US3590288A

US3590288A - Piezoelectric transducer utilizing a catenoidally tapered horn

Info

Publication number: US3590288A
Application number: US833099A
Authority: US
Inventors: Hildegard M Minchenko
Original assignee: Ohio State University
Current assignee: Ohio State University
Priority date: 1969-06-13
Filing date: 1969-06-13
Publication date: 1971-06-29
Anticipated expiration: 1988-06-29

Abstract

A piezoelectric transducer for converting high power electrical energy to high power mechanical energy at a sonic frequency range. Specifically, the invention is an electromechanical transducer capable of delivering high power levels ( 30 kw) in a high-Q, high efficiency construction utilizing ceramic polycrystalline driving elements. Reference is made to the claims for a legal definition of the invention.

Description

United States Patent [72] Inventor Hildegard M. Minchenko [56] References Cited Reynoldsburg, Ohio UNITED STATES PATENTS g f 333;? 1969 3,368,085 8/1968 Minchenko 310/82 3,396,8 196 M t :1 1 8. x 45 Patented June 29,1971 2 5 8 cMas ere a 3 0/ 2 [73} Assignee The Ohio State University Duggafl C b Ohi Assistant Examiner-Mark O. Budd AnorneyAnthony D. Cennamo [54] PIEZOELEC'IRIC TRANSDUCER UTILIZING A Tf 2 t Z t HORN ABSTRACT: A piezoelectric transducer for converting high a n g power electrical energy to high power mechanical energy at a [52] U.S.Cl. BIO/8.2, sonic frequency range. Specifically, the invention is an elec- 310/8.3, 310/87 tromechanical transducer capable of delivering high power [51] Int. Cl H04r 17/00 levels 30 kw) in a high-Q. high efficiency construction [50] Field of Search 310/82, utilizing ceramic polycrystalline driving elements. Reference 8.3, 8.7, 8.1, 8.0, 9.1 is made to the claims for a legal definition ofthe invention.

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\ 2 3 4 5 e 7 a 9 HILDEGARDMINCHENKO DIAMETER RATIO FIG. 6 B g ATTORNEY PIEZOELECTRIC TRANSDUCER UTILIZING A CATENOIDALLY TAPERED HORN BACKGROUND An electromechanical transducer such as a piezoelectric device is capable of transforming high frequency electrical impulses into high frequency mechanical impulses or vice versa. A sonic transducer of this type is disclosed by the present inventor, US. Pat. No. 3,396,285, dated Aug. 6, 1968, for Electromechanical Transducer, assigned to The Ohio State University.

The use of sonic energy has been suggested extensively in all fields of endeavor. Although the use of sonic energy has been at an increasing pace, realistically its use hasbeen limited by one primary factor, i.e., power. The prior art, in referring to high-power transducers, refers to transducers up to l watts." For such purposes as earth removal, concrete destruction (road wrecking) and other similar work efforts, this amount of power is negligible. A transducer to be economically acceptable in heavy work loads of this nature should have a power output of a horsepower or greater.

Until the above disclosure by the inventor, known transducers generally lacked the ability to handle large power levels continuously. Further, the permissible stresses and strains that could be endured by the electromechanical transducer material were so low that these materials were soon destroyed by the amounts of power than can be usefully employed in such processes. The only realistic solution hereintofore ,was the brute force method of employing more transducers or alternatively by increasing the number of horns. The cost of an installation of appreciable size is beyond economically acceptable limits. Simply then, the art awaited a transducer that at a reasonable cost would have a high energy output.

, both radially and longitudinally (axially). In this way the acoustic stresses in the piezoelectric elements are always compressive, never tensile, even under maximum voltage excitation.

SUMMARY The present invention is an improvement of the above-mentioned sonic transducer. The highest power rating that could be achieved in the past was kw. of sonic energy. The present invention provides approximately 30 kwf of sonic Another object of the invention is to provide an electromechanical sonic transducer with an efficiency of approximately 97.5 percent.

Another object of the invention is to provide an elec- BRIEF DESCRIPTION OF THE DRAWINGS FIG. I is a diagrammatic illustration of the cross section of an acoustic resonator showing longitudinal displacement and strain relations;

FIG. 2 is a diagrammatic illustration of a cylindrical sandwich type resonator;

FIG. 3 is a diagrammatic representation of the preferred embodiment of the invention;

FIG. 4 is a graphical representation of the resonant length of the preferred embodiment of FIG. 3 as a function of the diameter ratio for a 10.054-kI-Iz. catenoidal horn with a small energy which can be used economically for continuous duty operation in a number of industrial processes.

nant length, nodal position, and amplification factor as a function of diameter ratio.

OBJECTS Accordingly a principal object of the invention is to provide an improved electromechanical sonic transducer.

Another object of the invention is to provide an electromechanical sonic transducer with minimal losses and a very high-Q.

diameter of 2 inches;

FIG. 5 is a graphical representation of the nodal position with respect to the large diameter of the preferred embodiment of FIG. 3 as a function of the diameter ratio for a 10.054-

I kHz. catenoidal horn with a small diameter of 2 inches; and,

FIG. 6 is a graphical representation of the amplification factor of the preferred embodiment of FIG. 3 as a function of the diameter ratio for a 10.054-kI-Iz. catenoidal horn with a small diameter of 2 inches.

DESCRIPTION OF THE PREFERRED EMBODIMENT The following analysis provides a complete understanding of the theory underlying the design and operation of the transducer disclosed herein.

Referring to FIG. 1 the following assumptions are made: (I) the strain is uniformly distributed through the entire cross section, and (2) the wave front remains flat. It can be shown that this approximation is valid for longitudinal resonators whose length considerably exceeds the diameter.

Throughout the entire derivation, the units of measurement for all terms are given in the MKS (meter-kilogram-second) system. The symbols used are defined as follows:

x= longitudinal distance along X-axis (m) y radial distancefrom X-axis to the surface of the resonator (m) s= displacement from equilibrium position (m) s strain (m/m) A cross section area (m F and F =compressive force (newtons) Y= Youngs modulus (newton/m) p density of resonator material (kg/m t= time (seconds) o'= longitudinal stress (newtonslm L length of resonator (m) w=21rf= angular frequency (radians/sec) f frequency of exciting force (Hz) A wavelength of sound in resonator material (m Referring to FIG. 1, and utilizing the above approximations, only one component s N of the strain tensor will be in existence. If longitudinal compressive forces are applied, the plane at-x will be moved the distance s, the plane at dx the distance s ds. Since dx is very small the displacement at x dx can be written as:

s+ds =s+(6s/8x)dx. Eq. (1) The increase in length of the segment of length sx then is Now the strain is defined as the ratio of increase in length to original length or was; i

dx 6:r Eq. (3)

The longitudinal stress is defined as the compressive force per unit area:

a=(Force/Area)=(F/A) Eq. 4) Hookes law states that the elastic constant of the material or Young's modulus is equal to the negative ratio of stress over strain;

Rewriting Eq. (5) yields an expression for the internal longitudinal elastic force:

where A indicates that the cross-sectional area is a function of x.

it should be noted that, in the dynamic case, the elastic forces at the various cross sections will differ. If F is the force at x, the F=(8F/6x) represents the force at x=dx. Therefore the incremental elastic force in the segment of length dx can be expressed as:

Eq. (7) Substituting Eq. (6) into Eq. (7) yields:

a Os a os dF- [-AJ 8i)]dx=Y-az[11, )]d.7c

Siiice the mass of the segment is pA dx, the inertial force ofthe segment can be expressed as where 5/81 is the acceleration of the segment cit.

From equilibrium considerations, the elastic force increment dF has to equal the inertial force increment dF',

By introducing,( Y/5)=Ac,where c is the velocity of the resonator material, Eq. (lmcan also be written in the form:

as a d. a oz ox A, o, a, Eq. 11 Eq. (ll) is the basic wave or resonator equation for solid elastic materials.

The basic wave equation is alsoyalid for transmission lines. It is immaterial whether they are straight-through transmission lines, where the incoming signal amplitude is transmitted with a M ratio, or transmission lines with varying cross sectional area, where the signal amplitude is increased or decreased.

A sonic or ultrasonic force concentrator is a transmission line which accepts input energy at one end and delivers it concentrated into a smaller area at its output end.

Under the assumption of a half-wave force generator and a half-wave force concentrator, and ideal matching conditions, the following boundary values can be imposed:

at .t=0, strain s'=0 at x=L; strain s'=0.

Under the conditions the concentrator resonates and will not alter the mode of operation of the force generator. That means the displacement on the large end F80, where s is the amplitude of vibration of the force generator.

For the design of a useful force concentrator, various considerations must be taken: l) the desired frequency determines the overall length, and (2) the stress and strain distributions along the force concentrator have to be determined in order to design a horn for a particular application and to insure a reasonable life expectancy. Horns with large amplifica tion factors may physically break up when the stress in the material exceeds its fatigue strength. (3) The node locations planes of zero vibration amplitude in the axial directionmust be determined to provide locations where support structures can be attached without affecting the acoustical characteristics of the horn. (4) The amplification factorthe ratio of vibration amplitude at the output end to that at the input endis another factor of importance.

The force concentrator of the present invention is a catenoidal horn. This particular type concentrator has been found to be very effective and superior to stepped, conical, or exponential horns for high power applications.

in the following, the generating line" is defined as that partieular line, which rotated about the X-axis, will generate the surface under consideration. Subscripts 0" relate variables to the location x=0 along the X-axis and subscripts "L" refer to the location x=L. Also use is made of the following relation- Referring again to FIG. 1, the generating line for a catenoidal horn is given by:

y cosh[m(L-x)] Eq. (13) where the shape parameter m=( l/L) coshR, and the diameter ratio R=D,,/D,

The area relationship and its derivative for any point along the X-axis are given by:

A,= cosh [m(Lx)] Eq. (14 (A,) 2mA cosh [m(L-:z:)] sinh [m(La:)]

where C and C are constants to be evaluated by imposing the boundary conditions as stated previously. As a result, the displacement along the resonant catenoidal horn is determined as cosh (mL) cos [b(L-x)] cos (bL) cosh ML-m where b= /'];2: 2 The strain distribution is:

, cosh (mL) sin [b(L-:z:)]

cos (bL) cosh [m(L-:z:)] +m cos [b(L-:c)] tanh [m(L:c)]

cosh [m(La:)]

The frequency equation is:

(bL) tan (bL)= ff5 cosh R Eq. (20) The resonant length is:

where (bL) are the roots of the frequency equation (20). The nodes are located at:

(2n-- 1)1r x =L- -,n0, 1, 2 x L (22) The amplification factor M is: i

s L cosh (rnL) R 7 so cos (bL) cos (bL) Eq. (23

The above derivations are based on the thin-rod velocity of the material, without taking into account the radial displacements. Actually the presence of these displacements will somewhat alter the resonating length of the concentrator. One versed in the art can easily derive the necessary correction factor for small values of R /L and estimate the accuracy of the above formulas for different values of R /L.

The capability to convert electrical energy into vibratory output energy and the associated efficiency are important parameters in the design of the force generator. Therefore. piezoelectric materials are preferred over magnetostrictive materials. However. the fragility of the ceramic elements has to be considered. Only if that particular problem can be solved successfully. can a superior force generator be achieved.

The most efficient use can be made of the ceramic piezoelectric elements in a Langevin (sandwich) type force generator. The power conversion capability of the elements can be increased by the application of hydrostatic compression. However, it is important also to match the impedance of the ceramic elements to the impedance of the adjacent materials. The overall length of the compositeresonator depends upon the dimensions and acoustical properties of the piezoelectric element. If the natural resonance frequency of the unloaded element is known, the length of the loading sections can be calculated to reduce the frequency of the system to the desired natural frequency.

For the cylindrical force generator, the result can be summarized as follows: Referring to FIG. 2. the impedance Z of which should also equal the magnitude of the impedance 2 of the ceramic element at the interface, which is wnere: f,, natural resonant frequency of unloaded piezoelectric element f natural resonant frequency of system.

Subscripts l and c" refer to section I and piezoelectric elements respectively.

Equating Z to Z,., the length of

sections

1 or 2 can be determined for a certain frequency, or the frequency can be determined for a certain length.

The driving element of the force generator is of extreme importance. In the particular case, a very fragile ceramic constitutes the piezoelectric element. Due to this fact, conventional transducers have very low power ratings, unless a multiplicity of individual transducers are paralleled.

The solution to dealing with the fragility of the ceramic material is disclosed in my prior U.S. Pat. No. 3,396,285.

In theory and practice, the piezoelectric driving elements are under radial and axial pressure to assure that they do not operate in tension even under intense sonic action. Significantly, the structural design of the transducer that permits the extraordinary power output from the driving elements resides in a novel method of clamping the piezoelectric elements both radially and longitudinally. In this way, the acoustic stresses in the piezoelectric elements are always compressive, never tensile, even under maximum voltage excitation. At a frequency of kHz., the power conversion in the ceramic element is approximately twice as high as the manufacturers specifications indicate, and yet the elements do not destroy themselves even when operated over a long period of time.

Referring now to FIG. 3, there is illustrated the preferred embodiment of the invention. The force concentrator comprises a catenoidal horn 2 which is constructed in accordance with the preceding analysis. The large diameter of the horn is 6 inches and the small diameter is 2 inches. The 6-inch and 2- inch diameters give the force concentrator a diameter ratio of 3.

The constructed model of the preferred embodiment illustrated in FIG. 3. utilized four rings 8 of piezoelectric material in the force generator portion 6. These rings 8 and the associated clamping means 4 and conical section 50 comprise a half-wave driving structure or force generator section of the transducer.

Utilizing the small diameter (2 inches) of the catenoidal horn and the resonant frequency (10.054 kHz.) of the transducer as constant parameters a numerical computer analysis was performed. The resonant length, nodal position, and amplification factor were determined as a basis of the diameter ratio.

FIG. 4 shows in a graphical manner the resonant length of the transducer of the invention as a function of the diameter ratio. For the constructed embodiment ratio of 3, it is calculated and is shown that the resonant length is l0.0336 inches.

Referring now to FIG. 5, it is seen that the distance from the large end of the concentrator to the nodal plane is plotted as a function of the diameter ratio. A diameter ratio of 3 corresponds with a nodal location of 3.8939 inches from the force concentrators large diameter.

The graphical plot of the numerical analysis of the amplification factor's relation to the diameter ratio is illustrated in FIG. 6. An amplification factor of 3.57 results when the diameter ratio is chosen as 3.

As can be seen from the graphical representations in FIGS. 4 through 6, for large diameter ratios the relationships are almost linear, while for diameter ratios between one and three, radical changes in slope are found to occur.

Although certain and specific embodiments have been illustrated, it is to be understood that modifications may be made without departing from the true spirit and scope of the invention.

What I claim is:

l. A high power electromechanical transducer, the improvement comprising:

a resonant structure including a force generator section, a

force concentrator section, and a plurality of piezoelectric elements positioned intermediate said generator and said concentrator section in a region near the node of said resonant structure,

said force generator section including an elongated solid portion integral with said piezoelectric elements and said force concentrator section,

said force concentrator section comprising a catenoidally tapered bar having the larger diameter adjacent to said force generator section,

said resonant length and nodal location of said transducer being related to its said diameter ratio wherein a diameter ratio of 3 corresponds to l0.03+ inches in length and the nodal location is 3.8+ from the force concentrators large diameter.

2. A high power electromechanical transducer as set forth in claim 1 wherein the diameter ratios in excess of 3 are linearly related to the length of said transducer.

3. A high power electromechanical transducer as set forth in claim 1 wherein the diameter ratios in excess of 3 are linearly related to the nodal position of said transducer.

4. A high power electromechanical transducer as set forth in claim 1 wherein the amplification factor of said transducer is related to said diameter ratio wherein the diameter ratio of 3 corresponds to an amplification factor of 3.57.

6. A high power electromechanical transducer as set forth in claim 4 wherein the diameter ratios in excess of 3 are linearly related to the amplification factor of said transducer.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3, 590,288 Dated June 29, 1971 Hildegard M. Minchenko Inventor(s) It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Column 6, line 51, after "3.8+" insert inches Signed and sealed this 25th day of July 1972.

(SEAL) Attest:

EDWARD M.FLETCHER,JR. ROBERT GOTTSCHALK Attesting Officer Commissioner of Patents US. GOVERNMENT FRN'HNG OFFICE: 19.9 6-3i-JSI.

Claims

2. A high power electromechanical transducer as set forth in claim 1 wherein the diameter ratios in excess of 3 are linearly related to the length of said transducer.
3. A high power electromechanical transducer as set forth in claim 1 wherein the diameter ratios in excess of 3 are linearly related to the nodal position of said transducer.
4. A high power electromechanical transducer as set forth in claim 1 wherein the amplification factor of said transducer is related to said diameter ratio wherein the diameter ratio of 3 corresponds to an amplification factor of 3.57.
6. A high power electromechanical transducer as set forth in claim 4 wherein the diameter ratios in excess of 3 are linearly related to the amplification factor of said transducer.