# A New Approach on Vibrating Horns Design

Authors: Maria Violeta Guiman; Ioan Călin Roșca
Journal: Shock and Vibration
 (2017)

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## Abstract

An optimization method of the vibrating horns is presented considering the smallest action principle and the attached cutting tool mass. The model is based on Webster’s wave propagation equation and as an objective function the minimization of the volume in 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 horn’s cross section is obtained. The particularity of the new obtained shape is given by the nodal point position that is the same with the position of the maximum cross-sectional area. The obtained horn was analyzed from the modal point of view using theoretical and experimental methods. As theoretical methods, both the state-space method and the finite element method were used. An experimental setup for frequency response function determination was developed using a random input signal. The verification of the magnitude value was done considering a harmonic steady-state signal. The recorded values were compared with the predicted values. The numerical simulations and tests support the validity of the assumptions used in the horns optimization design.


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## Body

## 1. Introduction

Ultrasonic horns are axisymmetric devices that provide, with high efficiency, acoustic energy developed by a transducer that is working based on the magnetostriction effect, to a tool. These devices can be themselves tools or can be used with a tool attached at the small end, and thus they are tied with high-power applications of ultrasound, like cutting [1–3], drilling [4–6], turning [7], welding [8, 9], and so forth, or for machining different materials as carbon fiber reinforced composites [10–12].From a physical point of view, the work of such a system consists of an initial signal, emitted by a transducer at a given frequency, with an amplitudeu0, which is mechanically amplified by the horn, with an imposed magnitude coefficient q. The whole system is working in a resonance regime at an imposed frequency. The mechanical magnification of the input signal amplitude is generated by the horn varying cross section from an initial value S0, in the origin of the reference system, to an end one value SL (Figure 1).Figure 1
The solid horn geometry.Generally, the cross section variation is given by different mathematical functions (exponential [13], linear [14], tapered (or stepped) [15], catenoidal [14], Bézier [16], Gaussian, etc.). Studies concerning the wave propagation and basic concepts of horns design are presented in different papers [17–21].A problem that is associated with the horns design refers to the shape optimization. The optimization procedure is connected with design parameters like frequency working values [21], amplitude of the signal [22, 23], load transferred in the manufacturing area [24], the objective function [25, 26], the impact of different design variables on the objective function [27], combined signals transmitted in the horn [28], and so forth.In the present paper, we present an approach based on the principle of the least action that takes into consideration as an objective function the volume minimization. The considered input parameters are the working frequency, the maximum cross-sectional area, the magnification coefficient, and the attached mass.


## 2. Mathematical Model

In case of ultrasonic horns, a planar stress waves state is assumed. The general equation that describes one-dimensional propagation, in the longitudinal direction, of ultrasonic waves along a rigid horn with a variable cross-sectional area is Webster’s horn equation [29]:(1)∂2ux,t∂x2+∂ux,t∂x∂∂xln⁡Sx=1c2∂2ux,t∂t2,where u(x,t) is the signal, Sx=S(x) represents the cross-sectional area of the considered horn at the distance x from the reference (point 0), and c is the wave velocity in the horn’s material.One of the widely used techniques to find the explicit analytical solutions of the linear partial differential equations is the method of variables separation [30]. The method assumes that the solution of (1) can be written as a product of two functions, one in space domain u(x)=ux and the other one in time domain u(t)=ut:(2)ux,t=uxut=uxut.Introducing solution (2) in Webster’s equation (1), the new shape is obtained:(3)ux′′ux+ddxSxux′ux=1c2u¨tut,where the following notations were done: ux′′=d2ux/dx2, ux′=dux/dx, and u¨t=d2ut/dt2.This way, in (3), there were separate variables on each side of the equation.The congruency of both sides, left and right, of (3) is achieved only if the two functions are equal with the same separation constant β. It is considered as a constant value β=-ω2, where ω is the angular wave frequency (rad/s).Thus, from (3), we obtained the following ordinary differential equations:(4)u¨t+ω2ut=0;ux′′+ddxSxux′+k2ux=0,where k=ω/c is the wave number.The general solution of the time domain equation from (4) is given by(5)ut=D1cos⁡ωt+D2sin⁡ωtwith the initial conditions(6)utt=0=u0;dutdtt=0=0,where u0 is the initial amplitude of the ultrasonic wave.Thus, solution (5) becomes(7)ut=u0cos⁡ωt,and the total solution (2) becomes(8)ux,t=uxut=uxut=uxu0cos⁡ωt.In case of a steady-state mode, considering the assumption of an infinitesimal motion in a beam made of nondissipative medium and with a variable cross-sectional area, the second equation from (4), which represents the plane wave motion, can be rewritten as(9)ddxSxduxdx+k2Sxux=0.It is considered that the maximum diameter of the horn should not exceed one-quarter of the wavelengthλ and the maximum cross section is denoted as Sm. Thus, we can write the condition that, on the whole length of the horn, the cross section value has to be smaller than the maximum cross section:(10)Sx≤Sm.Considering relation (10), one can write the following mathematical relation:(11)Sm-Sx-a2x=0,where a(x) is an unspecified function of the variable x. In the following, the notations a(x)=ax and S(x)=Sx will be used.An optimization problem of the horn’s shape refers to minimizing the horn’s volume for a given initial set of design data. Based on the principle of the least action, one can write the action relation as(12)A=∫t1t2T-Vdt,where T is the kinetic energy and V represents the potential energy.Considering the wave shape as(13)u=ux,t=uxcos⁡ωt=uxcos⁡ωtfor a beam of length L with attached mass at one end, the case of an ultrasonic horn with the attached tool, the total energies of the system horn tool, mentioned in (12), are given by(14)T=12∫0Lmxu˙2dx+Mtu˙2L2,V=12∫0LEhmxρhu′2dx+σ22EtVt,where mx is the specific mass of the horn, Mt is the attached mass (toll mass), Eh is Young’s modulus of the horn material, Et is Young’s modulus of the tool material, Vt is the volume of the attached tool, ρh represents the horn material density, u˙=du/dt, and u′=du/dx.Considering a period, action (12) becomes(15)At1t2+2π/ω=A¯=πT¯-V¯with(16)T¯=ω22∫0Lρ·S·u2dx+Mtu2L,V¯=12∫0LEh·u′2·Sdx+σ·u′L·Vt.Based on [31, 32], the following functional is defined:(17)Ι=T¯-V¯+α1V+∫0LαxSm-Sx-ax2dx,where α1 and αx=αx are Lagrange multipliers and V is the volume of the horn, given by(18)V=∫0LSxdx.The optimization problem has a solution for the imposed limit values if the functionalΙ given by (17) has to be stationary related to the variations of ux and to the design parameter Sx. Thus, based on (18), it is necessary for Euler’s equations [33] to be satisfied:(19)∂Ι∂ux-ddx∂Ι∂ux′=0,∂Ι∂Sx-ddx∂Ι∂Sx′=0,∂Ι∂ax-ddx∂Ι∂ax′=0,where ux′=dux/dx, Sx′=dSx/dx, and ax′=dax/dx.Considering the functional given by (17) and (19), one can obtain the following differential equations:(a) From the first equation in (19) results(20)ω222uxρSx-ddx-Eh22ux′Sx=ω2uxρSx+EhSxux′=0which when divided by Eh becomes(21)Sxux′′+ρhω2EhSxux=0.Considering the velocity of the longitudinal wave in the horn ascl=Eh/ρh and the wave number k=ω/cl, (21) can be rewritten as(22)Sxux′′+k2Sxux=0.(b) From the second equation, one obtains(23)ω2ρhux2-Ehux′2+2α1-2αx=0,which when divided by Eh becomes(24)kux2-ux′2+2Ehα1-2Ehαx=0.(c) From the third equation results, it is found that(25)2αxax=0.Analyzing the previous equations, one can obtain the following remarks about the optimal design of the ultrasonic horn:(a)
Equation (21) is identical to the longitudinal wave propagation equation along a horn and it is available along the whole horn.(b)
Equation (25) leads to two other conditions:(26)αx=0orax=0.Conditions (26) involve the following aspect: the function ax is different from zero on the whole length of the horn except for one point, where Sx=Sm.Consequently, for (25) to be satisfied for any point in the range 0≤x≤L, it is necessary to have the first condition from (26) αx=0 in the abovementioned range. Moreover, all Lagrange’s coefficients have to be constants.(a)
From (24) and (26), the following is obtained:(27)αx=Eh2k2ux2-ux′2+2α1Eh=0or(28)ux′2-k2ux2-2α1Eh=0.


## 3. The Solution of the Differential Equation

The solution of (28) can be found considering the initial functional conditions of the ultrasonic horn. The following are considered as initial conditions:(a)
The length of the horn has to be equal to a wavelength.(b)
To achieve the assumption that the propagated waves are only longitudinal ones, a maximum value is considered for the magnification coefficientq=5 [17].(c)
The initial value of the signal amplitude, at the connection point between the transducer and the hornx=0, is u0 equal to the emitted signal by the transducer:(29)ux=u0=u0.(d)
Considering the initial signal and the magnification coefficient, the signal at the end of the horn (forx=L) is given by(30)ux=uL=-qu0.(e)
In (30), the negative value (−) is considered in order to obtain a nodal point in a position along the horn length.(f)
Between the two ends of the horn, there is no other extreme of the signal except for the maximum developed at the end of the horn, in the connection point with the tool.(g)
The following condition is considered [32]:(31)mux=L′-k2Mux=L=0,wherem is the mass on the unit length, at the end of the horn, ux=L′ is the strain at the end of the horn, M is the attached mass, and ux=L is the signal at the end of the horn, given by (30).Denoting(32)C=2Eα1,equation (28) can be rewritten as(33)u′2-k2u2-C=0,and integrating it with the derivative method, one can obtain the following equation [34]:(34)2u′u′′-k22uu′=0.Dividing (34) by u′≠0 results in the differential equation(35)u′′-k2u=0,with the solution(36)ux=C1ekx+C2e-kx.Introducing solution (36) in (33), after some calculations, the following is obtained:(37)-4k2C1C2=C,(38)C1=-C4k2C2.Considering (38) in solution (36) results in(39)ux=-C4k2C2ekx+C2e-kx.The constantsC and C2 can be found considering the initial conditions mentioned above, for x=0. Combining (38) and (30), the following relation is obtained:(40)u0=-C4k2C2+C2,which leads to the following relation for the two values of the constant C2 as a function of C:(41)C21,2=ku±k2u02+C2k.Considering (41) and (40), for both values C21,2, the same dependence is obtained between the constants C and C2:(42)C=4C2k2C2-u0.Introducing the constantC, given by (42), in solution (39) leads to the following relation of the amplitude u(x):(43)ux=u0ekx-C2ekx-e-kx.The constantC2 can be found considering conditions (39) and (40):(44)ρhShLu0ekL-C2ekL+e-kL+kLtρtStqu0=0,where ρh is the density of the horn’s material, ShL is the cross section at the end of the horn, St is the cross section of the tool (SL=St), Lt is the tool’s length, ρt is the density of the tool’s material, and q is the gain of the initial signal u0.From (44) results(45)C2=u0ρhShLekL+qkLtStρt2ρhShLcosh⁡kL.Considering dependence (42), one can define the function of the space dependent amplitude component of the signal along the horn:(46)ux=u0ekx-ρhShLekL+qkLtStρtρhShLekL+e-kLekx-e-kx.In the following, we use the notation(47)B=ρhShLekL+qkLtStρtρhShLekL+e-kL,and the signal function (46) becomes(48)ux=u01-Bekx+Be-kx.As it is known, an important aspect in horns design is to find the position of the neutral pointxn. The neutral point defines the position of the cross section where the signal is zero. Thus, the neutral point can be easily found considering relation (48). Equalizing function (48) with zero, it is obtained that(49)xnod=12klnBB-1.


## 4. Cross Section Shape Definition

From relation (48), we obtained the first two derivatives of the space component of the signal:(50)u′x=ku01-Bekx-Be-kx,u′′x=k2u01-Bekx+Be-kx.Equation (9) can be rewritten as(51)S′S=-u′′+k2·uu′.Considering relations (50) in (51) results in the new formula(52)Sx′Sx=-2u0k21-Bekx+Be-kxu0k1-Bekx-Be-kx.Defining the functiong(x)=gx as(53)gx=1-Bekx-Be-kx,(52) becomes(54)Sx′Sx=-2·gx′gx.Integrating (54) results in a constant, denoted by D:(55)ln⁡Sx=-2ln⁡gx+D,which depends on the initial geometrical conditions.At the end of the horn, forx=L, the cross section becomes Sx=SL and it is obtained that(56)D=ln⁡Sx+2ln⁡gL,where gL is the value of the function g(x)=gx, defined by (53), for x=L.Introducing the value ofD obtained by (56) in (55) results in(57)ln⁡Sx=-2ln⁡gx+ln⁡SL+2ln⁡gL,or(58)ln⁡SxSL=-2ln⁡gxgL,and the cross section at the distance x can be calculated as(59)Sx=SLgLgx2.An interesting design aspect is represented by the value and position of the maximum cross section. Thus, we considered the condition that the first derivative of function (59) is equal to zero. This condition leads to the relation(60)gx′=0with similar solution to (49).It can be concluded that the position of the maximum cross section is in the same place as the nodal point.The end cross section can be found as(61)SL=SmgngL2,and introducing (61) in (59) results in the relation of the cross section value in a point situated at the distance x from the system origin, as a function of the maximum cross section:(62)Sx=Smgngx2.Taking into consideration relation (62), a script in Matlab was made to generate the shape of the horn made of steel. The input parameters were the resonance frequency fr=19900 Hz, the wave velocity in steel c=50550 m/s, the horn’s material density ρh=7850 kg/m3, the end radius of the horn (connection surface with the tool) rhL=0.005 m, the magnification coefficient q=5, and tool radius equal to the end horn radius rt=rhL. The tool was considered to be made of steel.Based on the script written in Matlab, the shape presented in Figure2 was obtained, and in Figure 3 the physical model of the horn is presented.Figure 2
The half-horn shapes along the longitudinal axis.Figure 3
The physical model of the horn.Based on the same script, the following design data were found: the radius in the origin of the system (x=0) r0=29.256 mm, the largest radius rm=31.721 mm, and the position of the nodal point xnod=27.734 mm.Considering relation (48), using a script written in Matlab, we obtained the magnitude variation of the spatial components u(x) of the considered signal (13) (Figure 4).Figure 4
The signal magnitude along the horn.As it can be seen from Figure4, the nodal point is at the coordinate found by relation (49).


## 5. Theoretical Modal Analysis

### 5.1. State-Space Approach of the Optimized Model

The modal behavior can be described in many ways. Two of them are the state-space approach and transfer function representation. The differences between these two methods were described in [35, 36]. In the present paper, we considered as a theoretical modal analysis the state-space approach, and for the experimental modal analysis the FRF representation was made.The state-space method is often used in dynamic system analysis. This method can be applied in the case of lumped systems and it is based on the concept of state vector that consists of two components: the vector of displacement and the vector of the velocities. Practically, the “n” second-order coupled equations that describe the dynamic behavior of a lumped system are transformed in a set of 2n first-order coupled equations [37–39].The ultrasonic horn can be considered as a linear time-invariant (LTI) system that can be approximate with a lumped mass system. Thus, the obtained horn (Figure3) is divided into “n” elements with the same length li, where li=L/n, with L being the total length of the horn equal to a quarter of the wavelength. Considering the case of the free-free system, as is in the experimental modal analysis, the horn can be modeled as a lumped system made of “n” masses connected with springs (Figure 5).Figure 5
The lumped model of the horn.We considered 200 masses with a length ofli=6.344·10-4 m in each part. The obtained components are, taking into consideration the shape (Figure 3), small frusta of cone elements.The components of the modal model are established considering the following assumptions:(a)
The stiffness of the springs was considered to be equal to the equivalent stiffness of each element based on the relation(63)ki=ESaili,whereSai represents the average area of the region of length li:(64)Sai=Si+Si+12,with bothSi and Si+1 calculated using relation (62).(b)
The massmi of each small frustum of cone element was found considering the classical relation of the density using for volume the areas Si and Si+1 and the length li.Considering the d’Alembert principle, the motion equations for the system from Figure5 are given by(65)m1x¨1+k1x1-k1x2=0,m2x¨2+k1+k2x2-k1x1-k2x3=0,m3x¨3+k2+k3x3-k2x2-k3x4=0,⋮mix¨i+ki-1+kixi-ki-1xi-1-kixi+1=0,⋮mnx¨n+kn-1xn-kn-1xn-1=0.System (65) can be rewritten in matrix form as follows:(66)m100⋯⋯⋯00m20⋯⋯⋯000m3⋯⋯⋯0⋮⋮⋮⋮⋮⋮⋮000⋯mi⋯0⋮⋮⋮⋮⋮⋮0000⋯⋯⋯mnx¨1x¨2x¨3⋮x¨i⋮x¨n+k1-k10⋯⋯⋯0-k1k1+k2-k2⋯⋯⋯00-k2k2+k3-k3⋯⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯⋮⋯0⋮⋮⋮⋮⋮⋮0000⋯⋯-kn-1kn-1x1x2x3⋮xi⋮xn=000⋮0⋮0or(67)MX¨+KX=0,from which it is obtained that(68)X¨=-M-1KX.As the state vector, we considered the matrix vector(69)Δ=δ1,δ2,…,δ2qT=X,X˙T=x1,x2,…,xn,x˙1,x˙2,…,x˙nT,which, by derivation and considering (68), leads to the equation(70)Δ˙=X˙X¨=0I-M-1K0XX˙=AΔ,where I is the unit matrix and 0 is the zeros matrix, both of n×n type.The matrixA is defined as the state matrix or the transmission matrix and is a property of the system being defined by the structure of the system. The eigenvalues of the state matrix can be found based on the relation(71)λI-A=0that is developed in an equation with unknowns λ.If, in relation (71), the value λ=jωn is considered, where j=-1 and ωn is the natural frequency measured in rad/s, then the so-called characteristic equation is obtained, which gives the natural frequencies.Thus, considering the equivalent system from Figure5 with the associated motion equations (66), (67), and (70), one can find the natural frequencies of the system. The number and the precision of these frequencies depend on the number of the considered masses mi(i=1,n¯). Figure 6 presents the frequency values obtained for a model with 200 masses considering the method of eigenvalues of the transition matrix.Figure 6
The natural frequencies obtained for the lumped model.Figure6 also presents some of the obtained natural frequencies. The first natural frequency found was fn=19,837 Hz, which is close to the designed frequency of fr=19.9 kHz.


### 5.2. Finite Element Model

In the next step, the finite element model was considered. The used specialized software was ANSYS. The 3D model (Figure7) was made based on (59).Figure 7
The finite element model of the horn.For good convergence, we considered a mapped mesh consisting of first-order solid hexaelements with 1 mm length. We obtained 87,076 nodes and 75,254 elements. Mode extraction is carried out in the frequency range 18–24 kHz using Block Lanczos option with double precision. Two pure longitudinal modes were obtained with the natural frequencies off=19,817 Hz (Figure 8) and f=23,151 Hz (Figure 9).Figure 8
The mode shape for the natural frequency off=19,817 Hz.Figure 9
The mode shape for the natural frequency off=23,151 Hz.


## 5.1. State-Space Approach of the Optimized Model

The modal behavior can be described in many ways. Two of them are the state-space approach and transfer function representation. The differences between these two methods were described in [35, 36]. In the present paper, we considered as a theoretical modal analysis the state-space approach, and for the experimental modal analysis the FRF representation was made.The state-space method is often used in dynamic system analysis. This method can be applied in the case of lumped systems and it is based on the concept of state vector that consists of two components: the vector of displacement and the vector of the velocities. Practically, the “n” second-order coupled equations that describe the dynamic behavior of a lumped system are transformed in a set of 2n first-order coupled equations [37–39].The ultrasonic horn can be considered as a linear time-invariant (LTI) system that can be approximate with a lumped mass system. Thus, the obtained horn (Figure3) is divided into “n” elements with the same length li, where li=L/n, with L being the total length of the horn equal to a quarter of the wavelength. Considering the case of the free-free system, as is in the experimental modal analysis, the horn can be modeled as a lumped system made of “n” masses connected with springs (Figure 5).Figure 5
The lumped model of the horn.We considered 200 masses with a length ofli=6.344·10-4 m in each part. The obtained components are, taking into consideration the shape (Figure 3), small frusta of cone elements.The components of the modal model are established considering the following assumptions:(a)
The stiffness of the springs was considered to be equal to the equivalent stiffness of each element based on the relation(63)ki=ESaili,whereSai represents the average area of the region of length li:(64)Sai=Si+Si+12,with bothSi and Si+1 calculated using relation (62).(b)
The massmi of each small frustum of cone element was found considering the classical relation of the density using for volume the areas Si and Si+1 and the length li.Considering the d’Alembert principle, the motion equations for the system from Figure5 are given by(65)m1x¨1+k1x1-k1x2=0,m2x¨2+k1+k2x2-k1x1-k2x3=0,m3x¨3+k2+k3x3-k2x2-k3x4=0,⋮mix¨i+ki-1+kixi-ki-1xi-1-kixi+1=0,⋮mnx¨n+kn-1xn-kn-1xn-1=0.System (65) can be rewritten in matrix form as follows:(66)m100⋯⋯⋯00m20⋯⋯⋯000m3⋯⋯⋯0⋮⋮⋮⋮⋮⋮⋮000⋯mi⋯0⋮⋮⋮⋮⋮⋮0000⋯⋯⋯mnx¨1x¨2x¨3⋮x¨i⋮x¨n+k1-k10⋯⋯⋯0-k1k1+k2-k2⋯⋯⋯00-k2k2+k3-k3⋯⋯0⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯⋮⋯0⋮⋮⋮⋮⋮⋮0000⋯⋯-kn-1kn-1x1x2x3⋮xi⋮xn=000⋮0⋮0or(67)MX¨+KX=0,from which it is obtained that(68)X¨=-M-1KX.As the state vector, we considered the matrix vector(69)Δ=δ1,δ2,…,δ2qT=X,X˙T=x1,x2,…,xn,x˙1,x˙2,…,x˙nT,which, by derivation and considering (68), leads to the equation(70)Δ˙=X˙X¨=0I-M-1K0XX˙=AΔ,where I is the unit matrix and 0 is the zeros matrix, both of n×n type.The matrixA is defined as the state matrix or the transmission matrix and is a property of the system being defined by the structure of the system. The eigenvalues of the state matrix can be found based on the relation(71)λI-A=0that is developed in an equation with unknowns λ.If, in relation (71), the value λ=jωn is considered, where j=-1 and ωn is the natural frequency measured in rad/s, then the so-called characteristic equation is obtained, which gives the natural frequencies.Thus, considering the equivalent system from Figure5 with the associated motion equations (66), (67), and (70), one can find the natural frequencies of the system. The number and the precision of these frequencies depend on the number of the considered masses mi(i=1,n¯). Figure 6 presents the frequency values obtained for a model with 200 masses considering the method of eigenvalues of the transition matrix.Figure 6
The natural frequencies obtained for the lumped model.Figure6 also presents some of the obtained natural frequencies. The first natural frequency found was fn=19,837 Hz, which is close to the designed frequency of fr=19.9 kHz.


## 5.2. Finite Element Model

In the next step, the finite element model was considered. The used specialized software was ANSYS. The 3D model (Figure7) was made based on (59).Figure 7
The finite element model of the horn.For good convergence, we considered a mapped mesh consisting of first-order solid hexaelements with 1 mm length. We obtained 87,076 nodes and 75,254 elements. Mode extraction is carried out in the frequency range 18–24 kHz using Block Lanczos option with double precision. Two pure longitudinal modes were obtained with the natural frequencies off=19,817 Hz (Figure 8) and f=23,151 Hz (Figure 9).Figure 8
The mode shape for the natural frequency off=19,817 Hz.Figure 9
The mode shape for the natural frequency off=23,151 Hz.


## 6. Experimental Modal Analysis

The aim of the experimental modal analysis is to verify whether the considered imposed working frequency of 19.9 kHz is one of the natural frequencies of the considered horn 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 an electrodynamic shaker (Figure10) was considered [40]. The used equipment was a complete testing setup (Brüel & Kjær) made of an integrated platform, PULSE 12, with a force transducer (type 8230) and an accelerometer (type 4397). The excitation signal was generated by the PULSE 12 platform and was amplified using the power amplifier type 2718 and then transmitted to the tested horn by the shaker type 4810 (Figure 11).Figure 10
The modal testing setup.Figure 11
The detailed connection shaker horn.The random generated signal (the input signal) has a frequency span up to 25.6 kHz with an excitation time of 500 ms (Figure12). The Fourier spectrum of the input signal is shown in Figure 13.Figure 12
The input random signal in time domain.Figure 13
The Fourier spectrum of the input signal.As it is known, the frequency response function estimation from the measured data represents the main step of the experimental modal analysis. The analysis is based on nonparametric estimators. In the present study, we considered the often used estimatorH1. The obtained frequency response function, for the estimator H1, is presented in Figure 14.Figure 14
The frequency response function of the horn.The range of the measured frequencies was up to 20,000 Hz and the values are presented in Figure15.Figure 15
The frequencies obtained by experimental data.At the same time, based on the experimental modal analysis, we obtained the damping factorsζ for each natural frequency found. The distribution of these values is presented in Figure 16.Figure 16
The damping factor values distribution versus natural frequencies.Considering the facilities of the Matlab software, a power function was defined, which can approximate the damping factor for different values of the frequencyf:(72)ζf=321.9f-0.7427-0.07775.Function (72) can be used to study the influence of the damping in case of horn vibration at different frequencies.


## 7. Testing the Designed Horn

The test was designed to check whether the magnification of the designed horn is the initial consideredq=5 for the resonance frequency of 19.9 kHz. The used setup is made of the same components as the setup used in the experimental modal analysis with the exception of the accelerometers. For testing, two Brüel & Kjær accelerometers type 4517-002 were used. The signal was generated by a magnetostrictive transducer for a harmonic signal yt=Asin⁡ωt with a frequency f=19.9 kHz. The input signal, represented in time domain, is shown in Figure 17, and the output signal (response) also in time domain is designed in Figure 18.Figure 17
The input signal in time domain.Figure 18
The output signal in time domain.At the same time, for frequency control, we performed a check in frequency domain for both signals: the input (Figure19) and the output (Figure 20).Figure 19
The input signal in frequency domain.Figure 20
The output signal in frequency domain.Considering the values of the input and output signals (Figures17 and 18), the magnification obtained by experiment is given by the following relation:(73)qreal=uoutputuinput=42.789.32=4.59,which represents an error of about 8.2% between the considered value q=5 and the obtained value qreal.


## 8. Conclusions

Based on the principle of the smallest action, an optimization of the shape of an ultrasonic horn was performed considering Webster’s equation. The design was made considering a working resonance frequency offr=19.9 kHz and a magnification coefficient q=5, and the main assumption was that, through the horn and all attached coupling devices, only longitudinal plane waves propagate.The considered boundary conditions were the classical ones referring to the amplitude of the signal, relations (29) and (30), to which a new one was added, relation (31), which highlights the influence of the tool mass, added at the end of the 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 xn (49) being determined at the same time. Comparing the obtained shape with the classical shape, it can be seen that it is a different one, with the largest diameter in the neutral point. In the case of the other horns’ shape, the largest diameter is at one end.The natural frequencies were determined both in simulation (theoretically) and experimentally. In the case of theoretical modal analysis, the state-space representation was used. The horn was divided into 200 masses connected by springs (Figure5) and the eigenvalues of the transmission matrix of system (71) were calculated. The first obtained value of the natural frequency was the imposed resonant frequency f1,ss=19,982 Hz.The second considered theoretical approach in theoretical modal analysis was the finite element method. A model with 75,254 elements was designed, and based on ANSYS software facilities, two natural frequencies were found around the designed resonance frequency (Figures8 and 9), both for the longitudinal mode shape. In order to provide clarity to the results, a run was made in double precision mode.The experimental modal analysis offered the possibility of finding a set of natural frequencies (Figure16) and the corresponding values of the damping factor. Considering the facilities of the Matlab tool, a function of damping factor variation 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 frequency was found.At the end, the horn was tested to check whether the magnification coefficient is the same as the designed one. A harmonic signal with the frequency value equal to the resonance frequency was considered as the input, and the output signal was measured. The found magnification coefficient is close to the designed one (72).In the frame of the test, the attached tool was not considered. It is important for the assembly transducer-horn-tool to work in resonance regime and each part to work within the parameters for which it was designed.The aim of the test was to check whether the considered horn, designed in the proposed set of boundary conditions (29), (30), and (31), generates the initial designed mechanical magnification of the input signal.The influence of the attached tool mass is the particular design element that was taken into consideration compared with the other methods of optimization and design.Considering the abovementioned aspects, it can be concluded that the numerical simulations and tests that were done support the validity of the assumptions used in the horn’s optimization design.The proposed model offers as an advantage a more convenient placement of the nodal point that is situated in the same place as the largest cross section.The described design and optimization method offer the possibility of obtaining the shape based on the main working parameters.



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*Source: 8532021-2017-08-07.xml*