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),
International Journal of Mechanical Engineering
ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
and Technology (IJMET), ISSN 0976 – 6340(Print)
ISSN 0976 – 6359(Online) Volume 2
IJMET
Number 1, Jan - April (2011), pp. 36-59 ©IAEME
© IAEME, http://www.iaeme.com/ijmet.html

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
nghaulin@yahoo.fr, 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/cm2.

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.

36


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.
37


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 M fz = 9.5Nm et M t max = 5Nm . Using the maximum shear stress
max
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 d ≥ 12.73mm . 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.

38


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 E Pi = mgh i , 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:
'
• E Pi = mgXi (mass m) ;
1 1 &
• WSi = KX i 2 (spring) or Wai = CX mi 2 = Cgh i [12] (shock absorber) with
2 2
& the mass velocity at the beginning of the compression.
X mi

Applying the principle of mechanical energy conservation, we have:
 1 2  1 2
 E Pi + E ' Pi = Wi + WSi ⇔ mg (h i + X i ) = Wi + KX i  W = mg ( h i + X i ) − KX i
 2 ⇒ i 2
 E Pi + E ' Pi = Wi + Wai ⇔ mg(h i + X i ) = Wi + Cgh i  Wi = mg ( h i + X i ) − Cgh i



Equating two consecutive energies obtained at h j and h i with h j ≠ h i , we have:
 1 2 2 (h i + X i − h j − X j )
 mg(h i + X i − h j − X j ) = K ij (X i − X j ) ⇒ K ij = 2mg 2 2
 2 Xi − X j

 mg(h + X − h − X ) = C g(h − h ) ⇒ C = m (h i + X i − h j − X j )
 i i j j ij i j ij
hi − h j

with Kij and Cij respectively the spring stiffness and the absorption factor of the shock
absorber after two consecutive tests i and j.

39


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
n n 8

∑K
i , j=1
ij 2∑ K ij
i , j=1
∑K
i , j=1
ij

stiffness is finally K = = = 636.4.N / mm with i < j (table 4.2
=
n 2 − n n(n − 1) 27
2
has 27 values instead of 28). Applying the strength of materials formula,
Gd 4
K= = 625.862 N / mm with:
8D 3 n
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
n 8
2∑ Wij ∑W ij
i , j=1 i , j=1
of this absorbed energy is then W = = = 10.80J / cm 2 with i < j.
n (n − 1) 27
W
Generally, resilience K C = [1,2] with :
S
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
K CU = 2W = 21.60joules cm 2 [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.

40


4.3.1 Determination of the spring stiffness

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

∑K
i , j=1
ij

spring stiffness is finally K = = 9.95.N / mm with i < j; Applying the strength of
28
Gd 4
materials formula, K = = 10.084 N / mm with :
8D 3 n
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
1 2
spring is Wsi = mg ( h si + X si ) − KX si [2]. Moreover, according to Von Mises, the
2
P 2 L0
expression of the elastic strain energy in traction is U = [10] with :
2 ES 0
P = mg: tensile load;
L0 = 50mm: test specimen length;
E: Young’s modulus;
S0= 9mm2: cross square section of the test specimen.
P 2 L0
By analogy, the tensile failure energy is Wu = [12] where Suts is the tensile
2 S uts S 0
strength of the material used. Therefore, Wsi =
2
P L0
Wu ⇒ S utsi = . Table 4.6 gives the values of the tensile
[
S 0 2mg (hsi + X si ) − KX si
2
]
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
8

∑S
i =1
utsi
with a helical extension spring, is equal to S uts = = 37.82 MPa .
8

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

41


Table 4.8 gives the absorption factor matrix Cij of the shock absorber. The absorption
4

∑C
i , j=1
ij

factor of the shock absorber is finally C = = 5.176 N.s / m with i < j.
6

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
is Wai = mg( h ai + X ai ) − CgX ai 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
P2L
extension spring, is S utai = . Table 4.9 gives the values of
2gA[m(h ai + X ai ) − Ch ai ]
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
4

∑S
i =1
utai
to S uta = = 44.327 MPa .
4

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

42


1 n
are the sample mean t= ∑ t i and
n i =1
the sample standard deviation

1 8
S=
n − 1 i =1
(
∑ ti − t )
2
[10]. Since Kij and Wij are stochastic variables:
8 8

∑ K ij
i , j=1
∑K
i , j=1
ij

• The mean value of spring stiffness is K = K = = = 636.4 N / mm
n 27
where i < j. Its standard deviation is
8

∑ (K )
2
8 ij −K
1
S=
n − 1 i , j=1
(
∑ K ij − K )2
=
i , j=1

26
= 158.82 N / mm with i < j.

• The mean value of the ebony wood resilience is
8 8
2 ∑ Wij 2 ∑ Wij
i , j=1 i , j=1
K CU = K CU = 2 W = = = 21.6J / cm 2 with i < j. Its standard
n 27
2
8  K CU 
2 4 ∑  Wij −



4 8  K CU  i , j=1  2 
deviation is S = ∑1 Wij − 2  =

n − 1 i , j=   26
= 5.04J / cm 2

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
(f − Fi )2 where f = failures number within interval i.
give χ2 statistic: χ 2 = ∑ i i
Fi

43


Tabuled values of chi square statistic
Tables [14,15] give the critical value of χ 2 (d ) where the confidence level P = 1 - α =
P
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:
• χ 0.95 (2 ) = 5.99 > χ2 = 0.42 for the spring stiffness K. Therefore, the assumption of
2

normal distribution is verified with a confidence level 0.95;
• χ 0.95 (2 ) = 5.99 > χ2 = 4.13 for the resilience KCU of ebony wood. The hypothesis
2

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
k3
values [14]. Asymmetry factor is α 3 = 3 [15] where S 3 is the third power of
S
n
(
n ∑ R ri − R r )3

k4
i =1
standard deviation and k 3 = . Flattening factor is α 4 = [15] where
(n − 1)(n − 2) S4
S 4 is the fourth power of standard deviation and
2
 n
)
n
(
n (n + 1)∑ R ri − R r )4
 (
− 3(n − 1) ∑ R ri − R r
2


k4 =
i =1  i =1  ).
(n − 1)(n − 2)(n − 3)
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
44


 S S 
with unknown variance σ2 is given by S ut − t 1−−α
n 1 n 1
, S ut + t 1−−α  where
 2 n 2 n
1 n 1 n
S ut = ∑
n i =1
S uti , S =
n − 1 i =1
( )
∑ Suti − Sut and t 1n−−α1 2 read in the table of Student at
2

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

For ordinary glass and helical extension spring association: S ut = S uts = 37.818 MPa ,
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: S ut = S uta = 44.327 MPa , 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
S ut − S ut
t= must be less than the value tlimit read in the table of Student at 99%
S2
n −1
confidence and
n - 1 degrees of freedom. Indeed, for the combination of ordinary glass and helical
37.818 − 40
extension spring, t = = 0.656 < t lim it = 2.998 ; the combination of
8.552 2
7
44.327 − 40
ordinary glass and shock absorber gives t = = 3.661 < t lim it = 4.541 .
2.047 2
3
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

S uta − S uts
The test statistic t = of samples n1 and n2 must be less than the value
 1 1 
S2  +
n 

 1 n2 
tlimit read in the table of Student at 99% confidence and (n1 + n2 - 2) degrees of
freedom. Common variance to both samples is
1  n1 n2
2
S2 = ∑(
n 1 + n 2 − 2  i =1
) ( )
S utsi − S uts + ∑ S utai − S uta  [15]. Indeed,
2

i =1 

45


44.327 − 37.818
t= = 1.468 < 2.764 . Therefore, both samples are representative of
1 1
52.454 + 
8 4
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
n1 n2

∑ S utsi + ∑ Sutai
i =1 i =1
S ut = = 39.989 MPa .
n1 + n 2

5.2.6 Confidence interval of the tensile strength Sut

 n 1 S n 1 S 
At 99% confidence, this interval is given by S ut − t 1−−α , S ut + t 1−−α  .With
 2 n 2 n
S ut = 39 .989 MPa , 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.

46


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.

47


Table 4.1: Spring deflection used with Ebony wood test specimen

hi (mm) 300 320 340 370 420 450 480 500
Falling height
Xi (mm) Spring 4 4 4.5 5 5.5 6 6.5 7
deflection

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) 70 85 90 100 105 110 115 130
Falling height
Xri (mm) Spring 31 34 35 36,5 37 38 39 41
deflection

48


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 10.0 10.0 10.0 10.0 10.0 10.0 10.0
3 3 3 3 3 3 3 3
Sutsi(N/mm2 28.3 39.7 52.5 40.7 26.8 37.0 32.7 44.5
) 2 2 5 3 4 9 2 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 4.5 5 5.5 6
absorber deflection

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

49


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

N Interval Upper F(ti) F(ti)- Fi
limit F(ti-1)
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

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

N Interval Upper F(ti) F(ti)- Fi
limit F(ti-1)
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

Table 5.3: χ2 Statistics for spring stiffness K

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
2
Sum 27 27 χ = 0.42

Table 5.4: χ2 Statistics for ebony wood resilience KCU

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

50


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

51


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.

.

52


13

11

10
14 12
9

8
21
20
7
15 6
m
5
16
4
17
3
18

19 2 (2)

1

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)

53


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

54


Figure 2.3: Lifting System of the mass

55


Figure 3.1: Assembly drawing of the testing bench

56


Figure 3.2: Dimensions related to clearances

57


Figure 3.3: Crank Shaft detail drawing

58


Figure 4.1: Ebony wood test specimen

Figure 4.2: Ordinary glass test specimen

59

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