Advanced reliability analysis method for mechanisms based on uncertain measure

Abstract

In traditional mechanism reliability analysis, probability theory or statistical approaches are employed. However, these methods cannot be used under lack of data and great epistemic uncertainty. In this paper, an advanced mechanism reliability analysis method is put forward based on uncertain measure. To satisfy the subadditivity of epistemic uncertainties, a novel uncertainty quantification method based on uncertainty theory is proposed for mechanism reliability analysis. Then, a point kinematic reliability analysis method combined with uncertain measure is presented to calculate the kinematic uncertainty reliability of motion mechanism at each time instant. Three models are developed for estimating kinematic uncertainty reliability. Furthermore, first-order Taylor series expansion is used to solve nonlinear limit state functions. A new kinematic uncertainty reliability index (KURI) is presented based on normal uncertainty distribution. Finally, by applying the proposed method to a numerical experiment, the trend of uncertainty reliability was found to be consistent with the traditional method. The two practical engineering applications show that the presented method are more reasonable compared with the classical approaches when the information of design parameters is insufficient.

Keywords

Uncertainty quantification mechanism reliability reliability index uncertainty theory belief reliability

1 Introduction

Modeling and analysis of mechanism reliability play an important role in practical engineering problems. This is an intrinsically interdisciplinary field widely applied in many fields such as aviation industry, machine manufacturing, and astronautic engineering [1]. However, a mechanism reliability problem can be regarded as “uncertain” when lacking the information about the theoretical analysis model due to complex environmental factors, incomplete information, and inevitable measurement errors [2]. In practical reliability engineering, uncertainty quantification is a fundamental problem among all the reliability-related engineering activities. Quantifying the reliability of a mechanism or system using quantitative metrics has drawn much attention [3 –5].

In general, two types of uncertainty are distinguished: aleatory uncertainty originating from the inherent randomness of physical behavior and epistemic uncertainty caused by factors such as operational errors, experts’ judgment (subjective interpretation), and inevitable man-made mistake [5, 6]. Probability theory and statistical methods play a crucial role in traditional mechanism reliability analysis. This has been used to describe the aleatory uncertainty and estimate the kinematic reliability of the mechanism [1, 7]. Numerous probability theory based approaches have been developed, e.g., Monte Carlo simulation (MCS) method [8], first-order second moment method [1], Bayesian networks method [9], response surface method [10], and envelope function method [11, 12]. Although these methods are effective when the amount of statistical data is large, they cannot be used when only a few data or even no data are available. Therefore, the epistemic uncertainties cannot be well explained by random variables and probability theory [5, 13]. Without accounting for the effect of epistemic uncertainty, the probability-based reliability metric might yield infeasible solution with large differences and paradoxical results for a mechanism [4].

To solve this problem, numerous non-probabilistic approaches have been developed to measure epistemic uncertainty, such as interval analysis [14, 15], fuzzy interval analysis [15], evidence theory [6], fuzzy theory [16], possibility theory [17], and uncertainty theory [18]. According to the mathematical principle of six reliability analysis methods mentioned above, the first three methods are probability interval based metrics, whereas the latter three methods are based on subadditivity monotone measure [5]. Because the probability interval based methods might cause interval extension problem, the epistemic uncertainty may be exaggerated and thus leads to an over-conservative result [4, 5]. For completeness, the subadditivity monotone measures are utilized to compensate the over-conservative estimate results [5]. However, when subjective randomness appears in a mechanism, the fuzzy measure and possibility measure fail to satisfy the duality property, which can cause confusing results in engineering applications [5, 19]. From the above discussion, it can be concluded that a new metric is needed to satisfy the duality and subadditivity simultaneously in actual mechanism reliability problems.

To solve this problem and the limitations of the existing measures, a new mathematical theory known as uncertainty theory is introduced to describe the uncertainties of mechanism in this study; this theory was founded by Liu [20] in 2007 and refined by Liu [21] in 2010. Uncertainty theory is based on the uncertain measure to present the belief degrees of events determined by uncertainty factors [18 , 22]. By using the uncertain measure, uncertainty theory is regarded as a more reasonable mathematical system to explain epistemic uncertainty [22]. This is a subadditivity monotone measure based on the normality, duality, subadditivity, and product axioms. Because of the axioms of uncertainty theory, this metric can avoid interval extension problems and satisfy the duality property [4]. This theory provides a concrete mathematical description under various uncertain variables in the uncertainty space [21]. Liu [23] first applied this theory in system reliability and proposed the concept of uncertain reliability and uncertain risk. A belief reliability index was proposed by Zeng et al. [13, 18] as a new reliability metrics based on uncertainty theory to characterize the reliability of systems. Subsequently, Gao et al. [24] used the uncertain variable for the reliability analysis of k-out-of-n system and derived a formula to estimate the reliability index.

Bai [25] used the uncertainty theory to describe the reliability of series and parallel structure systems; a structural reliability metric was developed as the uncertain measure of the event. Wang [26] proposed the Cornell uncertainty reliability index for structural reliability analysis based on the uncertainty theory. However, this reliability index suffers from the following. First, it does not perform a theoretical derivation, and the Cornell uncertainty reliability index directly obtained by assuming the uncertain expected value divided by the uncertain standard deviation. Second, there is no functional relationship between Cornell uncertainty reliability index and reliability, so it cannot be used to calculate uncertainty reliability. Third, they only discuss the structural reliability, but in real mechanism reliability problem, it is usually necessary to calculate the kinematic reliability combined with the kinematic limit state function. Therefore, the aforementioned approaches may not be suitable for mechanism reliability analysis. To avoid the above disadvantages, in this study, the epistemic uncertainties are uniformly represented by uncertain variables. Then, a new quantification model and kinematic uncertainty reliability index (KURI) are presented based on the uncertainty theory and point kinematic reliability analysis method. For generalization, three models for kinematic uncertainty reliability estimation are proposed in this study.

The remainder of this paper is structured as follows: Section 2 briefly describes some useful mathematical concepts of uncertainty theory such as uncertain measure, uncertainty distribution, and uncertain expected value. Section 3 describes the derivation of some formulas based on the uncertain measure and uncertain variable to quantify the kinematic uncertainty reliability with the limit state function of mechanism. In Section 4, a new KURI is proposed based on normal uncertain variables. In Section 5, a numerical example is studied in detail to show the specific implementation of the presented method. Then, two practical engineering problems are provided to verify the accuracy and efficiency of the proposed method. Finally, some conclusions are drawn in Section 6.

2 Preliminaries of uncertainty theory

As a new branch of axiomatic mathematics, uncertainty theory has been widely applied as a new tool for modeling epistemic (especially human) uncertainties. This theory has been successfully applied in various fields such as maintenance optimization [27], decision making [28], and uncertain insurance [29]. In this section, some fundamental concepts concerning uncertainty theory are briefly introduced.

Definition 2.1 (Uncertain measure [21]) Let Γ be a nonempty set, and $L$ be a σ - algebra over Γ. Each element Λ in $L$ is regarded as an event. Then, a set function M is defined as an uncertain measure if it satisfies the following three axioms:

Axiom 1 (Normality axiom) M {Γ} = 1 for the universal set Γ.

Axiom 2 (Duality axiom) M {Λ} + M {Λ^c} =1 for any event $Λ \in L$ .

Axiom 3 (Subadditivity axiom) For each countable sequence of events Λ₁, Λ₂⋯, we have: $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {{Λ}_{i}}$ (1)

The following product axiom is used to calculate the uncertain measures of product events [30].

Axiom 4 (Product axiom) Let (Γ_k, $L_{k}$ , M_k) be uncertainty spaces, and Λ_k are arbitrarily selected events from $L_{k}$ for k = 1, 2, ⋯. The product uncertain measure M is an uncertain measure satisfying: $M {\prod_{k = 1}^{\infty} Λ_{k}} = \land_{k = 1}^{\infty} M_{k} {Λ_{k}}$ (2)

Definition 2.2 (Uncertain variable [32]) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, $L$ , M) to the set of real numbers, i.e., {ξ ∈ B} is an event for any Borel set B of real numbers.

Assuming that ξ₁, ξ₂, ⋯ , ξ_n are uncertain variables and f is a real-valued measurable function, ξ = f(ξ₁, ξ₂, ⋯ , ξ_n) is also an uncertain variable, i.e., for arbitrarily τ ∈ Γ, ξ(τ) can be expressed as follows: $ξ (τ) = f (ξ_{1} (τ), ξ_{2} (τ), \dots, ξ_{n} (τ))$ (3)

Definition 2.3 (Uncertainty distribution [32]) The uncertainty distribution Φ(x) of an uncertain variable ξ can be defined by Φ(x) = M {ξ ≤ x} for any real number x.

In general, an uncertainty distribution Φ(x) is regarded as regular if it is a continuous function and strictly increases with respect to x, with 0 < Φ(x) <1, and lim_x→-∞Φ(x) =0, lim_x→∞Φ(x) =1. Moreover, because the uncertainty distribution can describe the incomplete information of epistemic uncertainties, the human uncertainty, subjective random, and fuzzy variables can be uniformly characterized by uncertain variables and uncertainty distribution in the uncertainty space [21 , 30].

Example 2.1 An uncertain variable ξ is defined as a normal uncertain variable if it has a normal uncertainty distribution: $\begin{matrix} Φ (x) = (1 + exp (\frac{π (m - x)}{\sqrt{3} σ}))^{- 1}, & x \in R \end{matrix}$ (4)

It is denoted by $N (m, σ)$ , where m is the expected value and σ is the standard deviation. The derivative of normal uncertainty distribution can be expressed as follows: [31] $\begin{matrix} φ (x) = \frac{π}{\sqrt{3} σ} \frac{1}{2 + exp (- \frac{π}{\sqrt{3} σ} (x - m)) + exp (\frac{π}{\sqrt{3} σ} (x - m))}, & x \in R \end{matrix}$ (5)

Especially, a normal uncertainty distribution $N (m, σ)$ is defined as standard normal uncertainty distribution [32], if m = 0 and σ = 1, denoted by $N (0, 1)$ .

Definition 2.4 (Inverse uncertainty distribution [32]) Let ξ be an uncertain variable with regular uncertain distribution Φ(x). Then, the inverse function Φ^-1(α) is known as the inverse uncertainty distribution of ξ.

Definition 2.5 (Uncertain expected value [32]) Let ξ be an uncertain variable; the expected value of ξ can be defined as follows: $E (ξ) = \int_{0}^{+ \infty} M {ξ \geq x} d x - \int_{- \infty}^{0} M {ξ \leq x} d x$ (6)

Definition 2.6 (Uncertain variance [32]) Let ξ be an uncertain variable with finite expected value m, the variance of ξ can be defined as V(ξ) = E [(ξ - m) ²].

Theorem 2.1 Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with continuous uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f(ξ₁, ξ₂, ⋯ , ξ_n) is continuous, strictly increasing for ξ₁, ξ₂, ⋯ , ξ_m and strictly decreasing for ξ_m+1, ξ_m+2, ⋯ , ξ_n, then the uncertainty distribution of ξ = f(ξ₁, ξ₂, ⋯ , ξ_n) can be calculated as follows: [21]

$\begin{matrix} Ψ (x) = \sup_{f (ξ_{1}, ξ_{2}, \dots, ξ_{n}) = x} (\min_{1 \leq i \leq m} Φ_{i} (ξ_{i}) \\ \land \min_{m + 1 \leq i \leq n} (1 - Φ_{i} (ξ_{i}))) \end{matrix}$ (7)

In addition, assuming that Φ₁, Φ₂, ⋯ , Φ_n are regular uncertainty distributions, the inverse uncertainty distribution of ξ = f(ξ₁, ξ₂, ⋯ , ξ_n) can be expressed as follows: [23]

$\begin{matrix} Ψ^{- 1} (α) = f {(Φ}_{1}^{- 1} (α {), Φ}_{2}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), Φ_{m + 2}^{- 1} (1 - α), \dots, \\ Φ_{n}^{- 1} (1 - α)) \end{matrix}$ (8)

Theorem 2.2 Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. Assuming that ξ = f(ξ₁, ξ₂, ⋯ , ξ_n) is continuous, strictly increasing for ξ₁, ξ₂, ⋯ , ξ_m and strictly decreasing for ξ_m+1, ξ_m+2, ⋯ , ξ_n, the uncertain expected value of ξ = f(ξ₁, ξ₂, ⋯ , ξ_n) can be calculated as follows: [33] $\begin{matrix} E (ξ) = \int_{0}^{1} f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, \\ Φ_{m}^{- 1} (α), Φ_{m + 1}^{- 1} (1 - α), \\ \begin{matrix} \end{matrix} Φ_{m + 2}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) d α \end{matrix}$ (9)

Moreover, the uncertain variance of ξ = f(ξ₁, ξ₂, ⋯ , ξ_n) can be expressed as follows: [34] $\begin{matrix} V (ξ) = \int_{0}^{1} (f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, \\ Φ_{m}^{- 1} (α), Φ_{m + 1}^{- 1} (1 - α), \\ Φ_{m + 2}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) - m)^{2} d α \end{matrix}$ (10) where m is the expected value of ξ = f(ξ₁, ξ₂, ⋯ , ξ_n).

Theorem 2.3 Let ξ₁ and ξ₂ be independent normal uncertain variables $N (m_{1}, σ_{1})$ and $N (m_{2}, σ_{2})$ , respectively. Then, the sum ξ₁ + ξ₂ is also a normal uncertain variable $N (m_{1} + m_{2}, σ_{1} + σ_{2})$ , i.e., [32] $N (m_{1}, σ_{1}) + N (m_{2}, σ_{2}) = N (m_{1} + m_{2}, σ_{1} + σ_{2})$ (11)

3 Uncertain measure based quantification method for mechanism problems

In traditional mechanism reliability problems, probabilistic methods have been developed for the analysis of mechanism reliability under aleatory uncertainties. However, the epistemic uncertainties cannot be well explained by probability theory or statistical approaches. In this section, according to the limit state function of the mechanism under epistemic uncertainties, we redefine a new uncertain measure based quantification method and belief reliability [13]. A set of new metrics and their descriptions are shown as follows:

Let (Γ, $L$ , M) be an uncertainty space, and the mechanism system contains n-dimensional input factors x₁, x₂, ⋯ , x_n that affect the functioning of system. The factors in mechanism reliability problem usually consist of different epistemic uncertainties. The epistemic uncertainties can be described using an uncertain vector ξ =(ξ₁, ξ₂, ⋯ , ξ_n) [32].

If the main performance failure mode of a mechanism is that the stress is greater than the strength, then the performance reliability analysis is based on a stress-strength interference model. The stress here is generalized, such as load, temperature, and corrosion, etc. and the corresponding generalized strength can be fatigue strength, heat resistance, and corrosion resistance, respectively. Therefore, the kinematic limit state function of mechanism can be defined as follows: $\begin{matrix} Z (ξ, τ) = f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) \\ = R (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) - S (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) \end{matrix}$ (12) where S(ξ₁, ξ₂, ⋯ , ξ_n, τ) is the kinematic generalized stress, R(ξ₁, ξ₂, ⋯ , ξ_n, τ) is the kinematic overall strength, and τ is a motion input parameter associated with time.

Especially, if the main performance failure mode of a mechanism is regarded as an event where the motion error is greater than a specified allowable error threshold ɛ, then, the performance reliability analysis is based on motion error model.

The uncertain motion error can be defined as the difference between the actual motion and desired motion output, namely: $ϒ_{error} (ξ, τ) = ϒ (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) - ϒ_{desire} (τ)$ (13) where ϒ(ξ₁, ξ₂, ⋯ , ξ_n, τ) and ϒ_desire(τ) are actual motion and desired motion outputs at τ, respectively.

In this case, the kinematic limit state function at the τ of motion mechanism can be rewritten as follows: $Z (ξ, τ) = f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) = ɛ - | ϒ_{error} (ξ, τ) |$ (14)

Definition 3.1 The point kinematic uncertainty reliability of a motion mechanism can be defined as the belief degree of reliability event {Z( ξ , τ) > 0} at τ. $M_{reliability} (τ) = M {f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) > 0}$ (15)

Because of the duality of uncertain measure, the belief degree of the occurrence of a failure event {Z( ξ , τ) ≤ 0} at τ can be determined as follows: $\begin{matrix} M_{failure} (τ) = M {f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) \leq 0} \\ = 1 - M_{reliability} (τ) \\ = 1 - M {f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) > 0} \end{matrix}$ (16)

The uncertainty of a safety event at τ in mechanism can be quantified by M_reliability(τ) with a numerical value of [0, 1]. M_failure(τ) indicates the degree how a failure event will occur at τ. Obviously, it is believed that the failure event will occur when M_failure(τ) =1. The failure event will occur and its complementary has equal possibility when M_failure(τ) = M_reliability(τ) =0 . 5. Clearly, the higher the M_failure(τ), the more possible the occurrence of failure event at τ.

In particular, the M_reliability(τ) described here is different from the probabilistic reliability. Because when the probability distribution of mechanism parameters and data are insufficient, the probability distributions of random variables cannot be accurately obtained to calculate mechanism reliability. Therefore, a new kinematic uncertainty reliability is proposed under the uncertainty space.

Theorem 3.1 Assume that the performance reliability of a mechanism system is characterized by the kinematic limit state function Z( ξ , τ) = f(ξ₁, ξ₂, ⋯ , ξ_n, τ) with independent uncertain variables ξ₁, ξ₂, ⋯ , ξ_n, and f(ξ₁, ξ₂, ⋯ , ξ_n, τ) has an uncertainty distribution Ψ_Z(z, τ). Then, the belief degree of the occurrence of a failure event {Z( ξ , τ) ≤ 0} at τ can be derived as follows: $M_{failure} (τ) = M {Z (ξ, τ) \leq 0} = Ψ_{Z} (0, τ)$ (17)

The point kinematic uncertainty reliability Equation (15) can be rewritten as follows: $\begin{matrix} M_{reliability} (τ) = M {f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) > 0} \\ = 1 - M {f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) \leq 0} \\ = 1 - Ψ_{Z} (0, τ) \end{matrix}$ (18)

3.1 Kinematic uncertainty reliability (Model 1)

In practice, if the uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n of uncertain variables are provided, and the uncertainty distribution Ψ_Z(z, τ) is unknown, the kinematic uncertainty reliability of a system can be estimated using a new model as shown below:

Let a mechanism system has a kinematic limit state function Z( ξ , τ) = f(ξ₁, ξ₂, ⋯ , ξ_n, τ), and the limit state function contains independent uncertain variables ξ₁, ξ₂, ⋯ , ξ_n with continuous uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If the limit state function is continuous, strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n, then the kinematic uncertainty reliability at τ of the mechanism can be computed as follows: $\begin{matrix} M_{reliability} (τ) = 1 - M_{failure} (τ) \\ = 1 - \sup_{f (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) = 0} (\min_{1 \leq i \leq m} Φ_{i} (ξ_{i}) \\ \land \min_{m + 1 \leq i \leq n} (1 - Φ_{i} (ξ_{i}))) \end{matrix}$ (19)

Example 3.1 Assume that a mechanism has a kinematic limit state function Z( ξ , τ) = τξ₁ - ξ₂ with τ ≥ 0 and the limit state function contains independent uncertain variables ξ₁, ξ₂ with continuous uncertainty distributions Φ₁, Φ₂, respectively. Then, the kinematic uncertainty reliability at τ of the mechanism can be calculated as follows: $M_{reliability} (τ) = 1 - \sup_{τ ξ_{1} - ξ_{2} = 0} Φ_{1} (\frac{ξ_{2}}{τ}) \land (1 - Φ_{2} (ξ_{2}))$ (20)

Proof. Obviously, Z( ξ , τ) = τξ₁ - ξ₂ strictly increases with respect to ξ₁ and strictly decreases with respect to ξ₂. According to Theorem 2.1 and Equation (7), the uncertainty distribution Ψ_Z(z, τ) of Z( ξ , τ) can be derived as follows: $Ψ_{Z} (z, τ) = \sup_{τ ξ_{1} - ξ_{2} = z} Φ_{1} (\frac{ξ_{2} + z}{τ}) \land (1 - Φ_{2} (ξ_{2}))$ (21)

Based on Equation (21) and the duality of uncertain measure, the point kinematic uncertainty reliability can be computed as follows: $\begin{matrix} M_{reliability} (τ) = 1 - Ψ_{Z} (0, τ) \\ = 1 - \sup_{τ ξ_{1} - ξ_{2} = 0} Φ_{1} (\frac{ξ_{2}}{τ}) \land (1 - Φ_{2} (ξ_{2})) \end{matrix}$ (22)

3.2 Kinematic uncertainty reliability (Model 2)

If a mechanism system has a kinematic limit state function Z( ξ , τ) = f(ξ₁, ξ₂, ⋯ , ξ_n, τ), the limit state function contains independent uncertain variables ξ₁, ξ₂, ⋯ , ξ_n with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. Assuming that the limit state function is continuous, strictly increasing for ξ₁, ξ₂, ⋯ , ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n, the kinematic uncertainty failure M_failure(τ) = M {f(ξ₁, ξ₂, ⋯ , ξ_n, τ) ≤ 0} = α(τ) and kinematic uncertainty reliability M_reliability(τ) =1 - α(τ) at τ of the mechanism, where α(τ) is the root of the following equation: $\begin{matrix} f (Φ_{1}^{- 1} (α (τ)), Φ_{2}^{- 1} (α (τ)), \dots, Φ_{m}^{- 1} (α (τ)), \\ Φ_{m + 1}^{- 1} (1 - α (τ)), Φ_{m + 2}^{- 1} (1 - α (τ)), \dots, \\ Φ_{n}^{- 1} (1 - α (τ))) = 0 \end{matrix}$ (23)

Proof. It follows from Theorem 2.1 that Z( ξ , τ) is an uncertain variable with an inverse uncertainty distribution: $\begin{matrix} Ψ^{- 1} (α (τ)) = f (Φ_{1}^{- 1} (α (τ)), Φ_{2}^{- 1} (α (τ)), \dots, \\ Φ_{m}^{- 1} (α (τ)), \\ Φ_{m + 1}^{- 1} (1 - α (τ)), Φ_{m + 2}^{- 1} (1 - α (τ)), \dots, \\ Φ_{n}^{- 1} (1 - α (τ))) \end{matrix}$ (24) because M {f(ξ₁, ξ₂, ⋯ , ξ_n, τ) ≤0} = Ψ(0) = α(τ), α(τ) is the solution of the equation Ψ^-1(α(τ)) =0.

4 A novel uncertainty reliability index

In general, when the mechanism performance parameters are based on the subjective judgment of incomplete test data, the traditional kinematic probabilistic reliability index (KPRI) β_{probabilistic}(τ) cannot accurately measure the reliability. Furthermore, the Cornell uncertainty reliability index [26] cannot be used to calculate reliability, and it fails to analyze the motion mechanism reliability. To avoid the above shortcomings, a new uncertainty reliability index is proposed in this section based on the normal uncertainty distribution. It is an uncertain indicator that a reliability event will occur in the mechanism under epistemic uncertainties.

Assume that some epistemic uncertainty variables ξ₁, ξ₂, ⋯ , ξ_n coexist in a limit state function Z( ξ , τ). The epistemic uncertainty variables can be uniformly described by independent normal uncertain variables $N (m_{1}, σ_{1})$ , $N (m_{2}, σ_{2})$ , $N (m_{n}, σ_{n})$ with regular normal uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. According to Theorem 2.3, if the limit state function of the mechanism is a linear function, then the linear kinematic limit state function at τ is also a normal uncertain variable:

$\begin{matrix} N (m_{Z (ξ, τ)}, σ_{Z (ξ, τ)}) = N (f (m_{1}, m_{2}, \dots, m_{n}, τ), \\ f (σ_{1}, σ_{2}, \dots, σ_{n}, τ)) \end{matrix}$ (25) where m_{Z(
ξ
,τ)} is the uncertain expected value and σ_{Z(
ξ
,τ)} is the uncertain standard deviation of linear limit state function. m₁, m₂, ⋯ , m_n are the uncertain expected values. σ₁, σ₂, ⋯ , σ_n are the uncertain standard deviations of normal uncertain variables ξ₁, ξ₂, ⋯ , ξ_n, respectively.

Obviously, the uncertainty distribution function Ψ_Z(z, τ) of Z( ξ , τ) is a derivable function in the interval (- ∞ , 0]. According to the relationship between the function and its derivative, we can obtain: $Ψ_{Z} (0, τ) = \int_{- \infty}^{0} ψ_{Z} (z, τ) d z$ (26) where ψ_Z(z, τ) is the derivative of Ψ_Z(z, τ): $\begin{matrix} ψ_{Z} (z, τ) = \frac{π}{\sqrt{3} σ_{Z (ξ, τ)}} \times \\ \frac{1}{2 + \exp (- \frac{π}{\sqrt{3} σ_{Z (ξ, τ)}} (z - m_{Z (ξ, τ)})) + \exp (\frac{π}{\sqrt{3} σ_{Z (ξ, τ)}} (z - m_{Z (ξ, τ)}))} \end{matrix}$ (27)

Let Y(τ) =(Z( ξ , τ) - m_{Z(
ξ
,τ)})/ - σ_{Z(
ξ
,τ)}, then Y(τ) has a standard normal uncertainty distribution Φ_Y(y(τ)). Based on dy = dz/ - σ_{Z(
ξ
,τ)}, Equation (26) can be expressed as follows: $Ψ_{Z} (0, τ) = \int_{- \infty}^{- \frac{m_{Z (ξ, τ)}}{σ_{Z (ξ, τ)}}} φ (y (τ)) d y = Φ (- \frac{m_{Z (ξ, τ)}}{σ_{Z (ξ, τ)}})$ (28) where Φ(·) is the standard normal uncertainty distribution, and φ(y(τ)) is the derivative of Φ_Y(y(τ)): $φ (y (τ)) = \frac{π}{\sqrt{3}} \frac{1}{2 + \exp (- \frac{π y (τ)}{\sqrt{3}}) + \exp (\frac{π y (τ)}{\sqrt{3}})}$ (29)

According to Equations (17) and (28), the belief degree of the occurrence of a failure event {Z( ξ , τ) ≤0} at τ can be rewritten as follows: $M_{failure} (τ) = Φ (- \frac{m_{Z (ξ, τ)}}{σ_{Z (ξ, tau)}})$ (30)

4.1 Kinematic uncertainty reliability (Model 3)

Because of the duality of uncertain measure and Equation (30), the point kinematic uncertainty reliability can be computed as follows: $\begin{matrix} M_{reliability} (τ) = 1 - M_{failure} (τ) = 1 - Φ (- \frac{m_{Z (ξ, τ)}}{σ_{Z (ξ, τ)}}) \\ = Φ (\frac{m_{Z (ξ, τ)}}{σ_{Z (ξ, τ)}}) = Φ (\frac{E (Z (ξ, τ))}{\sqrt{V (Z (ξ, τ))}}) \end{matrix}$ (31) where m_{Z(
ξ
,τ)} = E(Z( ξ , τ)) and $σ_{Z (ξ, τ)} = \sqrt{V (Z (ξ, τ))}$ .

Therefore, given a limit state function of mechanism in the uncertainty space, a new KURI can be defined as shown in Section 4.2.

4.2 Uncertain measure based KURI

Definition 4.1 Let (Γ, $L$ , M) be an uncertainty space, and the linear kinematic limit state function f(ξ₁, ξ₂, ⋯ , ξ_n, τ) of the mechanism strictly increases with respect to ξ₁, ξ₂, ⋯ , ξ_m and strictly decreases with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n. According to Theorem 2.2 and Equation (31), the KURI β_uncertainty(τ) at τ can be expressed as follows: $\begin{array}{l} β_{u n c e r t a i n t y} (τ) = \frac{E (Z (ξ, τ))}{\sqrt{V (Z (ξ, τ))}} \\ = \frac{\int_{0}^{1} f (Φ_{1}^{- 1} (α (τ)), Φ_{2}^{- 1} (α (τ)), \dots, Φ_{m}^{- 1} (α (τ)), Φ_{m + 1}^{- 1} (1 - α (τ)), \dots, Φ_{n}^{- 1} (1 - α (τ))) d α}{\sqrt{\int_{0}^{1} (f (Φ_{1}^{- 1} (α (τ)), \dots, Φ_{m}^{- 1} (α (τ)), Φ_{m + 1}^{- 1} (1 - α (τ)), \dots, Φ_{n}^{- 1} (1 - α (τ))) - m_{Z (ξ, τ)})^{2} d α}} \end{array}$ (32) where E(Z( ξ , τ)) = m_{Z(
ξ
, τ}) is the uncertain expected value, and V(Z( ξ , τ)) is the uncertain variance at τ of limit state function Z( ξ , τ) = f(ξ₁, ξ₂, ⋯ , ξ_n, τ). τ is a motion input parameter associated with time.

When the limit state function is nonlinear in the mechanism, the KURI cannot be calculated using Equation (32). If the variance of uncertain variables is small, the nonlinear limit state function is then accurately approximated with the first-order Taylor series expression at the uncertain expected values m₁, m₂, ⋯ , m_n of input uncertain vector ξ $\begin{matrix} Z (ξ, τ) \approx Z_{L} (ξ, τ) = f_{L} (ξ_{1}, ξ_{2}, \dots, ξ_{n}, τ) \\ = f (m_{1}, m_{2}, \dots, m_{n}, τ) + \sum_{i = 1}^{n} \\ \frac{\partial f (m_{1}, m_{2}, \dots, m_{n}, τ)}{\partial ξ_{i}} (ξ_{i} - m_{i}) \end{matrix}$ (33)

Then, Equation (31) can be transformed into: $M_{reliability} (τ) = Φ (\frac{E (Z_{L} (ξ, τ))}{\sqrt{V (Z_{L} (ξ, τ))}})$ (34)

Similarly, Equation (32) can be derived as follows: $β_{u n c e r t a i n t y} (τ) = \frac{E (Z_{L} (ξ, τ))}{\sqrt{V (Z_{L} (ξ, τ))}} = \frac{\int_{0}^{1} f_{L} (Φ_{1}^{- 1} (α (τ)), Φ_{2}^{- 1} (α (τ)), \dots, Φ_{m}^{- 1} (α (τ)), Φ_{m + 1}^{- 1} (1 - α (τ)), \dots, Φ_{n}^{- 1} (1 - α (τ))) d α}{\sqrt{\int_{0}^{1} {(f_{L} (Φ_{1}^{- 1} (α (τ)), \dots, Φ_{m}^{- 1} (α (τ)), Φ_{m + 1}^{- 1} (1 - α (τ)), \dots, Φ_{n}^{- 1} (1 - α (τ))) - m_{Z_{L} (ξ, τ)})}^{2} d α}}$ (35)

The uncertainty theory based KURI is proposed to measure the likelihood that a reliability event will occur in the mechanism affected by epistemic uncertainties. A higher β_uncertainty(τ) indicates more possibility we believe that the reliability event will occur.

In summary, the flowchart of uncertainty theory based kinematic reliability analysis for mechanism is shown in Fig. 1. The steps in the flowchart are described below:

Fig. 1

Flowchart of uncertainty theory based kinematic reliability analysis.

Step 1. Set the initial parameters such as the uncertain expected values and standard deviation of uncertain variables, and the allowable error limit.

Step 2. Establish kinematic limit state function Z( ξ , τ) based on stress-strength interference or motion error model.

Step 3. Estimate kinematic uncertainty reliability using Equation (19) when the uncertain variables ξ₁, ξ₂, ⋯ , ξ_n have continuous uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. Estimate kinematic uncertainty reliability using Equation (23) when the uncertain variables ξ₁, ξ₂, ⋯ , ξ_n have regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively.

Step 4. Use the first-order Taylor series expression Z( ξ , τ) ≈ Z_L( ξ , τ) when the limit state function is nonlinear.

Step 5. Calculate KURI β_uncertainty(τ) at τ using Equation (32) or (35) when the epistemic uncertainties can be described by normal uncertain variables ξ₁, ξ₂, ⋯ , ξ_n with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively.

Step 6. Estimate kinematic uncertainty reliability at τ using Equation (31) or (34).

5 Case study

Section 5.1 describes a numerical experiment in detail to validate the presented method. Then, the proposed method is used to estimate uncertainty reliability and reliability index of two practical engineering problems in Sections 5.2 and 5.3, respectively. The effectiveness and accuracy of the presented method are illustrated and discussed in three examples.

5.1 A numerical example

Consider a numerical example as the nonlinear limit state function of a motion mechanism based on stress-strength interference model, which can be expressed as follows: $f (X, t) = 1 + (x_{1} + kt)^{3} - \exp (x_{2})$ (36) where X = [X₁, X₂] are input parameters, treated as uncertain variables; k = 1.0 × 10^-5 is a proportionality coefficient; t is the working time.

From the perspective of uncertainty theory, a motion mechanism reliability problem consists of uncertain variables X₁ and X₂. Assuming that X₁ follows normal uncertainty distribution: $Φ_{X_{1}} (x_{1}) = (1 + \exp (\frac{π (m - x_{1})}{\sqrt{3} σ} {))}^{- 1}$ (37) where x₁ ∈(- ∞ , + ∞). Let X₂ be an uncertain variable with linear uncertainty distribution $L (0.4, 0.9)$ $Φ_{X_{2}} (x_{2}) = {\begin{matrix} 0 & x_{2} \leq 0 . 4 \\ (x_{2} - 0 . 4) / 0 . 5 & 0 . 4 < x_{2} \leq 0 . 9 \\ 1 & 0 . 9 < x_{2} \end{matrix}$ (38)

Clearly, both Φ_{X
₁}(x₁) and Φ_{X
₂}(x₂) are regular functions. This indicates that the kinematic uncertainty reliability at t can be calculated using Model 2 proposed in Subsection 3.2. Then, the significance of reliability at t based on the uncertain measure can be explained by the change in expected value and variance of X₁.

Supposing that the variance σ² of X₁ is 0.64, the uncertainty reliability at t can be estimated with different values of m. For comparison, the probabilistic reliability based on MCS can be estimated when all the input parameters are considered as random variables, namely, X₁ ∼ N(m, σ²) and X₂ ∼ U(0 . 4, 0 . 9). The estimated results at t = 1 hour are shown in Table 1 and Fig. 2.

Table 1

Estimated results with different m (σ² = 0.64, t = 1.0)

m	Uncertainty reliability	Probabilistic reliability
	M_reliability(t)	MCS(t)
1.80	0.8359	0.8477
2.20	0.9222	0.9369
2.60	0.9660	0.9798
3.00	0.9858	0.9935
3.40	0.9942	0.9991
3.80	0.9976	0.9999

Fig. 2

Estimated results vs the mean value of X₁.

As shown in Table 1 and Fig. 2, when the mean value of X₁ increases with a constant variance and working time, both uncertainty reliability and probabilistic reliability increase. The trend of uncertainty reliability is consistent with probabilistic reliability. When the expected value of X₁ is larger, the reliability based on uncertain measure becomes larger, and the belief degree of mechanism reliability becomes higher. Therefore, Model 2 proposed in Subsection 3.2 is effective; it can be solved for nonlinear functions and truly describes the trend of mechanism reliability. Following the same method, let the expected value of X₁ is 3. Then, the uncertainty and probabilistic reliability can be estimated with different values (0.25, 0.49, 0.81, 1.21, 1.69, and 2.25) of σ². The estimated results at t = 1 hour are shown in Table 2 and Fig. 3.

Table 2

Estimated results with different σ² (m = 3, t = 1.0)

σ ²	Uncertainty reliability	Probabilistic reliability
	M_reliability(t)	MCS(t)
0.25	0.9989	1.0000
0.49	0.9922	0.9980
0.81	0.9776	0.9869
1.21	0.9569	0.9673
1.69	0.9329	0.9409
2.25	0.9082	0.9119

Fig. 3

Estimated results vs variance of X₁.

The results shown in Table 2 and Fig. 3 indicate that when the variances of X₁ increase with a constant variance and working time, the belief degree that failure event will increase with the dispersity of X₁ increases. Both the uncertainty and probabilistic reliability decrease. The estimated results show that the trend of uncertainty and probabilistic reliability is basically the same.

In addition, Tables 1 and 2 show that the reliability calculated using the proposed Model 2 is smaller than that calculated using the traditional statistical method. Therefore, using the probability measure based (PMB) method to estimate the reliability when the information about design parameters is insufficient will lead to over-optimistic results. It can be concluded that the uncertainty reliability estimated using the presented method in Subsection 3.2 is more conservative and can ensure the mechanism security to a larger extent.

The proposed method can be applied to help mechanism designers make the most reasonable choice when the performance parameters are obtained from the subjective judgment of incomplete test data.

5.2 Reliability assessment in a crank slider mechanism

In this subsection, a crank slider mechanism as shown in Fig. 4(a) was used for the uncertainty reliability analysis, and a horizontal load P was applied on the left direction of slider [35]. As shown in Fig. 4(b), the axial force of AB rod was computed using the decomposition of force at point B. The critical axial force of AB rod is shown in Fig. 5(a), represented by F_cr, and the cross-sectional size of AB rod is shown in Fig. 5(b).

Fig. 4

Sketch map of crank slider mechanism and force analysis of point B.

Fig. 5

Critical axial force and cross-sectional size of AB rod.

Obviously, from Fig. 4(b), we can obtain: $F_{AB} = \frac{P}{\cos (β)}$ (39)

According to the geometric relationship as shown in Fig. 5(a), the following geometric relationship equation holds: $\frac{R}{sin (β)} = \frac{L}{sin (α)}$ (40)

Because sin²(β) + cos²(β) =1, cos(β) can be expressed as follows: $cos (β) = \sqrt{1 - {(\frac{R}{L})}^{2} sin (α)}$ (41)

The trend of load P can be expressed as follows: $P = {\begin{matrix} f sin (α) & α \in [2 n π, (2 n + 1) π], n = {0, 1, 2, 3, \dots} \\ 0 & else \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \end{matrix}$ (42) where α is the turned angle of AO rod and shown in Fig. 5(a). Based on Equations (41) and (42), Equation (39) can be rewritten as follows: $F_{AB} = \frac{f sin (α)}{\sqrt{1 - {(\frac{R}{L})}^{2} sin (α)}}$ (43)

Based on the mechanics of materials, the moment of inertia cross-sectional for AB rod is I_z = hb³/12. Then, the critical force can be calculated as follows: $F_{cr} = \frac{π^{2} {EI}_{z}}{L^{2}} = \frac{π^{2} {Ehb}^{3}}{12 L^{2}}$ (44)

Assume that the main failure model of crank-slider mechanism is based on the fact that the axial force of AB rod is greater than the critical axial force. Therefore, the reliability analysis is based on stress-strength interference model. According to Equations (43) and (44), the kinematic limit state function at α of AB rod can be expressed as follows: $\begin{matrix} Z (b, E, f, R, h, L, α) = F_{cr} - F_{AB} \\ = \frac{π^{2} {Ehb}^{3}}{12 L^{2}} - \frac{f sin (α)}{\sqrt{1 - {(\frac{R}{L})}^{2} sin (α)}} \end{matrix}$ (45)

For simplicity, this study assumes that the length of AB rod L = 1m, and the range of α considers only one cycle [0, 2π], discrete as 72 time instants. The design parameters are determined according to actual experience or expert opinions. Hence, the horizontal force f, elastic modulus E, length of AO rod R, and cross-sectional b and h are defined as normal uncertain variables. The relevant distribution parameters of design variables are shown in Table 3.

Table 3

Distribution parameters of variables

Parameter	UMB method	PMB method
f (KN)	$N$ (2200,25)	N(2200,25)
E (Pa)	$N$ (2.0×10¹¹, 3.0×10¹⁰)	N(2.0×10¹¹,3.0×10¹⁰)
R (m)	$N$ (0.10,0.015)	N(0.10,0.015)
b (m)	$N$ (0.01,0.0001)	N(0.01,0.0001)
h (m)	$N$ (0.02,0.0001)	N(0.02,0.0001)

As shown in Equation (45), the limit state function is a nonlinear function and strictly increases with respect to b, h, and E, and strictly decreases with respect to f and R when the range of α is [0, 2π]. Then, the kinematic uncertainty reliability and KURI can be calculated using the uncertain measure based (UMB) method presented in Section 4. Similarly, the PMB method can be used as the reference when all the parameters are regarded to follow the normal distribution. The probabilistic reliability and KPRI can be estimated using MCS method and first-order second moment (FOSM) method, respectively. The results of MCS method were fitted using the polynomial fitting method. The estimated results are shown in Figs. 6 and 7.

Fig. 6

Kinematic reliability index of a crank slider mechanism.

The uncertainty reliability and KURI are 0.99993 and 5.4249 at α = 0, respectively. As shown in Figs. 6 and 7, the reliability gradually decreases as the limit state function strictly decreases for α when the range of α is [0, π/2]. The uncertainty reliability and KURI decrease to 0.96438 and 1.8186 when α = π/2, the lowest value in a rotation cycle. Then, the reliability gradually increases as the limit state function strictly increases with respect to α when the range of α is [π/2, π]. Because the load P is zero when the range of α is [π, 2π], both the probabilistic and uncertainty reliability are maintained close to 1, as shown in Fig. 7.

Figures 6 and 7 show that both the reliability and index estimated using the UMB method are smaller than those computed using the PMB method. This is because the PMB method neglects the influence of epistemic uncertainties. Thus, when the designers only know the mean value and variance of parameters, it is more reasonable to treat the parameters as uncertain variables. Designers use conservative estimation results with a lower risk, which is more beneficial to decision-making and future applications. The proposed method can provide a more reasonable result for practical engineering problems.

Fig. 7

Kinematic reliability of a crank slider mechanism.

5.3 Reliability assessment in a four-bar linkage mechanism

A four-bar function generator mechanism was used to verify the significance of the proposed approach [12]. The structure simplified model map of the mechanism is shown in Fig. 8.

Fig. 8

Four-rod function linkage mechanism.

The dimension parameters in this case are ξ =(L₁, L₂, L₃, L₄). Then, the loop equations of motion mechanism based on the kinematic analysis can be expressed as follows:

${\begin{matrix} L_{1} cos α + L_{2} cos δ - L_{3} cos ψ - L_{4} = 0 \\ L_{1} sin α + L_{2} sin δ - L_{3} sin ψ = 0 \begin{matrix} \end{matrix} \end{matrix}$ (46)

According to loop equations, the actual motion output angle ψ( ξ , α) can be computed as follows: $ψ_{actual} (ξ, α) = 2 arctan \frac{A \pm \sqrt{A^{2} + B^{2} - C^{2}}}{B + C}$ (47) where α is the input angle, and A = -2L₁L₃sinα

B = 2L₃(L₄ - L₁cosα)

$C = L_{2}^{2} - L_{1}^{2} - L_{3}^{2} - L_{4}^{2} + 2 L_{1} L_{4} \cos α$

Assuming that the mechanism is a sine function generator, a four-bar function generator mechanism is required to achieve the targeted function ψ_target(α) during the time interval [α₀, α_e], and the time interval will be discrete as 60 time instants in this study. $ψ_{target} (α) = 76 \deg + 60 \deg \times \sin [\frac{3}{4} (α - 95 . 5 \deg)]$ (48) where α₀ = 95deg and α_e = 215deg. The motion error function ψ_error( ξ , α) can be calculated as follows: $ψ_{error} (ξ, α) = ψ_{actual} (ξ, α) - ψ_{target} (α)$ (49)

Because of the influence of manufacturing and installation errors, the motion error can be processed as epistemic uncertainties. Assuming that the dimension parameters are uncertain variables with independent normal uncertainty distributions, caused by factors such as operational errors, experts’ judgment, and inevitable man-made mistake, the relevant distribution parameters of variables are shown in Table 4.

Table 4

Distribution parameters of variables

Parameter	UMB method	PMB method
L₁(mm)	$N$ (53.0, 0.1)	N(53.0, 0.1)
L₂(mm)	$N$ (122.0, 0.1)	N(122.0, 0.1)
L₃(mm)	$N$ (66.5, 0.1)	N(66.5, 0.1)
L₄(mm)	$N$ (100.0, 0.1)	N(100.0, 0.1)

With 20 samples of input uncertain variables, numerical realizations for the ψ_error( ξ , α) of four-bar function linkage mechanism were numerically simulated. Fig. 9 clearly shows that the motion error is an uncertain variable at each input angle α. Clearly, the motion error presents both positive and negative values. To ensure that the mechanism works correctly, the motion error must be less than a specified allowable error threshold ɛ, and the requirement may not be satisfied as the uncertainty of dimension variables. Therefore, the limit state function Z( ξ , α) can be established based on motion error function: $Z (ξ, α) = ɛ - | ψ_{error} (ξ, α) |$ (50)

Fig. 9

Motion error analysis of four-rod function linkage mechanism.

Considering that the limit state function is a nonlinear function, by solving the partial derivatives of each variable for the limit state function, we can obtain the following: $b_{1} (ξ, α) = \frac{\partial Z (ξ, α)}{\partial L_{1}} = - \frac{cos (δ (ξ, α) - α)}{L_{3} sin (δ (ξ, α) - ψ (ξ, α))}$ (51) $b_{2} (ξ, α) = \frac{\partial Z (ξ α)}{\partial L_{2}} = - \frac{1}{L_{3} sin (δ (ξ, α) - ψ (ξ, α))}$ (52) $b_{3} (ξ, α) = \frac{\partial Z (ξ, α)}{\partial L_{3}} = \frac{cos (ψ (ξ, α) - δ (ξ, α))}{L_{3} sin (δ (ξ, α) - ψ (ξ, α))}$ (53) $b_{4} (ξ, α) = \frac{\partial Z (ξ, α)}{\partial L_{4}} = - \frac{cos (δ (ξ, α))}{L_{3} sin (ψ (ξ, α) - δ (ξ, α))}$ (54) where $δ (ξ, α) = arctan \frac{L_{3} sin (ψ (ξ, α)) - L_{1} \sin (α)}{L_{4} + L_{3} \cos (ψ (ξ, α)) - L_{1} \cos (α)}$ (55)

According to Equation (33), with the first-order Taylor series expansion, the approximated limit state function can be expressed as follows: $Z_{L} (ξ, α) = Z (m, α) + \sum_{i = 1}^{4} b_{i} (m, α) (L_{i} - m_{i})$ (56) where m =(m₁, m₂, m₃, m₄) is the uncertain expectation vector.

By performing calculation, it was observed that when the range of α is [95deg, 215deg], the partial derivatives b₁( ξ , α), b₃( ξ , α) and b₄( ξ , α) show negative values, whereas b₂( ξ , α) shows a positive value. This indicates that the limit function strictly increases with respect to L₂ and strictly decreases with respect to L₁, L₃, L₄.

Let the specified allowable error threshold ɛ = 0.7deg. Then, the kinematic uncertainty reliability and index depending on the α can be computed using the proposed method shown in Section 4. In this study, we also performed MCS method with a sample size of 10⁷ when all the dimension parameters are considered to follow the normal distribution. Similarly, the kinematic reliability index can also be calculated using the FOSM method. The estimated results are shown in Fig. 10 and Table 5. The results of MCS method are fitted using the polynomial fitting method, as plotted in Fig. 11.

Fig. 10

Kinematic reliability index of a four-rod function linkage mechanism.

Table 5

Kinematic reliability at different values of α

α (Deg)	M _reliability	MCS	α (Deg)	M _reliability	MCS
95	0.99269	0.99940	155	0.99861	0.99998
97	0.99872	0.99998	157	0.99901	0.99999
99	0.99969	0.99999	159	0.99927	0.99999
101	0.99971	0.99999	161	0.99942	0.99999
103	0.99921	0.99999	163	0.99950	0.99999
105	0.99772	0.99996	165	0.99954	0.99999
107	0.99461	0.99973	167	0.99952	0.99999
109	0.98994	0.99924	169	0.99947	0.99999
111	0.98401	0.99772	171	0.99939	0.99999
113	0.97748	0.99463	173	0.99927	0.99999
115	0.97093	0.99191	175	0.99910	0.99999
117	0.96521	0.98710	177	0.99890	0.99999
119	0.96077	0.98342	179	0.99867	0.99993
121	0.95791	0.98126	181	0.99844	0.99991
123	0.95683	0.97972	183	0.99824	0.99996
125	0.95732	0.98011	185	0.99810	0.99993
127	0.95937	0.98146	187	0.99804	0.99989
129	0.96244	0.98484	189	0.99809	0.99999
131	0.96634	0.98785	191	0.99824	0.99996
133	0.97064	0.99056	193	0.99847	0.99991
135	0.97508	0.99331	195	0.99875	0.99995
137	0.97938	0.99547	197	0.99901	0.99997
139	0.98324	0.99679	199	0.99922	0.99999
141	0.98681	0.99782	201	0.99937	0.99999
143	0.98985	0.99891	203	0.99939	0.99999
145	0.99225	0.99923	205	0.99921	0.99998
147	0.99434	0.99969	207	0.99858	0.99997
149	0.99594	0.99972	209	0.99680	0.99988
151	0.99715	0.99988	211	0.99191	0.99925
153	0.99801	0.99992	213	0.98027	0.99462

Fig. 11

Kinematic reliability of a four-rod function linkage mechanism.

The trend of KURI is consistent with KPRI, as shown in Fig. 10. However, a clear gap exists between uncertainty and probabilistic reliability when α belongs to intervals [107deg, 147deg] and [179deg, 195deg]. The lowest value of uncertainty reliability in interval [107deg, 147deg] is 0.95683 when α = 123deg, as shown in Fig. 11 and Table 5. Furthermore, by comparing the uncertainty reliability M_reliability(α) for α = 101deg, α = 167deg, and α = 205deg, a slight difference of their reliabilities. However, the KURI for α = 101deg is 4.4861, and that for α = 167deg and α = 205deg is 4.2169, 3.9381. Thus, when the uncertainty reliability of some motion angles is close to 1, KURI can be used to distinguish the reliability differences of these motion angles.

Furthermore, the results shown in Figs. 10 and 11 indicate that the reliability and index estimated using the PMB method are both greater than those based on uncertain measure. In other words, when epistemic uncertainties exist in a four-bar function generator mechanism, if reliability and reliability index are computed using the probability measure, the accuracy and credibility of the calculated results will be adversely affected. Probability measure omits the factors of epistemic uncertainty and negatively affects the accuracy of calculated results. On one hand, the uncertain measure is more suitable for solving the reliability problem when the information of design parameters is insufficient; on the other hand, the probability measure is more accurate when the parameter information is sufficient.

The two practical engineering applications show that the uncertainty reliability based on uncertain measure is a better indicator to describe the epistemic uncertainties.

6 Conclusions

This study conducted the reliability analysis of mechanism by using uncertain measure. In this work, by analyzing the main failure model of mechanism, the kinematic limit state function was established based on stress-strength interference or motion error model. The following three conclusions are the main contributions of this work:

(1) A novel uncertainty quantification method is developed using uncertain measure for mechanism reliability analysis.

(2) Three models are proposed to estimate the kinematic uncertainty reliability for the mechanism.

(3) An advanced KURI for the mechanism is derived under the normal uncertainty distribution.

In summary, it can be concluded that the presented method can provide a more conservative result for mechanism reliability analysis when only a small amount of data are available. Therefore, the proposed estimation models and KURI can provide an important reference for mechanical designers to decision-making and future applications. Besides, when the designers continuously collect the parameter information and obtain the actual frequency, the reliability calculation with probability measure is more realistic.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 51675026 and No. 71671009), and Aeronautical Science Foundation of China under Grant (2018ZC74001).

References

Melchers

R.E.

and Beck

A.T.

, Structural reliability analysis and prediction, 3nd ed., New York, JohnWiley, 2008, 6–62.

Chakraborty

and Sam

P.C.

, Safety and risk modeling and its applications, Springer, 2011, 81–96.

Meng

, Pang

Y.S.

, Pu

Y.X.

and Wang

, New hybrid reliability-based topology optimization method combining fuzzy and probabilistic models for handling epistemic and aleatory uncertainties, Computer Methods in Applied Mechanics and Engineering 363 (2020), 112886.

Zhang

, Kang

and Wen

, Belief reliability for uncertain random systems, IEEE Transactions on Fuzzy Systems 26(6) (2018), 3605–3614.

Kang

, Zhang

, Zeng

, Zio

and Li

, Measuring reliability under epistemic uncertainty: review on non-probabilistic reliability metrics, Chinese Journal of Aeronautics 29(3) (2016), 571–579.

Zhang

, Ruan

X.X.

, Duan

M.F.

and Jiang

, An efficient epistemic uncertainty analysis method using evidence theory, Computer Methods in Applied Mechanics and Engineering 339 (2018), 443–466.

Lei

, Xufang

and Yangjunjian

, An effective approach for kinematic reliability analysis of steering mechanisms, Reliability Engineering and System Safety 180 (2018), 62–76.

Nirmala

, Babu

K.S.

and Reddy

K.H.

, Design and optimization of mechanical components and its Mechanism using Monte Carlo simulation, International Journal of Engineering Science 3(6) (2017), 675–680.

Sohag

and Yiannis

, Applications of Bayesian networks and Petri nets in safety, reliability, and risk assessments: A review, Safety Science 115 (2019), 154–175.

10.

Pan

C.Y.

, Wei

W.L.

, Zhang

C.Y.

, Song

L.K.

, Lu

and Liu

L.J.

, Reliability analysis of turbine blades based on fuzzy response surface method, Journal of Intelligent and Fuzzy Systems 29 (2015), 2467–2474.

11.

Wei

P.F.

, Song

J.W.

and Lu

Z.Z.

, Time dependent reliability sensitivity analysis of motion mechanisms, Reliability Engineering and System Safety 149 (2016), 107–120.

12.

Wang

Z.H.

, Wang

Z.L.

and Yu

, Time-dependent mechanism reliability analysis based on envelope function and vine-copula function, Mechanism and Machine Theory 134 (2019), 667–684.

13.

Zeng

Z.G.

, Wen

M.L.

and Kang

, Belief reliability: A new metrics for products’ reliability, Fuzzy Optimization and Decision Making 12(1) (2013), 15–27.

14.

F.Y.

, Liu

, Yan

Y.F.

, Rong

J.H.

, Yi

J.J.

and Wen

G.L.

, A time-variant reliability analysis method for non-linear limit-state functions with the mixture of random and interval variables, Engineering Structures 213 (2020), 110588.

15.

Wang

, Qiu

Z.P.

, Xu

M.H.

and Li

Y.L.

, Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters, Applied Mathematical Modelling 47 (2017), 573–586.

16.

Bagheri

, Miri

and Shabakhty

, Fuzzy time dependent structural reliability analysis using alpha level set optimization method based on genetic algorithm, Journal of Intelligent and Fuzzy Systems 32 (2017), 4173–4182.

17.

Marano

G.C.

and Quaranta

, A new possibilistic reliability index definition, Acta Mechanica 210(3) (2010), 291–303.

18.

Zeng

, Kang

, Wen

and Enrico

, Uncertainty theory as a basis for belief reliability, Information Sciences 429 (2018), 26–36.

19.

Zhang

, Zhang

J.G.

, You

and Zhou

, Reliability analysis of structures based on a probability uncertainty hybrid model, Quality and Reliability Engineering International 35 (2019), 263–279.

20.

Liu

, Uncertainty Theory. 2rd ed., Berlin: Germany, Springer, 2007, 8–15.

21.

Liu

, Uncertainty Theory. 3rd ed., A branch of mathematics for modeling human uncertainty, Berlin: Germany, Springer, 2010, 12–23.

22.

Liu

, Why is there a need for uncertainty theory?, Journal of Uncertain Systems 6(1) (2012), 3–10.

23.

Liu

, Uncertain risk analysis and uncertain reliability analysis, Journal of Uncertain System 4(3) (2010), 163–170.

24.

Gao

, Yao

, Zhou

and Ke

, Reliability analysis of uncertain weighted k-out-of-n systems, IEEE Transactions on Fuzzy Systems 26(5) (2018), 1–9.

25.

Bai

and Jin

W.L.

, Structural reliability analysis using uncertainty theory. In: Marine Structural Design. 2nd ed., Elsevier Butterworth-Heinemann, 2016, 603–614.

26.

Wang

, Zhang

J.G.

, Zhai

and Qiu

J.W.

, A new structural reliability index based on uncertainty theory, Chinese Journal of Aeronautics 30(4) (2017), 1451–1458.

27.

and Yao

, Block replacement policy with uncertain lifetimes, Reliability Engineering and System Safety 148 (2016), 119–124.

28.

Wen

, Han

, Yang

and Kang

, Uncertain optimization model for multi-echelon spare parts supply system, Applied Soft Computing 56(C) (2017), 646–654.

29.

Yao

and Zhou

, Ruin time of uncertain insurance risk process, IEEE Transactions on Fuzzy Systems 26(1) (2018), 19–28.

30.

Liu

, Some research problems in uncertainty theory, Journal of Uncertain System 2(1) (2009), 3–10.

31.

Guo

, Guo

and Thiart

, Liu’s uncertain normal distribution, In: Proceedings of the First International Conference on Uncertainty Theory; (2010), 191–207.

32.

Liu

, Uncertainty Theory. 5rd ed., Berlin: Germany, Springer, 2020, 9–112.

33.

Liu

Y.H.

and Ha

M.H.

, Expected value of function of uncertain variables, Journal of Uncertain System 4(3) (2010), 181–186.

34.

Yao

, A formula to calculate the variance of uncertain variable, Soft Computing 19(10) (2015), 2947–2953.

35.

Wang

W.X.

, Gao

H.S.

and Zhou

C.C.

, Reliability analysis of motion mechanism under three types of hybrid uncertainties, Mechanism and Machine Theory 121 (2018), 769–784.