researcharticle a new approach on vibrating horns design

Research ArticleA New Approach on Vibrating Horns Design

Maria Violeta Guiman and Ioan Cslin Rosca

Department of Mechanical Engineering Faculty of Mechanical Engineering Transilvania University of Brasov 29 Eroilor Av500036 Brasov Romania

Correspondence should be addressed to Ioan Calin Rosca icroscaunitbvro

Received 5 February 2017 Revised 1 June 2017 Accepted 19 June 2017 Published 7 August 2017

Academic Editor Tai Thai

Copyright copy 2017 Maria Violeta Guiman and Ioan Calin Rosca This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

An optimization method of the vibrating horns is presented considering the smallest action principle and the attached cutting toolmass The model is based on Websterrsquos wave propagation equation and as an objective function the minimization of the volumein structural equilibrium conditions was considered The considered input parameters were working frequency maximum cross-sectional area magnification coefficient and the attached mass At the end of the study a new shape function of the hornrsquos crosssection is obtained The particularity of the new obtained shape is given by the nodal point position that is the same with theposition of the maximum cross-sectional area The obtained horn was analyzed from the modal point of view using theoreticaland experimental methods As theoretical methods both the state-space method and the finite element method were used Anexperimental setup for frequency response function determination was developed using a random input signal The verification ofthe magnitude value was done considering a harmonic steady-state signal The recorded values were compared with the predictedvalues The numerical simulations and tests support the validity of the assumptions used in the horns optimization design

1 Introduction

Ultrasonic horns are axisymmetric devices that provide withhigh efficiency acoustic energy developed by a transducerthat is working based on the magnetostriction effect to atool These devices can be themselves tools or can be usedwith a tool attached at the small end and thus they are tiedwith high-power applications of ultrasound like cutting [1ndash3] drilling [4ndash6] turning [7] welding [8 9] and so forth orfor machining different materials as carbon fiber reinforcedcomposites [10ndash12]

From a physical point of view the work of such a systemconsists of an initial signal emitted by a transducer at agiven frequency with an amplitude 1199060 which is mechanicallyamplified by the horn with an imposedmagnitude coefficient119902 The whole system is working in a resonance regime atan imposed frequency The mechanical magnification of theinput signal amplitude is generated by the horn varying crosssection from an initial value 1198780 in the origin of the referencesystem to an end one value 119878119871 (Figure 1)

Generally the cross section variation is given by differentmathematical functions (exponential [13] linear [14] tapered

(or stepped) [15] catenoidal [14] Bezier [16] Gaussian etc)Studies concerning the wave propagation and basic conceptsof horns design are presented in different papers [17ndash21]

A problem that is associated with the horns design refersto the shape optimization The optimization procedure isconnected with design parameters like frequency workingvalues [21] amplitude of the signal [22 23] load transferredin the manufacturing area [24] the objective function [2526] the impact of different design variables on the objectivefunction [27] combined signals transmitted in the horn [28]and so forth

In the present paper we present an approach based on theprinciple of the least action that takes into consideration as anobjective function the volume minimization The consideredinput parameters are the working frequency the maximumcross-sectional area the magnification coefficient and theattached mass

2 Mathematical Model

In case of ultrasonic horns a planar stress waves state isassumedThegeneral equation that describes one-dimensional

HindawiShock and VibrationVolume 2017 Article ID 8532021 12 pageshttpsdoiorg10115520178532021

2 Shock and Vibration

x

x

y

L

SL

Sx

0

S0

Figure 1 The solid horn geometry

propagation in the longitudinal direction of ultrasonicwaves along a rigid horn with a variable cross-sectional areais Websterrsquos horn equation [29]

1205972119906 (119909 119905)1205971199092 + 120597119906 (119909 119905)120597119909 120597120597119909 (ln 119878119909) = 11198882 1205972119906 (119909 119905)1205971199052 (1)

where 119906(119909 119905) is the signal 119878119909 = 119878(119909) represents the cross-sectional area of the considered horn at the distance 119909 fromthe reference (point 0) and 119888 is the wave velocity in the hornrsquosmaterial

One of the widely used techniques to find the explicitanalytical solutions of the linear partial differential equationsis the method of variables separation [30] The methodassumes that the solution of (1) can be written as a product oftwo functions one in space domain 119906(119909) = 119906119909 and the otherone in time domain 119906(119905) = 119906119905

119906 (119909 119905) = 119906 (119909) 119906 (119905) = 119906119909119906119905 (2)

Introducing solution (2) in Websterrsquos equation (1) thenew shape is obtained

11990610158401015840119909119906119909 +119889119889119909 (119878119909)

1199061015840119909119906119909 =11198882 119905119906119905 (3)

where the following notations were done 11990610158401015840119909 = 119889211990611990911988911990921199061015840119909 = 119889119906119909119889119909 and 119905 = 11988921199061199051198891199052This way in (3) there were separate variables on each side

of the equationThe congruency of both sides left and right of (3) is

achieved only if the two functions are equal with the sameseparation constant 120573 It is considered as a constant value120573 = minus1205962 where 120596 is the angular wave frequency (rads)

Thus from (3) we obtained the following ordinarydifferential equations

119905 + 1205962119906119905 = 011990610158401015840119909 + 119889119889119909 (119878119909) 1199061015840119909 + 1198962119906119909 = 0

(4)

where 119896 = 120596119888 is the wave number

The general solution of the time domain equation from(4) is given by

119906119905 = 1198631 cos120596119905 + 1198632 sin120596119905 (5)

with the initial conditions

1199061199051003816100381610038161003816119905=0 = 119906011988911990611990511988911990510038161003816100381610038161003816100381610038161003816119905=0 = 0

(6)

where 1199060 is the initial amplitude of the ultrasonic waveThus solution (5) becomes

119906119905 = 1199060 cos120596119905 (7)

and the total solution (2) becomes

119906 (119909 119905) = 119906 (119909) 119906 (119905) = 119906119909119906119905 = 1199061199091199060 cos120596119905 (8)

In case of a steady-state mode considering the assump-tion of an infinitesimal motion in a beam made of nondissi-pative medium and with a variable cross-sectional area thesecond equation from (4) which represents the plane wavemotion can be rewritten as

119889119889119909 (119878119909 119889119906119909119889119909 ) + 1198962119878119909119906119909 = 0 (9)

It is considered that the maximum diameter of the hornshould not exceed one-quarter of the wavelength 120582 and themaximum cross section is denoted as 119878119898 Thus we can writethe condition that on the whole length of the horn the crosssection value has to be smaller than the maximum crosssection

119878 (119909) le 119878119898 (10)

Considering relation (10) one can write the followingmathematical relation

119878119898 minus 119878 (119909) minus 1198862 (119909) = 0 (11)

where 119886(119909) is an unspecified function of the variable 119909 In thefollowing the notations 119886(119909) = 119886119909 and 119878(119909) = 119878119909 will be used

An optimization problem of the hornrsquos shape refers tominimizing the hornrsquos volume for a given initial set of designdata Based on the principle of the least action one can writethe action relation as

119860 = int11990521199051

(119879 minus 119881) 119889119905 (12)

where 119879 is the kinetic energy and 119881 represents the potentialenergy

Considering the wave shape as

119906 = 119906 (119909 119905) = 119906 (119909) cos (120596119905) = 119906119909 cos (120596119905) (13)

Shock and Vibration 3

for a beam of length 119871with attachedmass at one end the caseof an ultrasonic horn with the attached tool the total energiesof the system horn tool mentioned in (12) are given by

119879 = 12 int119871

0(1198981199092) 119889119909 + 1198721199052 (119871)2

119881 = 12 int119871

0(119864ℎ119898119909120588ℎ ) (1199061015840)

2 119889119909 + 12059022119864119905119881119905(14)

where119898119909 is the specific mass of the horn119872119905 is the attachedmass (toll mass) 119864ℎ is Youngrsquos modulus of the horn material119864119905 is Youngrsquos modulus of the tool material 119881119905 is the volumeof the attached tool 120588ℎ represents the horn material density = 119889119906119889119905 and 1199061015840 = 119889119906119889119909

Considering a period action (12) becomes

119860|1199052+21205871205961199051

= 119860 = 120587 (119879 minus 119881) (15)

with

119879 = 12059622 [int119871

0120588 sdot 119878 sdot 1199062119889119909 +1198721199051199062 (119871)]

119881 = 12 [int119871

0119864ℎ sdot (1199061015840)2 sdot 119878 119889119909 + 120590 sdot 1199061015840 (119871) sdot 119881119905]

(16)

Based on [31 32] the following functional is defined

Ι = 119879 minus 119881 + 1205721119881 + int1198710120572119909 (119878119898 minus 119878119909 minus 1198862119909) 119889119909 (17)

where 1205721 and 120572119909 = 120572(119909) are Lagrange multipliers and119881 is thevolume of the horn given by

119881 = int1198710119878119909119889119909 (18)

The optimization problem has a solution for the imposedlimit values if the functional Ι given by (17) has to bestationary related to the variations of 119906119909 and to the designparameter 119878119909 Thus based on (18) it is necessary for Eulerrsquosequations [33] to be satisfied

120597Ι120597119906119909 minus119889119889119909 120597Ι1205971199061015840119909 = 0

120597Ι120597119878119909 minus119889119889119909 120597Ι1205971198781015840119909 = 0

120597Ι120597119886119909 minus119889119889119909 120597Ι1205971198861015840119909 = 0

(19)

where 1199061015840119909 = 119889119906119909119889119909 1198781015840119909 = 119889119878119909119889119909 and 1198861015840119909 = 119889119886119909119889119909Considering the functional given by (17) and (19) one can

obtain the following differential equations(a) From the first equation in (19) results

12059622 2119906119909120588119878119909 minus 119889119889119909 [minus119864ℎ2 21199061015840119909119878119909]= 1205962119906119909120588119878119909 + 119864ℎ (119878119909119906119909)1015840 = 0

(20)

which when divided by 119864ℎ becomes

(1198781199091199061015840119909)1015840 + 120588ℎ1205962119864ℎ 119878119909119906119909 = 0 (21)

Considering the velocity of the longitudinal wave in thehorn as 119888119897 = radic119864ℎ120588ℎ and the wave number 119896 = 120596119888119897 (21) canbe rewritten as

(1198781199091199061015840119909)1015840 + 1198962119878119909119906119909 = 0 (22)

(b) From the second equation one obtains

1205962 120588ℎ1199062119909 minus 119864ℎ (1199061015840119909)2 + 21205721 minus 2120572119909 = 0 (23)


1198961199062119909 minus (1199061015840119909)2 + 2119864ℎ 1205721 minus2119864ℎ 120572119909 = 0 (24)

(c) From the third equation results it is found that

2120572119909119886119909 = 0 (25)

Analyzing the previous equations one can obtain thefollowing remarks about the optimal design of the ultrasonichorn

(a) Equation (21) is identical to the longitudinal wavepropagation equation along a horn and it is availablealong the whole horn

(b) Equation (25) leads to two other conditions

120572119909 = 0or 119886119909 = 0 (26)

Conditions (26) involve the following aspect the function119886119909 is different from zero on the whole length of the hornexcept for one point where 119878119909 = 119878119898

Consequently for (25) to be satisfied for any point in therange 0 le 119909 le 119871 it is necessary to have the first conditionfrom (26) 120572119909 = 0 in the abovementioned range Moreover allLagrangersquos coefficients have to be constants

(a) From (24) and (26) the following is obtained

120572119909 = 119864ℎ2 [11989621199062119909 minus (1199061015840119909)2 + 21205721119864ℎ ] = 0 (27)

or

(1199061015840119909)2 minus 11989621199062119909 minus 21205721119864ℎ = 0 (28)

3 The Solution of the Differential Equation

The solution of (28) can be found considering the initialfunctional conditions of the ultrasonic horn The followingare considered as initial conditions


(a) The length of the horn has to be equal to awavelength(b) To achieve the assumption that the propagated waves

are only longitudinal ones a maximum value isconsidered for the magnification coefficient 119902 = 5[17]

(c) The initial value of the signal amplitude at theconnection point between the transducer and thehorn (119909 = 0) is 1199060 equal to the emitted signal by thetransducer

119906 (119909) = 119906 (0) = 1199060 (29)

(d) Considering the initial signal and the magnificationcoefficient the signal at the end of the horn (for119909 = 119871)is given by

119906 (119909) = 119906 (119871) = minus1199021199060 (30)

(e) In (30) the negative value (minus) is considered in orderto obtain a nodal point in a position along the hornlength

(f) Between the two ends of the horn there is noother extreme of the signal except for the maximumdeveloped at the end of the horn in the connectionpoint with the tool

(g) The following condition is considered [32]

1198981199061015840119909=119871 minus 1198962119872119906119909=119871 = 0 (31)

where 119898 is the mass on the unit length at the end ofthe horn 1199061015840119909=119871 is the strain at the end of the horn119872is the attached mass and 119906119909=119871 is the signal at the endof the horn given by (30)

Denoting

119862 = 21198641205721 (32)

equation (28) can be rewritten as

(1199061015840)2 minus 11989621199062 minus 119862 = 0 (33)

and integrating it with the derivative method one can obtainthe following equation [34]

2119906101584011990610158401015840 minus 119896221199061199061015840 = 0 (34)

Dividing (34) by 1199061015840 = 0 results in the differential equation11990610158401015840 minus 1198962119906 = 0 (35)

with the solution

119906 (119909) = 1198621119890119896119909 + 1198622119890minus119896119909 (36)

Introducing solution (36) in (33) after some calculationsthe following is obtained

minus4119896211986211198622 = 119862 (37)

1198621 = minus 119862411989621198622 (38)

Considering (38) in solution (36) results in

119906 (119909) = minus 119862411989621198622 119890119896119909 + 1198622119890minus119896119909 (39)

The constants 119862 and 1198622 can be found considering theinitial conditions mentioned above for 119909 = 0 Combining(38) and (30) the following relation is obtained

1199060 = minus 119862411989621198622 + 1198622 (40)

which leads to the following relation for the two values of theconstant 1198622 as a function of 119862

119862122 = 119896119906 plusmn radic119896211990620 + 1198622119896 (41)

Considering (41) and (40) for both values 119862122 the samedependence is obtained between the constants 119862 and 1198622

119862 = 411986221198962 (1198622 minus 1199060) (42)

Introducing the constant119862 given by (42) in solution (39)leads to the following relation of the amplitude 119906(119909)

119906 (119909) = 1199060119890119896119909 minus 1198622 (119890119896119909 minus 119890minus119896119909) (43)

The constant1198622 can be found considering conditions (39)and (40)

120588ℎ119878ℎ119871 [1199060119890119896119871 minus 1198622 (119890119896119871 + 119890minus119896119871)] + 119896119871 1199051205881199051198781199051199021199060 = 0 (44)

where 120588ℎ is the density of the hornrsquos material 119878ℎ119871 is the crosssection at the end of the horn 119878119905 is the cross section of thetool (119878119871 = 119878119905) 119871 119905 is the toolrsquos length 120588119905 is the density of thetoolrsquos material and 119902 is the gain of the initial signal 1199060

From (44) results

1198622 = 1199060 (120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905)2120588ℎ119878ℎ119871 cosh (119896119871) (45)

Considering dependence (42) one can define the function ofthe space dependent amplitude component of the signal alongthe horn

119906 (119909)= 1199060 [119890119896119909 minus 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (119890

119896119909 minus 119890minus119896119909)] (46)

In the following we use the notation

119861 = 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (47)

and the signal function (46) becomes

119906 (119909) = 1199060 [(1 minus 119861) 119890119896119909 + 119861119890minus119896119909] (48)


As it is known an important aspect in horns design isto find the position of the neutral point 119909119899 The neutralpoint defines the position of the cross section where thesignal is zero Thus the neutral point can be easily foundconsidering relation (48) Equalizing function (48) with zeroit is obtained that

119909nod = 12119896 ln( 119861119861 minus 1) (49)

4 Cross Section Shape Definition

From relation (48) we obtained the first two derivatives ofthe space component of the signal

1199061015840 (119909) = 1198961199060 [(1 minus 119861) 119890119896119909 minus 119861119890minus119896119909] 11990610158401015840 (119909) = 11989621199060 [(1 minus 119861) 119890119896119909 + 119861119890minus119896119909] (50)

Equation (9) can be rewritten as

1198781015840119878 = minus11990610158401015840 + 1198962 sdot 1199061199061015840 (51)

Considering relations (50) in (51) results in the new formula

1198781015840119909119878119909 = minus211990601198962 [(1 minus 119861) 119890119896119909 + 119861119890minus119896119909]1199060119896 [(1 minus 119861) 119890119896119909 minus 119861119890minus119896119909] (52)

Defining the function 119892(119909) = 119892119909 as119892 (119909) = (1 minus 119861) 119890119896119909 minus 119861119890minus119896119909 (53)

(52) becomes

1198781015840119909119878119909 = minus2 sdot1198921015840119909119892119909 (54)

Integrating (54) results in a constant denoted by119863ln (119878119909) = minus2 ln (119892119909) + 119863 (55)

which depends on the initial geometrical conditionsAt the end of the horn for 119909 = 119871 the cross section

becomes 119878119909 = 119878119871 and it is obtained that

119863 = ln (119878119909) + 2 ln (119892119871) (56)

where 119892119871 is the value of the function 119892(119909) = 119892119909 defined by(53) for 119909 = 119871

Introducing the value of119863 obtained by (56) in (55) resultsin

ln (119878119909) = minus2 ln (119892119909) + ln (119878119871) + 2 ln (119892119871) (57)

or

ln(119878119909119878119871) = minus2 ln(119892119909119892119871) (58)

and the cross section at the distance 119909 can be calculated as

119878119909 = 119878119871 (119892119871119892119909)2 (59)

0 20 40 60 80 100 120 1405

10

15

20

25

30

Horn length (mm)

Hor

n ra

dius

(mm

)

Figure 2 The half-horn shapes along the longitudinal axis

An interesting design aspect is represented by the valueand position of the maximum cross sectionThus we consid-ered the condition that the first derivative of function (59) isequal to zero This condition leads to the relation

1198921015840119909 = 0 (60)

with similar solution to (49)It can be concluded that the position of the maximum

cross section is in the same place as the nodal pointThe end cross section can be found as

119878119871 = 119878119898 [119892119899119892119871]2 (61)

and introducing (61) in (59) results in the relation of the crosssection value in a point situated at the distance 119909 from thesystem origin as a function of the maximum cross section

119878119909 = 119878119898 (119892119899119892119909)2 (62)

Taking into consideration relation (62) a script inMatlabwas made to generate the shape of the horn made of steelThe input parameters were the resonance frequency 119891119903 =19900Hz the wave velocity in steel 119888 = 50550ms the hornrsquosmaterial density 120588ℎ = 7850 kgm3 the end radius of thehorn (connection surface with the tool) 119903ℎ119871 = 0005m themagnification coefficient 119902 = 5 and tool radius equal to theend horn radius (119903119905 = 119903ℎ119871) The tool was considered to bemade of steel

Based on the script written inMatlab the shape presentedin Figure 2 was obtained and in Figure 3 the physical modelof the horn is presented

Based on the same script the following design data werefound the radius in the origin of the system (119909 = 0) 1199030 =29256mm the largest radius 119903119898 = 31721mm and theposition of the nodal point 119909nod = 27734mm

Considering relation (48) using a script written inMatlab we obtained the magnitude variation of the spatialcomponents 119906(119909) of the considered signal (13) (Figure 4)

As it can be seen from Figure 4 the nodal point is at thecoordinate found by relation (49)


Figure 3 The physical model of the horn

0 20 40 60 80 100 120 140minus5

minus4

minus3

minus2

minus1

0

1

Horn length (mm)

Sign

al m

agni

tude

u(x

)u0

Figure 4 The signal magnitude along the horn

5 Theoretical Modal Analysis

51 State-Space Approach of the OptimizedModel Themodalbehavior can be described inmany ways Two of them are thestate-space approach and transfer function representationThe differences between these twomethodswere described in[35 36] In the present paper we considered as a theoreticalmodal analysis the state-space approach and for the experi-mental modal analysis the FRF representation was made

The state-space method is often used in dynamic systemanalysis This method can be applied in the case of lumpedsystems and it is based on the concept of state vector thatconsists of two components the vector of displacement andthe vector of the velocities Practically the ldquo119899rdquo second-ordercoupled equations that describe the dynamic behavior of alumped system are transformed in a set of 2119899 first-ordercoupled equations [37ndash39]

The ultrasonic horn can be considered as a linear time-invariant (LTI) system that can be approximatewith a lumpedmass system Thus the obtained horn (Figure 3) is dividedinto ldquo119899rdquo elements with the same length 119897119894 where 119897119894 = 119871119899

x1 x2 xi xn

m1 m2 mi mn

k1 k2 kiminus1 knminus1ki

Figure 5 The lumped model of the horn

with 119871 being the total length of the horn equal to a quarter ofthe wavelength Considering the case of the free-free systemas is in the experimental modal analysis the horn can bemodeled as a lumped system made of ldquo119899rdquo masses connectedwith springs (Figure 5)

We considered 200 masses with a length of 119897119894 = 6344 sdot10minus4m in each part The obtained components are takinginto consideration the shape (Figure 3) small frusta of coneelements

The components of the modal model are establishedconsidering the following assumptions

(a) The stiffness of the springs was considered to be equalto the equivalent stiffness of each element based onthe relation

119896119894 = 119864119878119886119894119897119894 (63)

where 119878119886119894 represents the average area of the region oflength 119897119894

119878119886119894 = 119878119894 + 119878119894+12 (64)

with both 119878119894 and 119878119894+1 calculated using relation (62)

(b) The mass 119898119894 of each small frustum of cone elementwas found considering the classical relation of thedensity using for volume the areas 119878119894 and 119878119894+1 and thelength 119897119894

Considering the drsquoAlembert principle the motion equa-tions for the system from Figure 5 are given by

11989811 + 11989611199091 minus 11989611199092 = 011989822 + (1198961 + 1198962) 1199092 minus 11989611199091 minus 11989621199093 = 011989833 + (1198962 + 1198963) 1199093 minus 11989621199092 minus 11989631199094 = 0

119898119894119894 + (119896119894minus1 + 119896119894) 119909119894 minus 119896119894minus1119909119894minus1 minus 119896119894119909119894+1 = 0

119898119899119899 + 119896119899minus1119909119899 minus 119896119899minus1119909119899minus1 = 0

(65)


System (65) can be rewritten in matrix form as follows

[[[[[[[[[[[[[[[[[

1198981 0 0 sdot sdot sdot sdot sdot sdot sdot sdot sdot 00 1198982 0 sdot sdot sdot sdot sdot sdot sdot sdot sdot 00 0 1198983 sdot sdot sdot sdot sdot sdot sdot sdot sdot 0 0 0 0 sdot sdot sdot 119898119894 sdot sdot sdot 0 00 0 0 sdot sdot sdot sdot sdot sdot sdot sdot sdot 119898119899

]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[

1198961 minus1198961 0 sdot sdot sdot sdot sdot sdot sdot sdot sdot 0minus1198961 1198961 + 1198962 minus1198962 sdot sdot sdot sdot sdot sdot sdot sdot sdot 00 minus1198962 1198962 + 1198963 minus1198963 sdot sdot sdot sdot sdot sdot 0 sdot sdot sdot sdot sdot sdot 0 00 0 0 sdot sdot sdot sdot sdot sdot minus119896119899minus1 119896119899minus1

]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)

from which it is obtained that

X = minusMminus1KX (68)

As the state vector we considered the matrix vector

Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)

which by derivation and considering (68) leads to theequation

Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)

where I is the unit matrix and 0 is the zeros matrix both of119899 times 119899 typeThe matrix A is defined as the state matrix or the

transmission matrix and is a property of the system being

0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)

Number of frequencies

times106

Figure 6 The natural frequencies obtained for the lumped model

Figure 7 The finite element model of the horn

defined by the structure of the systemThe eigenvalues of thestate matrix can be found based on the relation

|120582I minus A| = 0 (71)

that is developed in an equation with unknowns 120582If in relation (71) the value 120582 = 119895120596119899 is considered where119895 = radicminus1 and 120596119899 is the natural frequency measured in rads

then the so-called characteristic equation is obtained whichgives the natural frequencies

Thus considering the equivalent system from Figure 5with the associated motion equations (66) (67) and (70)one can find the natural frequencies of the system Thenumber and the precision of these frequencies depend onthe number of the considered masses 119898119894 (119894 = 1 119899) Figure 6presents the frequency values obtained for a model with200 masses considering the method of eigenvalues of thetransition matrix

Figure 6 also presents some of the obtained naturalfrequencies The first natural frequency found was 119891119899 =19837Hz which is close to the designed frequency of 119891119903 =199 kHz

52 Finite Element Model In the next step the finite elementmodel was considered The used specialized software wasANSYS The 3D model (Figure 7) was made based on (59)

For good convergence we considered a mapped meshconsisting of first-order solid hexaelements with 1mm lengthWe obtained 87076 nodes and 75254 elements Mode extrac-tion is carried out in the frequency range 18ndash24 kHz usingBlock Lanczos option with double precision Two pure lon-gitudinal modes were obtained with the natural frequenciesof 119891 = 19817Hz (Figure 8) and 119891 = 23151Hz (Figure 9)


ResultantdisplacementContours of resultant displacement6934e minus 016241e minus 015548e minus 014854e minus 014161e minus 013467e minus 012774e minus 012080e minus 011387e minus 016934e minus 021185e minus 10

Freq = 19817

Figure 8 The mode shape for the natural frequency of 119891 = 19817Hz

Contours of resultant displacement6934e minus 016244e minus 015555e minus 014865e minus 014175e minus 013485e minus 012795e minus 012105e minus 011415e minus 017251e minus 023157e minus 03

Resultantdisplacement

Freq = 23151


Figure 10 The modal testing setup

6 Experimental Modal Analysis

The aim of the experimental modal analysis is to verifywhether the considered imposed working frequency of199 kHz is one of the natural frequencies of the consideredhorn and whether the natural frequencies found by state-space approach are close to the real natural frequencies

The method with random signal excitation given by anelectrodynamic shaker (Figure 10) was considered [40] Theused equipment was a complete testing setup (Bruel amp Kjaeligr)made of an integrated platform PULSE 12 with a forcetransducer (type 8230) and an accelerometer (type 4397)Theexcitation signal was generated by the PULSE 12 platformand was amplified using the power amplifier type 2718 andthen transmitted to the tested horn by the shaker type 4810(Figure 11)

The random generated signal (the input signal) has afrequency span up to 256 kHz with an excitation time of500ms (Figure 12) The Fourier spectrum of the input signalis shown in Figure 13

Figure 11 The detailed connection shaker horn

Compressed time (input)mdashinput

0 1

minus2minus16minus12

0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ

Working input input time capture analyzer

Figure 12 The input random signal in time domain


Fourier spectrum (input)mdashinput (magnitude)0

minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

Working input input FFT analyzer

Figure 13 The Fourier spectrum of the input signal

0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)

Working input input FFT analyzerFrequency response H1 (response input)mdashinput (magnitude)

Figure 14 The frequency response function of the horn

As it is known the frequency response function estima-tion from the measured data represents the main step of theexperimental modal analysis The analysis is based on non-parametric estimators In the present study we consideredthe oftenused estimator1198671Theobtained frequency responsefunction for the estimator1198671 is presented in Figure 14

The range of the measured frequencies was up to20000Hz and the values are presented in Figure 15

At the same time based on the experimental modalanalysis we obtained the damping factors 120577 for each naturalfrequency foundThe distribution of these values is presentedin Figure 16

Considering the facilities of the Matlab software a powerfunction was defined which can approximate the dampingfactor for different values of the frequency 119891

120577 (119891) = 3219119891minus07427 minus 007775 (72)

Function (72) can be used to study the influence of thedamping in case of horn vibration at different frequencies

0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104

Figure 15 The frequencies obtained by experimental data

0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104

Figure 16 The damping factor values distribution versus naturalfrequencies

7 Testing the Designed Horn

The test was designed to check whether the magnificationof the designed horn is the initial considered 119902 = 5 for theresonance frequency of 199 kHz The used setup is made ofthe same components as the setup used in the experimentalmodal analysis with the exception of the accelerometersFor testing two Bruel amp Kjaeligr accelerometers type 4517-002were used The signal was generated by a magnetostrictivetransducer for a harmonic signal 119910(119905) = 119860 sin120596119905 with afrequency 119891 = 199 kHz The input signal represented intime domain is shown in Figure 17 and the output signal(response) also in time domain is designed in Figure 18

At the same time for frequency control we performeda check in frequency domain for both signals the input(Figure 19) and the output (Figure 20)

Considering the values of the input and output signals(Figures 17 and 18) themagnification obtained by experimentis given by the following relation

119902real = 119906output119906input =4278932 = 459 (73)

which represents an error of about 82 between the consid-ered value 119902 = 5 and the obtained value 119902real


Time (input)mdashinput

minus20minus16minus12minus8minus4

048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)


Figure 17 The input signal in time domain

Time (output)mdashinput

minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)


Figure 18 The output signal in time domain

8 Conclusions

Based on the principle of the smallest action an optimizationof the shape of an ultrasonic hornwas performed consideringWebsterrsquos equation The design was made considering aworking resonance frequency of 119891119903 = 199 kHz and amagnification coefficient 119902 = 5 and themain assumptionwasthat through the horn and all attached coupling devices onlylongitudinal plane waves propagate

The considered boundary conditions were the classicalones referring to the amplitude of the signal relations (29)and (30) to which a new one was added relation (31) whichhighlights the influence of the tool mass added at the end ofthe horn This condition (31) is a particularity of the study

The optimization procedure led to a new shape function(62) with the space dependent signal component function(48) and the position of the neutral point 119909119899 (49) beingdetermined at the same time Comparing the obtained shapewith the classical shape it can be seen that it is a different onewith the largest diameter in the neutral point In the case ofthe other hornsrsquo shape the largest diameter is at one end

Fourier spectrum (input)mdashinput (magnitude)

0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)


Figure 19 The input signal in frequency domain

Fourier spectrum (output)mdashinput (magnitude)

0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)


Figure 20 The output signal in frequency domain

The natural frequencies were determined both in sim-ulation (theoretically) and experimentally In the case oftheoretical modal analysis the state-space representation wasused The horn was divided into 200 masses connected bysprings (Figure 5) and the eigenvalues of the transmissionmatrix of system (71) were calculatedThe first obtained valueof the natural frequency was the imposed resonant frequency1198911119904119904 = 19982Hz

The second considered theoretical approach in theoreticalmodal analysis was the finite element method A model with75254 elements was designed and based on ANSYS softwarefacilities two natural frequencies were found around thedesigned resonance frequency (Figures 8 and 9) both for thelongitudinal mode shape In order to provide clarity to theresults a run was made in double precision mode

The experimental modal analysis offered the possibilityof finding a set of natural frequencies (Figure 16) and thecorresponding values of the damping factor Considering thefacilities of the Matlab tool a function of damping factorvariation was obtained according to the frequencies values(72) useful in the dynamic analysis As it can be seen a


natural frequency around the resonance designed frequencywas found

At the end the horn was tested to check whether themagnification coefficient is the same as the designed one Aharmonic signal with the frequency value equal to the reso-nance frequency was considered as the input and the outputsignal was measured The found magnification coefficient isclose to the designed one (72)

In the frame of the test the attached tool was notconsidered It is important for the assembly transducer-horn-tool to work in resonance regime and each part to workwithin the parameters for which it was designed

The aim of the test was to check whether the consideredhorn designed in the proposed set of boundary conditions(29) (30) and (31) generates the initial designed mechanicalmagnification of the input signal

The influence of the attached tool mass is the particulardesign element that was taken into consideration comparedwith the other methods of optimization and design

Considering the abovementioned aspects it can be con-cluded that the numerical simulations and tests that weredone support the validity of the assumptions used in thehornrsquos optimization design

The proposed model offers as an advantage a moreconvenient placement of the nodal point that is situated inthe same place as the largest cross section

The described design and optimization method offer thepossibility of obtaining the shape based on the main workingparameters

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] V K Astashev and V I Babitsky ldquoUltrasonic cutting as anonlinear (vibro-impact) processrdquo Ultrasonics vol 36 no 1-5pp 89ndash96 1998

[2] R Singh and J S Khamba ldquoUltrasonic machining of titaniumand its alloys a reviewrdquo Journal of Materials Processing Technol-ogy vol 173 no 2 pp 125ndash135 2006

[3] T Asami and H Miura ldquoStudy of ultrasonic machin-ing by longitudinal-torsional vibration for processing brittlematerials-observation of machining marksrdquo Physics Procediavol 70 pp 118ndash121 2015

[4] V I Babitsky V K Astashev and A Meadows ldquoVibrationexcitation and energy transfer during ultrasonically assisteddrillingrdquo Journal of Sound and Vibration vol 308 no 3-5 pp805ndash814 2007

[5] L DeFu W L Cong Z J Pei and Y JunTang ldquoA cutting forcemodel for rotary ultrasonic machining of brittle materialsrdquoInternational Journal of Machine Tools amp Manufacture vol 52pp 77ndash84 2012

[6] J Wang P Feng J Zhang W Cai and H Shen ldquoInvestigationson the critical feed rate guaranteeing the effectiveness of rotaryultrasonic machiningrdquo Ultrasonics vol 74 pp 81ndash88 2017

[7] V Sharma and P M Pandey ldquoOptimization of machiningand vibration parameters for residual stresses minimization in

ultrasonic assisted turning of 4340 hardened steelrdquo Ultrasonicsvol 70 pp 172ndash182 2016

[8] M Roopa Rani K Prakasan and R Rudramoorthy ldquoStudieson thermo-elastic heating of horns used in ultrasonic plasticweldingrdquo Ultrasonics vol 55 no 1 pp 123ndash132 2015

[9] T V da Cunha and C E N Bohorquez ldquoUltrasound in arcwelding a reviewrdquo Ultrasonics vol 56 pp 201ndash209 2015

[10] W L Cong Z J Pei TW Deines A Srivastava L Riley and CTreadwell ldquoRotary ultrasonic machining of CFRP compositesa study on power consumptionrdquo Ultrasonics vol 52 no 8 pp1030ndash1037 2012

[11] P Feng J Wang J Zhang and J Zheng ldquoDrilling induced tear-ing defects in rotary ultrasonicmachining ofCSiC compositesrdquoCeramics International vol 43 no 1 pp 791ndash799 2017

[12] K Ding Y Fu H Su Y Chen X Yu and G Ding ldquoExperimen-tal studies on drilling tool load and machining quality of CSiCcomposites in rotary ultrasonicmachiningrdquo Journal ofMaterialsProcessing Technology vol 214 no 12 pp 2900ndash2907 2014

[13] E Eisner ldquoDesign of sonic amplitude transformers for largemagnificationrdquo in Proceedings of the IEEE vol 51 pp 512-512IEEE

[14] K Graff Wave Motion in Elastic Solids The Clarendon PressOxford UK 1975

[15] D Sindayihebura L Bolle A Cornet and L Joannes ldquoThe-oretical and experimental study of transducers aimed at low-frequency ultrasonic atomization of liquidsrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1442ndash14481998

[16] D-A Wang W-Y Chuang K Hsu and H-T Pham ldquoDesignof a Bezier-profile horn for high displacement amplificationrdquoUltrasonics vol 51 no 29 pp 148ndash156 2011

[17] G Amza and D Drimer ldquoThe design and construction of solidconcentrators for ultrasonic energyrdquo Ultrasonics vol 14 no 5pp 223ndash226 1976

[18] J J Coy and F S Tse ldquoSynthesis of solid elastic hornstransactions of ASMErdquo Journal of Engineering for Industry vol96 no 2 pp 627ndash632 1974

[19] X-P He and J Gao ldquoA review of ultrasonic solid horn designrdquoTechnical Acoustics vol 25 no 1 pp 82ndash86 2006

[20] V Salmon ldquoGeneralized plane wave horn theoryrdquo Journal of theAcoustical Society of America vol 17 no 3 pp 199ndash211 1946

[21] M P Kamat V B Venkayya and N S Khot ldquoOptimizationwith frequency constraints-limitationsrdquo Journal of Sound andVibration vol 91 no 1 pp 147ndash154 1983

[22] J-C Hung Y-P Tsai and C Hung ldquoOptimization of ultra-sonic plastic welding horns on amplitude uniformityrdquo AppliedMechanics and Materials vol 121 no 126 pp 278ndash282 2012

[23] T He X-Q Ye and Y Zhao ldquoOptimization design for ultra-sonic horn with large amplitude based on genetic algorithmrdquoJournal of Vibroengineering vol 17 no 3 pp 1157ndash1168 2015

[24] P Harkness A Mathieson C Murray and M Lucas ldquoOpti-mization of ultrasonic horns for momentum transfer andsurvivability in high-frequencylow frequency planetary drilltoolsrdquo in Proceedings of the AIAA SPACE Conference andExposition 2011 Long Beach Calif USA September 2011

[25] I C Rosca S T Chiriacescu andNC Cretu ldquoUltrasonic hornsoptimizationrdquo Physics Procedia vol 3 pp 1033ndash1040 2010

[26] I-C Rosca M-I Pop and N Cretu ldquoExperimental andnumerical study on anultrasonic hornwith shape designedwithan optimization algorithmrdquo Applied Acoustics vol 95 pp 60ndash69 2015


[27] K-M Shu Y-G Li C-C Chan and J-B Kuan ldquoOptimizeddesign of the horn of ultrasonic roll weldingrdquo Advanced Mate-rials Research vol 482-484 pp 2223ndash2226 2012

[28] L Xu S Lin and W Hu ldquoOptimization design of highpower ultrasonic circular ring radiator in coupled vibrationrdquoUltrasonics vol 51 no 7 pp 815ndash823 2011

[29] A G Webster ldquoAcoustical impedance and the theory of hornsand the phonographrdquo Proceedings of the National Academy ofScience of the United States of America vol 5 pp 275ndash282 1919

[30] E Eisner ldquoComplete Solutions of the ldquoWebsterrdquo Horn Equa-tionrdquo Journal of the Acoustical Society of America vol 41 no4B pp 1126ndash1146 1967

[31] H Popescu and V Chiroiu ldquoCalculul structurilor optimale(Calculation of optimal structures ndash in romanian)rdquo EdituraAcademiei Romane Bucuresti (Romanian Academy Printhouse)1981

[32] M J Turner ldquoDesign of minimum mass structures withspecified natural frequenciesrdquo AIAA Journal vol 5 no 3 pp406ndash412 1967

[33] M J Forray Variational Calculus in Science and EngineeringMcGraw Hill Inc New York NY USA 1968

[34] V A Ilyin and E G Poznyak ldquoFundamentals of MathematicalAnalysisrdquoMir Publishers vol I amp II 1982

[35] C T Chen Linear SystemTheory andDesign Holt Rinehart andWinston New York NY USA 1984

[36] L M Silverman ldquoRealization of linear dynamical systemsrdquoIEEE Transactions on Automatic Control vol 16 no 6 pp 554ndash567 1971

[37] BDe Schutter ldquoMinimal state-space realization in linear systemtheory an overviewrdquo Journal of Computational and AppliedMathematics vol 121 no 1-2 pp 331ndash354 2000

[38] D ENewlandMechanical VibrationAnalysis andComputationLongman Harlow and John Wiley New York NY USA 1989

[39] R Fazlollah Linear Spaces in Engineering Ginn amp CompanyWaltham Mass USA 1971

[40] J He and Z-F Fu Modal Analysis Butterworth-Heinemann2001

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



x

x

y

L

SL

Sx

0

S0

Figure 1 The solid horn geometry

propagation in the longitudinal direction of ultrasonicwaves along a rigid horn with a variable cross-sectional areais Websterrsquos horn equation [29]

1205972119906 (119909 119905)1205971199092 + 120597119906 (119909 119905)120597119909 120597120597119909 (ln 119878119909) = 11198882 1205972119906 (119909 119905)1205971199052 (1)

where 119906(119909 119905) is the signal 119878119909 = 119878(119909) represents the cross-sectional area of the considered horn at the distance 119909 fromthe reference (point 0) and 119888 is the wave velocity in the hornrsquosmaterial

One of the widely used techniques to find the explicitanalytical solutions of the linear partial differential equationsis the method of variables separation [30] The methodassumes that the solution of (1) can be written as a product oftwo functions one in space domain 119906(119909) = 119906119909 and the otherone in time domain 119906(119905) = 119906119905

119906 (119909 119905) = 119906 (119909) 119906 (119905) = 119906119909119906119905 (2)

Introducing solution (2) in Websterrsquos equation (1) thenew shape is obtained

11990610158401015840119909119906119909 +119889119889119909 (119878119909)

1199061015840119909119906119909 =11198882 119905119906119905 (3)

where the following notations were done 11990610158401015840119909 = 119889211990611990911988911990921199061015840119909 = 119889119906119909119889119909 and 119905 = 11988921199061199051198891199052This way in (3) there were separate variables on each side

of the equationThe congruency of both sides left and right of (3) is

achieved only if the two functions are equal with the sameseparation constant 120573 It is considered as a constant value120573 = minus1205962 where 120596 is the angular wave frequency (rads)

Thus from (3) we obtained the following ordinarydifferential equations

119905 + 1205962119906119905 = 011990610158401015840119909 + 119889119889119909 (119878119909) 1199061015840119909 + 1198962119906119909 = 0

(4)

where 119896 = 120596119888 is the wave number

The general solution of the time domain equation from(4) is given by

119906119905 = 1198631 cos120596119905 + 1198632 sin120596119905 (5)

with the initial conditions

1199061199051003816100381610038161003816119905=0 = 119906011988911990611990511988911990510038161003816100381610038161003816100381610038161003816119905=0 = 0

(6)

where 1199060 is the initial amplitude of the ultrasonic waveThus solution (5) becomes

119906119905 = 1199060 cos120596119905 (7)

and the total solution (2) becomes

119906 (119909 119905) = 119906 (119909) 119906 (119905) = 119906119909119906119905 = 1199061199091199060 cos120596119905 (8)

In case of a steady-state mode considering the assump-tion of an infinitesimal motion in a beam made of nondissi-pative medium and with a variable cross-sectional area thesecond equation from (4) which represents the plane wavemotion can be rewritten as

119889119889119909 (119878119909 119889119906119909119889119909 ) + 1198962119878119909119906119909 = 0 (9)

It is considered that the maximum diameter of the hornshould not exceed one-quarter of the wavelength 120582 and themaximum cross section is denoted as 119878119898 Thus we can writethe condition that on the whole length of the horn the crosssection value has to be smaller than the maximum crosssection

119878 (119909) le 119878119898 (10)

Considering relation (10) one can write the followingmathematical relation

119878119898 minus 119878 (119909) minus 1198862 (119909) = 0 (11)

where 119886(119909) is an unspecified function of the variable 119909 In thefollowing the notations 119886(119909) = 119886119909 and 119878(119909) = 119878119909 will be used

An optimization problem of the hornrsquos shape refers tominimizing the hornrsquos volume for a given initial set of designdata Based on the principle of the least action one can writethe action relation as

119860 = int11990521199051

(119879 minus 119881) 119889119905 (12)

where 119879 is the kinetic energy and 119881 represents the potentialenergy

Considering the wave shape as

119906 = 119906 (119909 119905) = 119906 (119909) cos (120596119905) = 119906119909 cos (120596119905) (13)



119879 = 12 int119871

0(1198981199092) 119889119909 + 1198721199052 (119871)2

119881 = 12 int119871

0(119864ℎ119898119909120588ℎ ) (1199061015840)

2 119889119909 + 12059022119864119905119881119905(14)



119860|1199052+21205871205961199051

= 119860 = 120587 (119879 minus 119881) (15)

with

119879 = 12059622 [int119871

0120588 sdot 119878 sdot 1199062119889119909 +1198721199051199062 (119871)]

119881 = 12 [int119871


(16)




119881 = int1198710119878119909119889119909 (18)


120597Ι120597119906119909 minus119889119889119909 120597Ι1205971199061015840119909 = 0

120597Ι120597119878119909 minus119889119889119909 120597Ι1205971198781015840119909 = 0

120597Ι120597119886119909 minus119889119889119909 120597Ι1205971198861015840119909 = 0

(19)



12059622 2119906119909120588119878119909 minus 119889119889119909 [minus119864ℎ2 21199061015840119909119878119909]= 1205962119906119909120588119878119909 + 119864ℎ (119878119909119906119909)1015840 = 0

(20)


(1198781199091199061015840119909)1015840 + 120588ℎ1205962119864ℎ 119878119909119906119909 = 0 (21)


(1198781199091199061015840119909)1015840 + 1198962119878119909119906119909 = 0 (22)


1205962 120588ℎ1199062119909 minus 119864ℎ (1199061015840119909)2 + 21205721 minus 2120572119909 = 0 (23)


1198961199062119909 minus (1199061015840119909)2 + 2119864ℎ 1205721 minus2119864ℎ 120572119909 = 0 (24)


2120572119909119886119909 = 0 (25)




120572119909 = 0or 119886119909 = 0 (26)




120572119909 = 119864ℎ2 [11989621199062119909 minus (1199061015840119909)2 + 21205721119864ℎ ] = 0 (27)

or

(1199061015840119909)2 minus 11989621199062119909 minus 21205721119864ℎ = 0 (28)







119906 (119909) = 119906 (0) = 1199060 (29)


119906 (119909) = 119906 (119871) = minus1199021199060 (30)




1198981199061015840119909=119871 minus 1198962119872119906119909=119871 = 0 (31)


Denoting

119862 = 21198641205721 (32)


(1199061015840)2 minus 11989621199062 minus 119862 = 0 (33)


2119906101584011990610158401015840 minus 119896221199061199061015840 = 0 (34)


with the solution

119906 (119909) = 1198621119890119896119909 + 1198622119890minus119896119909 (36)


minus4119896211986211198622 = 119862 (37)

1198621 = minus 119862411989621198622 (38)


119906 (119909) = minus 119862411989621198622 119890119896119909 + 1198622119890minus119896119909 (39)


1199060 = minus 119862411989621198622 + 1198622 (40)


119862122 = 119896119906 plusmn radic119896211990620 + 1198622119896 (41)


119862 = 411986221198962 (1198622 minus 1199060) (42)




120588ℎ119878ℎ119871 [1199060119890119896119871 minus 1198622 (119890119896119871 + 119890minus119896119871)] + 119896119871 1199051205881199051198781199051199021199060 = 0 (44)


From (44) results

1198622 = 1199060 (120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905)2120588ℎ119878ℎ119871 cosh (119896119871) (45)


119906 (119909)= 1199060 [119890119896119909 minus 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (119890

119896119909 minus 119890minus119896119909)] (46)


119861 = 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (47)


119906 (119909) = 1199060 [(1 minus 119861) 119890119896119909 + 119861119890minus119896119909] (48)



119909nod = 12119896 ln( 119861119861 minus 1) (49)





1198781015840119878 = minus11990610158401015840 + 1198962 sdot 1199061199061015840 (51)




(52) becomes

1198781015840119909119878119909 = minus2 sdot1198921015840119909119892119909 (54)




119863 = ln (119878119909) + 2 ln (119892119871) (56)



ln (119878119909) = minus2 ln (119892119909) + ln (119878119871) + 2 ln (119892119871) (57)

or

ln(119878119909119878119871) = minus2 ln(119892119909119892119871) (58)


119878119909 = 119878119871 (119892119871119892119909)2 (59)

0 20 40 60 80 100 120 1405

10

15

20

25

30

Horn length (mm)

Hor

n ra

dius

(mm

)



1198921015840119909 = 0 (60)



119878119871 = 119878119898 [119892119899119892119871]2 (61)


119878119909 = 119878119898 (119892119899119892119909)2 (62)








0 20 40 60 80 100 120 140minus5

minus4

minus3

minus2

minus1

0

1

Horn length (mm)

Sign

al m

agni

tude

u(x

)u0






x1 x2 xi xn

m1 m2 mi mn







119896119894 = 119864119878119886119894119897119894 (63)


119878119886119894 = 119878119894 + 119878119894+12 (64)







(65)



[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)




Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)


Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)



0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)


times106




|120582I minus A| = 0 (71)









Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119879 = 12 int119871

0(1198981199092) 119889119909 + 1198721199052 (119871)2

119881 = 12 int119871

0(119864ℎ119898119909120588ℎ ) (1199061015840)

2 119889119909 + 12059022119864119905119881119905(14)



119860|1199052+21205871205961199051

= 119860 = 120587 (119879 minus 119881) (15)

with

119879 = 12059622 [int119871

0120588 sdot 119878 sdot 1199062119889119909 +1198721199051199062 (119871)]

119881 = 12 [int119871


(16)




119881 = int1198710119878119909119889119909 (18)


120597Ι120597119906119909 minus119889119889119909 120597Ι1205971199061015840119909 = 0

120597Ι120597119878119909 minus119889119889119909 120597Ι1205971198781015840119909 = 0

120597Ι120597119886119909 minus119889119889119909 120597Ι1205971198861015840119909 = 0

(19)



12059622 2119906119909120588119878119909 minus 119889119889119909 [minus119864ℎ2 21199061015840119909119878119909]= 1205962119906119909120588119878119909 + 119864ℎ (119878119909119906119909)1015840 = 0

(20)


(1198781199091199061015840119909)1015840 + 120588ℎ1205962119864ℎ 119878119909119906119909 = 0 (21)


(1198781199091199061015840119909)1015840 + 1198962119878119909119906119909 = 0 (22)


1205962 120588ℎ1199062119909 minus 119864ℎ (1199061015840119909)2 + 21205721 minus 2120572119909 = 0 (23)


1198961199062119909 minus (1199061015840119909)2 + 2119864ℎ 1205721 minus2119864ℎ 120572119909 = 0 (24)


2120572119909119886119909 = 0 (25)




120572119909 = 0or 119886119909 = 0 (26)




120572119909 = 119864ℎ2 [11989621199062119909 minus (1199061015840119909)2 + 21205721119864ℎ ] = 0 (27)

or

(1199061015840119909)2 minus 11989621199062119909 minus 21205721119864ℎ = 0 (28)







119906 (119909) = 119906 (0) = 1199060 (29)


119906 (119909) = 119906 (119871) = minus1199021199060 (30)




1198981199061015840119909=119871 minus 1198962119872119906119909=119871 = 0 (31)


Denoting

119862 = 21198641205721 (32)


(1199061015840)2 minus 11989621199062 minus 119862 = 0 (33)


2119906101584011990610158401015840 minus 119896221199061199061015840 = 0 (34)


with the solution

119906 (119909) = 1198621119890119896119909 + 1198622119890minus119896119909 (36)


minus4119896211986211198622 = 119862 (37)

1198621 = minus 119862411989621198622 (38)


119906 (119909) = minus 119862411989621198622 119890119896119909 + 1198622119890minus119896119909 (39)


1199060 = minus 119862411989621198622 + 1198622 (40)


119862122 = 119896119906 plusmn radic119896211990620 + 1198622119896 (41)


119862 = 411986221198962 (1198622 minus 1199060) (42)




120588ℎ119878ℎ119871 [1199060119890119896119871 minus 1198622 (119890119896119871 + 119890minus119896119871)] + 119896119871 1199051205881199051198781199051199021199060 = 0 (44)


From (44) results

1198622 = 1199060 (120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905)2120588ℎ119878ℎ119871 cosh (119896119871) (45)


119906 (119909)= 1199060 [119890119896119909 minus 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (119890

119896119909 minus 119890minus119896119909)] (46)


119861 = 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (47)


119906 (119909) = 1199060 [(1 minus 119861) 119890119896119909 + 119861119890minus119896119909] (48)



119909nod = 12119896 ln( 119861119861 minus 1) (49)





1198781015840119878 = minus11990610158401015840 + 1198962 sdot 1199061199061015840 (51)




(52) becomes

1198781015840119909119878119909 = minus2 sdot1198921015840119909119892119909 (54)




119863 = ln (119878119909) + 2 ln (119892119871) (56)



ln (119878119909) = minus2 ln (119892119909) + ln (119878119871) + 2 ln (119892119871) (57)

or

ln(119878119909119878119871) = minus2 ln(119892119909119892119871) (58)


119878119909 = 119878119871 (119892119871119892119909)2 (59)

0 20 40 60 80 100 120 1405

10

15

20

25

30

Horn length (mm)

Hor

n ra

dius

(mm

)



1198921015840119909 = 0 (60)



119878119871 = 119878119898 [119892119899119892119871]2 (61)


119878119909 = 119878119898 (119892119899119892119909)2 (62)








0 20 40 60 80 100 120 140minus5

minus4

minus3

minus2

minus1

0

1

Horn length (mm)

Sign

al m

agni

tude

u(x

)u0






x1 x2 xi xn

m1 m2 mi mn







119896119894 = 119864119878119886119894119897119894 (63)


119878119886119894 = 119878119894 + 119878119894+12 (64)







(65)



[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)




Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)


Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)



0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)


times106




|120582I minus A| = 0 (71)









Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119906 (119909) = 119906 (0) = 1199060 (29)


119906 (119909) = 119906 (119871) = minus1199021199060 (30)




1198981199061015840119909=119871 minus 1198962119872119906119909=119871 = 0 (31)


Denoting

119862 = 21198641205721 (32)


(1199061015840)2 minus 11989621199062 minus 119862 = 0 (33)


2119906101584011990610158401015840 minus 119896221199061199061015840 = 0 (34)


with the solution

119906 (119909) = 1198621119890119896119909 + 1198622119890minus119896119909 (36)


minus4119896211986211198622 = 119862 (37)

1198621 = minus 119862411989621198622 (38)


119906 (119909) = minus 119862411989621198622 119890119896119909 + 1198622119890minus119896119909 (39)


1199060 = minus 119862411989621198622 + 1198622 (40)


119862122 = 119896119906 plusmn radic119896211990620 + 1198622119896 (41)


119862 = 411986221198962 (1198622 minus 1199060) (42)




120588ℎ119878ℎ119871 [1199060119890119896119871 minus 1198622 (119890119896119871 + 119890minus119896119871)] + 119896119871 1199051205881199051198781199051199021199060 = 0 (44)


From (44) results

1198622 = 1199060 (120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905)2120588ℎ119878ℎ119871 cosh (119896119871) (45)


119906 (119909)= 1199060 [119890119896119909 minus 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (119890

119896119909 minus 119890minus119896119909)] (46)


119861 = 120588ℎ119878ℎ119871119890119896119871 + 119902119896119871 119905119878119905120588119905120588ℎ119878ℎ119871 (119890119896119871 + 119890minus119896119871) (47)


119906 (119909) = 1199060 [(1 minus 119861) 119890119896119909 + 119861119890minus119896119909] (48)



119909nod = 12119896 ln( 119861119861 minus 1) (49)





1198781015840119878 = minus11990610158401015840 + 1198962 sdot 1199061199061015840 (51)




(52) becomes

1198781015840119909119878119909 = minus2 sdot1198921015840119909119892119909 (54)




119863 = ln (119878119909) + 2 ln (119892119871) (56)



ln (119878119909) = minus2 ln (119892119909) + ln (119878119871) + 2 ln (119892119871) (57)

or

ln(119878119909119878119871) = minus2 ln(119892119909119892119871) (58)


119878119909 = 119878119871 (119892119871119892119909)2 (59)

0 20 40 60 80 100 120 1405

10

15

20

25

30

Horn length (mm)

Hor

n ra

dius

(mm

)



1198921015840119909 = 0 (60)



119878119871 = 119878119898 [119892119899119892119871]2 (61)


119878119909 = 119878119898 (119892119899119892119909)2 (62)








0 20 40 60 80 100 120 140minus5

minus4

minus3

minus2

minus1

0

1

Horn length (mm)

Sign

al m

agni

tude

u(x

)u0






x1 x2 xi xn

m1 m2 mi mn







119896119894 = 119864119878119886119894119897119894 (63)


119878119886119894 = 119878119894 + 119878119894+12 (64)







(65)



[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)




Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)


Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)



0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)


times106




|120582I minus A| = 0 (71)









Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119909nod = 12119896 ln( 119861119861 minus 1) (49)





1198781015840119878 = minus11990610158401015840 + 1198962 sdot 1199061199061015840 (51)




(52) becomes

1198781015840119909119878119909 = minus2 sdot1198921015840119909119892119909 (54)




119863 = ln (119878119909) + 2 ln (119892119871) (56)



ln (119878119909) = minus2 ln (119892119909) + ln (119878119871) + 2 ln (119892119871) (57)

or

ln(119878119909119878119871) = minus2 ln(119892119909119892119871) (58)


119878119909 = 119878119871 (119892119871119892119909)2 (59)

0 20 40 60 80 100 120 1405

10

15

20

25

30

Horn length (mm)

Hor

n ra

dius

(mm

)



1198921015840119909 = 0 (60)



119878119871 = 119878119898 [119892119899119892119871]2 (61)


119878119909 = 119878119898 (119892119899119892119909)2 (62)








0 20 40 60 80 100 120 140minus5

minus4

minus3

minus2

minus1

0

1

Horn length (mm)

Sign

al m

agni

tude

u(x

)u0






x1 x2 xi xn

m1 m2 mi mn







119896119894 = 119864119878119886119894119897119894 (63)


119878119886119894 = 119878119894 + 119878119894+12 (64)







(65)



[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)




Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)


Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)



0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)


times106




|120582I minus A| = 0 (71)









Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 20 40 60 80 100 120 140minus5

minus4

minus3

minus2

minus1

0

1

Horn length (mm)

Sign

al m

agni

tude

u(x

)u0






x1 x2 xi xn

m1 m2 mi mn







119896119894 = 119864119878119886119894119897119894 (63)


119878119886119894 = 119878119894 + 119878119894+12 (64)







(65)



[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)




Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)


Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)



0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)


times106




|120582I minus A| = 0 (71)









Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]

123119894119899

+

[[[[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]]]]

119909111990921199093119909119894119909119899

=

00000

(66)

or

MX + KX = 0 (67)




Δ = 1205751 1205752 1205752119902119879 = X X119879= 1199091 1199092 119909119899 1 2 119899119879

(69)


Δ = XX = [ 0 I

minusMminus1K 0]X

X = AΔ (70)



0 10 20 30 40 50 60 700

05

1

15

2

25

Freq

uenc

y (H

z)


times106




|120582I minus A| = 0 (71)









Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Freq = 19817




Freq = 23151









0 1


0

1216

2

(s)

(N)

800Ｇ

400Ｇ

minus400Ｇ

minus800Ｇ

100Ｇ

200Ｇ

300

Ｇ

400

Ｇ

500

Ｇ

600

Ｇ

700

Ｇ

800Ｇ

900

Ｇ





minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

(Hz)

(dB

100

)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ



0200400600800

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

2Ｅ

18Ｅ

16Ｅ

14Ｅ

12Ｅ

1Ｅ

((m

Ｍ2)

N)







120577 (119891) = 3219119891minus07427 minus 007775 (72)


0 20 40 60 80 100 120 1400

02040608

112141618

2


Freq

uenc

y (H

z)

times104


0 02 04 06 08 1 12 14 16 18 20

1

2

3

4

5

6

7

8

Frequency (Hz)

Dam

ping

fact

or (

)

times104











048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












048

121620

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)




minus80

minus60

minus40

minus20

0

20

40

60

80

(s)

400u0

800u

12Ｇ

16Ｇ 2Ｇ

24Ｇ

28Ｇ

32Ｇ

36Ｇ 4Ｇ

44Ｇ

48Ｇ

(mＭ

2)



8 Conclusions





0123456789

10

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)




0

10

20

30

40

50

60

0

(Hz)

2Ｅ 4Ｅ 6Ｅ 8Ｅ 10Ｅ

12Ｅ

14Ｅ

16Ｅ

18Ｅ

20Ｅ

22Ｅ

24Ｅ

(mＭ

2)

















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















References











































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









researcharticle a new approach on vibrating horns design

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