Design of a testing bench, statistical and reliability analysis of some mechanical tests

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),

ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME

36

DESIGN OF A TESTING BENCH, STATISTICAL AND RELIABILITY ANALYSIS OF SOME MECHANICAL TESTS

Emmanuel NGALE HAULIN

Corresponding Author, University of Maroua, P.O. BOX 46 Maroua Cameroon

[email protected], Tel.: +237 77695790/96391889

Fax : +237 22291541/22293112

Ebénézer NJEUGNA

Kamtila WADOU University of Douala P.O. BOX 1872 Douala Cameroon

ABSTRACT

A testing bench was designed and manufactured in order to determine simultaneously

mechanical properties of materials and stiffness of helical extension springs or

absorption factor of shock absorbers.

The combination of one helical extension spring with eight ebony wood test

specimens enable to obtain, using the chi-square nonparametric statistical test at 95%

confidence with a reliability of 50%, the mean value of spring stiffness K or resilience

KCU of ebony wood and their standard deviation S :

• Spring stiffness: K= 636.4N/mm and S = 158.82N/mm;

• Ebony wood resilience: KCU = 21.6 J/cm2 and S = 4.5 J/cm

2.

The combination of four annealed and polished ordinary glass test specimens with one

helical extension spring led to obtain firstly the spring stiffness K = 9.95 N/mm and

secondly, using the parametric statistical test of Student-Fisher, the tensile strength of

annealed and polished ordinary glass Sut = 37.818 MPa within the confidence interval

Ic = [27.238, 48.398] MPa at 99% confidence and a standard deviation S = 8.552

MPa.

The combination of four annealed and polished ordinary glass test specimens with one

shock absorber led to obtain firstly the absorption factor of the shock absorber C =

5.176 N / mm and secondly, using the Fisher-Student test, the tensile strength of

polished and annealed ordinary glass Sut = 44.327 MPa within a confidence interval Ic

= [38.349, 50.248] MPa at 99% confidence and a standard deviation S = 2.047 MPa.

The final value of the tensile strength of polished and annealed ordinary glass

obtained, after an homogeneity statistical test applied to the two previous

combinations, is Sut = 39.989MPa within a confidence interval Ic = [33.495, 46.483]

MPa at 99% confidence and a standard deviation S = 7.2MPa.

Key words: mechanical design, testing bench, mechanical tests, statistics, reliability.

International Journal of Mechanical Engineering

and Technology (IJMET), ISSN 0976 – 6340(Print)

ISSN 0976 – 6359(Online) Volume 2

Number 1, Jan - April (2011), pp. 36-59

© IAEME, http://www.iaeme.com/ijmet.html

IJMET

© I A E M E



37

1. INTRODUCTION

Some testing benches were developed for the determination of stiffness of materials

and their resilience which are respectively the capacity of machine elements to avoid

excessive distortion under applied loads and their ability to absorb a certain amount of

energy (shock or dynamic loading) without damage.

Indeed, CHARPY pendulum [1], FREMONT resilience machine [2] and

NGOUAJOU machine [3] are used to determine resilience of materials. The

advantages of these three machines lie in their small dimensions and easy assembly.

FREMONT machine allows also a direct reading of the spring deflection linked with

specimen to be tested. Their common disadvantage is the determination of only one

mechanical property which is resilience of materials. The main disadvantages of

CHARPY machine are a lack of security during dynamic loading and a constant

potential energy (300J) [1]; those of FREMONT machine are friction and

deformations in the guides, fixed dimensions of the spring and a lack of back system

linked with the cursor used to read the deflection.

More over, springs with unknown stiffness are increasingly used in technical schools

and garages in Cameroon. However, two special devices are often used to determine

respectively stiffness and deflection of valve springs [4].

Some authors [5,6] used the coupling method in order to determine simultaneously

physical constants of more than one material. The aim of our study is then to

determine simultaneously, firstly resilience of materials and stiffness of helical

extension springs and secondly, tensile strength of materials and absorption factor of

shock absorbers or stiffness of helical extension springs by the means of a testing

bench designed and manufactured at the University of Douala, Cameroon.

This paper has four main parts. The first two parts concern conceptual and graphics

designs of the testing bench. The two last one deal firstly with mechanical tests and

secondly with statistical and reliability analysis of results obtained.

2. CONCEPTUAL DESIGN

2.1 MACHINE DESCRIPTION

2.1.1 Kinematic diagram

Figure 2.1 shows the kinematic diagram of the testing bench used for ebony wood test

specimens. After changing the fastening system of test specimen, machine

configuration is that of the figures 2.2 a) and b) and is used for polished and annealed

ordinary glass.

2.1.2 Functioning principle

Figure 2.1shows a 5 kg mass 15 which is in equilibrium at the height h from the test

specimen 4 by the means of a block 9 and is equipped with a knife 5 intended to strike

the specimen in the opposite direction of its notch.



38

Figure 2.2 shows mass 6 which is in equilibrium, by the means of a binding screw 5,

at the height h from the retaining plate 3 tied to the test specimen 4. Mass m is

intended to strike that retaining plate.

The mass, when released, is guided in translation on the frame 2 by the means of four

slides. An amount of its potential energy, when converted into kinetic energy, is

absorbed by the test specimen and the excess by the spring or shock absorber. The

cursor, with negligible friction, records the maximum deflection X of the spring or

shock absorber. Mass m is then raised up to a height h by the means of a cable which

winds round the pulley 13.

2. 2 MACHINE ELEMENTS DESIGN

The main elements of the testing bench have been designed according to the material

used and the applied loads. This paper presents only the design of the crank shaft 12

which is the main part of the lifting system of mass m and which brings this mass in

an equilibrium position before each test. The force F due to tension in cable 11 and

equal to 50N, will be used to design the crank shaft 12 subjected to bending and

torsion.

Determination of the crank shaft 12

The material used, 42CD4, has the following characteristics: yield strength Sy = 1500

N/mm², Young’s modulus E = 2.05 105 N/mm² [7]. The design is done during the

raising of mass m. Figure 2.3 shows the lifting system of the mass m.

The study of internal forces determines the critical section of the crank shaft 12 which

is in B where the maxima of bending moment and torque are

respectively Nm5MetNm5.9Mmaxtmaxfz == . Using the maximum shear stress

theory [8,9], stress concentration factors for normal and shear stresses kf = kts = 3

[8]and a factor of safety s = 3 [10], the maximum and minimum principal stresses

were determined and led to obtain a diameter 12.73mmd ≥ . Let us consider d = 20

mm.

3. GRAPHIC DESIGN

AutoCAD 2009 was used to draw the testing bench shown in the general assembly

drawing of figure 3.1. New machine elements references, different from those used in

the kinematic diagrams of figures 2.1 and 2.2, are taken into account and used later in

this study.

All necessary clearances [7] for the proper functioning of the testing bench were

defined and shown on its general assembly drawing. Dimensions of machine elements

related to those clearances were determined. Finally, each of these elements was

drawn.

This paper presents only the clearances related to the proper functioning of the crank

shaft 18 and its detail drawing respectively in figures 3.2 and 3.3.



39

4. MECHANICAL TESTS

4.1 TESTS METHODOLOGY

We present here the methodology of the experimentation which determines

simultaneously in the one hand the mechanical properties of materials and in the other

hand, the stiffness of helical extension springs or the absorption factor of shock

absorbers. Therefore, we will successively:

• Apply the energy transfer and the conservation of energy principles to express the

energy absorbed by the breaking or failure of a test specimen;

• Use the properties of homogeneous materials to deduce the values of spring

stiffness and absorption factor of the shock absorber used.

• Deduce the test specimen resilience or tensile strength.

First of all, we use n = 8 identical test specimens of ebony wood. Changing the

docking system of test specimens results in the use of n = 12 identical test specimens

of ordinary glass polished and annealed. The mass (m) is placed at a height h from the

point of impact. Its potential energy is iPi mghE ==== , i varies from 1 to n. A test

specimen is placed on its supports. Then mass (m) is released and falls freely. A

quantity Wi of its potential energy (converted into kinetic energy) is absorbed by the

failure of the test specimen and the excess by the spring or shock absorber. A cursor

registers the maximum deflection Xi of spring or shock absorber.

From the compression of spring or shock absorber, the following potential energies

can be obtained:

• i

'

Pi mgXE ==== (mass m) ;

• 2

iSi KX2

1W ==== (spring) or

i

2

miai CghXC2

1W ======== & [12] (shock absorber) with

miX& the mass velocity at the beginning of the compression.

Applying the principle of mechanical energy conservation, we have:

(((( ))))

(((( ))))

−−−−++++====

−−−−++++====⇒⇒⇒⇒

++++====++++⇔⇔⇔⇔++++====++++

++++====++++⇔⇔⇔⇔++++====++++

iiii

2

iiii

iiiiaiiPiPi

2

iiiiSiiPiPi

Cgh)Xh(mgW

KX2

1)Xh(mgW

CghWXhmgWW'EE

KX2

1WXhmgWW'EE

Equating two consecutive energies obtained at ijij hhwithhandh ≠≠≠≠ , we have:

−−−−

−−−−−−−−++++====⇒⇒⇒⇒−−−−====−−−−−−−−++++

−−−−

−−−−−−−−++++====⇒⇒⇒⇒−−−−====−−−−−−−−++++

ji

jjii

ijjiijjjii

2

j

2

i

jjii

ij

2

j

2

iijjjii

hh

)XhXh(mC)hh(gC)XhXh(mg

XX

)XhXh(mg2K)XX(K

2

1)XhXh(mg

with Kij and Cij respectively the spring stiffness and the absorption factor of the shock

absorber after two consecutive tests i and j.



40

4.2 DETERMINATION OF BOTH SPRING STIFFNESS AND RESILIENCE

OF EBONY WOOD

8 trials are performed using one helical extension spring and 8 identical test

specimens of ebony wood and. Table 4.1 presents the results obtained.

4.2.1 Determination of the spring stiffness

Table 4.2 shows the stiffness matrix Kij of spring used after 8 trials. The spring

stiffness is finally (((( ))))

mm/N.4.63627

K

1nn

K2

2

nn

K

K

8

1j,i

ij

n

1j,i

ij

2

n

1j,i

ij

========−−−−

====−−−−

====

∑∑∑∑∑∑∑∑∑∑∑∑============

with i < j (table 4.2

has 27 values instead of 28). Applying the strength of materials formula,

mm/N862.625nD8

GdK

3

4

======== with:

d (wire diameter) = 6.15mm;

D (average diameter of winding) = 20mm;

n (number of active coils) = 3;

G (shear modulus of elasticity) = 84000MPa (for spring made with steel [13]);

4.2.2 Determination of ebony wood resilience

Changing K by Kij in Wi or Wj expressions enables to obtain the energy Wij absorbed

by the failure of ebony wood test specimen and given in table 4.3. The average value

of this absorbed energy is then (((( ))))

2

8

1j,i

ij

n

1j,i

ij

cm/J80.1027

W

1nn

W2

W ========−−−−

====

∑∑∑∑∑∑∑∑========

with i < j.

Generally, resilience S

WK C ==== [1,2] with :

S (cm2) = cross section at the notch of the test specimen;

W (Joule) = energy absorbed by the test specimen;

KC (Joule/cm2) = material resilience

Taking into account the geometry of the notch of the test specimen, we have, for the

Charpy U-notch shown in figure 4.1, the average resilience of ebony wood 2

CU cms21.60joule2WK == [1,2];

4.3 DETERMINATION OF BOTH SPRING STIFFNESS AND TENSILE

STRENGTH OF ORDINARY GLASS POLISHED AND ANNEALED

It is performed using one helical extension spring and 8 identical test specimens of

polished and annealed glass. Table 4.4 presents the results obtained.



41

4.3.1 Determination of the spring stiffness

Table 4.5 shows the stiffness matrix Kij of helical extension spring for 8 trials. The

spring stiffness is finally mm/N.95.928

K

K

8

1j,i

ij

========

∑∑∑∑====

with i < j; Applying the strength of

materials formula, mm/N084.10nD8

GdK

3

4

======== with :

d (wire diameter) = 3.5mm;

D (average diameter of winding) = 25mm;

n (number of active coils) = 10;

G (shear modulus of elasticity) = 84000MPa (for spring made with steel [13]);

4.3.2 Determination of tensile strength of ordinary glass polished and annealed

The failure energy of the glass test specimen (figure 4.2) used with a helical extension

spring is 2

sisisisi KX2

1)Xh(mgW −−−−++++==== [2]. Moreover, according to Von Mises, the

expression of the elastic strain energy in traction is 0

0

2

2ES

LPU = [10] with :

P = mg: tensile load;

L0 = 50mm: test specimen length;

E: Young’s modulus;

S0= 9mm2: cross square section of the test specimen.

By analogy, the tensile failure energy is0

0

2

2 SS

LPW

uts

u = [12] where Suts is the tensile

strength of the material used. Therefore, Wsi =

Wu

( )[ ]2

0

0

2

2 sisisi

utsiKXXhmgS

LPS

−+=⇒ . Table 4.6 gives the values of the tensile

strength of glass when used with a helical extension spring for each of 8 trials.

The average value of the tensile strength of glass polished and annealed, when used

with a helical extension spring, is equal to MPa82.378

S

S

8

1i

utsi

uts ========

∑∑∑∑==== .

4.4 DETERMINATION OF BOTH ABSORPTION FACTOR OF SHOCK

ABSORBER AND TENSILE STRENGTH OF ORDINARY GLASS

POLISHED AND ANNEALED

It is performed using one shock absorber and 4 identical test specimens of polished

and annealed glass. Table 4.7 shows the results obtained.

4.4.1 Determination of the absorption factor of shock absorber



42

Table 4.8 gives the absorption factor matrix Cij of the shock absorber. The absorption

factor of the shock absorber is finally m/s.N176.56

C

C

4

1j,i

ij

========

∑∑∑∑====

with i < j.

4.4.2 Determination of tensile strength of ordinary glass polished and annealed

When using a shock absorber, the failure energy of the test specimen

isaiaiaiai CgX)Xh(mgW −−−−++++==== and the tensile strength of the ordinary glass when

using this shock absorber at a trial i, by analogy to that obtained when using a helical

extension spring, is(((( ))))[[[[ ]]]]aiaiai

2

utaiChXhmgA2

LPS

−−−−++++==== . Table 4.9 gives the values of

this tensile strength for each of 4 trials. The average value of the tensile strength of

glass polished and annealed, when used with a shock absorber, is equal

to MPa327.444

S

S

4

1i

utai

uta ========

∑∑∑∑==== .

5. STATISTICAL AND RELIABILITY STUDIES

The results obtained from the testing machine are subjected to two kinds of statistical

tests:

• the nonparametric chi-square test which verifies if all test values obtained for the

determination of both spring stiffness K and resilience KCU of ebony wood obey

to the statistical law chosen;

• the parametric Student-Fischer test which is used to compare the tensile strength

Sut known and published of the ordinary glass polished and annealed with the

average value obtained from a small sample (n < 30) used for:

o the determination of both helical extension spring stiffness K and tensile

strength Suts of ordinary glass polished and annealed;

o the determination of both absorption factor C of shock absorber and tensile

strength Suta of ordinary glass polished and annealed.

5.1 CHI-SQUARE (χ2) TEST

5.1.1 Choice of a statistical law followed by the experimental results

In the experimental results provided in tables 4.3, 4.4 and 4.5, we find a slow and

gradual change of parameters hi, Xi, Kij and Wij. Therefore, the statistical hypothesis

consists to assume that the normal distribution is the most likely parent to these

parameters [14].

5.1.2 Estimated parameters of the normal distribution

Considering a sample of n data, the estimated parameters of the normal distribution



43

are the sample mean ∑∑∑∑====

====n

1i

itn

1t and the sample standard deviation

(((( ))))∑∑∑∑====

−−−−−−−−

====8

1i

2

i tt1n

1S [10]. Since Kij and Wij are stochastic variables:

• The mean value of spring stiffness is mm/N4.63627

K

n

K

KK

8

1j,i

ij

8

1j,i

ij

================

∑∑∑∑∑∑∑∑========

where i < j. Its standard deviation is

(((( ))))(((( ))))

mm/N82.15826

KK

KK1n

1S

8

1j,i

2

ij8

1j,i

2

ij ====

−−−−

====−−−−−−−−

====

∑∑∑∑∑∑∑∑

====

====

with i < j.

• The mean value of the ebony wood resilience is

2

8

1j,i

ij

8

1j,i

ij

CUCU cm/J6.2127

W2

n

W2

W2KK ====================

∑∑∑∑∑∑∑∑========

with i < j. Its standard

deviation is 2

8

1j,i

2

CU

ij8

1j,i

2

CU

ij cm/J04.526

2

KW4

2

KW

1n

4S ====

−−−−

====

−−−−

−−−−====

∑∑∑∑∑∑∑∑

====

====

with i < j.

5.1.3 Verification of statistical hypothesis

It is now important to say whether the random variables that are the spring stiffness

and ebony wood resilience effectively obey the normal distribution with the

calculated parameters (mean and variance).

Number of intervals N of the chi-square (χ2) test The number of intervals is N = 1 +3.3 log n = 5.72. Given our sample n = 27, we

adopt N = 5. The restriction is that at least 5 theoretical failures must exist within each

interval.

Theoretical number of failures Fi for each interval i Fi = n x [F(ti)-F(ti-1)] where i = 1 , 2 , … N. and F(ti) - F(ti-1) = probability for a

failure to be in the interval i. The tables 5.1 and 5.2 give this theoretical number of

failures per interval respectively for the spring stiffness and the ebony wood

resilience.

Chi square (χ2) statistic Respectively for the spring stiffness and the ebony wood resilience, tables 5.3 and 5.4

give χ2 statistic:

( )∑

−=χ

i

2

ii2

F

Ff where fi = failures number within interval i.



44

Tabuled values of chi square statistic

Tables [14,15] give the critical value of (((( ))))dχ2

P where the confidence level P = 1 - α =

0.95 and the degree of freedom d = (N -1) - Z = 2 with Z the number of estimated

parameters (mean and variance). This critical value is compared to χ2 calculated:

• ( )2χ2

0.95 = 5.99 > χ2 = 0.42 for the spring stiffness K. Therefore, the assumption of

normal distribution is verified with a confidence level 0.95;

• ( )2χ2

0.95 = 5.99 > χ2 = 4.13 for the resilience KCU of ebony wood. The hypothesis

of normal distribution is verified with a confidence level 0.95;

5.1.4 Reliability of test results

The normal distribution is the most likely parent to random variables that are spring

stiffness K and resilience KCU of ebony wood. The failure probability F (t) represents

the probability that the random variables are less than the value ti. The reliability R (t)

= 1 - F (t) is the probability that these variables are greater than ti. Tables 5.5 and 5.6

show, for each failure ti observed, the failure probability and the reliability of these

variables.

These tables show that the reliability of mean values of spring stiffness (K = 636.4 N /

mm) and the resilience (KCU = 21.6 J/cm2) of ebony wood is equal to 0.5. Moreover, it

is higher than 0.5 below these values and less than 0.5 above them.

5.2 STUDENT-FISHER TEST

5.2.1 Checking of the normality assumption for the tensile strengths Suts and Suta

The normality assumption underlying the data is most often used for the Student-

Fisher test. From tables 4.7 and 4.8 giving respectively n1 = 8 values Suts of tensile

strength of ordinary glass polished and annealed and n2 = 4 values Suta of this

mechanical property, the asymmetry factor α3 and the flattening one α4 are virtually

nil. It follows that the normal distribution is the most likely parent to these two sets of

values [14]. Asymmetry factor is

3

3

3S

kα ==== [15] where

3S is the third power of

standard deviation and

(((( ))))

(((( ))))(((( ))))2n1n

RRnn

1i

3

rri

−−−−−−−−

−−−−

====

∑∑∑∑====

3k . Flattening factor is 4

4

4S

kα ==== [15] where

4S is the fourth power of standard deviation and

(((( )))) (((( )))) (((( )))) (((( ))))

(((( ))))(((( ))))(((( ))))3n2n1n

RR1n3RR1nn

2n

1i

2

rri

n

1i

4

rri

−−−−−−−−−−−−

−−−−−−−−−−−−−−−−++++

====

∑∑∑∑∑∑∑∑========

4k ).

5.2.2 Confidence intervals of Suts and Suta

At a confidence level P = 1 - α = 0.99, the confidence interval of tensile strength Sut



45

with unknown variance σ2 is given by

++++−−−−

−−−−

−−−−

−−−−

−−−− n

StS,

n

StS

1n

21ut

1n

21ut αα

where

∑∑∑∑====

====n

1i

utiut Sn

1S , (((( ))))∑∑∑∑

====

−−−−−−−−

====n

1i

2

ututi SS1n

1S and 1n

21

t −−−−

−−−−α read in the table of Student at

99% confidence and n - 1 degrees of freedom [13].

For ordinary glass and helical extension spring association: MPa818.37SS utsut ======== ,

n1 = 8, t = 3.499 and S1 = 8552. The confidence interval including Suts is [27.238,

48.398].

For ordinary glass and shock absorber association:

MPa327.44SS utaut ======== , n2 = 4, t

= 5.841 and S2 = 2.047. The confidence interval including Suta is [38.349, 50.248].

5.2.3 Test of conformity on the difference between two means

The standard value of the tensile strength of glass polished and annealed is Sut = 40

MPa [13]. This value must be compared to Suts and Suta. The test statistic

1n

S

SSt

2

utut

−−−−

−−−−==== must be less than the value tlimit read in the table of Student at 99%

confidence and

n - 1 degrees of freedom. Indeed, for the combination of ordinary glass and helical

extension spring, 998.2t656.0

7

552.8

40818.37t itlim

2====<<<<====

−−−−==== ; the combination of

ordinary glass and shock absorber gives 541.4t661.3

3

047.2

40327.44t itlim

2====<<<<====

−−−−==== .

Therefore, each of these two values Suts and Suta is representative of the tensile

strength of the polished and annealed ordinary glass.

5.2.4 Homogeneity test of two samples

The test statistic

++++

−−−−====

21

2

utsuta

n

1

n

1S

SSt of samples n1 and n2 must be less than the value

tlimit read in the table of Student at 99% confidence and (n1 + n2 - 2) degrees of

freedom. Common variance to both samples is

(((( )))) (((( ))))

−−−−++++−−−−

−−−−++++==== ∑∑∑∑∑∑∑∑

========

2n

1i

2

utautai

1n

1i

2

utsutsi

21

2SSSS

2nn

1S [15]. Indeed,



46

764.2468.1

4

1

8

1454.52

818.37327.44t <<<<====

++++

−−−−==== . Therefore, both samples are representative of

the same material. We can therefore assume that our sample size is n = n1 + n2 = 12,

and estimate the mean value of the corresponding tensile strength Sut.

5.2.5 Mean value of ordinary glass tensile strength Sut

The mean value of tensile strength of the sample size n = 12 is

MPa989.39nn

SS

S21

2n

1i

utai

1n

1i

utsi

ut ====++++

++++

====

∑∑∑∑∑∑∑∑======== .

5.2.6 Confidence interval of the tensile strength Sut

At 99% confidence, this interval is given by

++++−−−−

−−−−

−−−−

−−−−

−−−− n

StS,

n

StS

1n

21ut

1n

21ut αα .With

MPa989.39Sut ==== , n = 12, t = 3.106 and S = 7.243, the confidence interval is

[33.495, 46.483].

6. DISCUSSION

Ebony wood resilience obtained with 8 test specimens is KCU = 21.60 J/cm2with a

reliability of 50%. This value is very closed to that obtained by Sallenave [16] who

used 22 samples and recorded a mean value of 21 J/cm2.

The average value of fracture resistance Sut of polished and annealed ordinary glass,

with 99% confidence, is equal to 39.989 MPa and within the confidence interval

calculated. It is also very close to that obtained by standard bending tests and which is

4Kgf/mm2 or 40 MPa [13].

Mean values of springs stiffness used in combination with ebony wood and polished

and annealed ordinary glass are respectively 636.4N/mm and 9.95 N / mm. These

values are very close to the values 625.862 N / mm and 10.084 N / mm obtained using

the strength of materials formula.

Finally, the absorption factor of the shock absorber used to determine the tensile

strength of polished and annealed ordinary glass is C = 5.176 N / mm.

The methodology used and developed in this study is more interesting than the

standardized tests related to a single material. Indeed, it allows to determine

simultaneous, with a good reliability or a good confidence level, the mechanical

properties of two different materials: ebony wood and helical extension spring,

polished and annealed ordinary glass and helical extension spring, polished and

annealed ordinary glass and shock absorber.



47

In consideration of test values obtained, they should be considered as valid despite the

relatively small number of test specimens.

7. CONCLUSION

We have designed and manufactured a testing bench capable to determine

simultaneously:

• resilience of materials and stiffness of helical extension springs;

• tensile strength of brittle materials and absorption factor of shock absorbers or

helical extension springs stiffness.

First of all, tests on a helical extension spring and a ebony wood test specimen

permitted to conclude that, at 95% confidence, helical extension spring stiffness and

ebony wood resilience are normally distributed. The following results, with a

reliability of 0.5, were obtained:

• mean value of spring stiffness K = 636.4N/mm and its standard deviation S =

158.82N/mm;

• mean value of ebony wood resilience KCU = 21.6 J/cm2 and its standard deviation

S = 4.5 J/cm2.

Secondly, tests on polished and annealed ordinary glass in association with a helical

extension spring or a shock absorber led to:

• mean value of tensile strength of ordinary glass Suts = 37.82 MPa when used with

a helical extension spring;

• mean value of tensile strength of ordinary glass Suta = 44.327 MPa when used

with a shock absorber;

• mean value of tensile strength of ordinary glass Sut = 39.989 MPa when using a

single sample issued of the combination of samples of the two previous cases.

These values, compared to those published in the literature, highlight the reliability of

method used in this study. Therefore, our test bench can effectively serve as teaching

material for practical work in technical and engineering schools.



48

Table 4.1: Spring deflection used with Ebony wood test specimen

hi (mm)

Falling height

300 320 340 370 420 450 480 500

Xi (mm) Spring

deflection

4 4 4.5 5 5.5 6 6.5 7

Table 4.2: Spring stiffness Matrix Kij (used with Ebony wood test specimen)

W1 W2 W3 W4 W5 W6 W7 W8

W1 X ∞ 952.9 788.9 852.6 760 695.2 615.2

W2 # X 482.4 566.7 712.3 660 619.1 554.5

W3 # # X 642.1 810 707.9 645.5 565.2

W4 # # # X 961.9 736.4 646.4 550

W5 # # # # X 530.4 508.3 434.7

W6 # # # # # X 488 392.3

W7 # # # # # # X 303.7

W8 # # # # # # # X

Table 4.3: Energy absorbed Wij by the failure of ebony wood test specimen

W1 W2 W3 W4 W5 W6 W7 W8

W1 X ∞ 7.58 8.89 8.38 9.12 9.64 10.28

W2 # X 11.34 10.67 9.5 9.92 10.25 10.76

W3 # # X 10.72 9.02 10.06 10.69 11.57

W4 # # # X 6.73 7.55 10.67 11.88

W5 # # # # X 13.25 13.59 14.7

W6 # # # # # X 11.02 15.74

W7 # # # # # # X 17.98

W8 # # # # # # # X

Table 4.4: Spring deflection used with glass polished and annealed test specimen

Hri (mm)

Falling height

70 85 90 100 105 110 115 130

Xri (mm) Spring

deflection

31 34 35 36,5 37 38 39 41



49

Table 4.5: Spring stiffness Matrix Kij (used with glass polished and annealed test

specimen)

W1 W2 W3 W4 W5 W6 W7 W8

W1 X 9.25 9.11 9.58 10.07 9.75 9.48 9.74

W2 # X 8.17 9.95 10.82 12.18 9.61 9.92

W3 # # X 10.74 11.83 10.52 9.82 10.11

W4 # # # X 14.99 10.31 9.29 9.91

W5 # # # # X 8.02 7.91 9.31

W6 # # # # # X 7.81 9.72

W7 # # # # # # X 10.65

W8 # # # # # # # X

Table 4.6: Tensile strength of ordinary glass polished and annealed used with a helical

extension spring

K(N/mm) 10.0

3

10.0

3

10.0

3

10.0

3

10.0

3

10.0

3

10.0

3

10.0

3

Sutsi(N/mm2

)

28.3

2

39.7

2

52.5

5

40.7

3

26.8

4

37.0

9

32.7

2

44.5

7

Table 4.7: Shock absorber deflection when used with glass polished and annealed test

specimen

Hai (mm) 90 110 150 200

Xai (mm) Shock

absorber deflection

4.5 5 5.5 6

Table 4.8: Absorption factor Matrix Cij of Shock absorber used with glass polished

and annealed test specimen

W1 W2

W3 W4

W1 X 5.228 5.185 5.170

W2 # X 5.164 5.157

W3 # # X 5.151

W4 # # # X



50

Table 4.9: Tensile strength of ordinary glass polished and annealed used with shock

absorber

C (N.s/m) 5.176 5.176 5.176 5.176

Sutai (MPa) 44.847 42.153 43.393 46.916

Table 5.1: Theorical number of failures per interval for spring stiffness K

Table 5.2: Theorical number of failures per interval for ebony wood resilience

KCU

Table 5.3: χ2

Statistics for spring stiffness K

Table 5.4: χ2

Statistics for ebony wood resilience KCU

N Interval Upper limit

F(ti) F(ti)-F(ti-1)

Fi

1 0 - 500 500 303.7 392.3 434.7 482.4 488 0.19 0.19 5

2 501 – 600 600 508.3 530.4 550 554.5 565.2 566.7 0.41 0.22 6

3 601 – 650 650 615.2 619.1 642.1 645.5 646.4 0.54 0.13 4

4 651 – 750 750 660 695.2 707.9 712.3 736.4 0.76 0.22 6

5 751 - ∞ ∞ 760 788.9 810 852.6 952.9 961.9 1 0.24 6

N Interval Upper limit

F(ti) F(ti)-F(ti-1)

Fi

1 0 – 18 18 13.46 15.10 15.16 16.76 17.78 0.24 0.24 6

2 18.1–20 20 18.04 18.24 19.00 19.28 19.84 0.37 0.13 4

3 20.1– 1.5 21.40 20.12 20.50 20.56 21.34 21.34 21.38 0.48 0.11 3

4 21.41- 25 25 21.44 21.52 22.04 22.68 23.14 23.76 0.75 0.27 7

5 25.1– ∞ ∞ 26.50 27.18 29.40 31.48 35.96 1 0.25 7

Interval Upper limit Fi fi χi2

1 300 – 500 500 5 5 0

2 501 – 600 600 6 6 0

3 601 – 650 650 4 5 0.25

4 651 – 750 750 6 5 0.17

5 751 - ∞ ∞ 6 6 0

Sum 27 27 χ2 = 0.42

Interval Upper limit Fi fi χi2

1 0 – 18 18 6 5 0.17

2 18.1 – 20 20 4 5 0.25

3 20.1 – 21.40 21.4 3 6 3.00

4 21.41 - 25 25 7 6 0.14

5 25.1– ∞ ∞ 7 5 0.57

Sum 26 27 χ2 = 4.13



51

Table 5.5: Reliability of spring stiffness K

ti 303.7 392.3 434.7 482.4 488 508.3 530.4 550 554.5

F(ti) 0.0183 0.1618 0.1020 0.1660 0.1762 0.2090 0.2514 0.2946 0.3015

R(ti) 0.9817 0.8382 0.8980 0.8340 0.8238 0.7910 0.7486 0.7054 0.6985

ti 565.2 566.7 615.2 619.1 642.1 645.5 646.4 660 695.2

F(ti) 0.3264 0.3300 0.4483 0.4562 0.5180 0.5239 0.5239 0.5596 0.6443

R(ti) 0.6736 0.6700 0.5517 0.5438 0.4820 0.4761 0.4761 0.4404 0.3557

ti 707.9 712.3 736.4 760 788.9 810 852.6 952.9 961.9

F(ti) 0.6736 0.6844 0.7357 0.7823 0.8315 0.8521 0.9131 0.9767 0.9798

R(ti) 0.3263 0.3156 0.2643 0.2177 0.1685 0.1479 0.0869 0.0233 0.0202

Table 5.6: Reliability of resilience KCU of ebony wood

ti 13.46 15.10 15.16 16.76 17.78 18.04 18.24 19.00 19.28

F(ti) 0.0526 0.0985 0.1003 0.1685 0.2236 0.2389 0.2514 0.3015 0.3228

R(ti) 0.9474 0.9015 0.8997 0.8315 0.7764 0.7611 0.7486 0.6985 0.6772

ti 19.84 20.12 20.50 20.56 21.34 21.34 21.38 21.44 21.52

F(ti) 0.3632 0.3859 0.4129 0.4169 0.4801 0.4801 0.4840 0.4880 0.4920

R(ti) 0.6368 0.6141 0.5871 0.5831 0.5199 05199 0.5160 0.5120 0.5080

ti 22.04 22.68 23.14 23.76 26.50 27.18 29.40 31.48 35.96

F(ti) 0.5359 0.5832 0.6217 0.6664 0.8340 0.8665 0.9394 0.9750 0.9978

R(ti) 0.4641 0.4168 0.3783 0.3336 0.1660 0.1335 0.0606 0.0250 0.0022

FIGURE CAPTIONS

Figure 2.1: Kinematic diagram of the testing bench using ebony wood specimen

Figure 2.2: Kinematic diagram of the testing bench using ordinary glass specimen

Figure 2.3: Lifting System of the mass

Figure 3.1: Assembly drawing of the testing bench

Figure 3.2: Dimensions related to clearances

Figure 3.3: Crank Shaft detail drawing

Figure 4.1: Ebony wood test specimen

Figure 4.2: Ordinary glass test specimen



52

REFERENCES

1. Barralis J. G. (1997), « Précis de Métallurgie », AFNOR-Nathan, Paris.

2. Ballereau A. J. (1995), « Mécanique Industrielle, Tome 2, Approche système »,

Foucher, Paris.

3. Ngouajou (2004), « Conception et réalisation d’un mouton pendule pour la

Détermination de la résilience des coques de noix de coco et de palmiste »,

Mémoire de fin d’études, ENSET, Université de Douala, Cameroun, 2004.

4. Crouse H. W. (1979), « Mécanique automobile », 3e édition, traduit par Delucas,

J.

Bibliothèque nationale du Québec, Canada.

5. Breul P et al. (2004), « Diagnostic des ouvrages urbains en interaction avec le sol

par couplage de techniques rapides et complémentaires », 22ème

Rencontres

Universitaires de Génie Civil, Aubière Cedex.

6. DUBOZ R. et al. (2003), « Utiliser les modèles individus-centrés comme

laboratoires virtuels pour identifier les paramètres d’un modèle agrégé », 4ème

Conférence Francophone de Modélisation et Simulation, Toulouse.

7. Quatremer R. and Trotignon J. P. (1985), « Précis de construction mécanique 1.

Dessin conception et normalisation », 13ème

édition, AFNOR, Nathan, Paris.

8. Drouin G. et al. (1986), « Eléments de machines », Deuxième édition revue et

augmentée, Editions de l’Ecole Polytechnique de Montréal, Canada.

9. Bazergui A. et al. (1985), « Résistance des matériaux », Editions de l’Ecole

Polytechnique de Montréal, Canada.

10. Fanchon J. L. (1996), « Guide de Mécanique, Sciences et technologies

industrielles », Nathan, Paris.

11. Dietrich R. (1981), « Précis de méthodes d’usinage », 5ème

édition, AFNOR,

Nathan, Paris.

12. Wadou K. (2009), « Détermination expérimentale couplée de la résistance à la

Rupture en traction des matériaux fragiles et des rigidités des amortisseurs et

ressorts de compression », Mémoire de D.E.A, Université de Douala, Cameroun.

13. Bassino J. (1972), « Technologie en ouvrages métalliques : Tome I, Matériaux-

Usinages-Machines », Foucher, Paris.

14. Zdzislaw K. (1995), « Fiabilité et maintenabilité des systèmes mécaniques »,

Département de génie mécanique, Ecole Polytechnique de Montréal.

15. Pasquier A. (1969), « Eléments de calcul des probabilités et des théories de

sondage », Dunod , Paris.

16. Sallenave P. (1955), « Propriétés physiques et mécanique des bois tropicaux de

l’union française » Centre technique forestier tropical, France.

17. Agati P. and Mattera N. (1987), « Modélisation, Résistance des Matériaux,

Notion d’élasticité » Bordas, Paris.

.



53

m

1210

9

8

5

3

2 (2)

1

416

18

19

17

6

13

14

15

11

20

21

7

Figure 2.1: Kinematic diagram of the testing bench using ebony wood specimen

1 - machine stand 6 - slide 11 - rope 16 – push rod

2 - mounting (2) 7 - slide (2) 12 - crank shaft 17 - scale

3 - specimen support (2) 8 - column (2) 13 - pulley 18 - cursor

4 - test specimen 9 - block 14 - positioning rod 19 - spring

5 - knife 10 - hook 15 - mass 20 - slide bar (4)

21- stop pin (2)



54

Figure 2.2 (a)

Figure 2.2 (b)

Figure 2.2: Kinematic diagram of the testing bench using ordinary glass specimen

1 - machine stand 4 - test specimen 7 - specimen support 10 - scale

2 - mounting (2) 5 - binding screw 8 - slide 11 - cursor

3 - retaining plate 6 - mass 9 - push rod 12 - shock absorber

or spring



55

Figure 2.3: Lifting System of the mass



56

Figure 3.1: Assembly drawing of the testing bench



57

Figure 3.2: Dimensions related to clearances



58

Figure 3.3: Crank Shaft detail drawing



59

Figure 4.1: Ebony wood test specimen

Figure 4.2: Ordinary glass test specimen

Documents

Design of a testing bench, statistical and reliability analysis of some mechanical tests