Journal of the Balkan Tribological Association

Vol. 15, No 3, 309—322 (2009) Overall tribology

DETERMINATION OF RELIABILITY OF MOTOR VEHICLE STEERING SYSTEM TIE-ROD JOINT D. CatiC*, B. KrstiC, D. MiloradoviC Faculty of Mechanical Engineering, University of Kragujevac, 6 Sestre Janjic Street, 34 000 Kragujevac, Serbia E-mail:[email protected] ABSTRACT Procedures for planning of truncated tests for reliability assessment and processing of test results are presented in the paper, using the case of the tie-rod joint belonging to the steering system of light commercial vehicle. In order to select and quantify the optimum plan for testing the observed object, it is necessary to know in detail the structure, the way of operation, the causes of potential failure modes and mechanisms of their generation. Statistical data set related to the operation time until failure on tie-rod joint occurrence is gained by conducting tests in exploitation conditions. Test results processing is done using computer and corresponding software. Selection of optimum theoretical model of random variable distribution is conducted with taking into account all relevant identifiers. Keywords: reliability, planning of truncated tests, tie-rod joint, distribution model. aims and background Steering system is one of the vital parts of motor vehicle complex mechanical system1. Together with the braking system and with the tires, it has a crucial significance for safety of motor vehicles and people in traffic. Thus, a great attention is given to demands that are set before the steering system regarding the reliability. By analysis of modes, consequences and criticality of failures of the steering system elements built on light commercial vehicles, it has been established that the tie-rod joints are the most critical elements from the aspect of reliability and safety2. Application of procedures for accelerated testing in order to estimate reliability has a great importance from the aspect of reduction of the test costs and time necessary to obtain needed information on reliability. Description of a lar­ ger number of accelerated testing procedures for estimation of reliability may be * For correspondence.

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found in literature. Among them, a special place belongs to the plans for reduced testing3. Unlike other procedures, where acceleration of testing is obtained on account of forcing the operating regime or intensifying influence of environment, the plans of reduced testing enable faster obtaining of data on account of optimal statistical planning of tests. Determination of distribution law of operation time until failure occurrence is the final goal Fig. 1. Algorithm of the procedure of data processing in the area of reliability of the for determination of distribution observed object. law for operation time until failure This is a very ‘sensitive’ phase of calculations. occurrence Its results affect all further conclusions and decisions related to practical applications. The procedure for determination of the distribution law of operation time until failure occurrence based on empirical data is done in following three steps4 (Fig. 1). In the first step, estimation of correct operation indicators is done. Numerous characteristics of statistical ensemble (mean operation time until failure occurrence, median, mode, standard deviation, coefficient of asymmetry and coefficient of excess) are determined and estimation of values of complete characteristics of random variable (probability distribution function, reliability, distribution density and failure intensity) is done. In the second step, determination of theoretical model of distribution that may be used to approximate the empirical distribution is done based on the values obtained at the first step. Parameters of the selected model are determined afterwards. Verification of compatibility between the adopted theoretical model of distribution and the empirical distribution is performed in the third step. TIE-ROD JOINT AS A COMPONENT PART OF MOTOR VEHICLE STEERING SYSTEM The steering system of the wheeled vehicles contains the two basic subsystems: •  steering mechanism (group of steering wheel column), and •  steering linkage (group of tie-rods and steering arms). Steering linkage, besides connecting the steering mechanism and the steered wheels, has a very important task to provide proper kinematics of wheel turn. This means that the steering linkage must be completely in accord with the suspension system of the steered wheels, so the motion of the wheels relative to vehicle frame does not influence the safety of steering. Previously given task is obtained by designing the linkage system in the form of trapeze. 310

Fig. 2. Steering linkage of light truck steering system

The steering linkage of a light commercial vehicle steering system with onepiece cross tie-rod is presented in Fig. 2 (Ref. 5). The torque is transmitted from the output shaft of the steering gear, through the steering arm (1) and the drag arm (consisting of a drag arm joint (2), a clamp (3) and drag link with joint (4)), to a drag link steering arm (5) on the front left wheel. Drag link steering arm is connected to the wheel spindle by bolts. On the lower side of left wheel spindle, there is an arm (6) that transfers the force through a cross link (consisting of joints (7), clamps (3) and cross tube (8)), to the same link at the right wheel spindle. The front axle, wheel arms (6) and the cross tube form the steering trapeze. In order to achieve the basic function of the steering system – turning the wheels for a given angle, the linkage mechanisms must never be rigid constructions. Due to complex relative motion of the elements of the steering linkage system during turn, spherical joints are the most convenient links between the elements. The elements of the tie-rod joint used in light truck steering systems are presented in Fig. 3. Spherical joints provide mobility in all three planes. To preserve the proper steering kinematics, there must not be any clearance in tie-rod joint. Cancelation of clearance is achieved by designing Fig. 3. Tie-rod joint the joint cup in two parts, so the upper moving 1 – joint body, 2 – ball pin, 3 – cup, part of the cup presses the ball pin sphere with the 4 – spring, 5 – cover, 6 – sealing cap, 7 – nut help of a spring. Due to indisputable importance the tie-rod joint has in reliable and safe operation of the motor vehicle steering system, testing and determination of distribution law of operating time until failure occurrence were performed on the tie-rod joint of a light truck steering system. 311

RELIABILITY ESTIMATE RESEARCH In order to determine the distribution law for operation time until failure occurrence on the lateral tie-rod joint of a steering system for light trucks, the procedure of accelerated exploitation research tests is carried out, according to plans for reduced research for reliability estimate. Regarding possibility for recovering of operating ability after failure, the lateral tie-rod joint of motor vehicle steering system belongs to a group of irreparable objects. The final goal of testing for reliability evaluation is to determine the distribution law of operating time until failure of tie-rod joint. In this particular case, according to recommendations for selection of the optimal plan of reduced research6, the total number of research plans is reduced to plan [NRr], based on the object type chosen from the aspect of possibility for renewal of the operating ability after failure, on the final research objective and on the beginning of the object research. The procedure of planning the opened type research shown in Fig.  4 (Ref. 2), is used for quantitative definition of the optimal plan for reduced research. The results obtained after the first round of investigations must satisfy a condition specified in advance in procedures of planning an opened type research. It is usually assumed that dotted estimate of the variation coefficient of random variable must be smaller than the adopted value. If specified condition is not satisfied, pre-planning of plan parameters is conducted and research is continued. In the plan designation, N stands for sample quantity, R – for object replacement after failure and r – for a number of necessary data on operation time until failure occurrence or criterion for research termination. Numerical value of r depends on relative error, d, and the level of trust, g, of the results obtained. According to information approach to optimisation of characteristics of accuracy and the level of trust, the values of dopt and gopt depend on a model of hypothetical distribution, variation coefficient, n, of a random variable and on adopted initial values for r and dapr. In order to define quantitatively the plan of research, it is necessary to use expressions that determine the dependence between characteristics of accuracy and credibility and research results. For plan [NRr] and the assumed Weibull distribution of random variable, plan parameter r is determined from the following expression7: 2r b (1) (d + 1) = 2 , c1−g ,2 r where d is limit relative error of reliability index assessment; b – the shape parameter of the Weibull distribution; c21–g,2r – quantile of c-square distribution with 2r degrees of freedom and with corresponding probability of 1 – g, g – level of 312

Fig. 4. Algorithm of the procedure for opened type reduced research of reliability estimation according to plan [NRr]

confidence that the value of the observed index will be within the interval defined by relative error. The shape parameter of the Weibull distribution, b, is uniquely related to the variation coefficient through variation of random variable, according to the fol313

lowing expression:

v=

G (1 + 2 / b ) − G  (1 + 1/ b )  G (1 + 1/ b )

2

,

(2)

where G is Gamma function. Selection of characteristics d and g does not depend on mechanism of object failure occurrence. In contrast to them, the variation coefficient, n, is completely conditioned by the distribution of operating time until failure occurrence. The variation coefficient depends on a large number of factors, the most important of which are the level of technology of manufacturing, size and variability of load and mechanism of failure occurrence. Increase in level of manufacturing technology leads to decrease of the variation coefficient of operation time until failure. The variation coefficient is increased with the load reduction. Data given in Table 1 illustrate the influence of the mechanism of failure occurrence on the variation coefficient8. On the basis of data given in Ref. 2, failure of the tie-rod joint occurs in nearly 99% of cases due to occurrence of enhanced clearance in the joint that is due to maximum wear of a ball or a cup of the joint. According to Table 1, the variation coefficient of operation time until failure in the case of a failure due to maximum wear is equal to n = 0.3. Nevertheless, the intensity of wear in the concrete case depends on a large number of factors of stochastic nature, so a larger dissipation of data around the mean value may be expected. Thus, the coefficient of variation n = 0.5 is adopted. Based on expression (2), for n = 0.5, the shape parameter of the Weibull distribution is β ≈ 2.1. For initial number of necessary data, value of r = 50 was adopted. Since the tie-rod joint is a structure that influences the safety of the steering system funcTable 1. Recommended values for selection of the variation coefficient, n Law of Observed random variable distribution Normal product operation time until general repair machine operation time until general repair structure operation time until maximal wear occurrence structure operation time until limit condition provoked by combination of wear, fatigue and corrosion occurrence Lognormal operation time until fatigue fraction occurrence during bending or torsion operation time until damage to construction connections occurrence machine operating time between general repairs The Weibull’s operation time until damage due to contact fatigue occurrence law operation time until fatigue fraction during bending and torsion occurrence operation time until damage of construction connections

314

Variation coefficient, n 0.10 ÷ 0.20 0.30 0.30 0.30 0.40 ÷ 0.50 0.70 0.60 ÷ 0.80 0.70 0.30 ÷ 0.50 0.80

Table 2. Recommended values for selection of relative error, d and the confidence, g Testing object

Relative error, d

Confidence level, g

Product as a whole or a component that condition the outer shape of the product, that is its suitability for use Machine or basic component Machines, structures and elements that influence safety

0.15 ÷ 0.20

0.80 ÷ 0.90

0.10 ÷ 0.15 0.05

0.90 ÷ 0.95 0.95 ÷ 0.99

tioning, an a priori relative error of estimation of dapr = 0.05 was adopted, according to general recommendations for selection of values for d and g, given in Table  2 (Ref. 8). Figure 5 presents diagrams used for determination of dopt and gopt for [NRr] plan and for equal to 1, 2 and 3. These diagrams also may be used for [NRr] plan, since the equations for their forming have the same shape for [NUr] and [NRr] plans3. The variation coefficient of n = 0.523 corresponds to the value b = 2, which is approximate to adopted value. Based on Fig. 4,b, for b = 2, dapr = 0.05 and r = 50, optimal values of relative error and the confidence level are dopt = 0.08 and gopt = 0.87, respectively. Based on the adopted values of the parameters for quantitative definition of the research plan, the value r = 77 is obtained. Acquisition of data on operation time until failure is achieved by following the vehicle in exploitation. By measuring the operation time until failure of the first 77 joints of the lateral tie-rod, a statistical ensemble of data is acquired. Numerical values of random variable are in addition arranged in increasing variation series, as presented in Table 3. Operation time until failure occurrence is measured in km of the distance travelled.

Fig. 5. Optimum values of accuracy and confidence level of mean operation time until failure for [NUr] plan and the Weibull distribution with the shape parameter a – b = 1, b – b = 2, c – b = 3 (information criterion)

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Table 3. Operation times until failure of the lateral tie-rod joint of the light truck’s steering system Ordinal Distance Ordinal Distance Ordinal Distance Ordinal Distance Ordinal Distance numeral travelled numeral travelled numeral travelled numeral travelled numeral travelled until until until until until of of of of of failure failure failure failure failure failure failure failure failure failure (103 km) (103 km) (103 km) (103 km) (103 km) 1 16.051 17 53.912 33 102.676 49 117.552 65 151.800 2 27.100 18 62.837 34 102.676 50 118.582 66 155.929 3 27.987 19 63.500 35 103.041 51 118.685 67 163.628 4 31.606 20 69.522 36 103.041 52 121.110 68 173.670 5 32.151 21 69.522 37 103.649 53 121.110 69 176.720 6 35.810 22 69.530 38 103.649 54 124.920 70 185.841 7 36.005 23 80.329 39 103.908 55 129.860 71 187.225 8 40.120 24 82.302 40 104.018 56 129.960 72 188.695 9 40.216 25 85.491 41 104.018 57 134.880 73 202.940 10 40.715 26 85.902 42 109.584 58 134.880 74 204.790 11 46.116 27 90.278 43 114.090 59 138.627 75 218.182 12 47.910 28 90.278 44 114.253 60 139.543 76 241.174 13 49.356 29 93.416 45 114.370 61 142.517 77 247.327 14 50.866 30 94.101 46 115.080 62 147.512 15 52.268 31 94.298 47 115.080 63 148.590 16 53.652 32 98.753 48 117.201 64 148.590

Based on calculated mean value of tsr = 106  298.7 km and standard deviation of s = 51  129.34 km of the statistical ensemble of data, the variation coefficient of the obtained results of n = s /tsr = 0.481 is less than the adopted value of n = 0.5. In that way, the given condition of opened procedure for reduced testing for reliability assessment is met, so the data processing may be initiated in order to determine the law of distribution of operation time until failure of the observed object. Obtained variation coefficient of the statistical ensemble is smaller than the adopted value and, hence, additional condition for opened procedure of reduced reliability estimate research is satisfied. Since the sample was large, the number of intervals for grouping the values of random variable of operation time until failure occurs is obtained using the expression:

z = 1 + 3.3 lgn,

(3)

where n is the number of samples. For n = r = 77, the number of intervals rounded to integer is equal to 7. For the adopted values of number of intervals, z = 7, lower limit of the first interval, td1 = 16  000 km and the range of intervals, Dt = 34  000 km, the number of failures of the objects per each time interval is given in Table 4. 316

Table 4. Number of failures of the lateral tie-rod joint of the steering system in time intervals Distance travelled (103 km) 16÷50 Number of failures

13

50÷84 84÷118 118÷152 152÷186 186÷220 220÷254 11

25

16

5

5

2

DETERMINATION OF OPTIMAL THEORETICAL MODEL OF DISTRIBUTION Determination of the theoretical model of distribution of operation time until failure of lateral tie-rod joint of the steering system is conducted by using a computer program. Theoretical basis for formation of the program and examples of practical applications are given in Ref. 2. The program provides calculation of numerous characteristics of the statistical ensemble, assessing of values of the proper function indices, determination of parameters of theoretical approximate models and testing of given distribution models. In a concrete case, the initial part of the output list of the program contains the following data: Element name: Lateral tie-rod joint Total number of data: n = 77 Data processing is done by the use of large sample procedure. Minimum operation time until failure: tmin = 16 051.0 Maximum operation time until failure: tmax = 247 327.0 Calculated number of intervals: z = 7.225 Adopted number of intervals: z=7 Adopted lower limit of the first interval: td(1) = 16 000.0 Calculated range of interval: Δt = 33 007.285 Adopted range of interval: Δt = 34 000.0 Calculated numeric characteristics of statistical ensemble: – mean value: tsr =106 298.700 – standard deviation: σ = 51 129.340 – median: t50 =103 720.000 – mode: MO = 104 695.652 – coefficient of asymmetry: KA = 0.467 – coefficient of flatness: KE = – 0.129 In the next phase of the program and based on procedure for assessment of proper function indicators for a large sample2, estimated values of the number of proper functioning objects, n(t), reliability, R(t), unreliability, F(t), density of operation time until failure, f(t) and intensity of failure of lateral tie-rod joint, h(t), are obtained for the middles of the time intervals (Table 5). Graphic illustrations of evaluated values for density of operation time until failure, f(t) and intensity of failure, h(t), of lateral tie-rod joint are given in Fig.  6, 317

Table 5. Estimated values for indicators of proper operation of the tie-rod joint i 1 2 3 4 5 6 7

ti 33000.00 67000.00 101000.00 135000.00 169000.00 203000.00 237000.00

n(ti)

R(ti)

F(ti)

f(ti)

h(ti)

70.5 58.5 40.5 20.0 9.5 4.5 1.0

0.91558 0.75974 0.52597 0.25974 0.12338 0.05844 0.01299

0.08442 0.24026 0.47403 0.74026 0.87662 0.94156 0.98701

0.49656E-05 0.42017E-05 0.95493E-05 0.61115E-05 0.19099E-05 0.19099E-05 0.76394E-06

0.54234E-05 0.55304E-05 0.18155E-04 0.23529E-04 0.15480E-04 0.32680E-04 0.58824E-04

in the shape of polygons and histograms. During crude evaluation, these graphics may serve for determination of hypothetical models of empirical distribution. Based on diagrams of estimated values for steering system lateral tie-rod joint failure intensity for light trucks, a conclusion may be reached that these are failures in the ageing period of the observed object. Thus, exponential distribution should be eliminated from further discussions. Besides, on the basis of calculated values of statistical ensemble that represent the centre of dissipation of random variable (mean value, median and mode), it may be concluded that this is a normal distribution of random variable. Nevertheless, the value of the asymmetry coefficient and the graphic of empirical distribution density, given in Fig. 6, both point that this is positive asymmetrical distribution. For complete insight in the procedure for determination of optimum model of lateral tie-rod joint reliability and for making conclusions based on quantitative indicators, empiric distribution was approximated by four theoretical models most frequently used for modelling the reliability in the ageing period of machine elements. Table 6 shows parameters of approximate models and results of graphic tests for the Weibul, Rayleigh, normal and lognormal distribution.

Fig. 6. Diagrams of estimated values for distribution and intensity of lateral tie-rod joint failure

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Table 6. Comparative review of the results of graphical tests and deviations of approximate models from empiric failure distribution of lateral tie-rod joint Allocation of points of probability papers

Diagrams of deviation of approximate models from empiric distribution

lognormal distribution m = 11.367; s= 0.555

normal distribution m = 106 691.4; s = 57 418.2

the Rayleigh distribution s = 83 970

the Weibull distribution h = 118 755.9; b = 1.998

Distribution

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Graphical tests of theoretical approximate distribution models enable visual comparison between deviations of two or more theoretical models and empiric distribution. Nonparametric tests enable rejection of those theoretical models which do not meet criteria of used statistical tests. It is confirmed in practice that there are numerous situations where more theoretical distribution models satisfy statistical tests. At the same time, it is hard to give preference to only one of the used models, based on diagrams. In such situations, there is a problem of selection of theoretical model that ‘best describes’ behaviour of random variable. In nonparametric testing of hypotetical distirbution models, a large number of quantitative indicators of deviations of theoretical models from empirical distribution is gained. Suggestion of the authors of this paper is that calculated deviations should be used, not only as confirmation that theoretical model meets given test for adopted level of significance, but also for determination of the theoretical model by which all or majority of deviations are minimal. As illustration of the previous statement, a parallel review of characteristic values for the Kolmogorov, Pearson and Romanovsky tests, for four used theoretical approximate models of tie-rod joint reliability is presented in Table 7. Column for distributions in Table 7 contains approximate models ranked according to deviation values, from minimum to maximum, for the majority of the used tests. Although, the sequence of theoretical models is not the same for all used statistical tests, side-by-side review of these deviations gives a good basis for selection of the best solution. Shaded fields mark those deviations or theoretical models that do not meet the adopted criteria of nonparametric tests. In theory, if criteria of each test were tightened, with the level of significance with continual sequence of values, a level of significance would be reached that is met only by one theoretical model. In other words, theoretical model with minimum value of characteristic variation for some nonparametric test may be adopted with the highest level of significance as approximate. In concrete case, on the basis of values given in Table 7 and according to the Kolmogorov test, the Rayleigh and Weibull distribution have the minimum deviations, Dn. These deviations are equal and it might have been assumed on Table 7. Comparative review of quantitative indicators of deviation between theoretical distribution models and empirical distribution of operation time until tie-rod joint failure occurrence Test Distribution The Rayleigh distribution The Weibull distribution Normal distribution Lognormal distribution

320

The Kolmogorov test Dn

The Pearson test c2

The Romanovsky test RO

0.0409 0.0409 0.0513 0.1363

7.588 7.569 9.459 14.936

1.268 1.865 2.637 4.873

the basis of calculated values of the shape parameter of the Weibull distribution (b = 1.998 ≈ 2). For the Pearson test, the Weibull distribution has minimum value of c2. For the Romanovsky test, the comparative value of RO is the least for the Rayleigh distribution. On the basis of obtained results, it may be concluded that the Rayleigh or Weibull distribution may be used for approximation of the operation time until failure occurrence. If the Weibull distribution is adopted as approximate model, the expression for proper functioning probability of steering system tie-rod joint for a light truck is the following:

R (t ) = e

t −  h 

b

1.998

=e

  t −   118 755.9 

.

(4)

During determination of proper function probability of lateral tie-rod joint and all other functional indicators of reliability derived from expression (4), time, t, is expressed in kilometres of the distance passed. CONCLUSIONs Application of plans for reduced research of reliability estimation enables considerable savings of time and money. In order to use possibilities provided by plans for reduced research, it is necessary to know the observed object well. Adoption of initial values of parameters used for quantitative definition of plans is conducted based on gathered data. Authenticity of information obtained by such a procedure directly depends on the successful solution of this task. Determination of the theoretical model of distribution that best approximates the empiric distribution of random variable is done most easily and most efficiently using computer and the corresponding software support. Any other approach, like direct calculation without automation, would not only consume more time, but would imply, in general, smaller accuracy and considerably heavier conditions for selection of the optimum solution. Apart from enabling the rejection of those theoretical models that considerably deviate from empirical distribution, nonparametric testing of hypothetic distribution models gives the quantitative indicators whose comparison gives optimum distribution laws. The basis for selection of the distribution that best describes the behaviour of random variable is formed by approximation of empirical distribution with as many theoretical models as possible and by comparison of the obtained results. Presented procedure for determination of optimum theoretical model of empirical distribution is prevailing. It can be applied for mathematical modelling of empirical distribution of arbitrary continuous random variable. 321

Cognition of distribution laws of machine elements reliability enables determination of reliability of the entire system as a function of reliability of constitutive elements, the planning of maintenance measures, the determination of optimum retention period for constitutive parts or for the entire system, etc. REFERENCES 1. N. Janicijevic, D. Jankovic, J. Todorovic: Construction of Motor Vehicles. Faculty of Mechanical Engineering, Belgrade, 1987. 597 p. (in Serbian). 2.  D. catic: Development and Application of Reliability Theory Methods. Faculty of Mechanical Engineering from Kragujevac, Kragujevac, 2005. 241 p. (in Serbian). 3. I. Z. ARONOV, E. I. BURDASOV: Reliability Assessment Based on Results of Truncated Testing. Moscow, 1987, p. 184 (in Russian). 4.  J. Knezevic: Management of Maintenance Processes and Renewal of Technical Systems Based on the Theory of Reliability. OMO, Belgrade, 1988. 263 p. (in Serbian). 5.  Documentation, Catalogs and Service Manuals of the Factory of Commercial Vehicles ‘Zastava kamioni’ from Kragujevac. 6. R. S. SUDAKOV, O. I. TESKIN (Eds): Reliability and Efficiency in Technique. Manual T. 6, Experimental Processing and Testing, Mashinostroenie, Moscow, 1989, Vol. 6, p. 376 (in Russian). 7. A. I. KUBAREV, E. A. PANFILOV, B. I. HOHLOV: Reliability of Machines, Equipment and Devices of Vital Importance. Legprom-bitizdat, Moscow, 1987, Vol. 6, p. 336 (in Russian). 8. I. Z. ARONOV: Safety Indices Assessment in Planning of Testing of Operation Time until Failure. Reliability and Efficiency in Technique. Manual 6, Experimental Processing and Testing, Mashinostroenie, Moscow, 1989, 60–82 (in Russian). Received 5 May 2009 Revised 22 May 2009

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