cross-spring pivots can be formulated based on the finite element results by defining γ=R/T as the non-dimensional stress-concentrating parameter in the following subsections. Besides, as is well known, acute angle should be avoided as much as possible in engineering design, while it exists in most of the cross-spring pivots used in practical engineering applications. Therefore, the effects of parameter γ and
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acute angle φ on stress concentration factor are studied through finite element method. The parameter definitions of the
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improved cross-spring pivot are shown in Fig. 5.
Fig. 5. The parameter definitions of the cross-spring pivot. The ranges of the shape parameters of the cross-spring pivots are: 0
R
3.5mm, 0.3mm
T
0.6mm, ignoring the
impact of other parameters. For all cross-spring pivots designed, the linear elastic material properties are: the Young’s
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modulus E=7.2×1010 Pa, Poisson’s ratio υ=0.33, and the density ρ=2700 kg/m3. The FEA model of each cross-spring pivot is meshed with SOLID186 elements in ANSYS, which is a twenty-node three-dimensional hexahedral element with three degrees of freedom at each node. A typical mesh model is shown in Fig. 6. To ensure the validity of the FEA results, the mesh of each model is refined at the end of leaf-spring so that the size of mesh does not exceed 0.2mm at the fillet position. The bottom rigid body of leaf-spring is fixed, and loaded with unit load (i.e., bending moment and axial load) on the free end
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of leaf-spring to simplify the calculation. So the corresponding maximum stress σmax is recorded, which is then used to
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calculate theoretical stress concentration factor Ktb (or Ktt) using Eq. (16).
Fig. 6. Finite element mesh model. These results leave no doubt that the maximum stress of every cross-spring pivot model appears at the root of the leaf8
ACCEPTED MANUSCRIPT spring. When the fillets R becomes larger, the maximum stress in the cross-spring pivot model gradually decreases and tends to be stable. When the angle φ changes from an acute angle to a right angle, the maximum axial stress also decreases, while the maximum bending stress is almost constant. Therefore, the methods of reducing stress concentration are obtained. The FEA results for both Ktb and Ktt confirm the previous assumption that the stress concentration factors depend on the nondimensional parameter γ, but is almost independent of the other shape parameters (e.g., L and W). When the thickness T is fixed, the stress concentration factors can be reduced within a certain range by increasing the fillet R. In addition, when R
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increases to a certain value, the stress concentration factor tends to be a constant. As a result, a rational function of γ with first-order polynomials in both the numerator and denominator could fit the FEA results for Ktb and Ktt by using MATLAB program. Ultimately, the following equations are obtained, respectively:
Ktt =
1.018γ + 0.7319 γ + 0.433 1.269γ + 1.137 γ + 0.7288
(17)
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Ktb =
(18)
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Figs. 7 and 8 depict the fitting equations of stress concentration factors Ktb and Ktt and FEA results of leaf-spring with four thicknesses for the purpose of comparison. Compared with FEA results, the maximum errors of fitting equations for Ktb and Ktt are 2.25% and 2.1 %, respectively. Hence, Eqs. (17) and (18) are applicable to a wide range of the cross-spring pivots with acceptable error.
1.6
T=0.3mm T=0.4mm T=0.5mm T=0.6mm Fitting Equation
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γ
Fig. 7. Bending stress concentration factor versus γ.
Ktt
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Ktb
1.5
1.4
1.3
1.2 0
2
4
6
8
10
γ Fig. 8. Axial stress concentration factor versus γ.
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In order to study the relationships between the stress concentration factors and the angle φ (Fig. 5) in the cross-spring pivot, the maximum stresses of the cross-spring pivots with multiple φ values are obtained by virtue of FEA, and are used to calculate Ktb (or Ktt). Figs. 9 and 10 plot Ktb and Ktt calculated from the finite element results. It can be found in Figs. 9 and 10 that, as the angle φ increases, the bending stress concentration factors of the cross-spring pivots with different φ are almost the same, while the axial stress concentration factors decrease continuously. In addition, there is an important conclusion that the value of axial stress concentration factor reaches minimum when φ = 90°.
9
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=30° =39.23° =45° =60° =90°
1.6 1.5
Ktt
Ktb
1.4 1.3 1.2
1
0
2
4
6
8
RI PT
1.1 10
γ
γ Fig. 9. Bending stress concentration factor versus γ.
Fig. 10. Axial stress concentration factor versus γ.
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Finally, the equivalent stress amplitude and equivalent mean stress can be obtained by substituting Eqs. (15), (17) and (18) into Eq. (14). Furthermore, the equivalent fatigue stress of the cross-spring pivot can be solved according to Eq. (13), which
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can be used to predict the fatigue life of cross-spring pivots. Experimental evaluation
4.1 Stress-life model of the cross-spring pivots
Obviously, according to the above analysis, the nominal stress of the cross-spring pivot can be reduced by increasing the ratio of length to thickness of the leaf-spring. Moreover, stress concentration has great influences on the equivalent fatigue stress of the cross-spring pivot. Next, the geometric and shape parameters of the cross-spring pivots to be tested are determined to verify the correctness of the above theoretical analysis.
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According to the analysis in Section. 2, the sensitivity of the instrument should not be changed while investigating the ways to prolong the fatigue life of the cross-spring pivot. All the cross-spring pivots have the same rotational stiffness, that is, the relationship between the length L and thickness T of the leaf-spring satisfies Eq. (3). Additionally, the cross-spring pivots satisfy the two conditions: approximately constant rotational stiffness and minimum center shift. All the cross-spring pivots
Table 1. Table 1
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have the same geometric parameters, i.e., λ=1/3 and α=50.77°. The shape parameters of five cross-spring pivots are listed in
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The parameters and equivalent fatigue stresses of five cross-spring pivots.
Pivot L(mm) W(mm) T(mm) R(mm) φ(°) Ktb Ktt Seqv(MPa) a 45 10 0.4 0 39.23 1.82 1.49 57.26 b 45 10 0.4 0.5 39.23 1.19 1.38 40.40 c 45 10 0.4 1 39.23 1.12 1.33 38.25 d 45 10 0.4 1 90 1.09 1.10 35.81 e 64.07 10 0.45 0 39.23 1.82 1.49 49.90 The fatigue failures of most components occur in the stress concentration zones, and the stress concentration of the crossspring pivot locates at the end of the leaf-springs. To reduce the effects of the stress concentration on fatigue life, the method of designing fillets is the most common. Compared with pivot a, the pivots b and c have fillets with radius of 0.5mm and 1mm at the end of the leaf-springs, respectively, while the effective length L of pivots b and c is the same as that of pivot a. In addition, the rest of the geometric and shape parameters of pivots b and c are identical to that of pivot a. 10
ACCEPTED MANUSCRIPT The fatigue crack always appears at the end of the leaf-spring and is close to the moving stage. Based on the FEA results in Section. 3.3, the axial stress concentration factor is minimum when the φ is equal to 90°. To reduce the stress concentration, the φ is set to be 90° for pivot d. In addition, pivot d also has fillets with radius 1mm at the end of the leaf-springs. The rest of the shape parameters of the pivot d are the same as that of pivot a. Moreover, to investigate the influences of the ratio of length to thickness of the leaf-spring on the fatigue life, pivot e with a larger ratio of length to thickness is tested with reference to the pivot a. However, it also ensures that the relationship
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between length L and thickness T satisfies Eq. (3).
For the flexural static balancing instrument, the maximum rotational angle of the cross-spring pivot is 2°, and vertical load is P=30N. The ultimate tensile strength of aluminum alloy 7075-T6 is Su=538MPa [22]. The theoretical stress concentration factors of five pivots can be obtained by Eqs. (17) and (18). Ultimately, the equivalent fatigue stresses of these pivots can be solved. The results are tabulated in Table 1. Compared with pivot a, the equivalent fatigue stress of the pivots b-e are
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relatively small. Hence, the effectiveness of the two methods of reducing equivalent fatigue stress is verified.
These cross-spring pivots are machined by WEDM. The prototypes of five cross-spring pivots are shown in Fig. 11. The surface roughness Ra of the leaf-spring is measured by the surface roughness measuring instrument, and the result is
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Ra≈3.05um. Then, the surface reduction factor can be obtained, i.e., ka=0.75 [3]. The remaining Marin correction factors of the specific cross-spring pivot can also be calculated. The theoretical stress concentration factors Kt of the pivots a-e are obtained by the finite element simulation. So miscellaneous-effects factor km can be calculated. According to Eq. (12), the fatigue strengths of these cross-spring pivots are estimated. According to Eq. (7), S-N curves of the high-cycle fatigue life for these cross-spring pivots are estimated and expressed as Eqs. (19)-(23).
(19)
Sb = 9003.8 Nb −0.4231
(20)
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Sa = 8098.6 N a −0.4078
Sc = 8677.6 N c −0.4178
(21)
S d = 7689.6 N d −0.4003
(22)
Se = 8098.6 N e −0.4078
(23)
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where the subscript a-e represent the pivots a-e.
Fig. 11. Five cross-spring pivots used in the instrument.
4.2 Measurement set-up 11
ACCEPTED MANUSCRIPT In order to verify the validity of the above theoretical analysis, a fatigue test bench applying to the cross-spring pivot is implemented. There are three working principles that can be used to swing the cross-spring pivot back and forth, as shown in Fig. 12. Schoenen et al [5] and Ivanov et al [7] developed a platform of the fatigue life testing for notch type flexural pivot based on crank-slider mechanism and crank-rocker mechanism, respectively. Nevertheless, the crank-slider mechanism, which consists of too many components, is a relatively complicated system. In the crank-rocker mechanism, additional forces have an effect on the fatigue life of pivots. The crank-guide mechanism is adopted, because it consists of relatively few
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components. Accordingly, it is easy to assemble and maintain the mechanical system. Furthermore, there is no additional force on the fatigue life of pivot as in the crank-rocker mechanism.
ωA
slider
ω
B
D
C guide bar specimen
O
(a)
B crank ω
A
guide bar
C
link
crank
H
link
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A
B
θ
rocker
O
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crank
specimen
(b)
specimen
O
(c)
Fig. 12. The schematic diagram of fatigue test bench: (a) crank-slider mechanism, (b) crank-rocker mechanism and (c) crankguide mechanism.
The existing literatures studied the fatigue life of notch type pivot being subjected to bending stress only, ignoring the effects of vertical loads on the fatigue life of pivot [5,7]. This work explores how to prolong the fatigue life of the cross-
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spring pivot for a specific application. Since the load on the pivot have important influences on its fatigue life, the influences of the external forces on the fatigue test results should be avoided as much as possible. While testing the fatigue life of the cross-spring pivot, there are two challenges: test instruments are not damaged by loads, and test results are not affected by external forces. Additionally, in order to adapt to the cross-spring pivot with different shape parameters, the fatigue test bench with adjustable kinematic parameters must be designed.
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Referring to the working principle of crank-guide mechanism, a fatigue test bench is designed to swing the cross-spring pivot continuously at a predetermined angle. The prototype of the fatigue test bench for the cross-spring pivot is demonstrated in Fig. 13. A rod is fixed to the crank and deviates from the center of the crank. Meanwhile, the other end of the
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rod is located in the groove of the guide bar. The fixed stage of the cross-spring pivot is fixed to the base, and the moving stage is fixed to the guide bar with a groove. The swing of the guide bar is achieved by the rotation of the crank which is actuated by a stepper motor. The swing angle of the guide bar is equal to the rotational angle of the cross-spring pivot. As illustrated in Fig. 12(c), assuming that the rotational axis of the cross-spring pivot is fixed at O, the maximum value of the rotational angle of the cross-spring pivot can be expressed as
θmax = arcsin
l1 H
(24)
where l1 is the length of the crank; H is the distance between the center of crank and the rotational axis of the cross-spring pivot. The maximum rotational angle θmax of the pivot can be changed by adjusting the eccentric distance l1. Herein, the angle θmax is set to be 2°. 12
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Fig. 13. The photo of the fatigue test bench: (a) global graph and (b) partial view
There are some measures adopted to reduce test errors. In order to reduce the friction between the rod and the guide bar, there is a roller assembled at the end of the rod and rolling in the groove of the guide bar. In this way, the effects of friction
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on the results of fatigue test can be reduced.
Before starting the test, each pivot is kept in a static state by the mechanical limits which prevent the pivot from being damaged by excessive rotational angle. During the test, the pivot swings back and forth until the fracture occurs. To prevent payloads falling and protect other equipment from being damaged, thin wire ropes are used to protect the loads. The starting and stopping of the fatigue test are determined by a control unit. One of the photoelectric switches is fixed at the end of the guide bar and provides the motor with stopping signal when the pivot fractures in fatigue. The other photoelectric switch is used to record the number of the cross-spring pivot rotations, and the number is displayed in real time
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by a counter. At the same time, a camera monitors the whole process of fatigue test and the number of the counter. The control flow chart of the fatigue test is shown in Fig. 14.
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start
counter reset control unit motor turning
counter no photoelectric switch
end
yes
failure
motor stop
Fig. 14. The control flow charts of fatigue test
4.3 Experiments and results 13
ACCEPTED MANUSCRIPT In order to ensure machining accuracy of the pivots, five AL7075-T6 prototypes of the cross-spring pivot are processed by using Charmilles Robofil 380 WEDM machine. Furthermore, for the sake of reducing assembling error, two leaf-springs of the cross-spring pivot are cut by one process at the same time. Then, one of them is turned over and they are assembled by pins. Most of the cross-spring pivots for the flexural static balancing instrument have been subjected to the vertical loads p during rotation. Consequently, the fatigue life tests of the cross-spring pivots are carried out under the action of constant
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vertical load. The vertical load of 30N is applied on moving platform of each cross-spring pivot in the experiments. Besides, the maximum rotational angle of the cross-spring pivots is set to be 2°. The rotation of the pivot is generated by the horizontal force acting on the pivot by the test bench, and the moment created by the horizontal force is equal to the moment in the theoretical analysis. In addition, the rotational frequency of all the cross-spring pivots is set to be 2Hz according to the actual application.
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The fatigue testing results of five cross-spring pivots are listed in Table 2. Taking pivot a as the reference, the life cycles of the rest of the cross-spring pivots are relatively long, which confirms the effectiveness of the improved measures. Compared with pivot b, the fatigue life cycles of pivots c and d are improved, which supports the analysis about the stress
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concentration factors (Figs. 7-10), namely, the stress concentration factor (Ktb or Ktt) decreases with increasing γ, and the value of axial stress concentration factor reaches minimum when φ = 90°. By comparing the fatigue life cycles of pivots a and e, it can be proved that the larger ratio of length to thickness of the leaf-spring can prolong the fatigue life cycles of the cross-spring pivot. Based on what has been discussed above, the fatigue life of the cross-spring pivot can be maximized when φ is set to be 90° and fillets are designed. That is, stress concentration effects can be decreased by avoiding acute angle appearing at the root of the leaf-springs and designing fillet.
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Table 2 The fatigue life cycles of five cross-spring pivots.
Pivot
Experimental results
Error (%)
Improvement(%)
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187770 225106 19.88 354140 310573 12.30 435130 377052 13.35 669080 461698 31.00 263080 334281 27.06 Nevertheless, there are a certain amount of errors between the
37.97 67.50 105.10 48.50 fatigue test results and the actual values. The potential
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a b c d e
Analytical results
reasons are analyzed in the following:
a. On account of the existence of machining errors, the dimensions of the prototype are not exactly equal to that of the theoretical model. In particular, the thickness error of leaf-spring has much influences on fatigue life based on the above analysis.
b. For the sake of simplification, this paper assumes that the theoretical stress concentration factor is equal to the fatigue stress concentration factor. The former is actually less than the latter. Therefore, this introduces certain errors to the calculation of the equivalent fatigue stress and fatigue life cycles of the pivots. c. The effects of rotational frequency of the cross-spring pivot on the fatigue life are not considered in the theoretical analysis, but that cannot be ignored in practice. Mechanical limits are employed to prevent the pivots from being damaged by excessive rotational angle, but due to the presence of inertial forces, there are collisions when the cross-spring pivot reaches 14
ACCEPTED MANUSCRIPT the maximum angle. Furthermore, the higher the rotational frequency, the stronger the collisions. So the fatigue life of the cross-spring pivots may decrease to some extent, which will be studied in future work. d. Since the fatigue test process is time consuming and laborious, only one pivot sample of each type is tested in the experiments. As a result, there is some limitation in the test results. 5
Conclusion
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In order to establish the stress-life model of the cross-spring pivot, the fatigue strength of actual component is corrected using the Marin correction factors. There are two proposed methods to prolong the service life of the cross-spring pivot used in flexural static balancing instrument, i.e. increasing the ratio of length to thickness of leaf-springs and reducing stress concentration. The equations for theoretical stress concentration factors Kti of the cross-spring pivot which are applicable to the cross-spring pivots with acceptable errors, are obtained by fitting the FEA results. On these grounds, the equivalent
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fatigue stresses of five cross-spring pivots are solved to confirm the effectiveness of two methods. The experimental studies of pivots a-e with the same rotational stiffness are carried out to further verify the validity of the two methods. It is proved that the increase of the ratio of length to thickness of leaf-springs is beneficial to prolong the fatigue life of the cross-spring
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pivot without changing its rotational stiffness. In addition, designing fillet and avoiding acute angle appearing at the end of the leaf-springs can reduce the stress concentration effects and prolong the fatigue life of the cross-spring pivot. These conclusions provide an amount of theoretical supports for the preliminary design of the cross-spring pivot. The influences of two important factors (i.e., the ratio of length to thickness of leaf-springs and stress concentration) on the fatigue life of the cross-spring pivot are considered. Future studies will focus on other factors such as rotational frequency
Acknowledgments
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of the cross-spring pivot and surface roughness of leaf-springs.
The authors gratefully acknowledge the financial supports of National Natural Science Foundation of China (Grant Nos. 51675015, 91748205). References
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[1] Yan WX, Zhan ST, Qian ZY, Fu Z, Zhao YZ. Design of a measurement system for use in static balancing a two-axis gimbaled antenna. Journal of Aerospace Engineering 2014; 0: 1–12. [2] Boynton R, Wiener K, Kennedy P. Static balancing a device with two or more degrees of freedom (the key to
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obtaining high performance on gimbaled missile seekers). In: the 62nd Annual Conference of Society of Allied Weight Engineers, Inc. 2003. p. 24–8. Paper No. 3320. [3] Dirksen F, Anselmann M, Zohdi T I, et al. Incorporation of flexural hinge fatigue-life cycle criteria into the topological design of compliant small-scale devices. Precision Engineering, 2013, 37(3):531-541. [4] Henein S, Aymon C, Bottinelli S, et al. Fatigue failure of thin wire-electrodischarge-machined flexure flexures. // Photonics East. International Society for Optics and Photonics, 1999:110-121. [5] David Schoenen, Sascha Lersch Mathias Hüsing, et al. Development, design and application of a fatigue test bench for high precision flexure flexures. Proceedings of the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference.2016.p. 21-24. [6] Wang Q, Zhang X. Fatigue reliability based optimal design of planar compliant micro positioning stages. Review of Scientific Instruments, 2015, 86(10):105117. 15
ACCEPTED MANUSCRIPT [7] I. Ivanov, B. Corves. Fatigue testing of flexure flexures for the purpose of the development of a high-precision micro manipulator. Mech. Sci., 5, 59–66, 2014. [8] L.L. Howell, Compliant Mechanisms, Wiley, New York, 2001. [9] W.E. Young, An investigation of the cross-spring pivot, Journal of Applied Mechanics 11 (1944) 113–120. [10] S.T. Smith, Flexures: Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, New York, NY, 2000.
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[11] Zhao HZ, Bi SS. Stiffness and stress characteristics of the generalized cross-spring pivot. Mechanism & Machine Theory, 2010, 45(3):378-391.
[12] Gómez J F, Booker J D, Mellor P H. 2D shape optimization of leaf-type crossed flexure pivot springs for minimum stress. Precision Engineering, 2015, 42:6-21.
[13] Griffin S F. Stress optimization of leaf-spring crossed flexure pivots for an active Gurney flap mechanism// Society
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of Photo-optical Instrumentation Engineers Conference Series. International Society for Optics and Photonics, 2015. [14] Bi SS, Zhang SQ, Zhao HZ. Quasi-constant rotational stiffness characteristic for cross-spring pivots in high precision measurement of unbalance moment. Precision Engineering, 2016, 43:328-334.
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[15] Marin, J., Mechanical Behavior of Engineering Materials, Prentice Hall, Upper Saddle River, NJ, 1962. [16] Shigley J. E., Mitchell L. D. Mechanical Engineering Design, 1983.
[17] Howell L L, Rao S S, Midha A. Reliability-Based Optimal Design of a Bistable Compliant Mechanism. Journal of Mechanical Design, 1994, 116(4):1115-1121.
[18] Mischke C R. Prediction of Stochastic Endurance Strength. Journal of Vibration & Acoustics, 1987, 109(1):113. [19] J. Shigley, C. Mischke, T. Brown. Standard Handbook of Machine Design, McGraw-Hill Standard Handbooks (McGraw-Hill Education, 2004).
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[20] Pilkey D. Peterson’s stress concentration factors. New Jersey: Wiley; 2008. [21] Chen G. Generalized Equations for Estimating Stress Concentration Factors of Various Notch Flexure Hinges. Journal of Mechanical Design, 2014, 136(3):252-261.
[22] Zhao T, Zhang J, Jiang Y. A study of fatigue crack growth of 7075-T651 aluminum alloy. International Journal of
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Fatigue, 2008, 30(7):1169-1180.
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ACCEPTED MANUSCRIPT Highlights: This paper proposes two methods of prolonging the fatigue life of the cross-spring pivots (CSP) without changing another important performance, rotational stiffness. In order to establish the stress-life model, the equivalent fatigue stresses of the CSPs are derived in
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which the stress concentration factors are obtained by fitting the finite element results.
Reference to the crank-guide mechanism consisting of few components, a fatigue test bench applying to the CSP is implemented, in which there is no additional force on the fatigue life of pivot as
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in the crank-rocker mechanism.