research article flexural vibration test of a beam...

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 329530 6 pageshttpdxdoiorg1011552013329530

Research ArticleFlexural Vibration Test of a Beam Elastically Restrained at OneEnd A New Approach for Youngrsquos Modulus Determination

Rafael M Digilov and Haim Abramovich

Faculty of Aerospace Engineering TechnionmdashIsrael Institute of Technology 32000 Haifa Israel

Correspondence should be addressed to Rafael M Digilov edurafitechnionacil

Received 1 May 2013 Accepted 11 July 2013

Academic Editor Xing Chen

Copyright copy 2013 R M Digilov and H Abramovich This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A new vibration beam technique for the fast determination of the dynamic Young modulus is developed The method is based onmeasuring the resonant frequency of flexural vibrations of a partially restrained rectangular beamThe strip-shaped specimen fixedat one end to a force sensor and free at the other forms the Euler Bernoulli cantilever beam with linear and torsion spring on thefixed endThe beam is subjected to free bending vibrations by simply releasing it from a flexural position and its dynamic responsedetected by the force sensor is processed by FFT analysis Identified natural frequencies are initially used in the frequency equationto find the corresponding modal numbers and then to calculate the Young modulus The validity of the procedure was tested on anumber of industrial materials by comparing the measured modulus with known values from the literature and good agreementwas found

1 Introduction

The Young modulus is a fundamental material property andits determination is common in science and engineering[1 2] It is a key parameter in mechanical engineering designto predict the behavior of the material under deformationforces or more to get an idea of the quality of the materialYoungrsquosmoduli are determined from static and dynamic testsIn static measurements [3 4] such as the classical tensile orcompressive test a uniaxial stress is exerted on the materialand the elastic modulus is calculated from the transverseand axial deformations as the slope of the stress-strain curveat the origin Dynamic methods [5ndash12] are more preciseand versatile since they use very small strains far belowthe elastic limit and therefore are virtually nondestructiveallowing repeated testing of the same sample These includethe ultrasonic pulse-echo [6 7] or bar resonance methods[4 8ndash14] In the sonic pulse technique the dynamic Youngmodulus is determined by measuring the sound velocityin the sample In the resonance method the linear elasticuniform and isotropic material of density 120588 usually in theform of a bar of known dimensions is subjected to transverse

or flexural vibrations the natural frequency of 119899th mode ofwhich 119891

119899related to Youngrsquos modulus 119864 by the relation [15 16]

119891119899=

1205822

119899

21205871198972radic

119864119868

120588119860 119899 = 1 2 3 (1)

can be accurately measured In (1) 120582119899is the modal eigenvalue

that depends on boundary conditions 119897 is the vibrating lengthof the bar 119860 is its cross-sectional area and 119868 is the secondmoment of the cross-section equal to 120587119903

4

4 for a rod ofradius 119903 and 119889ℎ

3

12 for a rectangular beam with width119889 and depth ℎ Knowing the modal numbers by simplymeasuring the resonance frequencies geometry and densityof the specimen the Youngmodulus can be determined from(1) as

119864 =41205872

1198912

1198991198974

120588119860

1205824119899119868

(2)

The test sample is usually arranged in a manner to simulatefree-free or clamped-free end conditions [10ndash12] when 120582

119899

associated with the 119899th flexural mode is a constant

2 Advances in Materials Science and Engineering

KR

xl

z

z(x)KT

Figure 1 Model of a cantilever beam elastically restrained at 119909 = 0

In the present paper we develop a new approach inwhicha rectangular strip-shaped sample attached to a force sensorforms the Euler Bernoulli beamwith partial translational androtational restraints at the fixed end This feature expandsthe capabilities of the resonant beam method making itsuitable for materials with high stiffness and low density inwhich case it is difficult to ascertain the flexural resonancefrequencies with high certainty

2 Theoretical Background

Consider a rectangular bar of uniform density 120588 cross-section dimensions of which width 119889 and depth ℎ are muchless than length 119897 The bar is fixed at one end (119909 = 0)

to the force sensor with a linear (119870119879) and torsion (119870

119877)

springs constants (see Figure 1) and is otherwise free tomove in the transverse z-direction For small deflectionsthat is 120597119911(119909 119905)120597119909 ≪ 1 the effects of rotary inertia andshear deformation can be ignored In this case neglectingthe deflection due to the weight the flexural displacementof a bar 119911(119909 119905) at point 119909 is governed by the Euler-Bernoulliequation [10]

1205881198601205972

119911

1205971199052+ 119864119868

1205974

119911

1205971199094= 0 (3)

with boundary conditions at 119909 = 0

minus119870119879119911 (0 119905) = 119864119868

1205973

119911

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=0

119870119877

120597119911

120597119909

10038161003816100381610038161003816100381610038161003816119909=0= 119864119868

1205972

119911

1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=0

(4)

which correspond to the force and moment balance respec-tively At the free end 119909 = 119897 there are no moment and shearforce acting that is

1205972

119911

1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897

= 01205973

119911

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=119897

= 0 (5)

Equations (3)ndash(5) define completely the linear flexural vibra-tion problem in which the natural frequencies of the beamdepend on spring constants119870

119879and119870

119877 Applying the separa-

tion of variables method the solution of (3) can be cast in thefollowing form

119911 (119909 119905) = 119908 (119909) 119879 (119905) =

infin

sum

119899=1

119908119899(119909) 119890119894120596119899119905

(6)

where 119908119899(119909) describes the 119899th normal mode and 120596

119899

(=2120587119891119899 119891 resonant frequency) is the angular frequency of

the 119899th mode Substituting (6) into (3) gives an eigenvalueproblem in the form of a fourth-order ordinary differentialequation

1205974

119908119899(119909)

1205971199094minus 1205814

119899119908119899(119909) = 0 (7)

where 120581119899is related to the angular frequency120596

119899and themodal

number 120582119899by the dispersion relationship

1205814

119899= (

120582119899

119897)

4

= 1205962

119899

120588119860

119864119868 (8)

With the boundary condition (4) the solution of (7) admitsthe form of 119899th flexural eigenmode

119908119899(119909) = 119860

119899(cosh 120581

119899119909 + 119886119899sinh 120581

119899119909 + 119887119899sin 120581119899119909)

+ 119862119899(cos 120581

119899119909 minus 119887119899sinh 120581

119899119909 minus 119886119899sin 120581119899119909)

(9)

where related coefficients 119887119899and 119886119899are defined as

119886119899=

1 minus 1205824

119899119877119899119879119899

21205823119899119877119899

119887119899=

1 + 1205824

119899119877119899119879119899

21205823119899119877119899

(10)

with 119877119899and 119879

119899expressed through experimentally accessible

quantities

119877119899=

119870119877

(2120587119891119899)2

1205881198601198973 119879

119899=

119870119879

(2120587119891119899)2

120588119860119897

(11)

and integration constants119860119899and119862

119899are related by the modal

restriction equation (5) as119860119899

119862119899

=cos 120582119899+ 119887119899sinh 120582

119899minus 119886119899sin 120582119899

cosh 120582119899+ 119886119899sinh 120582

119899minus 119887119899sin 120582119899

= minussin 120582119899minus 119887119899cosh 120582

119899+ 119886119899cos 120582119899

sinh 120582119899+ 119886119899cosh 120582

119899minus 119887119899cos 120582119899

(12)

where the second equality implies the constraints on possiblevalues of 120582

119899 known as the frequency equation

(1 + 1205824

119899119877119899119879119899) minus (1 minus 120582

4

119899119877119899119879119899) cos 120582

119899cosh 120582

119899

minus 120582119899[(1205822

119899119877119899+ 119879119899) sin 120582

119899cosh 120582

119899

+ (1205822

119899119877119899minus 119879119899) sinh 120582

119899cos 120582119899]

= 0

(13)

For given values of 119877119899and 119879

119899 transcendental equation (13)

has infinite (iterative or graphical) solutions As 119879119899rarr 0 and

119877119899rarr 0 (13) becomes

1 minus cos 120582119899cosh 120582

119899= 0 (14)

that is the frequency equation for the free-free beam For119879119899≫ 119877119899(13) reduces to the frequency equation derived by

Chun [17] which in our case is written in the form

1198771198991205823

119899(1 + cos 120582

119899cosh 120582

119899)

minus sin 120582119899cosh 120582

119899minus sinh 120582

119899cos 120582119899= 0

(15)

which in turn at 119877119899rarr infin becomes the frequency equation

for the clamped-free beam1 + cos 120582

119899cosh 120582

119899= 0 (16)

Advances in Materials Science and Engineering 3

Sensor support

Specimens

Force sensor

Data logger

Figure 2 Equipment used for measuring Youngrsquos modulus with theforce sensor

3 Principles of Operation

The experimental setup consists of a commercial FourierForce Sensor DT 272 with rotation 119870

119877(=17837Nsdotm) and

translation 119870119879(=6400Nsdotmminus1) spring constants accuracy

plusmn2 and resolution (12 bit) 0005N for a scale range plusmn10NThe force sensor mounted on a support is connected throughdata acquisition system and data studio software to a personalcomputer (PC) as is shown in Figure 2

The strip-shaped specimen with a roughly mass of 119898 asymp

10 g is attached horizontally to a force sensor at one endand is free at the other end By simply displacing the freeend in the transverse direction and abruptly releasing it thesample is subjected to free flexural vibrations so that thecondition 120597119911120597119909 ≪ 1 was fulfilled At a given sample rate119891119904= 1000Hz the force sensor detects the dynamic response

and through the data acquisition system displays the dampedoscillations of the restoring force versus time (Figure 3(a))This vibration signal is analyzed and processed by an FFTimplemented in acquisition software like MultiLogPRO [18]providing the direct information about natural frequenciesup to 119891

1199042 which appears as single peaks in the frequency

spectrum (Figure 3(b)) At this point using data on thegeometrical dimensions of the sample its density and springconstants of the force sensor119870

119877and119870

119879 identified resonance

frequencies 119891119899are used in (11) to calculate dimensionless

parameters 119877119899and 119879

119899 Knowing 119877

119899and 119879

119899 modal number

120582119899was found from the graphical solution of (13) using the

mathematical packages MATHEMATICA [19] The Youngmodulus is then determined from

119864 = 4812058721205881198912

1198991198974

1205824119899ℎ2

(17)

The relative error of the method arises from the uncer-tainties in the measurement of the quantities in (17) Therelative error in the density 120588 determined by the hydrostaticmethod is |119889(ln 120588)| le 07 The uncertainty in the thicknessand length measurement are respectively |119889(ln ℎ)| le 04and |119889(ln 119897)| le 025The dispersion in the natural frequency

Table 1 Identified resonant frequencies 119891119899in Hz modal numbers

determined from (13) and Youngrsquos moduli in GPa evaluated by (17)for the brass specimens of different lengths119889 = 16mm ℎ = 15mmand 120588 = 8400 kgsdotmminus3

119897119889 119879119899119877119899

1198911

1205821

1198641

1198912

1205822

1198642

813 606 3782 1595 1116 23279 4068 100906 754 3114 1619 1103 19383 414 1009813 884 2684 1638 1074 17023 4179 1021069 1049 2307 1653 1079 14782 4215 10471225 1378 179 1679 1052 11524 4276 10371469 1982 130 1701 1089 8410 4326 109155 2207 1175 1709 1082 7625 4372 1065

determination is |119889(ln1198911)| le 025 By applying the error

propagation technique given by

|119889 (ln119864)| le 21003816100381610038161003816119889 (ln119891

1)1003816100381610038161003816 +

1003816100381610038161003816119889 (ln 120588)1003816100381610038161003816 + 4 |119889 (ln 119897)| + 2 |119889 (ln ℎ)|

(18)

we find that the relative error in Youngrsquos modulus doesnot exceed plusmn3 The greatest inaccuracies occurred in themeasurement of the spacemen dimensions

4 Results and Discussion

To test the accuracy and validity of the present method theeffect of the sample length on the resonance frequency andYoungrsquos modulus was studied A commercial brass strip ofwidth 119889 = 16mm thickness ℎ = 15mm and density 120588 =8400 kgsdotmminus3 was cut into samples of various length so thatone of the conditions of the Euler-Bernoulli beam theory119897ℎ gt 10 remained unchanged while the second 119897119889 rangedfrom 8 to 16 Since the ration119879

119899119877119899≫ 1 (see Table 1) we used

the approximated equation (15) to find the 1205821 The validity of

this approach is illustrated in Figure 4 which shows that thesolutions of both (15) and (13) practically coincide

Based on the results of measurements of the naturalfrequencies and modal numbers presented in Table 1 adouble logarithm plot of 119891

1198941205822

119894against 119897 predicted by (1) to

be linear with the gradient of minus2 shows that the slope of theline that best fits these data in a least-squares sense is minus202for the first mode and minus194 for the second one Based on thesame set of data we show in Figure 5 the plot of the ratio1198911198941198972

1205822

119894versus 119897119889 Most of the uncertainty in the ordinates

of this plot arises from uncertainty of 119897 rather than 119891119894 In the

range of 119897119889 gt 10 the value of 1198911198941198972

1205822

119894is constant to within

the experimental uncertainties showing that1198911198941205822

119894prop 119897minus2 is in

agreement with (1) For 119897119889 lt 10 that is smaller for validity ofthe Euler-Bernoulli beam theory the ratio 119891

1198941198972

1205822

119894decreases

(increases)with 119897 for the first (second)mode For 119897119889 ge 10 themean value of Youngrsquos modulus 119864 = 106 plusmn 2GPa lies withinthe range 95 divide 110GPa listed in an extensive table of ASTMtesting [20]

Below we compare the results of measurements of elasticmoduli at ambient temperature for a wide class of industrialmaterials with that accepted in the literature A set of testspecimens usedwere cut from the commercial sheetmaterials


minus2

minus15

minus1

minus05

0

05

1

15

2

0 2 4 6

Resto

ring

forc

e (N

)

Time (s)

(a)

0

005

01

015

02

025

03

0 50 100 150 200 250 300In

tens

ityFrequency (Hz)

(b)

Figure 3 The flexural vibration test of partially restrained aluminum beam with 119897 = 0282m 119889 = 135 cm and ℎ = 155mm (a) Dynamicresponse detected by the force sensor at sample rate 1 kHz (b) Identified resonance frequencies 119891

1= 147Hz 119891

2= 928Hz 119891

3= 2495Hz

minus6

minus4

minus2

0

2

4

6

minus3000

minus2000

minus1000

0

1000

2000

3000

155 165 175 185

Num

eric

al v

alue

of (

13)

Num

eric

al v

alue

of (

15)

(13)(15)

1205821

Figure 4 Graphical solutions of (13) and (15) for the first modeof flexural vibrations of partially restrained aluminum beam arerespectively 120582

1= 17752 and 120582

1= 17745 against 120582

1= 18751 for

ideal clamped-free case

into strips of the thickness (ℎ) from 05 to 33mm width(119889) from 5 to 165mm and length (119897) from 15 to 30 cmFor each specimen the length width and thickness werealtered and the value of the Young modulus calculatedby (17) for each set varied within the experimental errorTable 2 summarizes data of the specimen dimensionmaterial

density natural frequencies modal numbers and Youngrsquosmodulus calculated from the first resonant frequency It canbe seen that test results are in good agreement with theaccepted those in the literature data

In all cases identified natural frequencies of the partiallyrestrained cantilever beam are lower than those for theclamped-free case However this does not necessary meanthat the elastic modulus determined from these frequenciesshould be smaller than in ideal clamped-free case as themodal number being in the denominator equation (17) infourth power decreases as well Interestingly the plot of 1205824

1

versus 11198771in Figure 5 shows a linear dependence

1205824

1= 1205824

infin(1 minus

1

31198771

) (19)

where 120582infin

= 18751 is the first modal number for theclamped-free flexural vibrations of the beam Taking intoaccount (11) after substituting (19) in (17) we obtain theworking equation for the Young modulus determinationthrough the first resonance frequency of flexural vibrationsof the partially restrained cantilever beam (Figure 6)

119864 =481205872

1198912

11205881198974

ℎ21205824infin

(1 minus (41205872119891211205881198601198973

3119870119877))

(20)

5 Summary

A new technique for the fast determination of the dynamicYoungmodulus was developed yielding a substantial modifi-cation of the classical cantilever beammethodThe procedure


Table 2 Youngrsquos moduli of materials 119864 in GPa determined at room temperature by (17) and specimen parameters length 119897 in mm thicknessℎ in mm density 120588 in kgsdotmminus3 fundamental frequency 119891

1in Hz and modal number 120582

1calculated by (13)

Material 119897 ℎ 120588 1198911

1205821

119864 119864Literature

Al 6061 sheet 282 155 2715 1442 1778 700 70ndash72Zn coated steel 193 055 7820 120 1868 201 206Sheet steel 304 193 05 7970 110 1864 210 190ndash213Cooper alloy 214 10 8790 1189 1848 106 110ndash120Brass stripe 248 15 8400 1175 1709 108 96ndash110Perspex 190 30 1190 232 1798 42 24ndash46Wood oak 252 35 674 303 1664 126 11ndash12Compositea 232 25 1600 330 1429 920 36ndash150aGraphite carbon epoxy

0235

024

0245

025

0255

8 10 12 14 16

f1(l1205821)2

ld

First modeSecond modeThe mean value for ld gt 10

Figure 5 Plot of 119891119894(119897120582119894)2 versus 119897119889 based on the flexural vibration

test of a rectangular brass strip (119889 = 160mm and ℎ = 15mm) For119897119889 ge 10 the mean value 0245 (broken line) is constant to withinexperimental uncertainty

uses a rectangular beam partially restrained at one endflexural vibrations of which are detected with the aid of theforce sensor The relative experimental uncertainty is foundto be less than 3 which is mainly due to the uncertaintyin the samples dimensions The feasibility and accuracy ofa new experimental procedure has been demonstrated bymeasuring the Young modulus for a number of test materialswith different material properties Comparison of obtainedresults with those accepted in the literature data is good Therelative deviation of measured values from the cited data isless than 5Themethod has potential advantages over otherdynamicmethods of being very simple and fast and requiringno additional equipment to excite resonance frequencies Itis particularly suitable for composite materials having a high

3

5

7

9

11

13

0 05 1 15 2

1205821

1R1

Figure 6Themodal number of the first mode of flexural vibrationsof partially restrained cantilever beams in the fourth degree as afunction of the inverse of the dimensionless frequency parameter11198771 Symbols are experimental data taken fromTables 1 and 2 Solid

line is by (19)

stiffness and low density such as carbon fiber reinforcedplastic The accuracy can be significantly improved by moreprecise determination of specimen dimensions

Acknowledgment

This work was supported by the Ministry of Absorptionand Immigration of Israel through the KAMEA ScienceFoundation

References

[1] J D Lord and R Morrell ldquoldquoElastic modulus measurementrdquoMeasurement good practice guide No 98rdquo Febraury 2007httpwwwnplcoukpublicationsguidesguides-by-title


[2] J D Lord Review of Methods and Analysis Software for theDetermination of Modulus from Tensile Tests vol 41 NPLMeasurement Note MATC (MN) 2002

[3] P E Armstrong Measurement of Mechanical Properties Tech-niques of Metals Research vol 5 part 2 of Edited by R FBunshan John Wiley amp Sons New York NY USA 1971

[4] M Radovic E Lara-Curzio and L Riester ldquoComparison ofdifferent experimental techniques for determination of elasticproperties of solidsrdquo Materials Science and Engineering A vol368 no 1-2 pp 56ndash70 2004

[5] A Wolfenden Ed Dynamic Elastic Modulus Measurements inMaterials American Society for Testing and Materials 1990

[6] A S Birks and R E Green Nondestructive Testing Handbookvol 7 of Ultrasonic Testing American Society for Nondestruc-tive Testing 1991

[7] A Migliori and J L Sarrao Resonant Ultrasound SpectroscopyApplications to Physics Materials Measurements and Nonde-structive Evaluation John Wiley amp Sons New York NY USA1997

[8] AWolfendenMRHarmoucheGV Blessing et al ldquoDynamicYoungrsquos modulus measurements in metallic materials resultsof an interlaboratory testing programmrdquo Journal of Testing andEvaluation 1989

[9] G Roebben B Bollen A Brebels J van Humbeeck and O vander Biest ldquoImpulse excitation apparatus to measure resonantfrequencies elastic moduli and internal friction at room andhigh temperaturerdquo Review of Scientific Instruments vol 68 no12 pp 4511ndash4515 1997

[10] W Lins G Kaindl H Peterlik and K Kromp ldquoA novelresonant beam technique to determine the elastic moduli independence on orientation and temperature up to 2000∘CrdquoReview of Scientific Instruments vol 70 no 7 pp 3052ndash30581999

[11] K Heritage C Frisby and A Wolfenden ldquoImpulse excitationtechnique for dynamic flexural measurements at moderatetemperaturerdquo Review of Scientific Instruments vol 59 no 6 pp973ndash974 1988

[12] R M Digilov ldquoFlexural vibration test of a cantilever beamwith a force sensor fast determination of Youngrsquos modulusrdquoEuropean Journal of Physics vol 29 no 3 p 589 2008

[13] American Society for Testing and Materials ldquoStandard testmethod for dynamic Youngrsquos modulus Shear modulus andpoissonrsquos ratio for advanced ceramics by impulse excitation ofvibrationrdquo Standard C 1259-01 April 2001

[14] Comite Europeen de Normalisation ldquoDetermination ofdynamic elastic modulus by measuring the fundamentalresonant frequencyrdquo Standard EN 14146 2004

[15] D J Inman Engineering Vibration Prentice-Hall EnglewoodCliffs NJ USA 1994

[16] S S RaoMechanical Vibrations Addison-Wesley Menlo ParkCalif USA 3rd edition 1995

[17] K R Chun ldquoFree vibration of a beam with one end spring-hinged and the other freerdquo ASME Journal of Applied Mechanicsvol 39 no 4 pp 1154ndash1155 1972

[18] ldquoMultilab software for MultiLogPROrdquo Fourier System httpfouriereducom

[19] Mathematica 7 Wolfram Research Champaign Ill USA 2008[20] J R Davis EdMetals Handbook ASM International Materials

Park Ohio USA 1990

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KR

xl

z

z(x)KT

Figure 1 Model of a cantilever beam elastically restrained at 119909 = 0

In the present paper we develop a new approach inwhicha rectangular strip-shaped sample attached to a force sensorforms the Euler Bernoulli beamwith partial translational androtational restraints at the fixed end This feature expandsthe capabilities of the resonant beam method making itsuitable for materials with high stiffness and low density inwhich case it is difficult to ascertain the flexural resonancefrequencies with high certainty

2 Theoretical Background

Consider a rectangular bar of uniform density 120588 cross-section dimensions of which width 119889 and depth ℎ are muchless than length 119897 The bar is fixed at one end (119909 = 0)

to the force sensor with a linear (119870119879) and torsion (119870

119877)

springs constants (see Figure 1) and is otherwise free tomove in the transverse z-direction For small deflectionsthat is 120597119911(119909 119905)120597119909 ≪ 1 the effects of rotary inertia andshear deformation can be ignored In this case neglectingthe deflection due to the weight the flexural displacementof a bar 119911(119909 119905) at point 119909 is governed by the Euler-Bernoulliequation [10]

1205881198601205972

119911

1205971199052+ 119864119868

1205974

119911

1205971199094= 0 (3)

with boundary conditions at 119909 = 0

minus119870119879119911 (0 119905) = 119864119868

1205973

119911

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=0

119870119877

120597119911

120597119909

10038161003816100381610038161003816100381610038161003816119909=0= 119864119868

1205972

119911

1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=0

(4)

which correspond to the force and moment balance respec-tively At the free end 119909 = 119897 there are no moment and shearforce acting that is

1205972

119911

1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897

= 01205973

119911

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=119897

= 0 (5)

Equations (3)ndash(5) define completely the linear flexural vibra-tion problem in which the natural frequencies of the beamdepend on spring constants119870

119879and119870

119877 Applying the separa-

tion of variables method the solution of (3) can be cast in thefollowing form

119911 (119909 119905) = 119908 (119909) 119879 (119905) =

infin

sum

119899=1

119908119899(119909) 119890119894120596119899119905

(6)

where 119908119899(119909) describes the 119899th normal mode and 120596

119899

(=2120587119891119899 119891 resonant frequency) is the angular frequency of

the 119899th mode Substituting (6) into (3) gives an eigenvalueproblem in the form of a fourth-order ordinary differentialequation

1205974

119908119899(119909)

1205971199094minus 1205814

119899119908119899(119909) = 0 (7)

where 120581119899is related to the angular frequency120596

119899and themodal

number 120582119899by the dispersion relationship

1205814

119899= (

120582119899

119897)

4

= 1205962

119899

120588119860

119864119868 (8)

With the boundary condition (4) the solution of (7) admitsthe form of 119899th flexural eigenmode

119908119899(119909) = 119860

119899(cosh 120581

119899119909 + 119886119899sinh 120581

119899119909 + 119887119899sin 120581119899119909)

+ 119862119899(cos 120581

119899119909 minus 119887119899sinh 120581

119899119909 minus 119886119899sin 120581119899119909)

(9)

where related coefficients 119887119899and 119886119899are defined as

119886119899=

1 minus 1205824

119899119877119899119879119899

21205823119899119877119899

119887119899=

1 + 1205824

119899119877119899119879119899

21205823119899119877119899

(10)

with 119877119899and 119879

119899expressed through experimentally accessible

quantities

119877119899=

119870119877

(2120587119891119899)2

1205881198601198973 119879

119899=

119870119879

(2120587119891119899)2

120588119860119897

(11)

and integration constants119860119899and119862

119899are related by the modal

restriction equation (5) as119860119899

119862119899

=cos 120582119899+ 119887119899sinh 120582

119899minus 119886119899sin 120582119899

cosh 120582119899+ 119886119899sinh 120582

119899minus 119887119899sin 120582119899

= minussin 120582119899minus 119887119899cosh 120582

119899+ 119886119899cos 120582119899

sinh 120582119899+ 119886119899cosh 120582

119899minus 119887119899cos 120582119899

(12)

where the second equality implies the constraints on possiblevalues of 120582

119899 known as the frequency equation

(1 + 1205824

119899119877119899119879119899) minus (1 minus 120582

4

119899119877119899119879119899) cos 120582

119899cosh 120582

119899

minus 120582119899[(1205822

119899119877119899+ 119879119899) sin 120582

119899cosh 120582

119899

+ (1205822

119899119877119899minus 119879119899) sinh 120582

119899cos 120582119899]

= 0

(13)

For given values of 119877119899and 119879

119899 transcendental equation (13)

has infinite (iterative or graphical) solutions As 119879119899rarr 0 and

119877119899rarr 0 (13) becomes

1 minus cos 120582119899cosh 120582

119899= 0 (14)

that is the frequency equation for the free-free beam For119879119899≫ 119877119899(13) reduces to the frequency equation derived by

Chun [17] which in our case is written in the form

1198771198991205823

119899(1 + cos 120582

119899cosh 120582

119899)

minus sin 120582119899cosh 120582

119899minus sinh 120582

119899cos 120582119899= 0

(15)

which in turn at 119877119899rarr infin becomes the frequency equation

for the clamped-free beam1 + cos 120582

119899cosh 120582

119899= 0 (16)


Sensor support

Specimens

Force sensor

Data logger




119877(=17837Nsdotm) and








119877and119870



parameters 119877119899and 119879

119899 Knowing 119877

119899and 119879

119899 modal number



119864 = 4812058721205881198912

1198991198974

1205824119899ℎ2

(17)




119897119889 119879119899119877119899

1198911

1205821

1198641

1198912

1205822

1198642

813 606 3782 1595 1116 23279 4068 100906 754 3114 1619 1103 19383 414 1009813 884 2684 1638 1074 17023 4179 1021069 1049 2307 1653 1079 14782 4215 10471225 1378 179 1679 1052 11524 4276 10371469 1982 130 1701 1089 8410 4326 109155 2207 1175 1709 1082 7625 4372 1065



|119889 (ln119864)| le 21003816100381610038161003816119889 (ln119891

1)1003816100381610038161003816 +

1003816100381610038161003816119889 (ln 120588)1003816100381610038161003816 + 4 |119889 (ln 119897)| + 2 |119889 (ln ℎ)|

(18)




119899119877119899≫ 1 (see Table 1) we used




1198941205822



1205822




1205822





1198941198972

1205822

119894decreases




minus2

minus15

minus1

minus05

0

05

1

15

2

0 2 4 6

Resto

ring

forc

e (N

)

Time (s)

(a)

0

005

01

015

02

025

03

0 50 100 150 200 250 300In

tens

ityFrequency (Hz)

(b)


1= 147Hz 119891

2= 928Hz 119891

3= 2495Hz

minus6

minus4

minus2

0

2

4

6

minus3000

minus2000

minus1000

0

1000

2000

3000

155 165 175 185

Num

eric

al v

alue

of (

13)

Num

eric

al v

alue

of (

15)

(13)(15)

1205821


1= 17752 and 120582

1= 17745 against 120582

1= 18751 for





1


1205824

1= 1205824

infin(1 minus

1

31198771

) (19)

where 120582infin


119864 =481205872

1198912

11205881198974

ℎ21205824infin

(1 minus (41205872119891211205881198601198973

3119870119877))

(20)

5 Summary





1calculated by (13)

Material 119897 ℎ 120588 1198911

1205821



0235

024

0245

025

0255

8 10 12 14 16

f1(l1205821)2

ld





3

5

7

9

11

13

0 05 1 15 2

1205821

1R1


line is by (19)


Acknowledgment


References





















Park Ohio USA 1990








CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Sensor support

Specimens

Force sensor

Data logger




119877(=17837Nsdotm) and








119877and119870



parameters 119877119899and 119879

119899 Knowing 119877

119899and 119879

119899 modal number



119864 = 4812058721205881198912

1198991198974

1205824119899ℎ2

(17)




119897119889 119879119899119877119899

1198911

1205821

1198641

1198912

1205822

1198642

813 606 3782 1595 1116 23279 4068 100906 754 3114 1619 1103 19383 414 1009813 884 2684 1638 1074 17023 4179 1021069 1049 2307 1653 1079 14782 4215 10471225 1378 179 1679 1052 11524 4276 10371469 1982 130 1701 1089 8410 4326 109155 2207 1175 1709 1082 7625 4372 1065



|119889 (ln119864)| le 21003816100381610038161003816119889 (ln119891

1)1003816100381610038161003816 +

1003816100381610038161003816119889 (ln 120588)1003816100381610038161003816 + 4 |119889 (ln 119897)| + 2 |119889 (ln ℎ)|

(18)




119899119877119899≫ 1 (see Table 1) we used




1198941205822



1205822




1205822





1198941198972

1205822

119894decreases




minus2

minus15

minus1

minus05

0

05

1

15

2

0 2 4 6

Resto

ring

forc

e (N

)

Time (s)

(a)

0

005

01

015

02

025

03

0 50 100 150 200 250 300In

tens

ityFrequency (Hz)

(b)


1= 147Hz 119891

2= 928Hz 119891

3= 2495Hz

minus6

minus4

minus2

0

2

4

6

minus3000

minus2000

minus1000

0

1000

2000

3000

155 165 175 185

Num

eric

al v

alue

of (

13)

Num

eric

al v

alue

of (

15)

(13)(15)

1205821


1= 17752 and 120582

1= 17745 against 120582

1= 18751 for





1


1205824

1= 1205824

infin(1 minus

1

31198771

) (19)

where 120582infin


119864 =481205872

1198912

11205881198974

ℎ21205824infin

(1 minus (41205872119891211205881198601198973

3119870119877))

(20)

5 Summary





1calculated by (13)

Material 119897 ℎ 120588 1198911

1205821



0235

024

0245

025

0255

8 10 12 14 16

f1(l1205821)2

ld





3

5

7

9

11

13

0 05 1 15 2

1205821

1R1


line is by (19)


Acknowledgment


References





















Park Ohio USA 1990








CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




minus2

minus15

minus1

minus05

0

05

1

15

2

0 2 4 6

Resto

ring

forc

e (N

)

Time (s)

(a)

0

005

01

015

02

025

03

0 50 100 150 200 250 300In

tens

ityFrequency (Hz)

(b)


1= 147Hz 119891

2= 928Hz 119891

3= 2495Hz

minus6

minus4

minus2

0

2

4

6

minus3000

minus2000

minus1000

0

1000

2000

3000

155 165 175 185

Num

eric

al v

alue

of (

13)

Num

eric

al v

alue

of (

15)

(13)(15)

1205821


1= 17752 and 120582

1= 17745 against 120582

1= 18751 for





1


1205824

1= 1205824

infin(1 minus

1

31198771

) (19)

where 120582infin


119864 =481205872

1198912

11205881198974

ℎ21205824infin

(1 minus (41205872119891211205881198601198973

3119870119877))

(20)

5 Summary





1calculated by (13)

Material 119897 ℎ 120588 1198911

1205821



0235

024

0245

025

0255

8 10 12 14 16

f1(l1205821)2

ld





3

5

7

9

11

13

0 05 1 15 2

1205821

1R1


line is by (19)


Acknowledgment


References





















Park Ohio USA 1990








CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






1calculated by (13)

Material 119897 ℎ 120588 1198911

1205821



0235

024

0245

025

0255

8 10 12 14 16

f1(l1205821)2

ld





3

5

7

9

11

13

0 05 1 15 2

1205821

1R1


line is by (19)


Acknowledgment


References





















Park Ohio USA 1990








CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






















Park Ohio USA 1990








CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



research article flexural vibration test of a beam...

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