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

    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),International Journal of Mechanical EngineeringISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEand Technology (IJMET), ISSN 0976 – 6340(Print)ISSN 0976 – 6359(Online) Volume 2 IJMETNumber 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 CameroonABSTRACTA testing bench was designed and manufactured in order to determine simultaneouslymechanical properties of materials and stiffness of helical extension springs orabsorption factor of shock absorbers.The combination of one helical extension spring with eight ebony wood testspecimens 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 resilienceKCU 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 onehelical extension spring led to obtain firstly the spring stiffness K = 9.95 N/mm andsecondly, using the parametric statistical test of Student-Fisher, the tensile strength ofannealed and polished ordinary glass Sut = 37.818 MPa within the confidence intervalIc = [27.238, 48.398] MPa at 99% confidence and a standard deviation S = 8.552MPa.The combination of four annealed and polished ordinary glass test specimens with oneshock 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 ofpolished 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 glassobtained, after an homogeneity statistical test applied to the two previouscombinations, 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
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME1. INTRODUCTIONSome testing benches were developed for the determination of stiffness of materialsand their resilience which are respectively the capacity of machine elements to avoidexcessive distortion under applied loads and their ability to absorb a certain amount ofenergy (shock or dynamic loading) without damage.Indeed, CHARPY pendulum [1], FREMONT resilience machine [2] andNGOUAJOU machine [3] are used to determine resilience of materials. Theadvantages of these three machines lie in their small dimensions and easy assembly.FREMONT machine allows also a direct reading of the spring deflection linked withspecimen to be tested. Their common disadvantage is the determination of only onemechanical property which is resilience of materials. The main disadvantages ofCHARPY machine are a lack of security during dynamic loading and a constantpotential energy (300J) [1]; those of FREMONT machine are friction anddeformations in the guides, fixed dimensions of the spring and a lack of back systemlinked with the cursor used to read the deflection.More over, springs with unknown stiffness are increasingly used in technical schoolsand garages in Cameroon. However, two special devices are often used to determinerespectively stiffness and deflection of valve springs [4].Some authors [5,6] used the coupling method in order to determine simultaneouslyphysical constants of more than one material. The aim of our study is then todetermine simultaneously, firstly resilience of materials and stiffness of helicalextension springs and secondly, tensile strength of materials and absorption factor ofshock absorbers or stiffness of helical extension springs by the means of a testingbench designed and manufactured at the University of Douala, Cameroon.This paper has four main parts. The first two parts concern conceptual and graphicsdesigns of the testing bench. The two last one deal firstly with mechanical tests andsecondly with statistical and reliability analysis of results obtained.2. CONCEPTUAL DESIGN2.1 MACHINE DESCRIPTION2.1.1 Kinematic diagramFigure 2.1 shows the kinematic diagram of the testing bench used for ebony wood testspecimens. After changing the fastening system of test specimen, machineconfiguration is that of the figures 2.2 a) and b) and is used for polished and annealedordinary glass.2.1.2 Functioning principleFigure 2.1shows a 5 kg mass 15 which is in equilibrium at the height h from the testspecimen 4 by the means of a block 9 and is equipped with a knife 5 intended to strikethe specimen in the opposite direction of its notch. 37
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEFigure 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 isintended to strike that retaining plate.The mass, when released, is guided in translation on the frame 2 by the means of fourslides. An amount of its potential energy, when converted into kinetic energy, isabsorbed by the test specimen and the excess by the spring or shock absorber. Thecursor, with negligible friction, records the maximum deflection X of the spring orshock absorber. Mass m is then raised up to a height h by the means of a cable whichwinds round the pulley 13.2. 2 MACHINE ELEMENTS DESIGNThe main elements of the testing bench have been designed according to the materialused and the applied loads. This paper presents only the design of the crank shaft 12which is the main part of the lifting system of mass m and which brings this mass inan equilibrium position before each test. The force F due to tension in cable 11 andequal to 50N, will be used to design the crank shaft 12 subjected to bending andtorsion.Determination of the crank shaft 12The material used, 42CD4, has the following characteristics: yield strength Sy = 1500N/mm², Young’s modulus E = 2.05 105 N/mm² [7]. The design is done during theraising 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 whichis in B where the maxima of bending moment and torque arerespectively M fz = 9.5Nm et M t max = 5Nm . Using the maximum shear stress maxtheory [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 stresseswere determined and led to obtain a diameter d ≥ 12.73mm . Let us consider d = 20mm.3. GRAPHIC DESIGNAutoCAD 2009 was used to draw the testing bench shown in the general assemblydrawing of figure 3.1. New machine elements references, different from those used inthe kinematic diagrams of figures 2.1 and 2.2, are taken into account and used later inthis study.All necessary clearances [7] for the proper functioning of the testing bench weredefined and shown on its general assembly drawing. Dimensions of machine elementsrelated to those clearances were determined. Finally, each of these elements wasdrawn.This paper presents only the clearances related to the proper functioning of the crankshaft 18 and its detail drawing respectively in figures 3.2 and 3.3. 38
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME4. MECHANICAL TESTS4.1 TESTS METHODOLOGYWe present here the methodology of the experimentation which determinessimultaneously in the one hand the mechanical properties of materials and in the otherhand, the stiffness of helical extension springs or the absorption factor of shockabsorbers. 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 thedocking system of test specimens results in the use of n = 12 identical test specimensof ordinary glass polished and annealed. The mass (m) is placed at a height h from thepoint of impact. Its potential energy is E Pi = mgh i , i varies from 1 to n. A testspecimen is placed on its supports. Then mass (m) is released and falls freely. Aquantity Wi of its potential energy (converted into kinetic energy) is absorbed by thefailure of the test specimen and the excess by the spring or shock absorber. A cursorregisters the maximum deflection Xi of spring or shock absorber.From the compression of spring or shock absorber, the following potential energiescan 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 miApplying 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 shockabsorber after two consecutive tests i and j. 39
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME4.2 DETERMINATION OF BOTH SPRING STIFFNESS AND RESILIENCEOF EBONY WOOD8 trials are performed using one helical extension spring and 8 identical testspecimens of ebony wood and. Table 4.1 presents the results obtained.4.2.1 Determination of the spring stiffnessTable 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 ijstiffness is finally K = = = 636.4.N / mm with i < j (table 4.2 = n 2 − n n(n − 1) 27 2has 27 values instead of 28). Applying the strength of materials formula, Gd 4K= = 625.862 N / mm with: 8D 3 nd (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 resilienceChanging K by Kij in Wi or Wj expressions enables to obtain the energy Wij absorbedby 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=1of this absorbed energy is then W = = = 10.80J / cm 2 with i < j. n (n − 1) 27 WGenerally, resilience K C = [1,2] with : SS (cm2) = cross section at the notch of the test specimen;W (Joule) = energy absorbed by the test specimen;KC (Joule/cm2) = material resilienceTaking into account the geometry of the notch of the test specimen, we have, for theCharpy U-notch shown in figure 4.1, the average resilience of ebony woodK CU = 2W = 21.60joules cm 2 [1,2];4.3 DETERMINATION OF BOTH SPRING STIFFNESS AND TENSILE STRENGTH OF ORDINARY GLASS POLISHED AND ANNEALEDIt is performed using one helical extension spring and 8 identical test specimens ofpolished and annealed glass. Table 4.4 presents the results obtained. 40
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME4.3.1 Determination of the spring stiffnessTable 4.5 shows the stiffness matrix Kij of helical extension spring for 8 trials. The 8 ∑K i , j=1 ijspring stiffness is finally K = = 9.95.N / mm with i < j; Applying the strength of 28 Gd 4materials formula, K = = 10.084 N / mm with : 8D 3 nd (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 annealedThe failure energy of the glass test specimen (figure 4.2) used with a helical extension 1 2spring is Wsi = mg ( h si + X si ) − KX si [2]. Moreover, according to Von Mises, the 2 P 2 L0expression of the elastic strain energy in traction is U = [10] with : 2 ES 0P = mg: tensile load;L0 = 50mm: test specimen length;E: Young’s modulus;S0= 9mm2: cross square section of the test specimen. P 2 L0By analogy, the tensile failure energy is Wu = [12] where Suts is the tensile 2 S uts S 0strength of the material used. Therefore, Wsi = 2 P L0Wu ⇒ 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 utsiwith a helical extension spring, is equal to S uts = = 37.82 MPa . 84.4 DETERMINATION OF BOTH ABSORPTION FACTOR OF SHOCK ABSORBER AND TENSILE STRENGTH OF ORDINARY GLASS POLISHED AND ANNEALEDIt is performed using one shock absorber and 4 identical test specimens of polishedand annealed glass. Table 4.7 shows the results obtained.4.4.1 Determination of the absorption factor of shock absorber 41
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEMETable 4.8 gives the absorption factor matrix Cij of the shock absorber. The absorption 4 ∑C i , j=1 ijfactor of the shock absorber is finally C = = 5.176 N.s / m with i < j. 64.4.2 Determination of tensile strength of ordinary glass polished and annealedWhen using a shock absorber, the failure energy of the test specimenis Wai = mg( h ai + X ai ) − CgX ai and the tensile strength of the ordinary glass whenusing this shock absorber at a trial i, by analogy to that obtained when using a helical P2Lextension 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 ofglass polished and annealed, when used with a shock absorber, is equal 4 ∑S i =1 utaito S uta = = 44.327 MPa . 45. STATISTICAL AND RELIABILITY STUDIESThe results obtained from the testing machine are subjected to two kinds of statisticaltests:• 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) TEST5.1.1 Choice of a statistical law followed by the experimental resultsIn the experimental results provided in tables 4.3, 4.4 and 4.5, we find a slow andgradual change of parameters hi, Xi, Kij and Wij. Therefore, the statistical hypothesisconsists to assume that the normal distribution is the most likely parent to theseparameters [14].5.1.2 Estimated parameters of the normal distributionConsidering a sample of n data, the estimated parameters of the normal distribution 42
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME 1 nare the sample mean t= ∑ t i and n i =1 the sample standard deviation 1 8S= 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 hypothesisIt is now important to say whether the random variables that are the spring stiffnessand ebony wood resilience effectively obey the normal distribution with thecalculated parameters (mean and variance).Number of intervals N of the chi-square (χ2) testThe number of intervals is N = 1 +3.3 log n = 5.72. Given our sample n = 27, weadopt N = 5. The restriction is that at least 5 theoretical failures must exist within eachinterval.Theoretical number of failures Fi for each interval iFi = n x [F(ti)-F(ti-1)] where i = 1 , 2 , … N. and F(ti) - F(ti-1) = probability for afailure to be in the interval i. The tables 5.1 and 5.2 give this theoretical number offailures per interval respectively for the spring stiffness and the ebony woodresilience.Chi square (χ2) statisticRespectively 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
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEMETabuled values of chi square statisticTables [14,15] give the critical value of χ 2 (d ) where the confidence level P = 1 - α = P0.95 and the degree of freedom d = (N -1) - Z = 2 with Z the number of estimatedparameters (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 resultsThe normal distribution is the most likely parent to random variables that are springstiffness K and resilience KCU of ebony wood. The failure probability F (t) representsthe 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 5.6show, for each failure ti observed, the failure probability and the reliability of thesevariables.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, itis higher than 0.5 below these values and less than 0.5 above them.5.2 STUDENT-FISHER TEST5.2.1 Checking of the normality assumption for the tensile strengths Suts and SutaThe 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 tensilestrength of ordinary glass polished and annealed and n2 = 4 values Suta of thismechanical property, the asymmetry factor α3 and the flattening one α4 are virtuallynil. It follows that the normal distribution is the most likely parent to these two sets of k3values [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 =1standard deviation and k 3 = . Flattening factor is α 4 = [15] where (n − 1)(n − 2) S4S 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 SutaAt a confidence level P = 1 - α = 0.99, the confidence interval of tensile strength Sut 44
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME  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 nS 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 299% 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 meansThe standard value of the tensile strength of glass polished and annealed is Sut = 40MPa [13]. This value must be compared to Suts and Suta. The test statistic S ut − S utt= must be less than the value tlimit read in the table of Student at 99% S2 n −1confidence andn - 1 degrees of freedom. Indeed, for the combination of ordinary glass and helical 37.818 − 40extension spring, t = = 0.656 < t lim it = 2.998 ; the combination of 8.552 2 7 44.327 − 40ordinary glass and shock absorber gives t = = 3.661 < t lim it = 4.541 . 2.047 2 3Therefore, each of these two values Suts and Suta is representative of the tensilestrength of the polished and annealed ordinary glass.5.2.4 Homogeneity test of two samples S uta − S utsThe 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 offreedom. 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
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME 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 SutThe mean value of tensile strength of the sample size n = 12 is n1 n2 ∑ S utsi + ∑ Sutai i =1 i =1S ut = = 39.989 MPa . n1 + n 25.2.6 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. DISCUSSIONEbony wood resilience obtained with 8 test specimens is KCU = 21.60 J/cm2with areliability of 50%. This value is very closed to that obtained by Sallenave [16] whoused 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 intervalcalculated. It is also very close to that obtained by standard bending tests and which is4Kgf/mm2 or 40 MPa [13].Mean values of springs stiffness used in combination with ebony wood and polishedand annealed ordinary glass are respectively 636.4N/mm and 9.95 N / mm. Thesevalues are very close to the values 625.862 N / mm and 10.084 N / mm obtained usingthe strength of materials formula.Finally, the absorption factor of the shock absorber used to determine the tensilestrength of polished and annealed ordinary glass is C = 5.176 N / mm.The methodology used and developed in this study is more interesting than thestandardized tests related to a single material. Indeed, it allows to determinesimultaneous, with a good reliability or a good confidence level, the mechanicalproperties of two different materials: ebony wood and helical extension spring,polished and annealed ordinary glass and helical extension spring, polished andannealed ordinary glass and shock absorber. 46
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEIn consideration of test values obtained, they should be considered as valid despite therelatively small number of test specimens.7. CONCLUSIONWe have designed and manufactured a testing bench capable to determinesimultaneously:• 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 specimenpermitted to conclude that, at 95% confidence, helical extension spring stiffness andebony wood resilience are normally distributed. The following results, with areliability 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 helicalextension 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 ofmethod used in this study. Therefore, our test bench can effectively serve as teachingmaterial for practical work in technical and engineering schools. 47
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Table 4.1: Spring deflection used with Ebony wood test specimenhi (mm) 300 320 340 370 420 450 480 500Falling heightXi (mm) Spring 4 4 4.5 5 5.5 6 6.5 7deflection 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 specimenHri (mm) 70 85 90 100 105 110 115 130Falling heightXri (mm) Spring 31 34 35 36,5 37 38 39 41deflection 48
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME 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 # # # # # # # XTable 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 7Table 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
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME 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 KN 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 52 501 – 600 600 508.3 530.4 550 554.5 565.2 566.7 0.41 0.22 63 601 – 650 650 615.2 619.1 642.1 645.5 646.4 0.54 0.13 44 651 – 750 750 660 695.2 707.9 712.3 736.4 0.76 0.22 65 751 - ∞ ∞ 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 KCUN 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 62 18.1–20 20 18.04 18.24 19.00 19.28 19.84 0.37 0.13 43 20.1– 1.5 21.40 20.12 20.50 20.56 21.34 21.34 21.38 0.48 0.11 34 21.41- 25 25 21.44 21.52 22.04 22.68 23.14 23.76 0.75 0.27 75 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
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Table 5.5: Reliability of spring stiffness K ti 303.7 392.3 434.7 482.4 488 508.3 530.4 550 554.5F(ti) 0.0183 0.1618 0.1020 0.1660 0.1762 0.2090 0.2514 0.2946 0.3015R(ti) 0.9817 0.8382 0.8980 0.8340 0.8238 0.7910 0.7486 0.7054 0.6985 ti 565.2 566.7 615.2 619.1 642.1 645.5 646.4 660 695.2F(ti) 0.3264 0.3300 0.4483 0.4562 0.5180 0.5239 0.5239 0.5596 0.6443R(ti) 0.6736 0.6700 0.5517 0.5438 0.4820 0.4761 0.4761 0.4404 0.3557 ti 707.9 712.3 736.4 760 788.9 810 852.6 952.9 961.9F(ti) 0.6736 0.6844 0.7357 0.7823 0.8315 0.8521 0.9131 0.9767 0.9798R(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 19.28F(ti) 0.0526 0.0985 0.1003 0.1685 0.2236 0.2389 0.2514 0.3015 0.3228R(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 21.52F(ti) 0.3632 0.3859 0.4129 0.4169 0.4801 0.4801 0.4840 0.4880 0.4920R(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 35.96F(ti) 0.5359 0.5832 0.6217 0.6664 0.8340 0.8665 0.9394 0.9750 0.9978R(ti) 0.4641 0.4168 0.3783 0.3336 0.1660 0.1335 0.0606 0.0250 0.0022FIGURE CAPTIONSFigure 2.1: Kinematic diagram of the testing bench using ebony wood specimenFigure 2.2: Kinematic diagram of the testing bench using ordinary glass specimenFigure 2.3: Lifting System of the massFigure 3.1: Assembly drawing of the testing benchFigure 3.2: Dimensions related to clearancesFigure 3.3: Crank Shaft detail drawingFigure 4.1: Ebony wood test specimenFigure 4.2: Ordinary glass test specimen 51
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEMEREFERENCES1. 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
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME 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 specimen1 - machine stand 6 - slide 11 - rope 16 – push rod2 - mounting (2) 7 - slide (2) 12 - crank shaft 17 - scale3 - specimen support (2) 8 - column (2) 13 - pulley 18 - cursor4 - test specimen 9 - block 14 - positioning rod 19 - spring5 - knife 10 - hook 15 - mass 20 - slide bar (4)21- stop pin (2) 53
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Figure 2.2 (a) Figure 2.2 (b) Figure 2.2: Kinematic diagram of the testing bench using ordinary glass specimen1 - machine stand 4 - test specimen 7 - specimen support 10 - scale2 - mounting (2) 5 - binding screw 8 - slide 11 - cursor3 - retaining plate 6 - mass 9 - push rod 12 - shock absorberor spring 54
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Figure 2.3: Lifting System of the mass 55
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Figure 3.1: Assembly drawing of the testing bench 56
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Figure 3.2: Dimensions related to clearances 57
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Figure 3.3: Crank Shaft detail drawing 58
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),ISSN 0976 – 6359(Online) Volume 2, Number 1, Jan - April (2011), © IAEME Figure 4.1: Ebony wood test specimen Figure 4.2: Ordinary glass test specimen 59