FaizAhmadandTakasarHussaindownloads.hindawi.com/journals/aav/2013/903934.pdfAdvances in Acoustics and Vibration kT h c/c T 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 0 5 10 15 20 25 30 35

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2013 Article ID 903934 6 pageshttpdxdoiorg1011552013903934

Research ArticleAnomalous Dispersion of the 119878

1Lamb Mode

Faiz Ahmad and Takasar Hussain

Centre for Advanced Mathematics and Physics National University of Sciences and Technology H-12 CampusIslamabad 44000 Pakistan

Correspondence should be addressed to Takasar Hussain htakasaryahoocom

Received 30 April 2013 Revised 1 July 2013 Accepted 16 July 2013

Academic Editor Abdelkrim Khelif

Copyright copy 2013 F Ahmad and T Hussain This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The 1198781mode of the Lamb spectrum of an isotropic plate exhibits negative group velocity in a narrow frequency domain This

anomalous behavior is explained analytically by examining the slope of each mode first in its initial state and then near its turningpoints

1 Introduction

The dispersion relation for symmetric Lamb modes propa-gating in an infinite isotropic plate of thickness 2ℎ is given bythe well known Rayleigh-Lamb equation [1]

tan (119901ℎ)tan (119902ℎ)

= minus41199011199021198962

(1199012minus 1198962)2 (1)

where

119901 = radic1205962

1198882

119879

minus 1198962

119902 = radic1205962

1198882

119871

minus 1198962

(2)

In (1) 119888119879and 119888119871 respectively denote the phase speeds of

the transverse and longitudinal bulk waves in the materialAlso120596 and 119896 respectively denote the frequency and thewavenumber of the modeThe phase velocity 119888 of a mode is givenby

119888 =120596

119896 (3)

If 119888 is plotted as a function of the frequency the spectrumappears as in Figure 1 which depicts the spectrum for a steelplate with 119888

119879= 324 kms and 119888

119871= 595 kms

Themost striking feature in Figure 1 is the shape of the 1198781

mode which has a turning point at 119896119879= 2686 and the phase

velocity becomes double valued for 119896119879ℎ in [2686 2873]This

phenomenon of negative group velocity is of great technicalsignificance and has been observed in a large number ofexperiments [2ndash8]

The afore mentioned feature of 1198781mode was first noticed

by Tolstoy and Usdin [9] in 1957 In all isotropic materialswith 120581 = 2 (] = 13) only the 119878

1mode has this ldquoanomalous

behaviorrdquo and other modes behave normally We will call itthe 1198781anomaly An explanation of this peculiar shape ofmode

1198781has posed a challenge since its discovery in 1957For the special case of a material with ] = 13 that is

119888119871= 2119888119879 each of the modes 119878

3119899+1 119899 = 0 1 2 3 exhibits

anomalous behavior [10] Anomalous pairs of modes mayalso occur for certain special values of the Poisson ratio Wewill call it the pair anomaly

Although (1) governs the behavior of all modes anoma-lous or otherwise no simple theory seems to exist whichshould provide a satisfactory explanation of why certainmodes in the spectrum should possess a bulge while othersproceed in a normal manner However certain physicalexplanations of this phenomenon exist In 1983 Whitakerand Haus [11] noted the fact that ldquopropagation of waves withdispersion of this sort has been experimentally verified [2]but the reason for their appearance is not well understoodrdquoThey used the coupled mode theory to argue that when thefundamental 119875 mode and the first harmonic 119878 mode are

2 Advances in Acoustics and Vibration

kTh

cc T

25 3 35 4 45 5 55 6 65 7 75 80

5

10

15

20

25

30

35

40

Figure 1 Symmetric Lamb modes on a steel plate (120581 = 183)showing phase velocity as a function of normalized frequency

nearly degenerate at cutoff a coupling effect can occur at theboundaries Uberall et al [12] hypothesize that ldquoone observesa repulsion phenomenon between neighboring dispersioncurves similar to that encountered in atomic physics forquasidegenerate energy levels of atoms when combining intomoleculesrdquo Prada et al [3] express the same view in thewords ldquothis phenomenon leads to a strong repulsion betweenthe dispersion curves of the neighbouring modesrdquo

It is clear that all of these authors focused on the pairanomaly only because in the 119878

1anomaly the 119878

1mode

remains distinct and it does not coincide with any othermode at cutoff Mode repulsion cannot explain 119878

1anomaly

To the best of our knowledge the 1198781anomaly still remains

an unsolved mystery Recently Hussain and Ahmad [13]considered ZGV points in the spectrum of Lamb modes incompressible orthotropic plate It was found that in additionto modes with a single ZGV point a large number of modesexist with multiple such points

In this paper we will examine Rayleigh-Lamb spectrumfor the symmetric modes of an isotropic material We willanalyze (1) and derive mathematical expressions which willexplain both types of anomalies

2 The Mode Spectrum

Let 119896 and 119888 respectively denote the wave number and phasespeed of the mode Define the dimensionless speed 119888119888

119879by 119910

and the dimensionless frequency 120596ℎ119888119879by 119906

Then

119901ℎ = ℎradic1205962

1198882

119879

minus 1198962

= ℎ119896radic1199102minus 1 =

ℎ119896119888119888119879

119888119888119879

radic1199102minus 1 =

119906

119910

radic1199102minus 1

(4)

119902ℎ =119906

119910

radic1199102

1205812minus 1 (5)

With respect to the variables 119906 and 119910 (1) becomes

119891 (119906 119910)

= (1199102minus 2)2

sin(radic1199102 minus 1119906119910) cos(radic

1199102

1205812minus 1

119906

119910)

+ 4radic1199102minus 1radic

1199102

1205812minus 1 sin(radic

1199102

1205812minus 1

119906

119910)

times cos(radic1199102 minus 1119906119910) = 0

(6)

In this section we will consider 120581 = 2 We will calculatethe derivative 119889119910119889119906 at two positions of the spectrum Wewrite

119863119899120581=119889119910

119889119906

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=120581

119863119899119871=119889119910

119889119906

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=119871≫120581

(7)

where the subscript 119899 refers to themode under considerationFor 2radic3 lt 120581 lt 339 (minus1 lt ] lt 0452) we will show that119863119899120581lt 0 for all modes On the other hand119863

1119871gt 0 for the 119878

1

mode while119863119899119871lt 0 for all other modes Since the derivative

for the 1198781mode changes from positive to negative it must

exhibit a bulge before 119910 = 120581 No other mode undergoes areversal of the slope hence all other modes continue theirdownward journey until they asymptotically approach theline 119910 = 1

In Appendix A we give expressions for 120597119891120597119906 and 120597119891120597119910The derivative 119889119910119889119906 is found as

119889119910

119889119906= minus

120597119891120597119906

120597119891120597119910 (8)

When 119910 = 120581 (6) becomes

sin(119906radic1205812minus 1

120581) = 0 (9)

Therefore

119906119899=

119899120587120581

radic1205812minus 1

(10)

Equation (10) shows that the line 119910 = 120581 intersects themodes at infinitely many points At 119910 = 120581 and 119906 = 119906

119899 partial

derivatives (A1) become

120597119891

120597119906

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=120581

=

(1205812minus 2)2radic1205812minus 1

120581

120597119891

120597119910

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=120581

= [

[

(1205812minus 2)2

1205812radic1205812minus 1

+ 4

radic1205812minus 1

1205812

]

]

119906119899

(11)

Advances in Acoustics and Vibration 3

Hence

119863119899120581= minus

(1205812minus 1)32

(1205812minus 2)2

[(1205812minus 2)2

+ 4 (1205812minus 1)] 119899120587

(12)

It is clear that119863119899120581lt 0 for all modes Also |119863

119899120581| becomes

progressively smaller as 119899 increases and 119863119899120581

rarr 0 as 119899 rarr

infin This means that for 119910 = 120581 there is a plateau region andthis plateau is flatter for higher modes

Next we find 119863119899119871 For 119910 ≫ 120581 the expressions for 120597119891120597119906

and 120597119891120597119910 are given by

120597119891

120597119906≃ 1199104[cos 119906 cos 119906

120581minus1

120581sin 119906 sin 119906

120581] (13)

120597119891

120597119910≃ 119906119910 cos 119906 cos 119906

120581+ 41199103 cos 119906

120581sin 119906

+8119910

120581cos 119906 sin 119906

120581minus 120581119906119910 sin 119906 sin 119906

120581

(14)

Thus

119889119910

119889119906

≃ minus

1199104minus (1199104120581) tan 119906 tan (119906120581)

119906119910 + 41199103 tan 119906 + (8119910120581) tan (119906120581) minus 120581119906119910 tan 119906 tan (119906120581)

(15)

For 119910 ≫ 120581 (6) gives

1199102≃ minus

4 tan (119906120581)120581 tan 119906

(16)

Also from (15) and (16) we have

119863119899119871≃ minus

1199103+ (11991054) tan2119906

119899

119906119899+ 1199102 tan 119906

119899(2 + (120581

24) 119906119899tan 119906119899) (17)

or equivalently

119863119899119871≃ minus

1199103+ (11991054) tan2119906

119899

119906119899minus (4120581) tan (119906

119899120581) (2 + (120581

24) 119906119899tan 119906119899) (18)

We have replaced 119906 by 119906119899in (17) and (18) since for a fixed

119910 (16) yields infinitely many roots 119906119899 119899 = 1 2 3

To fix ideas we consider the case of a steel plate for which120581 = 183 The general case follows on similar lines

From (16) we see that for the 1198781mode 119906

1120581 should be

slightly less than 1205872 so that 1199061120581 is in the first quadrant and

the corresponding 1199061is in the second quadrant to yield a large

positive 1199102Hence for the mode 119878

1 we have

1198631119871

≃ minus

1199103+ (11991054) tan2 (1205811205872)

(1205811205872) + 1199102 tan (1205811205872) (2 + (12058124) sdot (1205811205872) tan (1205811205872))

(19)

For 120581 = 183 and for large 119910 1198631119871gt 0

Table 1 Approximate and exact values of 119906119899when 119910 = 20 for the

first five modes

Modes Approximate 119906119899

Exact value at 119910 = 201198781

2875 28561198782

3142 31801198783

6283 62891198784

8624 86501198785

9425 9449

For 2radic3 lt 120581 lt 2 and large 119910 approximate values of 119906119899

from (16) for the first few modes are 1205811205872 120587 31205811205872 2120587 and51205811205872 For the steel plate these values are compared inTable 1with the exact values found from (6) when 119910 = 20

We have seen previously that 1198631119871

gt 0 for the 1198781mode

Now for the 1198782mode 119906

2≃ 120587 and from (18) we get

1198632119871≃ minus

1199103

120587 minus (8120581) tan (120587120581) (20)

Here 120581 = 183 and tan(120587120581) lt 0 thus 1198632119871

lt 0 Ina similar fashion by successively using (17) and (18) we canshow that119863

119899119871lt 0 for all 119899 ge 2

From the previous we conclude that for steel the 1198781is the

only mode which reverses its slope while going from largevalues of 119910 to 119910 = 120581 The previous analysis applies to allmaterials with 120581 lt 2

If 120581 gt 2 the 1198781mode occurs when 119906

1≃ 120587 and from (18)

the 1198781mode will have positive slope as long as

120587 minus8

120581tan 120587

120581lt 0 (21)

or 2 lt 120581 lt 339 which corresponds to 13 lt ] lt 0452Thus we have established that the 119878

1mode will be

anomalous for all materials falling in the range 2radic3 lt 120581 lt339 Since the slope of 119878

1becomes negative for all 119910 when

120581 gt 339 the mode will lose its anomalous character beyond120581 = 339 which corresponds to ] = 0452

3 The Exceptional Case 120581 = 2

The case 120581 = 2 merits special treatment The spectrum ofsymmetric modes appears as in Figure 2

The pairs 1198781-1198782 1198784-1198785 1198787-1198788 appear to merge for large

119910 and then bifurcate as they descend to lower values ofthe phase speed On the other hand 119878

3 1198786 1198789 appear to

behave normallyThis phenomenon was first reported by Mindlin [14]

Each of the modes 1198781 1198784 1198787 shows anomalous dispersion

With 120581 = 2 (6) becomes for 119910 ≫ 1

1199104 cos 119906

2sin 119906 + 1199102 sin 119906

2cos 119906 = 0 (22)

which is satisfied for 119906119899= 2119899120587 119899 = 1 2 3 It is shown in

Appendix B that a more accurate solution of (6) is

119906119899= 119899120587(2 +

1

1199102) (23)


0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

kTh

cc T

Figure 2 Symmetric Lamb modes on a nickel plate (120581 = 2)

In addition to (23) (16) has roots given by

minus1

21199102≃tan (1199062)tan 119906

=1

2(1 minus tan2 119906

2) (24)

Ignoring 1 in comparison with 1199102 we get

tan2 1199062≃ 1199102 (25)

or

cot2 1199062≃1

1199102 (26)

Let

119906

2=(2119899 + 1) 120587

2+ 120598 (27)

Equation (26) becomes

cot2 (1205872+ 120598) ≃

1

1199102 (28)

or

tan2120598 ≃ 1

1199102 (29)

which leads to

120598 ≃ plusmn1

119910 (30)

Thus for large 119910 1199061= 120587 minus 2120598 119906

2= 120587 + 2120598 119906

4= 3120587 minus 2120598

1199065= 3120587 + 2120598 and so forthSince 120598 rarr 0 as 119910 rarr infin the modes 119906

1 1199062 1199064 1199065

1199067 1199068 appear to coalesce for large 119910 Now with 120581 = 2 (18)

gives 1198631119871

gt 0 1198634119871

gt 0 and 1198637119871

gt 0 while the slopes forall other modes are negative This argument establishes theanomalous dispersion of the modes 119878

3119899+1 119899 = 0 1 2 3 As

1199063 1199066 1199069 occur slightly above 2120587 4120587 6120587 (17) gives a

negative value for the slope of each of these modes Hencethese modes behave in a normal manner

4 Conclusion

We have explained analytically the anomalous behavior ofLamb modes for an isotropic material by looking at the slopeof each mode for large as well as small 119910 For small 119910 slope isfound at 119910 = 120581 This simple technique explains in an analyticmanner theoretical results given by several authors about theanomalous dispersion of the 119878

1mode

Appendices

A The Partial Derivatives

Expressions for the partial derivatives are as follows

120597119891

120597119906

= ((1199102minus 2)2

radic1199102minus 1 cos

119906radic1199102minus 1

119910

times cos119906radic11991021205812minus 1

119910)(119910)

minus1

+(4(1199102

1205812minus 1)radic119910

2minus 1 cos


119910


119910)(119910)

minus1

minus((1199102minus 2)2radic1199102

1205812minus 1 sin


119910

times sin119906radic11991021205812minus 1

119910)(119910)

minus1

minus(4 (1199102minus 1)radic

1199102

1205812minus 1 sin


119910


119910)(119910)

minus1

120597119891

120597119910

= (1199102minus 2)2

(119906

radic1199102minus 1

minus


1199102

)



119910cos

119906radic11991021205812minus 1

119910

+ 4radic1199102minus 1radic119910

21205812minus 1

times(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910cos

119906radic11991021205812minus 1

119910

+ 4119910 (1199102minus 2) sin


119910cos

119906radic11991021205812minus 1

119910

+(4119910radic1199102minus 1 cos


119910


119910)(1205812radic11991021205812minus 1)

minus1

+(4119910radic11991021205812minus 1 cos


119910


119910)(radic119910

2minus 1)

minus1

minus 4radic1199102minus 1radic119910

21205812minus 1(

119906

radic1199102minus 1

minus


1199102

)


119910sin

119906radic11991021205812minus 1

119910

minus (1199102minus 2)2

(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910sin

119906radic11991021205812minus 1

119910

(A1)

B The Mode 1198783119899

for 120581 = 2

For 119910 ≫ 1 radic1199102 minus 1119910 ≃ 1 minus 1(21199102) and radic(11991024) minus 1119910 ≃12 minus 1119910

2

Let 119906119899≃ 2119899120587+(120572

1198991199102) and ignore the terms of order 11199104

or higher Then

cos(radic1199102minus 1

119910119906119899) ≃ 1

cos(radic11991024 minus 1

119910119906119899) ≃ minus1

sin(radic1199102minus 1

119910119906119899) ≃ sin

120572119899minus 119899120587

1199102

≃120572119899minus 119899120587

1199102

sin(radic11991024 minus 1

119910119906119899) ≃ ∓ sin

120572119899minus 2119899120587

1199102

≃ ∓120572119899minus 2119899120587

1199102

(B1)

Putting these expressions in (6) we have

minus1199104120572119899 minus 119899120587

1199102

∓ 2 (120572119899minus 2119899120587) = 0 (B2)

or

(1199102plusmn 2) 120572

119899= 119899120587119910

2∓ 4119899120587 (B3)

or

120572119899=1198991205871199102∓ 4119899120587

1199102plusmn 2

≃ 119899120587 (B4)

for large 119910This result shows that the modes 119878

3119899 119899 = 1 2 3

intersect the line 119910 = 1199100 for large 119910

0 at points slightly to

the right of 119906119899= 2119899120587 119899 = 1 2 3

Acknowledgment

Faiz Ahmad is grateful to the Higher Education Commissionof Pakistan for financial support

References

[1] J D Achenbach Wave Propagation in Elastic Solids chapter 6North-Holland Amsterdam The Netherlands 1980

[2] A H Meitzler ldquoBackward wave tranmission stress pulses inelstic cylinders and platesrdquo Journal of the Acoustical Society ofAmerica vol 38 pp 835ndash842 1965

[3] C Prada D Clorennec and D Royer ldquoLocal vibration of anelastic plate and zero-group velocity Lamb modesrdquo Journal ofthe Acoustical Society of America vol 124 no 1 pp 203ndash2122008

[4] K Nishimiya K Mizutani N Wakatsuki and K YamamotoldquoDetermination of 12 condition for fastest NGV of Lamb-Typewaves under each density ratio of solid and liquid layersrdquoAcoustics 08 Paris pp 3613ndash3618


[5] J Wolf T D K Nook R Kille andW G Mayer ldquoInvestigationof lamb waves having negative group velocityrdquo Journal of theAcoustical Society of America vol 83 pp 122ndash126 1988

[6] O Balogun T W Murray and C Prada ldquoSimulation andmeasurement of the optical excitation of the S1 zero groupvelocity Lamb wave resonance in platesrdquo Journal of AppliedPhysics vol 102 no 6 Article ID 064914 2007

[7] K Negishi ldquoNegative group velocities of Lamb wavesrdquo Journalof the Acoustical Society of America vol 64 no 1 p S63 1978

[8] M Germano A Alippi A Bettucci and G Mancuso ldquoAnoma-lous and negative reflection of Lamb waves in mode conver-sionrdquo Physical Review B vol 85 no 1 Article ID 012102 2012

[9] I Tolstoy and E Usdin ldquoWave propagation in elastic plates lowand high mode dispersionrdquo Journal of the Acoustical Society ofAmerica vol 29 pp 37ndash42 1957

[10] M F Werby and H Uberall ldquoThe analysis and interpretation ofsome special properties of higher order symmetric Lambwavesthe case for platesrdquo Journal of the Acoustical Society of Americavol 111 no 6 pp 2686ndash2691 2002

[11] N A Whitaker Jr and H A Haus ldquoBackward wave effectin acoustic scattering measurementsrdquo IEEE Ultrasonics Sympo-sium pp 891ndash894 1983

[12] H Uberall B Hosten M Deschamps and A Gerard ldquoRepul-sion of phase-velocity dispersion curves and the nature of platevibrationsrdquo Journal of the Acoustical Society of America vol 96no 2 pp 908ndash917 1994

[13] T Hussain and F Ahmad ldquoLamb modes with multiple zero-group velocity points in an orthotropic platerdquo Journal of theAcoustical Society of America vol 32 pp 641ndash645 2012

[14] R D Mindlin An Introduction To the Mathematical Theory ofVibrations of Elastic Plates Monograph Sec 211 US ArmySignal Corps Eng Lab Ft Monmouth NJ USA 1995 Editedby J Yang World Scientific Singapore 2006

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kTh

cc T

25 3 35 4 45 5 55 6 65 7 75 80

5

10

15

20

25

30

35

40

Figure 1 Symmetric Lamb modes on a steel plate (120581 = 183)showing phase velocity as a function of normalized frequency

nearly degenerate at cutoff a coupling effect can occur at theboundaries Uberall et al [12] hypothesize that ldquoone observesa repulsion phenomenon between neighboring dispersioncurves similar to that encountered in atomic physics forquasidegenerate energy levels of atoms when combining intomoleculesrdquo Prada et al [3] express the same view in thewords ldquothis phenomenon leads to a strong repulsion betweenthe dispersion curves of the neighbouring modesrdquo

It is clear that all of these authors focused on the pairanomaly only because in the 119878

1anomaly the 119878

1mode

remains distinct and it does not coincide with any othermode at cutoff Mode repulsion cannot explain 119878

1anomaly

To the best of our knowledge the 1198781anomaly still remains

an unsolved mystery Recently Hussain and Ahmad [13]considered ZGV points in the spectrum of Lamb modes incompressible orthotropic plate It was found that in additionto modes with a single ZGV point a large number of modesexist with multiple such points

In this paper we will examine Rayleigh-Lamb spectrumfor the symmetric modes of an isotropic material We willanalyze (1) and derive mathematical expressions which willexplain both types of anomalies

2 The Mode Spectrum

Let 119896 and 119888 respectively denote the wave number and phasespeed of the mode Define the dimensionless speed 119888119888

119879by 119910

and the dimensionless frequency 120596ℎ119888119879by 119906

Then

119901ℎ = ℎradic1205962

1198882

119879

minus 1198962

= ℎ119896radic1199102minus 1 =

ℎ119896119888119888119879

119888119888119879

radic1199102minus 1 =

119906

119910

radic1199102minus 1

(4)

119902ℎ =119906

119910

radic1199102

1205812minus 1 (5)

With respect to the variables 119906 and 119910 (1) becomes

119891 (119906 119910)

= (1199102minus 2)2

sin(radic1199102 minus 1119906119910) cos(radic

1199102

1205812minus 1

119906

119910)

+ 4radic1199102minus 1radic

1199102

1205812minus 1 sin(radic

1199102

1205812minus 1

119906

119910)

times cos(radic1199102 minus 1119906119910) = 0

(6)

In this section we will consider 120581 = 2 We will calculatethe derivative 119889119910119889119906 at two positions of the spectrum Wewrite

119863119899120581=119889119910

119889119906

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=120581

119863119899119871=119889119910

119889119906

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=119871≫120581

(7)

where the subscript 119899 refers to themode under considerationFor 2radic3 lt 120581 lt 339 (minus1 lt ] lt 0452) we will show that119863119899120581lt 0 for all modes On the other hand119863

1119871gt 0 for the 119878

1

mode while119863119899119871lt 0 for all other modes Since the derivative

for the 1198781mode changes from positive to negative it must

exhibit a bulge before 119910 = 120581 No other mode undergoes areversal of the slope hence all other modes continue theirdownward journey until they asymptotically approach theline 119910 = 1

In Appendix A we give expressions for 120597119891120597119906 and 120597119891120597119910The derivative 119889119910119889119906 is found as

119889119910

119889119906= minus

120597119891120597119906

120597119891120597119910 (8)

When 119910 = 120581 (6) becomes

sin(119906radic1205812minus 1

120581) = 0 (9)

Therefore

119906119899=

119899120587120581

radic1205812minus 1

(10)

Equation (10) shows that the line 119910 = 120581 intersects themodes at infinitely many points At 119910 = 120581 and 119906 = 119906

119899 partial

derivatives (A1) become

120597119891

120597119906

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=120581

=

(1205812minus 2)2radic1205812minus 1

120581

120597119891

120597119910

10038161003816100381610038161003816100381610038161003816119906=119906119899119910=120581

= [

[

(1205812minus 2)2

1205812radic1205812minus 1

+ 4

radic1205812minus 1

1205812

]

]

119906119899

(11)


Hence

119863119899120581= minus

(1205812minus 1)32

(1205812minus 2)2

[(1205812minus 2)2

+ 4 (1205812minus 1)] 119899120587

(12)


119899120581| becomes





and 120597119891120597119910 are given by

120597119891

120597119906≃ 1199104[cos 119906 cos 119906

120581minus1

120581sin 119906 sin 119906

120581] (13)

120597119891

120597119910≃ 119906119910 cos 119906 cos 119906

120581+ 41199103 cos 119906

120581sin 119906

+8119910

120581cos 119906 sin 119906

120581minus 120581119906119910 sin 119906 sin 119906

120581

(14)

Thus

119889119910

119889119906

≃ minus

1199104minus (1199104120581) tan 119906 tan (119906120581)


(15)

For 119910 ≫ 120581 (6) gives

1199102≃ minus

4 tan (119906120581)120581 tan 119906

(16)


119863119899119871≃ minus

1199103+ (11991054) tan2119906

119899

119906119899+ 1199102 tan 119906

119899(2 + (120581

24) 119906119899tan 119906119899) (17)

or equivalently

119863119899119871≃ minus

1199103+ (11991054) tan2119906

119899

119906119899minus (4120581) tan (119906

119899120581) (2 + (120581

24) 119906119899tan 119906119899) (18)





1120581 should be




1 we have

1198631119871

≃ minus

1199103+ (11991054) tan2 (1205811205872)

(1205811205872) + 1199102 tan (1205811205872) (2 + (12058124) sdot (1205811205872) tan (1205811205872))

(19)



first five modes


Exact value at 119910 = 201198781

2875 28561198782

3142 31801198783

6283 62891198784

8624 86501198785

9425 9449







1198632119871≃ minus

1199103

120587 minus (8120581) tan (120587120581) (20)



119899119871lt 0 for all 119899 ge 2




1≃ 120587 and from (18)


120587 minus8

120581tan 120587

120581lt 0 (21)


1mode will be








3 1198786 1198789 appear to



With 120581 = 2 (6) becomes for 119910 ≫ 1

1199104 cos 119906

2sin 119906 + 1199102 sin 119906

2cos 119906 = 0 (22)



119906119899= 119899120587(2 +

1

1199102) (23)


0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

kTh

cc T



minus1

21199102≃tan (1199062)tan 119906

=1

2(1 minus tan2 119906

2) (24)


tan2 1199062≃ 1199102 (25)

or

cot2 1199062≃1

1199102 (26)

Let

119906

2=(2119899 + 1) 120587

2+ 120598 (27)


cot2 (1205872+ 120598) ≃

1

1199102 (28)

or

tan2120598 ≃ 1

1199102 (29)

which leads to

120598 ≃ plusmn1

119910 (30)


2= 120587 + 2120598 119906

4= 3120587 minus 2120598


1 1199062 1199064 1199065


gives 1198631119871

gt 0 1198634119871

gt 0 and 1198637119871


3119899+1 119899 = 0 1 2 3 As



4 Conclusion


1mode

Appendices



120597119891

120597119906

= ((1199102minus 2)2



119910


119910)(119910)

minus1

+(4(1199102

1205812minus 1)radic119910

2minus 1 cos


119910


119910)(119910)

minus1


1205812minus 1 sin


119910


119910)(119910)

minus1


1199102

1205812minus 1 sin


119910


119910)(119910)

minus1

120597119891

120597119910

= (1199102minus 2)2

(119906

radic1199102minus 1

minus


1199102

)



119910cos

119906radic11991021205812minus 1

119910


21205812minus 1

times(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910cos

119906radic11991021205812minus 1

119910

+ 4119910 (1199102minus 2) sin


119910cos

119906radic11991021205812minus 1

119910



119910


119910)(1205812radic11991021205812minus 1)

minus1

+(4119910radic11991021205812minus 1 cos


119910


119910)(radic119910

2minus 1)

minus1


21205812minus 1(

119906

radic1199102minus 1

minus


1199102

)


119910sin

119906radic11991021205812minus 1

119910


(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910sin

119906radic11991021205812minus 1

119910

(A1)

B The Mode 1198783119899

for 120581 = 2


2

Let 119906119899≃ 2119899120587+(120572


or higher Then


119910119906119899) ≃ 1


119910119906119899) ≃ minus1


119910119906119899) ≃ sin

120572119899minus 119899120587

1199102

≃120572119899minus 119899120587

1199102


119910119906119899) ≃ ∓ sin

120572119899minus 2119899120587

1199102

≃ ∓120572119899minus 2119899120587

1199102

(B1)


minus1199104120572119899 minus 119899120587

1199102

∓ 2 (120572119899minus 2119899120587) = 0 (B2)

or

(1199102plusmn 2) 120572

119899= 119899120587119910

2∓ 4119899120587 (B3)

or

120572119899=1198991205871199102∓ 4119899120587

1199102plusmn 2

≃ 119899120587 (B4)


3119899 119899 = 1 2 3



the right of 119906119899= 2119899120587 119899 = 1 2 3

Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Hence

119863119899120581= minus

(1205812minus 1)32

(1205812minus 2)2

[(1205812minus 2)2

+ 4 (1205812minus 1)] 119899120587

(12)


119899120581| becomes





and 120597119891120597119910 are given by

120597119891

120597119906≃ 1199104[cos 119906 cos 119906

120581minus1

120581sin 119906 sin 119906

120581] (13)

120597119891

120597119910≃ 119906119910 cos 119906 cos 119906

120581+ 41199103 cos 119906

120581sin 119906

+8119910

120581cos 119906 sin 119906

120581minus 120581119906119910 sin 119906 sin 119906

120581

(14)

Thus

119889119910

119889119906

≃ minus

1199104minus (1199104120581) tan 119906 tan (119906120581)


(15)

For 119910 ≫ 120581 (6) gives

1199102≃ minus

4 tan (119906120581)120581 tan 119906

(16)


119863119899119871≃ minus

1199103+ (11991054) tan2119906

119899

119906119899+ 1199102 tan 119906

119899(2 + (120581

24) 119906119899tan 119906119899) (17)

or equivalently

119863119899119871≃ minus

1199103+ (11991054) tan2119906

119899

119906119899minus (4120581) tan (119906

119899120581) (2 + (120581

24) 119906119899tan 119906119899) (18)





1120581 should be




1 we have

1198631119871

≃ minus

1199103+ (11991054) tan2 (1205811205872)

(1205811205872) + 1199102 tan (1205811205872) (2 + (12058124) sdot (1205811205872) tan (1205811205872))

(19)



first five modes


Exact value at 119910 = 201198781

2875 28561198782

3142 31801198783

6283 62891198784

8624 86501198785

9425 9449







1198632119871≃ minus

1199103

120587 minus (8120581) tan (120587120581) (20)



119899119871lt 0 for all 119899 ge 2




1≃ 120587 and from (18)


120587 minus8

120581tan 120587

120581lt 0 (21)


1mode will be








3 1198786 1198789 appear to



With 120581 = 2 (6) becomes for 119910 ≫ 1

1199104 cos 119906

2sin 119906 + 1199102 sin 119906

2cos 119906 = 0 (22)



119906119899= 119899120587(2 +

1

1199102) (23)


0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

kTh

cc T



minus1

21199102≃tan (1199062)tan 119906

=1

2(1 minus tan2 119906

2) (24)


tan2 1199062≃ 1199102 (25)

or

cot2 1199062≃1

1199102 (26)

Let

119906

2=(2119899 + 1) 120587

2+ 120598 (27)


cot2 (1205872+ 120598) ≃

1

1199102 (28)

or

tan2120598 ≃ 1

1199102 (29)

which leads to

120598 ≃ plusmn1

119910 (30)


2= 120587 + 2120598 119906

4= 3120587 minus 2120598


1 1199062 1199064 1199065


gives 1198631119871

gt 0 1198634119871

gt 0 and 1198637119871


3119899+1 119899 = 0 1 2 3 As



4 Conclusion


1mode

Appendices



120597119891

120597119906

= ((1199102minus 2)2



119910


119910)(119910)

minus1

+(4(1199102

1205812minus 1)radic119910

2minus 1 cos


119910


119910)(119910)

minus1


1205812minus 1 sin


119910


119910)(119910)

minus1


1199102

1205812minus 1 sin


119910


119910)(119910)

minus1

120597119891

120597119910

= (1199102minus 2)2

(119906

radic1199102minus 1

minus


1199102

)



119910cos

119906radic11991021205812minus 1

119910


21205812minus 1

times(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910cos

119906radic11991021205812minus 1

119910

+ 4119910 (1199102minus 2) sin


119910cos

119906radic11991021205812minus 1

119910



119910


119910)(1205812radic11991021205812minus 1)

minus1

+(4119910radic11991021205812minus 1 cos


119910


119910)(radic119910

2minus 1)

minus1


21205812minus 1(

119906

radic1199102minus 1

minus


1199102

)


119910sin

119906radic11991021205812minus 1

119910


(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910sin

119906radic11991021205812minus 1

119910

(A1)

B The Mode 1198783119899

for 120581 = 2


2

Let 119906119899≃ 2119899120587+(120572


or higher Then


119910119906119899) ≃ 1


119910119906119899) ≃ minus1


119910119906119899) ≃ sin

120572119899minus 119899120587

1199102

≃120572119899minus 119899120587

1199102


119910119906119899) ≃ ∓ sin

120572119899minus 2119899120587

1199102

≃ ∓120572119899minus 2119899120587

1199102

(B1)


minus1199104120572119899 minus 119899120587

1199102

∓ 2 (120572119899minus 2119899120587) = 0 (B2)

or

(1199102plusmn 2) 120572

119899= 119899120587119910

2∓ 4119899120587 (B3)

or

120572119899=1198991205871199102∓ 4119899120587

1199102plusmn 2

≃ 119899120587 (B4)


3119899 119899 = 1 2 3



the right of 119906119899= 2119899120587 119899 = 1 2 3

Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

kTh

cc T



minus1

21199102≃tan (1199062)tan 119906

=1

2(1 minus tan2 119906

2) (24)


tan2 1199062≃ 1199102 (25)

or

cot2 1199062≃1

1199102 (26)

Let

119906

2=(2119899 + 1) 120587

2+ 120598 (27)


cot2 (1205872+ 120598) ≃

1

1199102 (28)

or

tan2120598 ≃ 1

1199102 (29)

which leads to

120598 ≃ plusmn1

119910 (30)


2= 120587 + 2120598 119906

4= 3120587 minus 2120598


1 1199062 1199064 1199065


gives 1198631119871

gt 0 1198634119871

gt 0 and 1198637119871


3119899+1 119899 = 0 1 2 3 As



4 Conclusion


1mode

Appendices



120597119891

120597119906

= ((1199102minus 2)2



119910


119910)(119910)

minus1

+(4(1199102

1205812minus 1)radic119910

2minus 1 cos


119910


119910)(119910)

minus1


1205812minus 1 sin


119910


119910)(119910)

minus1


1199102

1205812minus 1 sin


119910


119910)(119910)

minus1

120597119891

120597119910

= (1199102minus 2)2

(119906

radic1199102minus 1

minus


1199102

)



119910cos

119906radic11991021205812minus 1

119910


21205812minus 1

times(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910cos

119906radic11991021205812minus 1

119910

+ 4119910 (1199102minus 2) sin


119910cos

119906radic11991021205812minus 1

119910



119910


119910)(1205812radic11991021205812minus 1)

minus1

+(4119910radic11991021205812minus 1 cos


119910


119910)(radic119910

2minus 1)

minus1


21205812minus 1(

119906

radic1199102minus 1

minus


1199102

)


119910sin

119906radic11991021205812minus 1

119910


(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910sin

119906radic11991021205812minus 1

119910

(A1)

B The Mode 1198783119899

for 120581 = 2


2

Let 119906119899≃ 2119899120587+(120572


or higher Then


119910119906119899) ≃ 1


119910119906119899) ≃ minus1


119910119906119899) ≃ sin

120572119899minus 119899120587

1199102

≃120572119899minus 119899120587

1199102


119910119906119899) ≃ ∓ sin

120572119899minus 2119899120587

1199102

≃ ∓120572119899minus 2119899120587

1199102

(B1)


minus1199104120572119899 minus 119899120587

1199102

∓ 2 (120572119899minus 2119899120587) = 0 (B2)

or

(1199102plusmn 2) 120572

119899= 119899120587119910

2∓ 4119899120587 (B3)

or

120572119899=1198991205871199102∓ 4119899120587

1199102plusmn 2

≃ 119899120587 (B4)


3119899 119899 = 1 2 3



the right of 119906119899= 2119899120587 119899 = 1 2 3

Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910cos

119906radic11991021205812minus 1

119910


21205812minus 1

times(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910cos

119906radic11991021205812minus 1

119910

+ 4119910 (1199102minus 2) sin


119910cos

119906radic11991021205812minus 1

119910



119910


119910)(1205812radic11991021205812minus 1)

minus1

+(4119910radic11991021205812minus 1 cos


119910


119910)(radic119910

2minus 1)

minus1


21205812minus 1(

119906

radic1199102minus 1

minus


1199102

)


119910sin

119906radic11991021205812minus 1

119910


(119906

1205812radic11991021205812minus 1

minus

119906radic11991021205812minus 1

1199102

)


119910sin

119906radic11991021205812minus 1

119910

(A1)

B The Mode 1198783119899

for 120581 = 2


2

Let 119906119899≃ 2119899120587+(120572


or higher Then


119910119906119899) ≃ 1


119910119906119899) ≃ minus1


119910119906119899) ≃ sin

120572119899minus 119899120587

1199102

≃120572119899minus 119899120587

1199102


119910119906119899) ≃ ∓ sin

120572119899minus 2119899120587

1199102

≃ ∓120572119899minus 2119899120587

1199102

(B1)


minus1199104120572119899 minus 119899120587

1199102

∓ 2 (120572119899minus 2119899120587) = 0 (B2)

or

(1199102plusmn 2) 120572

119899= 119899120587119910

2∓ 4119899120587 (B3)

or

120572119899=1198991205871199102∓ 4119899120587

1199102plusmn 2

≃ 119899120587 (B4)


3119899 119899 = 1 2 3



the right of 119906119899= 2119899120587 119899 = 1 2 3

Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

FaizAhmadandTakasarHussaindownloads.hindawi.com/journals/aav/2013/903934.pdfAdvances in Acoustics and Vibration kT h c/c T 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 0 5 10 15 20 25 30 35