自行走輪椅結(jié)構(gòu)設(shè)計【電動輪椅設(shè)計】,電動輪椅設(shè)計,自行走輪椅結(jié)構(gòu)設(shè)計【電動輪椅設(shè)計】,行走,輪椅,結(jié)構(gòu)設(shè)計,電動,設(shè)計 CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 22,aNo. 4,a2009 594 DOI: 10.3901/CJME.2009.04.594, available online at ; Reliability Simulation and Design Optimization for Mechanical Maintenance LIU Deshun * , HUANG Liangpei, YUE Wenhui, and XU Xiaoyan Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment Hunan University of Science and Technology, Xiangtan 411201, China Received September 8, 2008; revised April 16, 2009; accepted April 30, 2009; published electronically May 5, 2009 Abstract: Reliability model of a mechanical product system will be newly reconstructed and maintenance cost will increase because failed parts can be replaced with new components during service, which should be accounted for in system design. In this paper, a reliability model and reliability-based design optimization methodology for maintenance are presented. First, based on the time-to-failure density function of the part of the system, the age distributions of all parts of the system during service are investigated, a reliability model of the mechanical system for maintenance is developed. Then, reliability simulations of the systems with Weibull probability density functions are performed, the system minimum reliability and steady reliability for maintenance are defined based on reliability simulation during the life cycle of the system. Thirdly, a maintenance cost model is developed based on replacement rates of the parts, a reliability-based design optimization model for maintenance is presented, in which total life cycle cost is considered as design objective and system reliability as design constrain. Finally, the reliability-based design optimization methodology for maintenance is used to design of a link ring for the chain conveyor, which shows that optimal design with the lowest maintenance cost can be obtained, and minimum reliability and steady reliability of the system can satisfy requirement of system reliability during service of the chain conveyor. Key words: maintenance, reliability, simulation, design optimization 1 Introduction During the life cycle of a mechanical product, maintenance, which is implemented on the judgment of practical states, preservation and reconstruction of some certain states for the product, is very important to keep the product available and prolong its life. Studies on maintenance for mechanical products are roughly classified into the following three catalogs. (1) How to formulate maintenance policy or (and) how to optimize maintenance periods considering system reliability and maintenance cost, e.g., when system reliability is subjected to some certain conditions, maintenance policy and optimal maintenance interval are determined to make maintenance cost lowest 14 . (2) To develop maintenance methods and tools to ensure system maintenance to both low cost and short repair time, such as special maintenance toolboxes developed 59 . (3) To design for maintenance(DFM), namely during design procedure, system maintainability is evaluated and * Corresponding author. E-mail: This project is supported by National Basic Research Program of China (973 Program, Grant No. 2003CB317001), Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 07A018), Hunan Provincial Natural Science Foundation of China (Grant No. 07JJ5074), and National Natural Science Foundation of China (Grant No. 50875082) is improved 1014 . Maintenance starts at design. Obviously, design methodology for maintenance, which is one of best effective maintenance means in the life-cycle of a product, attracts many researchers interests. However, research on design for maintenance is mainly centralized on two fields. One is maintainability evaluation on product design alternatives, the other is some peculiar structures of parts designed for convenient maintenance. For example, computer-aided maintainability evaluation tools for product design 11 , product assembly and disassembly simulation programs for maintenance 12 ,airplane design for maintenance 13 , and so on. But studies on design methodologies considering product reliability, maintenance cost and maintenance policy are seldom reported. SHU and FLOWER once pointed out that reckoning in labor cost and production interval cost, design decision of alternatives of the part would be influenced. However, subsequent research reports have not been presented 15 . In this paper, based on the time-to-failure density function of the part, distributions of service age of parts for a mechanical system that undergoes maintenance are investigated. Then the reliability model of the mechanical system is reconstructed and simulated. Finally, a novel design optimization methodology for maintenance is developed and illustrated by means of design of a link ring for the chain conveyor. CHINESE JOURNAL OF MECHANICAL ENGINEERING 595 2 Reconstruction of Reliability Model of Mechanical System for Maintenance 2.1 Model assumptions After a mechanical system runs some time, due to replacement of fail parts, primary reliability model is inapplicable to changed system, thus the reliability model should be reconstructed. The mechanical system discussed in this paper has following characteristics. (1) System consists of a large number of same type parts, in which the number of parts is constant during the whole life cycle of the system. (2) The time-to-failure density distribution functions of all parts are the same, also, replacement parts have the same failure distribution functions as the original parts (3) Failure of each part is a random independent event, i.e., failure of one part does not affect failure of other parts in the system. For example, a chain conveyor widely used in many industries consists of a large number of same round rings, same link sheets and same scrape boards. Their respective numbers are constant after the chain conveyor is put into the service. Also, each part, being subjected to similar work conditions and similar failure states, has the same or identical density distribution of time to failure. Moreover, replacement parts have failure time density function same or identical to the original parts during the service of the chain conveyor. 2.2 Reliability modeling for maintenance Reliability of a mechanical system depends on its parts, yet reliability and failure probability of which rest on their service ages. Herein, according to the density distribution function of time to failure of the part, part service age distribution of the mechanical system is calculated, then reliability model of the mechanical system for maintenance is developed. During the service of a mechanical system, some parts that fail require to be replaced in time, hence age distribution of parts of the mechanical system undergoing maintenance has been changed. Supposed that after the mechanical system runs some time n tn= , where is time between maintenance activities, i.e., maintenance interval, the unit of can be hours, days, months, or years. If () in pt represents age proportion of parts at n t with age i , thus age distribution of parts at time n t denotes matrix 01 (),(), nn pt pt (), , in pt () nn pt .The failure density function of parts and current age distribution of parts in the system determine age distribution at next time, or the portion of the contents of each bin that survive to the next time step. An age distribution obtained at each time step for each part population determines failure rate for the following time step. To find failure probability of parts the failure density function is integrated from zero to n t .The portion of the population that survives advances to the next age box, and the portion that fail is replaced by new parts to become zero age to reenter the first box. Initially, all parts are new and zero age in the first box. That is, at 0 0t = , the portion in the first box is 00 () 1pt = . (1) At 1 t = , age fractions of the first box and the second box are represented as 11 0 0 0 01 00 0 () ()1 ()d , () () ()d. pt pt fx x pt pt fxx = = (2) Portions of both age boxes survive and advance to the next age box, and portions of failed parts from both boxes replaced by new parts appear in the first box. At 2 2t = , the proportions of the first three boxes are calculated as follows: 2 22 11 0 12 01 0 2 02 11 01 0 0 () ()1 ()d , () ()1 ()d , () () ()d () ()d, pt pt fxx pt pt fx x pt pt fxx pt fxx = = =+ # (3) So, at n tn= , portions of parts in each box are calculated by using the following equations: 11 0 (1) 121 0 (2) 231 0 3 321 0 2 211 0 () ( )1 ()d , () ( )1 ()d , () ( )1 ()d , () ( )1 ()d , () ( )1 ()d , n nn n n n nn n n n nn nn nn nn pt p t fxx pt pt fxx pt pt fxx pt pt fxx pt pt fxx p = = = = = # 101 0 1 (1) 01 0 0 () ( )1 ()d , () ( ) ()d. nn n i nin i tpt fxx pt pt fxx + = = = (4) Where 0 () n pt is the fraction of population of parts with age 0 at n t , representing parts that have just been put into service. It means that 0 () n pt is failure rate of parts, or replacement rate of failed parts. In other word， the fractions of parts in the first box at 01 , , n tt t are new parts that replace these failed parts. A series system consists of N parts that have the same failure density distribution, each part is just a series unit, and each unit is relatively independent. In series system the YLIU Deshun, et al: Reliability Simulation and Design Optimization for Mechanical MaintenanceY 596 failure of any one unit results in system failure, in according to the principle of probability multiplication, the reliability of series systems becomes () 0 0 () 1 ()d . in pt Nn i n i Rt fx x = = (5) Since the number of parts that comprise the system is constant, here, the system reliability of the mechanical system for maintenance is defined as () 0 0 () () 1()d in N nn pt Nn i N i Rt R t fx x = = = () 0 0 1()d. in ptn i i fx x = (6) From above to see, as long as the time-to-failure density function and maintenance interval are given, service age distributions of parts and system reliability could be obtained by simulation. 3 Replacement Rate and Reliability Simulation for Maintenance 3.1 Weibull distribution of time to failure The Weibull probability density function is widely used in failure modeling in mechanical parts and electronic components. Here the Weibull distribution with two parameters is used to simulate reliability of the system that is undergoing maintenance, that is, the time-to-failure density function of systems constituted parts is 1 () exp ,0 xx fx x = . (7) In Eq. (7), is the shape parameter, is the scale parameter. x is time, whose unite can be hours, days, or years. Five failure density functions with their Weibull parameters 10, 1,2,3,4,5 = are described in Fig. 1. It is shown that is large, before service age of parts arrives at the expected value, failure probability of parts is extremely low. Whereas, is small, many parts fails in short time of service. 3.2 Reliability simulation Different maintenance interval of the mechanical system and different time-to-failure density function of its parts are selected to simulate reliability of the system shown as Fig. 2Fig. 4. Fig. 2 shows how simulation time step (maintenance interval) affects system reliability, the plots shown correspond to maintenance interval 0.5,1,2 = , and with Weibull distribution parameters 4, 10 =. Fig. 3 plots the influence of the scale parameter of Weibull distribution on system reliability, and four curves represent four different type parts corresponding to a constant value of equal to 4 paired with value of 8,10,12,15 respectively. Fig. 4 reveals how the shape parameter of Weibull distribution affects system reliability, and Weibull distribution parameters of five curves are 10, = 1, 2,3, 4,5 = . Correspondingly, their replacement rate curves of systems parts for these time-to-failure density distribution functions are plotted in Fig. 5. Additionally, in Fig. 3Fig. 5, maintenance interval is 1 = . Fig. 1. Weibull probability distributions Fig. 2. System reliability R(t) with Fig. 3. System reliability R(t) with Several characteristics of these figures are of interest. First, the reliability and replacement rate eventually reaches steady state. This agrees with Drenicks Theorem, which CHINESE JOURNAL OF MECHANICAL ENGINEERING 597 states the superposition of an infinite number of independent Fig. 4. System reliability R(t) with Fig. 5. Part replacement rate p 0 (t) equilibrium renewal process is homogeneous Poisson process. During the initial stage of system service, parts of the system are “new”, then, become “old”. The portion of parts that fail gradually increases, thus the part replacement rate increases and system reliability will drop monotonically. With the replacement of a significant portion of the population, portion of parts that fail will decrease, thus the part replacement rate will drop and the system reliability will rise until this oscillation is over and next oscillation begins. After some oscillations, the population becomes more age-diversified with each oscillation, and the age distribution approaches steady. At that time, the oscillations in replacement rate and system reliability diminish. Compared Fig. 4 with Fig. 5, it is shown that the trend of replacement rate is contrary to the change of system reliability. When system reliability increases, part replacement rate reduces. Otherwise, as system reliability reduces, part replacement rate increases. Secondly, the steady state value and the degree of oscillation of the system reliability depend on maintenance interval. As Fig. 2 shows, the reliability rises as maintenance interval decreases since parts that fail are being replaced more quickly. The shorter the maintenance interval is, the higher reliability is, and the smaller oscillations are. However, frequent repairs will result in higher maintenance cost. Thirdly, the steady state value of the system reliability depends on the parameters of Weibull distribution. The dependence on is not surprising, higher values of for a given set of yield higher values for expect time to failure and thus lower replacement rate and higher reliability. More interestingly, with the increase of the value of , the steady values of replacement rate decrease and the steady values of reliability increase. Fourth, the degree of oscillation of system reliability depends on the parameters of Weibull distribution. Although the influence of on oscillations can be neglected, the influence of on oscillations should be paid special attention to. Bigger value of denotes that failure rate of parts is lower before service time of parts reaches expected life time, and the majority of parts prolong use time, thus, the steady value of system reliability becomes higher. However, in this case, the majority of parts fail at quite centralized time, so minimum value of system reliability is lower. It is suggested that , denoting concentrative degree of failure time distribution, is a sensitive parameter. The influence of on steady value of reliability is different from and contrary to that of on minimum value of reliability. Therefore, selection of appropriate should be paid special attention to in design, because both steady value and minimum reliability coincidentally meet design requirements. 3.3 Definitions Simulation results show that system reliability varies during service. The reliability of a system experiences several oscillations, sometimes is maximum value and then minimum value, finally reaches steady value. Oscillations of system reliability periodically decay, and the period is about the expected life time of parts (for Weibull distribution, the parameter approximates expected life at big ). For design and maintenance of mechanical systems, minimum value and steady value of system reliability are of importance. Minimum reliability of the system appears at beginning stage, but steady reliability value of the system appears after running a long time. Here, to conveniently discuss later, minimum reliability and steady reliability of the system for maintenance are defined based on simulation results of system reliability shown as in Fig. 6. Fig. 6. System reliability parameters definition YLIU Deshun, et al: Reliability Simulation and Design Optimization for Mechanical MaintenanceY 598 As it appears at initial phase, minimum reliability of the system can be found in discrete reliability values of simulation results from 0t = to 2t = . At this time, minimum reliability m R is defined as ()min ( ) , 0,1, , . mi RRti n= (8) Supposed that some simulation time is 0 T , and max min ,RR represent maximum value and minimum value of 00 , 2tTT + respectively. Once when ratio of maximum reliability value and minimum reliability value min max /RR is satisfied, system reliability is regarded as arriving at steady value at time 0 T . Thus system reliability, or called as steady reliability, is defined as max min ()/2 s RR R=+ , (9) 1 is the stabilization requirement, which could usually be 98%. If 0 T does not exist, system reliability will be unsteady. 4 Reliability-based Design and Optimization Modeling for Maintenance A reliability-based design optimization model for maintenance is presented to make a trade-off between the system reliability and life-cycle cost of parts that includes maintenance cost, in which the above models are helpful to calculate part replacement rate of the system, minimum reliability and system reliability. In the model, the cost of life cycle is considered as a design objective, and the reliability of the system is considered as design constraint. The task is to find a design having the minimum cost and satisfying the constraints. 4.1 Model of life cycle cost Life cycle costs of mechanical systems include production costs and maintenance costs. System maintenance costs are from items listed as follows: (1) cost of parts replacement, (2) operation cost including cost of resources spent (i.e. labor, equipment) for replacing parts, (3) indirect cost resulting from production interrupt caused by replacing parts, and (4) preparation work cost for replacing parts 16 .The foregoing three items are concerned with the number of replacing parts every time of maintenance. The more parts replaced will consume more resource, occupy more production time, thus bring tremendous loss and increase maintenance cost. The last item is not concerned with the number of replacing parts but times of maintenance or replacement. As a result, maintenance costs of mechanical systems are classified as cost considering part replacement number and cost considering maintenance times. In this way, for a mechanical system with a constant number of parts N , after it runs for time M , its life cycle cost model, including production cost and maintenance cost, is represent as 0102 1 () m i i Cc cpt c = =+ + . (10) In Eq. (10), C is total cost of life cycle of the system for per part in the system. 012 ,ccc denote coefficient of part production cost, coefficient of replacement cost and coefficient of preparation cost respectively, and these coefficients can be confirmed by statistical analysis of field datum. /mM= , where M represents life of the system. The first term of right-hand side of Eq. (10) represents production cost of the system, the second term of right-hand side of Eq. (10) represents maintenance cost of the system. In Eq. (9), 10 cc , because part replacement cost includes not only production cost of the part that replaces the failed part, but also costs that are spent for resources, and indirect cost caused by replacement. Obviously, the cost that Eq. (10) denotes is not absolute cost, but relative cost. Eq. (10) is also represented as 0210 1 () m i i Cc mc c pt = =+ + . (11) 4.2 Model of reliability-based design and optimization Supposed that a type part of the system has n design alternatives, 12 (, , , ) n X xx x= , their failure density functions are e