Holooly Plus Logo
Holooly Plus Logo

Question 24.4: The tolerance specified by the designer for the diameters of...

The tolerance specified by the designer for the diameters of a transmission shaft is 25.000 ± 0.025 mm. The shafts are machined on three different machines. It was observed from the sample of shafts that the diameters are normally distributed with a standard deviation of 0.015 mm for each of the three machines. However, the mean diameter of shafts fabricated on the three machines is found to be 24.99, 25.00 and 25.01 mm respectively. Determine the percentage of rejected shafts in each case and comment on the result.

The Blue Check Mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

\text { Given } \hat{\sigma}=0.015 mm \quad X_{1}=25.000-0.025 mm .

X_{2}=25.000+0.025 mm \quad \mu_{1}=24.99 mm .

\mu_{2}=25 mm \quad \mu_{3}=25.01 mm .

Step I Percentage of rejected shafts in each machine
The lower and upper limits specified by the designer are,

X_{1}=25.000-0.025=24.975 mm .

X_{2}=25.000+0.025=25.025 mm .

The results are tabulated in the following way:

Machine C Machine B Machine A
25.01 25.00 24.99 mean μ (mm)
–2.33 -1.67 -1 Z_{1}=\frac{X_{1}-\mu}{\hat{\sigma}}
+1 +1.67 +2.33 Z_{2}=\frac{X_{2}-\mu}{\hat{\sigma}}
0.4901 0.4525 0.3413 \text { Area } A_{1}

\left(Z=0 \text { to } Z_{1}\right)
0.3413 0.4525 0.4901 \text { Area } A_{2}

\left(Z=0 \text { to } Z_{2}\right)
0.8314 0.9050 0.8314 \left(A_{1}+A_{2}\right)
16.86 9.5 16.86 Percentage
rejection

Step II Comments on result
The above data is illustrated in Fig. 24.13. It is observed from the figure that in case of machines A or C, the process is not centred, which results in a large percentage of rejection. With the same magnitude of tolerance, the percentage of rejection is small in case of the machine B, because the process is centred.

24.13

Related Answered Questions

It has been observed from a sample of 200 bearing bushes that the internal diameters are normally distributed with a mean of 30.010 mm and a standard deviation of 0.008 mm. The upper and lower limits for the internal diameter, as specified by the designer, are 30.02 and 30.00 mm respectively.

Verified Answer:
\text { Given } \quad \mu=30.010 mm \quad ...

The life of a ball bearing is a normally distributed random variable, with a mean of 10 000 h and a standard deviation of 500 h. The manufacturer of the bearings wants to give a guarantee that 90% of the bearings will reach or exceed the rated life published in his catalogue. What should be the

Verified Answer:
\text { Given } \quad \mu=10000 h \quad \h...

Tension test is carried out on 120 specimens made of grey cast iron of grade FG300. It is observed that the ultimate tensile strength (UTS) is normally distributed with a mean of 300 N/mm^2 and a standard deviation of 25 N/mm^2. (i) How many specimens have UTS less than 275 N/mm^2? (ii) How many

Verified Answer:
\text { Given } \mu=300 N / mm ^{2} \quad ...

A mechanical component is subjected to a mean stress of 100 N/mm^2 and a standard deviation of 10 N/mm^2. The material of the component has a mean strength of 130 N/mm^2 and a standard deviation of 15 N/mm^2. (i) Find the probability of failure for the component. (ii) If better manufacturing

Verified Answer:
\text { Given } \mu_{S}=130 N / mm ^{2} \q...

A mechanical component is subjected to a normally distributed force with a mean of 1000 N and a standard deviation of 200 N. The designer has used a factor of safety of 1.5 based on mean values. However due to variations in dimensions, the strength of the component is normally distributed with a

Verified Answer:
\text { Given } \mu_{S}=1500 N \quad \hat{...

One hundred test specimens made of grey cast iron FG 300 are tested on a universal testing machine to determine the ultimate tensile strength (Sut) of the material. The results are tabulated as follows: Calculate: (i) the mean; (ii) the variance; and (iii) the standard deviation for this sample.

Verified Answer:
Step I Mean f_{i} X_{i}^{2} [la...

Five bolts with internal cracks are accidentally mixed with 95 bolts without any defects. What is the probability that the assembly shop will use a defective bolt? Also, fi nd out the possibility of not using the defective bolts.

Verified Answer:
Given Number of bolts without defect = 95 Number o...

The recommended class of fit for the journal and the bush of a hydrodynamic bearing is 40H6-e7. The dimensions of the journal and the bush are normally distributed and the natural tolerance is equal to the design tolerance. From the considerations of hydrodynamic action and bearing stability, the maximum

Verified Answer:
Given Class of fi t = 40H6-e7. X_{1}=0.08 ...

The recommended class of transition fit between the recess and the spigot of a rigid coupling is 60H6-j5. Assuming that the dimensions of the two components are normally distributed and that the specified tolerance is equal to the natural tolerance, determine the probability of interference fit

Verified Answer:
Given Class of fi t = 60H6-j5 There are two popula...

An assembly of three components A, B and C is shown in Fig. 24.15. The dimensions of the three components are normally distributed and natural tolerance is equal to design tolerance as shown in the figure. Determine the percentage of assemblies where interference is likely to occur

Verified Answer:
Step I Population of component-A (A) For the popul...