Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method

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Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method Jianing Wu a, Shaoze Yan a,∗, Junlan Li a,b, Yongxia Gu c a

State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, PR China Key Laboratory of Mechanism and Equipment Design of Ministry of Education, School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China c School of Material and Mechanical Engineering, Beijing Technology and Business University, Beijing 1000487, PR China b

a r t i c l e

i n f o

Article history: Received 18 October 2015 Revised 6 July 2016 Accepted 21 July 2016 Available online xxx Keywords: Mechanism reliability evaluation Bistable compliant mechanism Degradation Uncertainties

a b s t r a c t The bistable compliant mechanism has been widely used in the field of microelectromechanical systems, industrial lines, and daily necessities. Enhancing mechanism reliability is helpful to prolong product life and to reduce the maintenance costs of bistable compliant mechanisms. This paper proposes one method to evaluate the mechanism reliability of the bistable compliant mechanism considering degradation and uncertainties of the parameters. The new method can be subdivided as follows: first, the pseudo-rigid body model is used to calculate the motion precision of the mechanism. Second, by introducing the degradation of stiffness and the randomness of the bending line into the kinematic equations, the other equilibrium position can be calculated with respect to the service duration. Third, by using the health index to judge the status of the mechanism, a large sample of failure time is generated and processed by the Kaplan–Meier estimator to characterize the mechanism reliability of the bistable compliant mechanism. A case studied in this paper reveals the mechanism reliability of a typical bistable compliant mechanism under different health indices and provides clues for the reliability evaluation and performance improvement of bistable compliant mechanisms. © 2016 Published by Elsevier Inc.

1. Introduction In the research field of mechanical engineering, compliant mechanisms are flexible mechanisms that transfer an input force or (angular) displacement to another point through elastic body deformation [1–6]. Notably, these mechanisms are usually monolithic (single-piece) or jointless structures with certain advantages over the rigid-body or jointed mechanisms, such as reduced manufacturing and assembly time and cost [6–8]. Considering the lack of joints, the universal “friction” embedded in the conventional joints between two parts of a rigid body is absent, thereby making the compliant mechanism expanding to more areas. Currently, compliant mechanisms are widely employed in micro-electromechanical

∗

Corresponding author. E-mail addresses: [email protected] (J. Wu), [email protected] (S. Yan), [email protected] (J. Li), [email protected] (Y. Gu).

http://dx.doi.org/10.1016/j.apm.2016.07.006 0307-904X/© 2016 Published by Elsevier Inc.

Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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systems, including the micro-accelerometers and electro-thermal micro-actuators [7]. The concept and synthesis method of the bistable compliant mechanism was first proposed by Howell [3]. The mechanism has two equilibrium positions in the potential mapping with respect to the angle. As a main branch of the compliant mechanism, the bistable compliant mechanism is useful in switches, gates, and other applications as it provides two stable configurations for actual use. A number of papers have discussed the design and analysis method of the bistable compliant mechanism, and some of them are involved in reliability-based issues [4]. Thus, the bistable compliant mechanism, regarded as the general bistable mechanism, is selected as the main objective in this paper. Typically, two approaches known in the literature for the systematic syntheses of compliant mechanisms are the kinematics based approach and the structural optimization based approach [5–8]. Generally, the pseudo-rigid model is most widely used in the kinematic-based approach [5]. Now the modeling method has evolved into the more accurate ones, such as the finite element modeling, the corrected finite element modeling and other modeling methods [9,10]. Given the increasing requirement of tasks and the complexity of missions, more emphases are placed on research related to the stability and maintainability of bistable compliant mechanisms [4]. Scholars first focused on the problem of fatigue and devoted themselves to seeking effective ways to prolong the longevity of the compliant mechanism by controlling the process of fatigue. Howell et al. [4] considered fatigue as the major concern in many compliant mechanisms because of the cyclic stresses induced on the flexible members, and they proposed one method for the probabilistic design of the bistable compliant mechanism. Xu examined the design of compliant hinges for compliant joints and concluded that elliptical profiles have the advantage of achieving a long fatigue life, but the corner-filleted design offers the highest flexibility [2]. Research related to reliability evaluation and health management has also been conducted [5–8]. Although bistable mechanisms are eliciting increasing attention from engineers, failures have occurred during application, and thus designers have been impelled to extensively investigate the failure mechanism, reliability evaluation, and maintenance methods of bistable compliant mechanisms [11–12]. With the improvement of material properties and the enhancement of manufacturing technology, fracture caused by fatigue rarely occurs within the anticipated life of the bistable compliant mechanism. The bistable compliant mechanism has two stable working states, and the main concern of its performance properties is its ability to accurately form the two operating configurations. On the one hand, to meet the requirement of more accurate output, the ability of exporting precision kinematic output is the most important performance indicator of the bistable compliant mechanism. On the other hand, the motion accuracy decreases when cracks grow on the compliant hinges. When the bistable compliant mechanism cannot meet the requirement of motion accuracy, the structure may become intact without the failure of fracture in the hinges. Therefore, at present, a general direction of engineers’ main concern is how to improve the reliability of motion accuracy caused by stiffness degradation rather than the problems associated with fatigue fracture. This concern can be referred to as the theme of the so-called “mechanism reliability”. Mechanism reliability is the ability of a certain mechanism to maintain the output accuracy under consistent conditions. Mechanism reliability is a comprehensive indicator that reflects the operating characteristics of one mechanism used in the machinery industry [13]. As far as we know, the concept of mechanism reliability was first defined by Feng [13]. He indicated that two parts of research should be focused on: the reliability of structures and the reliability of output properties. The reliability of output properties is concerned with mechanism reliability, which is the main topic of this paper. Although certain papers have introduced the methodologies of traditional mechanism reliability evaluation [13–15], the methodologies of evaluating the mechanism reliability of bistable compliant mechanisms have been scarcely reported. As regards mechanical engineering facing practical problems, the characteristics of uncertainties and degradation have drawn much attention [14–16]. Many scholars have focused on system performance to evaluate reliability rather than the traditional binary states, namely, normal and abnormal [17–22]. Based on the abovementioned literature, certain obstacles must be overcome in the mechanism reliability evaluation of the bistable compliant mechanism. First, a dynamic modeling that considers parametric uncertainties and degradation must be conducted to obtain the output motion property of the mechanism. However, the mathematical modeling and solution method have yet to be reported. To examine the methods of improving the kinematic properties of the mechanism, Luo and Sun [23] proposed the direct probability method (DPM) to evaluate motion reliability. Given the law of motion, this method can provide the kinematic properties of the displacement, velocity, and acceleration (as well as the angular displacement, angular velocities, and angular accelerations) of a particular component. In addition, DPM can be used to calculate the probability that kinematic parameters lie in a specified interval, and it has been used in the probabilistic design of the bistable compliant mechanism [8]. Particularly, the widely used DPM can only be applied when the output properties have the explicit analytical form. Second, the classic method for reliability evaluation needs to obtain a certain amount of failure samples. Targeted at the mechanism reliability evaluation of the bistable compliant mechanism, the approach on how to obtain these samples has not been clearly identified in previous studies. Third, how to explore the failure data to obtain the reliability properties is not presented. Consequently, the main objective of this paper is to evaluate the mechanism reliability of the bistable compliant mechanism using a large sample of simulation data considering the uncertainties and degradation of the parameters. This paper is structured as follows. Section 2 introduces the operational principle of the bistable compliant mechanism, and the basic equations of geometry, force, and potential energy are examined. The concrete method of evaluating the mechanism reliability of the bistable compliant mechanism is proposed in Section 3. Section 4 presents a case study of evaluating the mechanism reliability of a typical bistable compliant mechanism. The last section concludes the paper. Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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3

Fig. 1. The Young bistable compliant mechanism and the joints.

2. Formulations of the bistable compliant mechanism A typical compliant mechanism is shown in Fig. 1(a). The most obvious feature of the compliant mechanism is that it uses compliant joints. In a classical rotary joint, the relative rotation takes place between a shaft and its housing, mating parts that are concentrically located. The rotation can be limited to a specific angular sector as indicated in Fig. 1(b). A compliant joint can provide a similar rotary output. The “centers” of the two adjacent members undergoing the relative rotation are no longer collocated as shown in the lower right corner of Fig. 1(b) [3]. Many types of bistable compliant mechanisms exist, including the bistable compliant cam mechanism, bistable compliant link mechanism, and bistable Geneva mechanism. In this paper, the Young mechanism, regarded as a typical form of a lever bistable compliant mechanism, is selected to investigate the characteristics of the mechanism reliability because the Young mechanism has some significant advantages. First, the Young mechanism has two stable states that can satisfy the requirement of bistability for actual use in the situation in which the mechanism needs to reconfigure its profile in only two states. Second, generated from the traditional four-bar link mechanism, the Young mechanism is easy to be designed and manufactured, and it provides possibilities of wider interests in engineering application. Third, the modeling of the Young mechanism is based on the pseudo-rigid model, which can be easily represented in mathematical formulae. Based on the considerable merits listed above, the Young mechanism has become the most widely used bistable compliant mechanism in some engineering applications, especially in the field of micro-electromechanical systems [24–26]. Howell [3] defined the Young mechanism as follows: (1) has two revolute joints and therefore two links (a link is defined as the continuum between two rigid-body joints); (2) has two compliant segments that are part of the same link; and (3) has a pseudo-rigid body model that resembles a four-bar mechanism. As shown in Fig. 1(a), if the mechanism is the Young bistable compliant mechanism, it must satisfy the following: (1) joints Q1 and Q4 should apply the revolute joints, and (2) joints Q2 and Q3 should apply the compliant joints. The lengths of the rods Q1 Q2 , Q2 Q3 , Q3 Q4 , and Q1 Q4 are defined as r1 , r2 , r3 , and r4 respectively. The pseudo-rigid body model of QM contains two torsion springs in points Q2 and Q3 to simulate the compliant joints. Moreover, the equivalent spring stiffnesses are k2 and k3 , respectively (Fig. 1(a)). To obtain the bistable properties of this mechanism, the analysis is subdivided into two parts: geometry-based analysis and elastic potential energy-based analysis [3]. 2.1. Geometry-based analysis The model shown in Fig. 1 indicates that by using the coordinates of points Q1 (0, 0), Q2 (r2 cos θ2 , r2 sin θ2 ) and Q4 (r1 , 0 ), the coordinate of Q3 (xQ3 , yQ3 ) can be calculated by the geometric constraint.



(xQ3 − r2 cos θ2 )2 + (xQ3 − r2 sin θ2 )2 = r32 . (xQ3 − r1 )2 + y2Q3 = r42

(1)

As shown in Fig. 1(a), the angles of θ3 and θ4 yield:

θ3 =

 π /2, xQ3 − r2 sin θ2 = 0 arctan

 θ4 =

yQ3 −r2 cos θ2 , xQ3 xQ3 −r2 sin θ2

− r2 sin θ2 = 0

π /2, xQ3 − r1 = 0 arctan

−yQ3 , xQ3 r1 −xQ3

− r1 = 0

.

,

(2)

(3)

Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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Fig. 2. Two equilibrium positions of the bistable compliant mechanism.

2.2. Elastic potential energy-based analysis The mechanism has two stable positions which can hold a presupposed configuration to complete tasks in actual engineering applications. As depicted in Fig. 1(b), the stiffness of the compliant joint can be expressed by Howell [3]:

K = EI/w,

(4)

in which E is Young’s modulus, I is the area moment of inertia of the flexible segment, and w is the width of the compliant joint. And we have:

I = ht 3 /64,

(5)

in which h and t are the length and thickness of the joint. In an arbitrary position of the bistable compliant mechanism, the angles of the mechanism denote to θ2 , θ3 , θ4 (Fig. 1). The elastic potential energy is described by the generalized coordinates [3]:

V =

1 (k2 ψ22 + k3 ψ32 ), 2

(6)

where,

ψ2 = θ2 − θ20 − (θ3 − θ30 ), ψ3 = θ4 − θ40 − (θ3 − θ30 ),

(7)

where the “0” subscript denotes the initial value of each angle in the first position. As illustrated in Fig. 2, when the mechanism provides a bistable state, the first deviation of elastic potential energy is zero and the second deviation of elastic potential energy is positive. The bistable state is expressed as [3]:

 ⎧ dV  ⎪ =0 ⎪ ⎨ d θ2  θ2 =P1 ,P3 . ⎪ d 2V  ⎪ >0 ⎩ 2  d θ2 θ =  ,  2 P1 P3

(8)

By solving Eq. (8), we can get the two equilibrium positions of the bistable compliant mechanism and the positions

P1 (the first equilibrium position) and P3 (the second equilibrium position), so these two positions correspond to the respective positions where the mechanism can hold a stable structure [3,27]. Let ddV θ = 0, the parameters satisfy: 2





 θ4 − θ40 − θ3 + θ30 r2 [r4 sin(θ4 − θ2 )−r3 sin(θ3 − θ2 )]  k2 . =  · k3 θ2 − θ20 − θ3 + θ30 r4 [r3 sin(θ3 − θ4 )−r2 sin(θ4 − θ2 )] 

(9)

Notably, Eq. (9) is suitable for calculating the arbitrary stable position because it is a single-degree-of-freedom system, and the geometric constraint leads to the uniqueness of the geometric configuration. 3. Method of evaluating mechanism reliability The Young mechanism has certain advantages. However, given its special geometrical structure and physical property, this type of mechanism introduces uncertainties into practical application that may affect the kinematic and dynamic performances and even cause massive loss in precision. Two aspects of uncertainties or degradation must be emphasized: degradation of the stiffness in the compliant joints and random position of the bending hinge line. The new approach for mechanism reliability evaluation of the Young mechanism is illustrated in Fig. 3, in which the critical parts are highlighted in red font as “Degradation” and “Uncertainties.” First, the degradation law of the stiffness is Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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Fig. 3. Flow chart of the new method.

Fig. 4. The cross-ply laminate model and degradation of Young modulus.

investigated by the material properties and geometrical features. Second, the method of representing the random position of the bending hinge is proposed. Then, we obtain the dynamic output by combining the uncertainties of the parameters with the motion equations. A large sample of output parameters can be computed by the above steps. The distribution function of the mechanism reliability is not preliminarily known in this case, and thus we cannot use the parametric method for reliability evaluation. A feasible way of solving such a problem is to consider the nonparametric methods. Two methods for the nonparametric reliability evaluation can be used: the empirical reliability function and the Kaplan–Meier estimator [28]. By comparing these two methods, the Kaplan–Meier estimator exhibits jumps that vary with the number of censored units between two consecutive failures rather than the jumps in the empirical reliability function (equal to 1/n). This variation improves the accuracy for reliability evaluation [27]. Thus, we select the Kaplan–Meier estimator to evaluate the mechanism reliability. Through the Kaplan–Meier estimator, we can obtain the mechanism reliability with respect to service cycles, which provide a useful reference for reliability prediction and maintenance management. The following section introduces the method in detail. 3.1. Modeling of the stiffness degradation In this paper, the transverse cross-ply laminate is applied in the compliant joints. The stiffness degradation continues in the process of application. The law of stiffness reduction is defined by the modified shear lag analysis considering the concept of the interlaminar shear layer. Fig. 4(a) shows the cross-ply laminate and its analytical model [27]. The time-dependent remaining stiffness of the material, i.e., Ex , is expressed as:

Ex = Ex0

ε0 +

d abQ11

(2 ) 1

ε0

σx0 L [β1 (eαL − 1 ) − β2 (e−αL − 1 )]

,

(10)

where Ex0 is the initial Young modulus of the material, ε0 is the comprehensive strain of the plate element, and σx(02 ) is the initial thermal stress components of the undamaged cross-ply laminate in the x direction. Notably, b is the thickness of Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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Fig. 5. Compliant hinge model of the bistable compliant mechanism. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

two outside layers, 2d is the thickness of the central 90° layer, L is the dimensional parameter, and α is the intermediate variable [27]. By applying the data provided in [27], we can obtain the degradation of the stiffness in the bistable compliant mechanism, which is shown in Fig. 4(b). For instance, at the c1 cycle, the Young modulus decreases to Ex = 0.5Ex0 , and then it decreases to 0.3Ex0 when the compliant joint serves the c2 cycle. 3.2. Modeling of the randomness of the bending hinge The bending hinge is not deterministic, and it introduces uncertainties into the system. To analyze the uncertainties of the lengths, we establish the following coordinates in the beginning. As shown in Fig. 5(a), the origin of the coordinate is arranged in the middle of the flexure part as Point O [1,3]. The work space of the hinge is limited to the planes parallel to xOy. The coordinate of point Pi on the x axis is defined by:

Pi ((1/2 − ε ) · l, 0 ), ε ∼ N (0, 1 ),

(11)

where the scale factor ε follows the Gaussian distribution, illustrated in Fig. 5(a) as the red curve [28–32]. For a typical bistable compliant mechanism of the four-bar link mechanism, the real lengths of the rods, identified as ri∗ (i = 1, 2, 3, 4 ), can be expressed as (Fig. 5(b)):

⎧∗ r1 = r1 ⎪ ⎪ ⎨∗

r2 = r2 + ε2 l , r3∗ = (1 − ε2 )l + r3 + ε3 l ⎪ ⎪ ⎩∗ r4 = ( 1 − ε3 )l + r4

(12)

where ri (i = 1, 2, 3, 4 ) are the ideal lengths of the rods with ideal joints with no clearances. The second equilibrium position indicates the degree of mechanism reliability especially in the aspect of motion precision. If the second position corresponding to the second angle of equilibrium lies in the interval that is ±δ (0 < δ < 1 ) around the ideal value of angle, the motion accuracy of the mechanism can be accepted for further use. Otherwise, the bistable compliant mechanism is identified as that in failure. Fig. 6 presents the concept of the “safe zone” for the motion accuracy of the bistable compliant mechanism. The health index, which describes the state of the bistable compliant mechanism, yields the following expression:



H=

1, θ2t ∈ ((1 − δ )θ2X , (1 + δ )θ2X ) 0, θ2t ∈ / ((1 − δ )θ2X , (1 + δ )θ2X )

.

(13)

Eq. (13) shows that the status of the bistable compliant mechanism is regarded as a normal state if H is equal to 1. If the status of the bistable compliant mechanism is abnormal, H equals 0. 3.3. Extraction of the failure data The output of the mechanism, namely, the second stable position and the failure time can be calculated by the following steps (Fig. 7): (1) Initialization: set the sample size to N (N ≥ 1) and the variable that records the number of samples to n(1 ≤ n ≤ N ). Initialize the stiffness and the lengths of the bistable compliant mechanism as ki (i = 2, 3 ) and r j ( j = 1, 2, 3, 4 ), respectively. The first stable position is set to θm0 (m = 2, 3, 4 ). Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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Fig. 6. Potential energy of the bistable compliant mechanism in the course of serving.

Fig. 7. The method of calculating the output.

(2) Set the initial time t to 0. Through the geometry and the potential energy equations, the ideal output angle θmX (m = 2, 3, 4 ) can be calculated. (3) Considering the degradation of stiffness and the randomness of the geometrical parameters, we use the potential energy equation to calculate the output angle θmi (m = 2, 3, 4 ) as well as the information of the second stable position. (4) Calculate the health index H by referring to the condition of whether or not θ2t lies on the interval ((1 − δ )θ2X , (1 + δ )θ2X ). If H = 1, let t = t + t and jump to step (3). Otherwise, we record the failure time as ti (i = 1, 2 . . . N ), and let n = n + 1. Then, we proceed to step (3). (5) Determine whether or not n is equal to N. If it is equal to N, the program attains termination. Otherwise, proceed to step (2). (6) After the calculation, a large sample of failure time data can be obtained as the vector of failure data T = {ti }i=1,2...N . Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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Fig. 8. Mechanism reliability by the Kaplan–Meier estimator. Table 1 The parameters of the ideal bistable compliant mechanism. Parameter

r1

r2

r3

r4

θ20

θ30

θ40

k2

k3

Value Unit

100 mm

250 mm

225 mm

250 mm

7 deg

−87.69 deg

−51.01 deg

35.88 N mm/degree

1.00 N mm/degree

3.4. Reliability evaluation The samples of failure data can be organized in ascending order:

t (1 ) ≤ t (2 ) ≤ · · · t (w ) .

(14)

Using the Kaplan–Meier estimator, the probability of failure can be defined as [28]:

P (TF > t(i ) ) = P (TF > δt ) · P (TF > t(1) +δt |TF > t(1) )

·P (TF > t(2) +δt |TF > t(2) ) . . . P (TF > t(i ) +δt |TF > t(i ) ),

(15)

where δt is an arbitrary small time interval in which no censoring or failure occurs. Thus the Kaplan–Meier estimator of the reliability function with censored data yields:

R(t ) =



all i such that t(i ) ≤ t

pˆ i =



all i such that t(i ) ≤ t

ni − 1 , ni

(16)

where ni is the number of operational units right before t( i ) . The basic principle of computing the mechanism reliability for the bistable compliant mechanism is graphically illustrated in Fig. 8. 4. Case study A case study is proposed in this section. The parameters, including the rod length and the angles, are initialized. According to the method shown in Section 3, the bistable property of the bistable compliant mechanism is examined by introducing the degradation of the stiffness and the uncertainties of the rod length. The reliability with respect to cycle time can be obtained using the Kaplan–Meier estimator. 4.1. Parameter initialization The structure of the bistable compliant mechanism and parameters of the ideal bistable compliant mechanism are shown in Fig. 9 and Table 1, respectively. The actual use requirements indicate that the angles of θ2 and θ20 are organized as 83° and 7°, respectively. The angle between the two equilibrium positions is 76°. The other angles, including θ3 , θ30 , θ4 and θ40 , can be calculated by Eqs. (1)–(3) following the geometric constraints. Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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9

Fig. 9. Structure of the bistable compliant mechanism for case study. Table 2 Degradation of the stiffness with respect to service cycles. Service cycle (×106)

E x /Ex 0

Service cycle

Ex /Ex 0

Service cycle

Ex /Ex 0

0 0.0500 0.10 0 0 0.1500 0.20 0 0 0.2500 0.30 0 0

0.9999 0.9929 0.9904 0.9859 0.9796 0.9715 0.9679

0.3500 0.40 0 0 0.4500 0.50 0 0 0.5500 0.60 0 0 0.6500

0.9618 0.9579 0.9521 0.9490 0.9471 0.9424 0.9351

0.70 0 0 0.7500 0.80 0 0 0.8500 0.90 0 0 0.9500 0.9900

0.9301 0.9258 0.9228 0.9195 0.9137 0.9086 0.9029

Fig. 10. Elastic energy in different cycles.

Polypropylene is applied to the compliant joints. The degradation of the Young modulus is shown in Table 2 [29]. In consideration of the loss of Young modulus during manufacturing, the initial stiffness is set to 0.9999. By Eq. (10), the Young modulus decreases with the passage of time, and it reaches 0.9029 when the compliant joint is used for 0.99 × 106 service cycles. Referring to Eq. (10), the length of bars with randomness can be calculated by the method proposed in Section 3 through simulation. 4.2. Simulation The simulation time is set to the cycles of service as 106 service cycles, and the sample size is set to 100. Four samples of elastic energy in different cycles is shown in Fig. 10. Fig. 10(a) illustrates the peaks of elastic energy when it is used for 0 × 106 , 0.25 × 106 , 0.5 × 106 , and 1 × 106 service cycles, respectively. As shown in Table 3, the peaks of energy decrease Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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J. Wu et al. / Applied Mathematical Modelling 000 (2016) 1–12 Table 3 Energy peaks and second stable positions. Service cycle: T (× 106 )

0

0.25

0.5

1.0

Average

Marker: T EPT (N × mm × deg ree ) d1 T (%) θ2T (deg) d2 T (%)

1 2063.91 0.00 83.00 0.00

2 1760.82 14.69 87.00 4.82

3 1750.33 5.96 91.00 4.60

4 1680.24 4.00 94.50 3.85

– 1813.83 5.16 88.88 3.32

Fig. 11. Mechanism reliability of the bistable compliant mechanism.

because of the Young modulus loss. The relative decreasing rate of energy peak (d1 T (%)) and the relative decreasing rate of the second stable position θ2 (d2 T (%)) are defined as:

⎧ d1T = [E pT − E p(T −1) ]/E p(T −1) , T = 1, 2, 3 ⎪ ⎪ ⎨d = [θ − θ 2T 2T 2(T −1 ) ]/θ2(T −1 ) . d = 0 ⎪ 10 ⎪ ⎩

(17)

d20 = 0

The maximum value of the elastic energy decreases at 14.69% in the time interval [0 × 106 , 0.25 × 106 ], faster than that of any other interval of service cycles at 5.69% and 4.00%, respectively. By contrast, the change in the second stable position is comparatively minimal with the average value of 3.32%. Clearly, the elastic energy degrades more than θ2X , which represents the ability of holding a stable configuration of the bistable compliant mechanism. As shown in the flow chart in Fig. 7, a total of 100 samples of failure time are obtained. The mechanism reliability is calculated by ranking the failure time using the Kaplan–Meier estimator. In this case, the index δ is selected as 10%, 5%, and 2.5%. The Kaplan–Meier estimator exhibits jumps that vary with the number of censored units between two consecutive failures [28]. Fig. 11 indicates that the mechanism reliability decreases at different paces when different health indices are selected. Taking the strictest criterion of the three indices, i.e., δ = 2.5%, as an example, the reliability decreases to 0.9301 when it is applied to the service cycle of 1.0 × 106 . However, it only degrades to 0.9903 when δ is defined as 10%. Therefore, the characteristics of the failure may not be completely identical to each other when different indices δ are selected. 4.3. Results and discussion The results in Fig. 11 and Table 4 indicate that the mechanism reliability of the bistable compliant mechanism decreases with respect to the service cycle. By applying the index at 10%, 5%, and 2.5%, the mechanism reliability reaches 0.9903, 0.9502, and 0.9301 when the bistable compliant mechanism is used for 1.0 × 106 cycles. For engineering use, two routes are practically useful as shown in Fig. 11 and Table 4. The first route is used to determine the mechanism reliability when it is adopted for certain service cycles, i.e., for reliability evaluation. For instance, the mechanism reliability decreases to 0.96 when it is used for 0.8 × 106 service cycles with the index δ = 5%. The function of the second route is to develop strategies for maintenance. We take the curve of δ = 2.5% as an example. If the engineer would like to effectively and safely Please cite this article as: J. Wu et al., Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.07.006

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11

Table 4 Mechanism reliability against service cycle. Service cycle: T ( × 106 )

0

5

δ = 10% δ = 5% δ = 2.5%

1 1 1

0.99954 0.99701 0.99631

0.99881 0.99578 0.99323

0.99851 0.9934 0.99068

0.99803 0.99189 0.98802

0.99745 0.9908 0.98664

0.99714 0.988 0.98143

10

15

20

25

30

Service cycle: T ( × 106 ) δ = 10% δ = 5% δ = 2.5%

35 0.99648 0.98518 0.97811

40 0.99561 0.98234 0.97366

45 0.9951 0.97991 0.97076

50 0.99453 0.97787 0.96973

55 0.99415 0.97554 0.9637

60 0.99345 0.97357 0.96072

65 0.99301 0.97101 0.9568

Service cycle: T ( × 106 ) δ = 10% δ = 5% δ = 2.5%

70 0.99246 0.96754 0.95383

75 0.99221 0.96277 0.95113

80 0.99185 0.96099 0.94398

85 0.99109 0.95912 0.94229

90 0.99071 0.95654 0.93827

95 0.99042 0.95353 0.93189

100 0.99029 0.95019 0.93006

use the bistable compliant mechanism with the reliability at above 0.95, it could only be used for 0.8 × 106 service cycles. Therefore, the related components should be updated when the duration reaches the service cycles of 0.8 × 106 . 5. Conclusion This paper proposes a method to evaluate the mechanism reliability of the bistable compliant mechanism considering the degradation of stiffness in compliant hinges and the uncertainties of the positions of bending hinges. By combining the input parameters with the output properties using dynamic equations, we can obtain a large sample size of the failure data of the bistable compliant mechanism under the preset health index. The Kaplan–Meier estimator, which is considered a tool for the nonparametric reliability evaluation, is employed to calculate the mechanism reliability. The results indicate that the mechanism reliability degrades with respect to the service cycle. For the bistable compliant mechanism case given in this paper, mechanism reliability decreases to 0.9301 when it is used for 1.0 × 106 cycles by adopting ± 2.5% as the range of the safety zone. In the future, the bistable compliant mechanism may be widely applied in the field of aerospace engineering systems [30–32]. Consequently, to improve the applicability of the bistable compliant mechanism, future works should focus on the harsh space environment, for instance, considering the degradation caused by the variable thermal heating and long-term load. Specifically, in-depth research should focus on the cumulated damage induced by the alternating thermal environment in space, microgravity, and atomic oxygen corrosion from the perspective of the failure mechanism to predict the mechanism reliability of the bistable compliant mechanism. Even better, engineers should determine a series of effective strategies for reliability improvement and maintenance of the bistable compliant mechanism based on reliability evaluation. Acknowledgments This work was supported by the National Science Foundation of China under Contract No. 51475258, Beijing Natural Science Foundation under Contract No. 3132030, and the foundation of the Key Laboratory of Mechanism Theory and Equipment Design of the Ministry of Education. National Natural Science Foundation of China under Contact No. 51305294. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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