Keywords

1 Introduction

Vehicle reliability and maintenance is one of the essential functions in the logistics sector, involving routine checkup and preventive maintenance. It can be a costly function when it does not perform on time, in turn, results in various vehicle system failure. Chisa et al. [1] considered a statistical model to perform the reliability analysis to predict the performance parameters of the automobile subsystem. The results reflected that preventive maintenance plays a significant role to avoid instant system failure. Jbili et al. [2] also carried out a similar study to identify the reliability of heavy vehicles and suggested that the vehicles travelling over long distances must undergo frequent maintenance. It is always necessary to conduct studies pertaining to assistance to vehicle maintenance perform and incorporate new technologies to avoid car system failure [3, 4]. In a similar context, the study of Dobromirov et al. [5] reviewed and examined the systematization of different automobile sector. Authors concluded and suggested to enhance the use of new technologies associated with of latest devices equipped for regular maintenance. Another factor which is also essential to avoid sudden system failure is the vehicle to vehicle communication [6]. Minimal literature was reported to examine the reliability of the automobile subsystem of cars. The present study aims to analyse the reliability of car subsystem Wagon-R. The reliability of different systems was calculated on the basis of distance travelled by various cars.

2 Methodology

Data of car model Maruti Suzuki Wagon-R were collected from a private service station and represented in Table 1. The categorization of data is done as per the travelled distance by the respective cars such as up to 25,000 km, between 25,001–50,000 and 50,001–75,000 km. In addition to this, the collected details were analysed, respectively, car systems like the clutch, front brake, suspension and rear brake system. The collected data were analysed using the Weibull distribution method (WDM), as the WDM results reasonably more accurate and also forecast with extremely small sample size.

Table 1 Collected sample data of Maruti Suzuki Wagon-R
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The following procedure is adopted to calculate the reliability of Wagon-R,

  1. 1.

    Plot a bar graph for each system of collected data and take abscissa as the number of cars taken and ordinate as a distance before failure.

  2. 2.

    Then, we convert this distance before failure to time before failure by assuming speed of vehicles as 60 km/h.

  3. 3.

    Find total average distance between failure and total average time to failure by simply taking average of distance between failure by number of car failed, and also, arrange in ascending order.

  4. 4.

    Find median rank using following equation

    $${\text{M.R.}}\% = \frac{{i - 3}}{{N + 4}} \times 100$$
    (1)

where i = number of sample arranged in ascending order as failure, N = number of sample taken.

  1. 5.

    For analysing the data, we use two parameter Weibull distribution and find reliability as a relation

    $$R(t) = {\text{e}}^{{\left( {\frac{{ - t}}{\eta }} \right)^{\beta } }}$$
    (2)

where R(t) = reliability function, t = time, η = scale parameter, β = slope parameter

$$F(t) = 1 - R(t)$$
(3)

where F(t) = unreliability function.

To calculate reliability, we have to estimate the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood. For our work, we use rank regression on Y on principle of least square method. Performing rank regression on Y requires that straight line mathematically be fitted to be a set of data points such that the sum of squares of vertical deviations from the points to be minimized. The first step is to bring our function linear form. For two parameter Weibull distribution, the cumulative density function is

$$F(t) = - \beta \ln (\eta ) + \beta \ln (t)$$
(4)
$$y = \ln \left[ { - \ln \left( {1 + F(t)} \right)} \right]$$
(5)
$$a = - \beta \ln (\eta )$$
(6)
$$b = \beta$$
(7)
$$x = \ln (t)$$
(8)

Which results in the linear equation of y = a + bx, the least square parameter estimation method (also known as regression analysis) was used, and following equation for regression on Y was used:

$$\hat{a} = \frac{{\sum\nolimits_{i = 1}^{N} {y_{i} } }}{{N}} - \hat{b}\frac{{\sum\nolimits_{i = 1}^{N} {y_{i} } }}{{N}} = \overline{y} - \hat{b}\overline{x}$$
(9)
$$\hat{b} = \frac{{\sum\nolimits_{i = 1}^{N} {{x_{i} y_{i} }} - \frac{{\sum\nolimits_{i = 1}^{N} {x_{i} } \sum\nolimits_{i = 1}^{N} {y_{i} } }}{N}}}{{\sum\nolimits_{i = 1}^{N} {x_{i}^{2} } - \frac{{\left( {\sum\nolimits_{i = 1}^{N} {x_{i} } } \right)^{2} }}{{N}}}}$$
(10)

In this equations for yi and xi are: where yi = y for i sample in (v), xi = x for i sample in Eq. (3) and f(t) for i sample is calculated from median ranks (M.R.) and construct a table as per respective figure and calculate β and η with the help of Eqs. (4)–(9) and calculate N, t, x = ln(t), F(t), y, x2, y2, x * y.

  1. 6.

    After calculating the parameter, we find the reliability at the point of average time to failure for selected car system. The aim of using the traditional technique for car maintenance is to calculate reliability function of time R(t) of car subsystem. For calculating the reliability function R(t) for each subsystem, the collected data were converted from average distance to failure to average time to failure by assuming the sample-A cars travelled the distance by average speed 60 km/h. This is because the reliability function which was used in this study is a function of time, where the reliability decreases as time increases.

  2. 7.

    Assuming the time and compare the reliability at this time for the samples subsystem and find which is more reliable system of Wagon-R.

The limitations of the study include no cost analysis were performed as this study mainly focused to identify the failure sequence of automotive subsystem.

3 Results and Discussions

3.1 Clutch System

Figure 1 represents the graph between the number of cars failed for a clutch system according to the distance travelled. It can be observed from Table 2, the car serial numbers 2 and 5 failed after 40,100 km for running of 668.33 h. and 42,000 km for running of 700 h., respectively. Similarly, for car serial numbers 8, 11 and 12 failed after 50,000 km for a running of 833.33 h, 12,000 km for running of 200 h. and 58,000 km for a running of 966.67 h., respectively. It can be inferred that the failure of the clutch system started after the initial running of 12,000 km and ranges up to 58,000 km for an initial run. Moreover, a new Wagon-R car clutch system failure ranges from 200–966.67 h of running. Tables 3 and 4 show the different calculation for reliability analysis.

Fig. 1
figure 1

Failure of clutch system in various cars

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Table 2 Details of time to failure for clutch system
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Table 3 Median ranks of clutch system
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Table 4 Calculation table of clutch system
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3.2 Rear Brake System

Figure 2 depicts the graph of distance travelled before failure of rear brake system for different cars serial number, and Table 5 shows the actual details of distance travelled with failure time. It is clear from Table 5 rear brakes of car serial number 4 failed at 29,000 km after running of around 483.33 h. For serial number 5 and 7, the distances and time are 74,000 with 12,333.33 h and 40,000 km with 666.67 h, respectively. For others, serial numbers such as 9, 11 and 13 rear brake failed at 39,000 km after 650 h, 60,000 km after 1000 h and 45,000 km after 750 h. The range of distance travelled starts with 29,000–74,000 km with a time frame of 483.33–12,333 h. running. Tables 6 and 7 represent the various calculation values for rear brake reliability.

Fig. 2
figure 2

Rear brake system

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Table 5 Details of time to failure for rear brake of system
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Table 6 Median ranks of rear brake system
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Table 7 Calculation table of rear brake system
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3.3 Suspension System

Figure 3 shows the failure graph of various car suspension system, and Table 8 depicts the distance of failure along with time to failure. It can be seen that the failure of suspension started after running of 27,000 km for 450 h. For serial no. 9, for car no. 2, 5, 6, 7 and 14, suspension system failed after running of 48,000 km for 800 h, 33,500 km for 558.33 h, 54,000 km for 900 h, 45,000 km for 750 h and 35,000 km for 583.33 h running. It may be noted that the car serial number 4 suspension system failed two times after travelling of 42,000 and 51,000 km. From Table 8, it can be inferred that suspension system is more liable to failure after clutch system and front brake system (Tables 9 and 10).

Fig. 3
figure 3

Suspension system

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Table 8 Details of time to failure for suspension system
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Table 9 Median ranks of suspension system
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Table 10 Median ranks of suspension system
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3.4 Front Brake System

Figure 4 shows graph of failure of front brake system for various cars travelled up to 70,000 km, and Table 11 represents the time of failure along with distance. The front brake failure for car no. 2, 6, 12, 13, 14 and 15 has travelled 50,000 km at around 833.33 h, 38,000 km at around 633.33 h, 44,000 km at 733.33 h, 58,000 km at 966.67 h, 21,000 km at around 350 h and 32,000 km at around 533.33 h, respectively. It can be noted that the car serial no. 5 front brake failed two times after travelling of 60,000 and 64,000 km at around 1000 h and 1066.67 h, respectively (Tables 12 and 13).

Fig. 4
figure 4

Front brake system

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Table 11 Details of time to failure for front brake system
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Table 12 Median ranks of front brake system
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Table 13 Calculation table of front brake system
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4 Conclusion

In the present work, the reliability of automotive segment Wagon-R was calculated for various associated systems such as clutch, suspension, rear and front brake based on the travelled distance. The results showed that most of the vehicles fails in the sequence of the clutch, front brake, suspension followed by rear brake and the concerned system failed at 12,000, 21,000, 27,000 and 29,000 km of travelling. As, the condition of failure depends upon the various factors such as driving conditions, regular checkup or servicing, condition of car, etc. Results prevails that, it is advisable to carry out the routine maintenance after achieving an initial distance of 10,000 km. Moreover, to maintain the reliability of every associated system of car, it is essential to perform the maintenance after every 3000–5000 km only after running of initial 10,000 km. The results and approach may be further utilized to perform the reliability analysis for various other automotive systems to avoid any failure.