Catenoidal Horn Research Articles

Design and Analysis of Hollow Catenoidal Horn Profile for Ultrasonic Machining of Composite Materials

The composite materials Si/AlC, crystal quartz, and ceramic glass are becoming an important part of the present society in many engineering and non-engineering fields. Conventional methods available for machining these materials have many flaws due to which their application on large scale is restricted. A non-conventional process known as ultrasonic machining (USM), can be implemented for machining of these materials effectively. Anyhow machining efficiency of USM greatly depends upon its horn design and therefore in the present study a horn based on a catenoidal profile with aluminum and titanium material for USM was designed and developed by using solid works and ANSYS. Modal and harmonic analysis was done on the horn for computation of various parameters of interest such as resonant frequency, amplitude vibration and equivalent stresses. After the computation of the results, they were analyzed and compared with those available in the literature in terms of stresses and magnification factor for their validation comparison with literature, it was found that an aluminum catenoidal horn shows higher magnification with the least stress magnitude as compared to horns avail-able in the literature and hence can be used in USM as a replacement for existing horns.

Using the Design Environment for Low-amplitude Thermoacoustic Energy Conversion in the Classroom.

The Design Environment for Low-amplitude Thermoacoustics Energy Conversion (DeltaEC) is a free software package that is capable of modeling quasi-one-dimensional, multi-domain acoustical networks and includes a 300-page users' guide. The program is a versatile differential equation solver that analyzes a series of "segments" representing ducts, compliances, speakers, etc., that are specified using MKS units. DeltaEC can be a useful pedagogical tool for an introductory course on acoustics in fluids. Examples of its instructional applications include illustration of effective length and quality factor for a Helmholtz resonator and the modes of coupled Helmholtz resonators, as well as standing waves within a resonator of variable cross section and within a catenoidal horn driven by an electrodynamic loudspeaker. Segments representing electrodynamic loudspeakers, radiation loading, and flow resistance in porous media will be used to demonstrate the coupled-oscillatory behavior of a bass-reflex enclosure's complex electrical impedance versus frequency. DeltaEC's plotting features allow immediate visualization of the pressure and velocity fields as well as the power flow or temperature distribution through a network or within a resonator, horn, or waveguide. The "thermo-physical property" feature provides fluid and solid properties at the students' choice of pressure, temperature, and frequency, making it useful as a "handbook" for other assignments.

A nonrational B-spline profiled horn with high displacement amplification for ultrasonic welding

A new horn with high displacement amplification for ultrasonic welding is developed. The profile of the horn is a nonrational B-spline curve with an open uniform knot vector. The ultrasonic actuation of the horn exploits the first longitudinal displacement mode of the horn. The horn is designed by an optimization scheme and finite element analyses. Performances of the proposed horn have been evaluated by experiments. The displacement amplification of the proposed horn is 41.4% and 8.6% higher than that of the traditional catenoidal horn and a Bézier-profile horn, respectively, with the same length and end surface diameters. The developed horn has a lower displacement amplification than the nonuniform rational B-spline profiled horn but a much smoother stress distribution. The developed horn, the catenoidal horn, and the Bézier horn are fabricated and used for ultrasonic welding of lap-shear specimens. The bonding strength of the joints welded by the open uniform nonrational B-spline (OUNBS) horn is the highest among the three horns for the various welding parameters considered. The locations of the failure mode and the distribution of the voids of the specimens are investigated to explain the reason of the high bonding strength achieved by the OUNBS horn.

Rapid dissolution of switchgrass in 1-butyl-3-methylimidazolium chloride by ultrasonication

The utilization of ultrasonics to rapidly dissolve switchgrass in ionic liquid, 1-butyl-3-methylimidazolium chloride ([Bmim][Cl]) was evaluated in this work. The objective of the study focused on determining the effects of various ultrasonic conditions on the recovery of carbohydrate from biomass, lignin removal, and enzymatic hydrolysis. Dried ground switchgrass was added to ionic liquid, then sonicated at a frequency of 20kHz. The experiments were conducted using a catenoidal horn at varying amplitudes of 96μm, 128μm, and 160μm and sonication times of 2, 3, and 4min. Similarly, ground switchgrass was dissolved in ionic liquid assisted by conventional heat treatment at 130°C for 12 and 24h. The results showed good delignification results of 53% for the 24h heat pretreated samples and of 50.8% for ultrasonic assisted samples at 160μm amplitude and 4min. Even in the presence of lignin in the recovered biopolymer, both of heat treated and ultrasonicated samples obtained 100% glucan digestibility after only 3h of enzymatic hydrolysis. Heat pretreated samples exhibited 44–59% lower xylan digestibility compared to ultrasonic pretreated samples (160μm amplitude and 4min sonication time). Scanning electron microscope images displayed significant changes in biomass structure from intact and crystalline of the untreated biomass to disintegrated and amorphous of the treated biomass (heat treated and ultrasonicated). With increasing ultrasonic amplitude the carbohydrate recovery decreased. Also, more than 50% of the hemicellulose fraction was lost during biomass recovery. Overall, it was concluded that ultrasonication was a promising technology to enhance dissolution of lignocellulose in ionic liquid.

Simultaneous saccharification and fermentation and economic evaluation of ultrasonic and jet cooking pretreatment of corn slurry

The potential of ultrasonics to replace hydrocooking in corn-to-ethanol plants was examined in this study. Batch and continuous experiments were conducted on corn slurry with sonication at a frequency of 20 kHz. Batch mode used a catenoidal horn operated at an amplitude of 144 μm peak-to-peak (p–p) for 90 s. Continuous experiments used a donut horn operating at inner radius amplitude of 12 μm p–p. Jet-cooked samples from the same ethanol plant were compared with ultrasonicated samples. The highest starch-to-ethanol conversion was obtained by the jet-cooked samples with a yield of 74% of the theoretical yield. Batch and continuous sonication achieved 71.2% and 68% conversion, respectively, however, statistical analysis showed no significant difference between the jet cooking and ultrasonication. On the basis of the similar performance, an economic analysis was conducted comparing jet cooking and ultrasonic pretreatment. The analysis showed that the capital cost for the ultrasonics system was ~10 times higher compared to the capital cost of a hydrocooker. However,due to the large energy requirements of hydrocookers, the analysis showed lower total overall costs for continuous ultrasonication than that for jet cooking, assuming the current energy prices. Because of the high utility cost calculated for jet cooking, it is concluded that ultrasonication poses as a more economical option than jet cooking. Overall, the study shows that ultrasonics is a technically and economically viable alternative to jet cooking in dry-grind corn ethanol plant.

Numerical Investigation of the Mean Flow Effect on the Acoustic Reflection at the Open End of Clarinet-Like Instruments

This article investigates the influence of a mean flow on the magnitude of the reflection coefficient |R| and on the length correction I at the end of clarinet-like instruments. The investigations are conducted on a cylindrical pipe terminated by a catenoidal horn using an axisymmetric numerical model based on the lattice Boltzmann method. The simulations are carried out using input values of the mean flow found experimentally for different dynamic playing levels (pianissimo, mezzo forte and fortissimo) and different reed types (soft and hard). The results show that |R| is unaffected in the range of mean flows corresponding to the different dynamic levels analyzed. Conversely, the end correction l appears to be considerably more sensitive to variations of the dynamic level, particularly in the low frequency region. Nevertheless, the results obtained indicate a small effect for musical contexts and thus generally substantiate the zero mean flow assumption commonly used to predict the acoustic transmission and the dynamic response of clarinet-like systems.

Design of a Bézier-profile horn for high displacement amplification

A new horn for high displacement amplification is developed. The profile of the horn is a cubic Bézier curve. The ultrasonic actuation of the horn exploits the first longitudinal displacement mode of the horn. A design method of the horn using an optimization scheme and finite element analyses is developed. Prototypes of the horns are manufactured by a numerical control machining process. Performances of the proposed horn have been evaluated by experiments. Experimental results of the harmonic response of the fabricated horn confirm the effectiveness of the design method. The displacement amplification of the proposed horn is 71% higher than that of the traditional catenoidal horn with the same length and end surface diameters.

Ultrasonic pretreatment of corn slurry for saccharification: A comparison of batch and continuous systems

The effects of ultrasound on corn slurry saccharification yield and particle size distribution was studied in both batch and continuous-flow ultrasonic systems operating at a frequency of 20 kHz. Ground corn slurry (28% w/v) was prepared and sonicated in batches at various amplitudes (192–320 μm peak-to-peak (p–p)) for 20 or 40 s using a catenoidal horn. Continuous flow experiments were conducted by pumping corn slurry at various flow rates (10–28 l/min) through an ultrasonic reactor at constant amplitude of 12 μm p–p. The reactor was equipped with a donut shaped horn. After ultrasonic treatment, commercial alpha- and gluco-amylases (STARGEN TM 001) were added to the samples, and liquefaction and saccharification proceeded for 3 h. The sonicated samples were found to yield 2–3 times more reducing sugars than unsonicated controls. Although the continuous flow treatments released less reducing sugar compared to the batch systems, the continuous flow process was more energy efficient. The reduction of particle size due to sonication was approximately proportional to the dissipated ultrasonic energy regardless of the type of system used. Scanning electron microscopy (SEM) images were also used to observe the disruption of corn particles after sonication. Overall, the study suggests that both batch and continuous ultrasonication enhanced saccharification yields and reduced the particle size of corn slurry. However, due to the large volume involve in full scale processes, an ultrasonic continuous system is recommended.

Corrections to the Plane-Wave Approximation in Rapidly Flaring Horns

The input impedance of an air column (horn) with a slowly varying cross section can be calculated approximately by solving the wave equation assuming plane waves (Webster equation). In some cases analytical solutions are possible. For an arbitrary horn shape described as a succession of short cones the one-dimensional transmission-line (TL) method can be used. This fails for horns with a large flare, where transverse flow demands kinetic energy, effectively increasing the local inertance. The quantities determining the magnitude of this correction appear from a variational calculation based on Rayleigh's principle. The magnitude of these quantities was determined by numerically solving the wave equation at low frequency in a number of horn shapes by a finite difference method. Studied are conical, hyperboloid and catenoidal horns. Since horns, in particular those found in wind instruments, are found in many other shapes, general use of this method is not practical. In this paper a simple formula is proposed useful for any horn shape to calculate approximately the additional inertance. It is introduced as a correction in the TL method and is applicable to any shape, even for a stepwise diameter change and for a flange. Its accuracy for most geometries found on wind instruments is within the detection threshold of the ear, a 0.2% frequency shift. It is not applicable any more when cross-dimensions become comparable to the wavelength.

DELTAEC is also an acoustics teaching tool

A major revision of the Los Alamos thermoacoustics code, renamed DELTAEC, was released in 2007. It takes advantage of a user‐friendly, menu‐driven Windows environment and has indigenous plotting capabilities. DELTAEC is a differential equation solver that analyzes one‐dimensional acoustical networks defined by a series of "segments" representing ducts, compliances, speakers, etc. This talk will relate experiences using this software to teach a first‐year graduate core course on acoustics in fluids. Examples include illustration of effective length and quality factor of Helmholtz resonators, as well as the more challenging standing wave solutions within a resonator of variable cross‐section. The plotting feature allows immediate illustration of the pressure and velocity fields as well as power flow within the resonator. DELTAEC will also adjust gas mixture concentration to match a specified frequency. Segments representing electrodynamic loudspeakers, radiation loading, and flow resistance in porous media will be used to demonstrate the coupled‐oscillatory behavior of a bass‐reflex enclosure's complex electrical impedance vs. frequency. Calculation of the modes of a catenoidal horn of finite length will be presented. The "thermo‐physical property" feature provides fluid and solid properties at the students' choice of pressure, temperature and frequency, making it useful as a "handbook" for other assignments.

Temperature Rise and Welding Characteristics of Various-Frequency Ultrasonic Plastic Welding Systems

Welding characteristics and temperature increases of ultrasonic plastic welding parts over a frequency range from 27 to 94 kHz are studied. Using 27, 40, 67, and 94 kHz ultrasonic plastic welding systems, temperature increases at welding surfaces of lapped 1.0-, 2.0-, and 3.0-mm-thick polypropylene plates and polymethyl methacrylate plates are measured using 0.1- and 0.2-mm-diameter thermocouples inserted between plates, and temperature distributions at cross sections of lapped plate specimens are measured using a thermotracer. The 94 kHz vibration system used for ultrasonic plastic welding consists of a bolt-clamped Langevin-type longitudinal vibration source using four 30-mm-diameter piezoelectric ceramic (PZT) rings, a stepped horn (vibration velocity transform ratio N=3.0) and a catenoidal horn (N=3.13) with an 8-mm-diameter welding tip. The other vibration systems have similar configurations. In the case of using a higher-frequency system, increases in temperature measured at the welding parts are larger. Temperature rises are larger for lapped plate specimens than that for a one-piece specimen owing to the vibration loss of welding surfaces.

Frequency Characteristics of Ultrasonic Plastic Welding: (27 kHz to 180 kHz Ultrasonic Plastic Welding Systems)

Welding characteristics and temperature rises at plastic welding part over frequency range from 27kHz to 180kHz were studied. Ultrasonic plastic welding systems of 27, 40, 67, 94, 150 and 180kHz were developed and direct welding characteristic of lapped plastic sheets were measured. The temperature rises at welding surfaces of lapped 1.0-mm-thick polypropylene and polymethyl methacrylate plates were measured using thermocouples inserted between plate specimens. And also, temperature rise distributions in cross sections of the lapped plate specimens were measured using a thermotracer. The longitudinal vibration systems used for ultrasonic plastic welding consisted of a bolt-clamped Langevin type longitudinal vibration source using two or four piezoelectric ceramic (PZT) rings, a stepped horn and a catenoidal horn with a 4-mm- or 8-mm-diameter welding tip. Using higher frequency system, temperature rise measured at the welding part were higher. Maximum temperature rise measured using thermocouples and a thermotracer was over 350°C under excessive condition.

Temperature Distributions in Welding Specimens for Ultrasonic Plastic Welding Systems Using Various Frequencies

Welding characteristics and temperature increases of plastic welding parts over a frequency range of 27 kHz to 94 kHz are studied. Using 27 kHz, 40 kHz, 67 kHz and 94 kHz ultrasonic plastic welding systems, temperature increase at welded surfaces of 1.0-mm-thick polypropylene plates and polymethyl methacrylate plates is measured using thermocouples inserted between plates, and temperature distributions at cross-sections of lapped plate specimens are measured using a thermotracer. The 94 kHz vibration system used for ultrasonic plastic welding consists of a bolt-clamped Langevin type longitudinal vibration source using four 30-mm-diameter piezoelectric ceramic (PZT) rings, a stepped horn (vibration velocity transform ratio N=3.0) and a catenoidal horn (N=3.13) with an 8-mm-diameter welding tip. The other vibration systems have similar configurations. Using a higher frequency system, increases in temperature measured at the welding parts are higher. The maximum temperature increase measured using thermocouples and a thermotracer was over 330°C.

Configuration of a 30-mm-diameter 94 kHz Ultrasonic Longitudinal Vibration System for Plastic Welding

Vibration and welding characteristics of a 94 kHz ultrasonic plastic welding system are studied. The 94 kHz ultrasonic plastic welding systems consist of a 30-mm-diameter bolt-clamped Langevin-type PZT longitudinal transducers with four PZT rings, a stepped horn (vibration transform ratio N = 3.0) with a supporting flange at a nodal position and a catenoidal horn (N = 3.13) with an 8-mm-diameter welding tip. Maximum vibration velocity of the 94 kHz welding tip was 3.2 m/s (peak-to-zero value) at loaded condition. The welding characteristics of the 1.0-mm-thick polypropylene sheet specimens using the 94 kHz welding system were studied. Using the 94 kHz system, a weld strength of more than 370 N per one welded area was obtained at a vibration velocity of 2.7 m/sp-0 (peak-to-zero value), welding time of 0.8 s and static pressure of 600 kPa.

Characterization of a cosine-shaped horn

Several approaches to the problem of the acoustic wave propagation in a duct with a varying cross section have been described in the literature. It has been shown that a simple one-dimensional analysis gives accurate predictions of the sound propagation. In this research, the impedance of a cosine-shaped horn is obtained using the Wentzel–Kramers–Brillouin (WKB) approximation to solve the Webster wave equation and the results are compared with those obtained from a numerical solution of the equivalent system of nonlinear equations. A comparison of the throat impedance ratio is presented for similar exponential, conical, parabolic, and catenoidal horns with the same overall dimensions. The mouth of the horn is assumed to be terminated in an infinite baffle. In addition, since the device may be used as an impedance transformer, the results for the impedance ratio for two pipes of different areas joined by a cosine-shaped connector are compared with several well-known couplings. Such a cosine-shaped connector will act as a simple discontinuity when its length is short compared with a wavelength and as a transformer for acoustic impedance for higher frequencies.

On the acoustic of inverse sinusoidal and catenoidal nozzles

The acoustic wave equation, for quasi-one-dimensional propagation, along a duct of varying cross section, containing a low Mach number mean flow, is obtained using as variables either the potential or the velocity; the ray approximation, which holds only for wavelength short compared with the lengthscales or variation of cross section and mean flow velocity, is used as a factor to reduce the wave equation to a Schrödinger form. It is shown that the latter, reduced from, for the potential, is the most convenient to study the acoustics of catenoidal and sinusoidal nozzles; it is found that these inherit respectively the filtering properties of catenoidal horns, and transparency properties of sinusoidal horns. This approach also applies to the exponential nozzle whereas, for the Gaussian nozzle, the sound field can be expressed in terms of Hermite functions, using a semi-reduced form of the wave equations. The exact solutions of the nozzle wave equation, for the four families of ducts, are plotted as amplitude and phase versus distance, for several combinations of frequency and low Mach number. The nozzle families considered include the catenoidal in the sinusoidal and exponential and the first six eigenfunctions of Gaussian nozzles. The acoustics of inverse sinusoidal and catenoidal nozzles can also be studied, without use of the preceding transformations, via the solution of modified Mathieu equation.

An efficient numerical method for solving the horn equation

For lossless horns, there are many contours for which exact solutions of the so-called ‘‘Webster’’ horn equation are known. For quantitative work on brass instruments, however, it may be necessary to solve the equation for arbitrary contours, perhaps known only numerically from measurements of existing instruments. The present paper describes a numerical technique that is efficient and accurate. The effects of viscous and thermal damping at the walls of the horn are easily incorporated. Like most numerical methods for solving differential equations, this breaks the length of the horn into a number of smaller pieces. However, the pieces are not quasi-infinitesimal, but are initially as long as a quarter wavelength. The number of pieces is then increased and the error is estimated and reduced in a series of consecutive approximations using a method called ‘‘the deferred approach to the limit.’’ Where an exact solution is known (e.g., lossless exponential or catenoidal horn), results accurate to about six or seven decimal places are easily obtainable when the calculations are carried out with a resolution of about 15 significant figures.

ON THE ACOUSTICS OF LOW MACH NUMBER BULGED, THROATED AND BAFFLED NOZZLES

The acoustic wave equation, for quasi-one-dimensional propagation, along a duct of varying cross-section, containing a low Mach number mean flow, is obtained (Section 2), by using as variables either the potential or the velocity (Section 2.1);the ray approximation, that holds only for wavelengths which are short compared with the length scales of the variation of the cross-section and mean flow velocity, is used as a factor to reduce the wave equation to a Schrödinger form (Section 2.2). It is shown that the latter, reduced form, for the potential, is the most convenient for studying the acoustics of catenoidal (Section 2.3) and sinusoidal (Section 3.1) nozzles; it is found that these inherit respectively the filtering properties of catenoidal horns, and transparency properties of sinusoidal horns. This approach also applies to the exponential nozzle (Section 3.3), whereas for the Gaussian nozzle, the sound field can be expressed in terms of Hermite functions, by using a semi-reduced form (Section 3.2) of the wave equations. The exact solutions of the nozzle wave equation (Section 3), for the four families of ducts, are plotted (Section 4) as amplitude and phase versus distance, for several combinations of frequency and low Mach number. The nozzle families considered include the catenoidal in (Figures 1, 2, 6–9) in Section 4.1, the sinusoidal (Figures 3, 10, 11) and exponential (Figures 5, 12, 13) in Section 4.2, and the first six eigenfunctions of Gaussian nozzles (Figures 4, 14–17) in Section 4.3. The various cases presented here are applications of a technique which derives the acoustics of low Mach number nozzles from that of the corresponding horns: i.e., ducts of the same cross-section but without mean flow.

Silicon microfabricated horns for power ultrasonics

As part of a study of silicon surgical tools, silicon ultrasonic horns for impedance transformation have been made with using bulk micromachining. The ultrasonic amplification of silicon stepped, linear, exponential, and catenoidal half-wavelength horns actuated by piezoelectric lead-zirconate-titanate (PZT) plates is measured and found to agree well with predictions from a numerical model. With the catenoidal horn, a maximum ultrasonic velocity of 14.2 m s −1 is obtained with a displacement of 35 μm measured at 64.5 kHz at 150 V pp drive. By way of comparison, in cataract surgery a particle velocity of only 11.9 m s −1 is used.

Some general properties of the exact acoustic fields in horns and baffles

The propagation of the fundamental, longitudinal acoustic mode in a duct of variable cross-section is considered, and the “Webster” wave equations for the sound pressure and velocity are used to establish some general properties of the exact acoustic fields. The equipartition of kinetic and compression energies is shown (section 2.1) to hold at all stations only for (i) a duct of constant cross-section and (ii) an exponential horn; these are the two cases for which the wave equations for the acoustic velocity and pressure coincide. It is proved (section 2.3) that there are only five duct shapes, forming two dual families, which have constant cut-off frequency(ies): namely, (I) the exponential duct, which is self-dual, and is the only shape with constant (and coincident) cut-offs both for the velocity and pressure; (II) the catenoidal horns, of cross-section S∼cosh 2, sinh 2, which, with their duals (III) the inverse catenoidal ducts S∼sech 2, csch 2, have one constant cut-off frequency, respectively, for the acoustic pressure and velocity. The existence of at least one constant cut-off frequency implies that the corresponding wave equation can be transformed into one with constant coefficients, and thus the acoustic fields calculated exactly in terms of elementary (exponential, circular and hyperbolic) functions; this property also applies to the imaginary transformations of the above shapes, viz., the sinusoidal S∼sin 2 and inverse sinusoidal S∼csc 2 ducts, that have no cut-off frequency, i.e., are acoustically “transparent”. It is shown that elementary exact solutions of the Webster equation exist only (section 3.1) for these seven shapes: namely, the exponential, catenoidal, sinusoidal and inverse ducts; it is implied that for all other duct shapes the exact acoustic fields involve special functions, in infinite or finite terms, e.g., Bessel and Hermite functions respectively for power-law and Gaussian horns. Examples of the method of analysis are given by calculating, in elementary form, the exact acoustic fields in inverse catenoidal ducts, for all cases of (a) propagating waves above, (b) non-oscillating modes below and (c) transition fields at the cut-off frequency. The inverse catenoidal ducts consist of (A) the horn of cross-section S∼sech 2, ressembling the “soliton” of non-linear water wave fame, and (B) the baffle of cross-section S∼csch 2, which also matches two exponentially converging ducts, but has infinite, instead of finite, flare at the origin. The geometrical and acoustic properties of these ducts are illustrated by sets of six plots, in Figure 1(a) for the sech-horn and in Figure 1(b) for the csch-baffle; the exact acoustic fields are described by amplitude and phase decompositions of the sound velocity and pressure, plotted as functions of position along the duct, for four frequencies ranging from the cut-off condition to the ray limit (or W.K.B.J. approximation).