design of a testing bench, statistical and reliability analysis of some mechanical tests
TRANSCRIPT
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
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
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.
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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.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
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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.
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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.
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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
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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
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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.
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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
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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,
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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.
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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.
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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
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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
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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
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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
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Usinages-Machines », Foucher, Paris.
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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.
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.
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ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
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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)
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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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
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Figure 2.3: Lifting System of the mass
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Figure 3.1: Assembly drawing of the testing bench
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Figure 3.2: Dimensions related to clearances
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Figure 3.3: Crank Shaft detail drawing
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Figure 4.1: Ebony wood test specimen
Figure 4.2: Ordinary glass test specimen