Filtry
wszystkich: 1046
wybranych: 940
Wyniki wyszukiwania dla: HULL HYDRODYNAMIC DERIVATIVES
-
π-Indenyl substituted zirconium compounds containing terminal bondedphosphanylphosphido ligands [Ind2Zr(Cl){(Me3Si)P-PR2-jP1}]. Synthesis,X-ray analysis and NMR studies
PublikacjaPraca prezentuję reakcję kompleksów indenylowych cyrkonu z solami litowymi difosfanów o ogólnym wzorze R2P-P(SiMe3)Li, gdzie R = tBu, Et2N, iPr2N. W wyniku tych reakcji otrzymano 3 nowe kompleksy fosfanylofosfidowe.Otrzymane nowe kompleksy zbadano przy pomocy technik rentgenowskiej analizy strukturalnej i NMR.Dichloro-bis(indenyl)zirconium(IV) reacts with lithium derivatives of diphosphanes R2P-P(SiMe3)Li (R = tBu, Et2N, iPr2N)...
-
A headspace solid-phase microextraction method development and itsapplication in the determination of volatiles in honeys by gas chromatography
PublikacjaThe first part of the paper presents the detailed optimisation of all steps of the determination procedureof volatiles from Polish honeys by headspace solid-phase microextraction (HS-SPME) as the sample preparationmethod and gas chromatography/mass spectrometry (GC-MS) as the method of final determination.The methodology of semi-quantative analysis allowed the isolation and identification of volatile fractioncompounds from popular...
-
Relativity of arithmetic as a fundamental symmetry of physics
PublikacjaArithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters ‘plus’ or ‘times’ one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in...
-
Laplace domain BEM for anisotropic transient elastodynamics
PublikacjaIn this paper, we describe Laplace domain boundary element method (BEM) for transient dynamic problems of three-dimensional finite homogeneous anisotropic linearly elastic solids. The employed boundary integral equations for displacements are regularized using the static traction fundamental solution. Modified integral expressions for the dynamic parts of anisotropic fundamental solutions and their first derivatives are obtained....
-
Chromogenic azomacrocycles with imidazole residue: Structure vs. properties
PublikacjaNew diazo macrocycles linked by hydrocarbon chain bearing imidazole or 4-methylimidazole residue have been synthetized with satisfactory yield (24–55%). The structure of macrocycles was confirmed by X-ray analysis and spectroscopic methods (1H NMR, MS, FTIR). Metal cation complexation studies were carried out in acetonitrile and acetonitrile-water system. It was found that azomacrocyles form triple-decker complexes with lead(II)....
-
Unassisted formation of hemiaminal ether from 4-aminopyridine and o-vanillin - experimental and theoretical study
PublikacjaThe reactions between o-vanillin and three isomeric aminopyridines lead to imines of diverse spatial conformation and reactivity. The direct products of these simple reactions carried out in methanol are either imine compounds formed in the reactions of 2- amino- and 3-aminopyridine with o-vanillin or the α-aminoether formed in the reaction of o-vanillin with 4-aminopyridine. The Schiff-type derivative of 4-aminopyridine and o-vanillin,...
-
Excitation of Non-Wave Modes by Sound of Arbitrary Frequency in a Chemically Reacting Gas
PublikacjaThe nonlinear phenomena in the field of high intensity sound propagating in a gas with a chemical reaction, are considered. A chemical reaction of A → B type is followed by dispersion and attenuation of sound which may be atypical during irreversible thermodynamic processes under some conditions. The first and second order derivatives of heat produced in the chemical reaction evaluated at the equilibrium temperature, density and...
-
Collagen and hyaluronic acid hydrogel in water-in-oil microemulsion delivery systems
PublikacjaThe efficacy of microemul-sion in dermal and transdermal delivery of active ingredients. Here, we assessed the water-in-oilmicroemulsion capacity to incorporate two natural polymers, i.e. collagen and hyaluronic acid withlow and high molecular weight. Systems were extensively characterized in terms of conductivity, phaseinversion studies, droplet diameter, polydispersity index and rheological properties. The results of thisresearch...
-
Identification and non-integer order modelling of synchronous machines operating as generator
PublikacjaThis paper presents an original mathematical model of a synchronous generator using derivatives of fractional order. In contrast to classical models composed of a large number of R-L ladders, it comprises half-order impedances, which enable the accurate description of the electromagnetic induction phenomena in a wide frequency range, while minimizing the order and number of model parameters. The proposed model takes into account...
-
Fundamental properties of solutions to fractional-order Maxwell's equations
PublikacjaIn this paper, fundamental properties of solutions to fractional-order (FO) Maxwell's equations are analysed. As a starting point, FO Maxwell's equations are introduced in both time and frequency domains. Then, we introduce and prove the fundamental properties of electromagnetic field in FO electromagnetics, i.e. energy conservation, uniqueness of solutions, and reciprocity. Furthermore, the algorithm of the plane wave simulation...
-
About Unusual Diffraction and Thermal Self-Action of Magnetosonic Beam
PublikacjaThe dynamics of slightly diverging two-dimensional beams whose direction forms a constant angle θ with the equilibrium straight magnetic strength is considered. The approximate dispersion relations and corresponding links which specify hydrodynamic perturbations in confined beams are derived. The study is dedicated to the diffraction of a magnetosonic beam and nonlinear thermal self-action of a beam in a thermoconducting gaseous plasma....
-
Assembly of 1D Granular Structures from Sulfonated Polystyrene Microparticles
PublikacjaBeing able to systematically modify the electric properties of nano- and microparticles opens up new possibilities for the bottom-up fabrication of advanced materials such as the fabrication of one-dimensional (1D) colloidal and granular materials. Fabricating 1D structures from individual particles offers plenty of applications ranging from electronic sensors and photovoltaics to artificial flagella for hydrodynamic propulsion....
-
Post processing and selecting data obtain with parametric sub-bottom profiler SES-2000 Standard during sounding the Gulf of Gdansk
PublikacjaThe main goal of the paper is to describe the results of sounding the Gulf of Gdansk seabed using a parametric sub-bottom profiler SES-2000 Standard. Quality of obtained during trials data depends inter alia on proper location of antenna to reduce influence of pitch, roll, heave and ship noise (bubbles from propeller and a hull flow, vibration from main engine and peripheral devices). Furthermore calibration of complementary units...
-
Magnetic Signature Description of Ellipsoid-Shape Vessel Using 3D Multi-Dipole Model Fitted on Cardinal Directions
PublikacjaThe article presents a continuation of the research on the 3D multi-dipole model applied to the reproduction of magnetic signatures of ferromagnetic objects. The model structure has been modified to improve its flexibility - model parameters determined by optimization can now be located in the cuboid contour representing the object's hull. To stiffen the model, the training dataset was expanded to data collected from all four cardinal...
-
A Concept for Safe and Less Expensive Acceleration of a Marine Steam Turbine Start-up
PublikacjaThis paper analyses the issue of accelerated start-up of a marine steam turbine, which is an important problem because the start-up of a steam machine involves the combustion of fuel that is not transformed into useful energy. To find novel technologies that offer improvements in this aspect is essential due to restrictions on reducing ship emissions. Thus, the shorter the start-up time, the better for the environment and economy....
-
Determination of benzodiazepines and Z-hypnotic drugs in whole blood samples by GC–MS/MS: Method development, validation and application
PublikacjaBenzodiazepines (BZDs) and Z-drugs are widely used as anxiolytics, sedative hypnotics, anticonvulsants and muscle relaxants. “Designer benzodiazepines” (DBZDs) are a new psychoactive substance class consisting of benzodiazepine derivatives that are not allowed for medical use and are known for being used recreationally. From a toxicologist standpoint, the huge number of such substances implicate a necessity for developing fast...
-
Synthesis of CoFe2O4 Nanoparticles: The Effect of Ionic Strength, Concentration, and Precursor Type on Morphology and Magnetic Properties
PublikacjaThe present study highlights the effect of metal precursor types (SO4 2¯, Cl¯, and NO3¯), their concentration, and the influence of ionic strength of reaction environment on the morphology, surface, and magnetic properties of CoFe2O4 particles. The magnetic nanoparticles were obtained by chemical coprecipitation in alkaline medium at increasing metal concentration in the range of 0.0425 mol·dm-3 to 0.17 mol·dm-3 and calcination...
-
Analytical aspects of model compounds toxicity assessment using MTT and Microtox bioanalytica assays
PublikacjaMore and more increased intensity of anthropopressure processes can be observed, among other things, in the release of great quantities of synthetic substances into the environment, including the Endocrine Disrupting Compounds. The effects of the above-mentioned human impact on the environment not only can be noticed in the homeostasis disturbance in ecosystems, but they are also visible with respect to human. The high incidence...
-
Synthesis and biological evaluation of novel analogues of batracylin with synthetic amino acids and adenosine: an unexpected effect on centromere segregation in tumor cells through a dual inhibition of topoisomerase IIa and Aurora B
PublikacjaIn the search for new anticancer agents we designed and synthesized batracylin derivatives with linking synthetic amino acid side chains of different lengths and adenosine. Unexpectedly, we have found that in water and the culture media adenosine–amino acid–BAT conjugates form supramolecular structures and this prevents these compounds from entering cells. Consequently, these compounds exerted no biological activity when tested...
-
Optical recognition elements: macrocyclic imidazole chromoionophores entrapped in silica xerogel
PublikacjaMaterials containing new chromoionophores consisting of crown residue and azole moiety as partsof macrocycles were encapsulated by the sol-gel procedure in silica xerogel matrices and proposed aschemical recognition elements especially for such metal ions as Li+, Cs+ and Cu2+. Action of these recognition elements is in principle based on changes of reflectance. The recognition elements containing 21-membered chromogenic...
-
Topochemical, Single‐Crystal‐to‐Single‐Crystal [2+2] Photocycloadditions Driven by Chalcogen‐Bonding Interactions
PublikacjaThe face-to-face association of (E)-1,2-di(4-pyridyl)ethylene (bpen) molecules into rectangular motifs stabilized for the first time by chalcogen bonding (ChB) interactions is shown to provide photoreactive systems leading to cyclobutane formation through single-crystal-to-single-crystal [2+2] photodimerizations. The chelating chalcogen bond donors are based on original aromatic, ortho-substituted bis(selenocyanato)benzene derivatives...
-
Manganese Phosphatizing Coatings: The Effects of Preparation Conditions on Surface Properties
PublikacjaManganese phosphate coating could be used to protect the surface of steel products. However, it is essential to determine the effects which process parameters, as well as the types of additives used, have on the efficiency of coating deposition. Thus, we present here a process of phosphatization of low-alloy steel (for 15 min at 95 ◦C) in manganese/nickel baths followed by a passivation process with the use of a silicon and zircon...
-
Synthesis, structural and spectroscopic properties of asymmetric schiff bases derived from 2,3-diaminopyridine
PublikacjaTwo Schiff base derivatives, 4-(2-amino-3-pyridyliminomethyl)phenol (I) and 3-(2-amino-3- pyridyliminomethyl)nitrobenzene (II ), were synthesised and characterised by spectroscopy. The structure of I was determined by single crystal X-ray diffraction studies. The asymmetric Schiff base derived from 2,3-diaminopyridine selectively recognise transition and heavy metal cations, and some anion. Ligands I and II form stable complexes...
-
Comparative Study of Integer and Non-Integer Order Models of Synchronous Generator
PublikacjaThis article presents a comparison between integer and non-integer order modelling of a synchronous generator, in the frequency domain as well as in the time domain. The classical integer order model was compared to one containing half -order systems. The half-order systems are represented in a Park d-q axis equivalent circuit as impedances modelled by half-order transmittances. Using a direct method based on the approximation...
-
Wpływ wybranych właściwości maszyny badawczej na wynik eksperymentu tribologicznego z tarciem ślizgowym
PublikacjaNiniejsza monografia stanowi podsumowanie przekrojowych badań związanych z wpływem właściwości stanowiska badawczego (tribometru) na przebieg i rejestrowane wyniki eksperymentu tribologicznego z tarciem ślizgowym ciał stałych smarowanych cieczą w warunkach tarcia bez efektów smarowania hydrodynamicznego. Autor przedstawia wyniki kompleksowych analiz właściwości dynamicznych stanowiska badawczego w kontekście efektów obserwowanych...
-
Calculations of notch stress factor of a thin-walled spreader bracket fillet weld with the use of a local stress approach
PublikacjaPresence of geometric notches in welded joints causes concentration of strains and stresses, therefore reducing fatigue strength of such joints. This article presents an analysis of stress concentrations in a fillet weld of a spreader mounting bracket on a small sailing yacht. The aim of this article is to direct the attention of designers, manufacturers and regulatory bodies to issues of fatigue cracks that form in brackets fastening...
-
Understanding the Electronic Structure and Optical Properties of Vacancy-Ordered Double Perovskite A2BX6 for Optoelectronic Applications
PublikacjaOver the past few years, metal halide perovskite solar cells have made significant advances. Currently, the single-junction perovskite solar cells reach a conversion efficiency of 25.7%. Perovskite solar cells with a wide band gap can also be used as top absorber layers in multi-junction tandem solar cells. We examined the dynamical and thermal stability, electronic structure, and optical features of In2PtX 6 (X = Cl, Br, and I)...
-
Conditions for increasing the recognition of degradation in thermal-flow diagnostics, taking into account environmental legal aspects
PublikacjaThe ever-increasing demand for electricity and the need for conventional sources to cooperate with renewable ones generates the need to increase the efficiency and safety of the generation sources. Therefore, it is necessary to find a way to operate existing facilities more efficiently with full detection of emerging faults. These are the requirements of Polish, European and International law, which demands that energy facilities...
-
Effect of presence of lifting pocket on the THD performance of a large tilting-pad thrust bearing
PublikacjaHydrostatic assistance is a commonly used method to improve limited load carrying ability of tiltingpad thrust bearings during transient states of operation of vertical shaft hydro-generators. Despite of special hydraulic equipment (as pumps, valves, etc.), it also requires manufacturing of special recesses/pockets at pad sliding surfaces, into which oil is injected under high pressure. It allows to lift the rotor before start-up...
-
Experimental study on the feasibility of alternative materials for tilting pad thrust bearings operating in transition to mixed friction
PublikacjaIn hydrodynamic bearings traditional bearing alloys: Babbitts and bronzes are most frequently utilized. Polymer sliding layers are sometimes applied as a valuable alternative. Hard diamond-like carbon (DLC) coatings, which are also considered for certain applications may show some advantages, as well. Although material selection is of secondary importance in a full film lubrication regime it becomes important in mixed friction...
-
Application of different building representation techniques in HEC-RAS 2-D for urban flood modeling using the Toce River experimental case
PublikacjaThis paper presents the impact of the choice of building representation techniques and hydrodynamic models on urban flood simulations using HEC-RAS 2-D for the Toce River physical model. To this end, eight numerical models based on previous laboratory experiments were prepared to simulate unsteady urban flooding on each side of building units. Two simplified building layouts (aligned and staggered) were examined, where models were...
-
Sample preparation procedure using extraction and derivatization of carboxylic acids from aqueous samples by means of deep eutectic solvents for gas chromatographic-mass spectrometric analysis
PublikacjaThe paper presents a new procedure for the determination of organic acids in a complex aqueous matrixusing ultrasound-assisted dispersive liquid–liquid microextraction followed by injection port derivati-zation and GC–MS analysis. A deep eutectic solvent (choline chloride: 4-methylphenol in a 1:2 mol ratio)was used both as an extracting solvent and as a derivatizing agent to yield ion pairs which were next con-verted to methyl...
-
Hydraulic analysis of causes of washout of Gdynia-Orłowo seashore during the flood in the Kacza river estuary
PublikacjaIn July 2016 in the Three-city agglomeration a rainfall episode of over a day duration and 150 mm summary rainfall height, occurred. This situation, extreme as for Polish conditions, caused significant freshets in rivers and streams running into Gdansk Bay, the Baltic Sea, and serving as collectors of rainfall waters for the sea-coast towns. In many areas of the Three-city flood phenomena and overflows took place. The flood also...
-
Enhancing colloidal stability of nanodiamond via surface modification with dendritic molecules for optical sensing in physiological environments
PublikacjaPre-treatment of diamond surface in low-temperature plasma for oxygenation and in acids for carboxylation was hypothesized to promote the branching density of the hyperbranched glycidol polymer. This was expected to increase the homogeneity of the branching level and suppress interactions with proteins. As a result, composite nanodiamonds with reduced hydrodynamic diameters that are maintained in physiological environments were...
-
Products of Photo- and Thermochemical Rearrangement of 19-Membered di-tert-Butyl-Azoxybenzocrown
PublikacjaThe preparation and characterization of products of the photochemical and thermochemical rearrangements of 19-membered azoxybenzocrowns with two, bulky, tert-butyl substituents in benzene rings in the para positions to oligooxyethylene fragments (meta positions to azoxy group, i.e., t-Bu-19-Azo-O have been presented. In photochemical rearrangement, two colored typical products were expected, i.e., 19-membered o-hydroxy-m,m′-di-tert-butyl-azobenzocrown...
-
Golden artefacts, resin figurines, body adhesives and tomb sediments from the pre-Columbian burial site El Caño (Gran Coclé, Panamá): tracing organic contents using molecular archaeometry
PublikacjaThis research aimed to determine the origin of organic residues from funerary contexts in the El Ca~no settlement (Gran Cocl�e area, Panam�a, Central America) by means of multiple molecular probing techniques (GC-MS of organic solvent extracts and pyrolysis-GC-MS, THM-GC-MS and FTIR of solid samples). The samples include particles of precious resin figurines, fillings of golden objects, tomb sediments, plant exudates from extant...
-
Adsorption behavior and corrosion inhibitive characteristics of newly synthesized cyano-benzylidene xanthenes on copper/sodium hydroxide interface: Electrochemical, X-ray photoelectron spectroscopy and theoretical studies
PublikacjaElegant process for synthesis of 3-(7H-dibenzo[c,h]xanthen-7-yl)benzaldehyde (3), as new starting material to create a set of novel xanthene analogues, 2-(3-(7H-dibenzo[c,h]xanthen-7-yl)benzylidene)malononitrile (4), 3-(3-(7H-dibenzo[c,h]xanthen-7-yl)phenyl)-2-cyanoacrylic acid (5), and Ethyl-3-(3-(7H-dibenzo[c,h]xanthen-7-yl)phenyl)-2-cyanoacrylate (6), was achieved starting with available materials under mild conditions. Various...
-
Synteza, struktura chemiczna i właściwości bezizocyjanianowych poliuretanów otrzymywanych z wykorzystaniem CO2 oraz surowców pochodzenia naturalnego.
PublikacjaZ uwagi na wciąż rosnącą świadomość proekologiczną, politykę zrównoważonego rozwoju, a także nurt zielonej chemii wiodącym trendem w przemyśle tworzyw sztucznych jest poszukiwanie alternatywnych metod otrzymywania materiałów polimerowych. Komercyjnie poliuretany otrzymywane są w wyniku reakcji poliaddycji di- lub poliizocyjanianów z poliolami i małocząsteczkowymi przedłużaczami łańcucha. Z uwagi na niekorzystne właściwości diizocyjanianów,...
-
The innovative exploitation of Brassica vegetables in the health quality food production chain
PublikacjaIn this overview the multidirectional opportunities of innovative exploitation of chemical and biological properties of vegetables from Brassiceae family are discussed. The specific chemical composition of brassicas, as regard both nutrients and non-nutrients, renders them particularly suitable for the health food production chain, literally from field to fork. the ability of these plants to accumulate heavy metals may be utilised...
-
The innovative Exploitation of Brassica Vegetables in the Health Quality Food Production Chain
PublikacjaIn this overview the multidirectional opportunities of innovative exploitation of chemical and biological properties of vegetables from Brassiceae family are discussed. The specific chemical composition of brassicas, as regard both nutrients and non-nutrients, renders them particularly suitable for the health food production chain, literally from field to fork. the ability of these plants to accumulate heavy metals may be utilised...
-
Measurement of activity coefficients at infinite dilution of organic solutes in the ionic liquid 1-hexyl-1,4-diaza[2.2.2]bicyclooctanium bis(trifluoromethylsulfonyl)imide using gas–liquid chromatography
PublikacjaBased on gas–liquid chromatography (GLC) retention data, activity coefficients at infinite dilution of 39 different solutes in the ionic liquid 1-hexyl-1,4-diaza[2.2.2]bicyclooctanium bis(trifluoromethylsulfonyl)imide ([HDABCO][NTf2]) have been determined. Measurements covered a broad range of types of organic solutes including alkanes, alcohols, ketones, benzene derivatives and McReynold’s compounds. Activity coefficients were...
-
Fractional Spectral and Fractional Finite Element Methods: A Comprehensive Review and Future Prospects
PublikacjaIn this article, we will discuss the applications of the Spectral element method (SEM) and Finite element Method (FEM) for fractional calculusThe so-called fractional Spectral element method (f-SEM) and fractional Finite element method (f-FEM) are crucial in various branches of science and play a significant role. In this review, we discuss the advantages and adaptability of FEM and SEM, which provide the simulations of fractional...
-
Polydopamine Biomaterials for Skin Regeneration
PublikacjaDesigning biomaterials capable of biomimicking wound healing and skin regeneration has been receiving increasing attention recently. Some biopolymers behave similarly to the extracellular matrix (ECM), supporting biointerfacial adhesion and intrinsic cellular interactions. Polydopamine (PDA) is a natural bioadhesive and bioactive polymer that endows high chemical versatility, making it an exciting candidate for a wide range of...
-
Theoretical Design of the Molecular Structure of Bent-Core Mesogens with Large Second-Order Nonlinear Optical Properties
PublikacjaABSTRACT: The first hyperpolarizability of two series of moleculeswith bent-shaped structures has been calculated at the ab initio level.The two series consist of carboxyl derivatives for which somemolecules are known to exhibit banana phases and of their sulfinatehomologues that have not been synthesized yet. The final purpose isto reveal the relevance or not in synthesizing these latter molecules.The strategy is based on reporting...
-
The Production Possibility of the Antimicrobial Filaments by Co-Extrusion of the PLA Pellet with Chitosan Powder for FDM 3D Printing Technology
PublikacjaThe last decades have witnessed a major advancement and development in three-dimensional (3D) printing technology. In the future, the trend’s utilization of 3D printing is expected to play an important role in the biomedical field. This work presents co-extrusion of the polylactic acid (PLA), its derivatives (sPLA), and chitosan with the aim of achieving filaments for printing 3D objects, such as biomedical tools or implants. The...
-
Synthesis and steroid sulfatase inhibitory activities of N-alkanoyl tyramine phosphates and thiophosphates
PublikacjaA series of phosphate and thiophosphate analogs based on the frameworks of N-alkanoyl tyramines have been synthesized and biologically evaluated. Their binding modes have been modeled using docking techniques. The inhibitory effects of the synthesized compounds were tested on STS isolated from the human placenta as well as the MCF-7, MDA-MB-231 and SkBr3 cancer cell lines. Most of the new STS inhibitors possessed potent activity...
-
Bezczujnikowe sterowanie trakcyjnym silnikiem IPMSM małej mocy
PublikacjaThis paper describes an algorithm for estimation of IPMSM angular rotor position. The algorithm uses derivatives of motor phase currents resulting from PWM modulation to obtain the rotor position. Control of the IPMSM electromagnetic torque requires a precise estimation of the rotor angular position throughout the wide speed range. This involves using a set of estimation methods switched with the dependence on the actual rotor...
-
Influence of Osmolytes on Protein and Water Structure: A Step To Understanding the Mechanism of Protein Stabilization
PublikacjaResults concerning the thermostability of hen egg white lysozyme in aqueous solutions with stabilizing osmolytes, trimethylamine-N-oxide (TMAO), glycine (Gly), and its N-methyl derivatives, N-methylglycine (NMG), N,N-dimethylglycine (DMG), and N,N,N-trimethylglycine (betaine, TMG), have been presented. The combination of spectroscopic (IR) and calorimetric (DSC) data allowed us to establish a link between osmolytes’ influence on...
-
Electron-Induced Decomposition of Uracil-5-yl O-(N,N-dimethylsulfamate): Role of Methylation in Molecular Stability
PublikacjaThe incorporation of modified uracil derivatives into DNA leads to the formation of radical species that induce DNA damage. Molecules of this class have been suggested as radiosensitizers and are still under investigation. In this study, we present the results of dissociative electron attachment to uracil-5-yl O-(N,N-dimethylsulfamate) in the gas phase. We observed the formation of 10 fragment anions in the studied range of electron...
-
Using FreeFEM open software for modelling the vibrations of piezoelectric devices
PublikacjaModelling vibrations of piezoelectric transducers has been a topic discussed in the literature for many decades. The first models - so-called one-dimensional - describe the vibrations only near operating frequency and near its harmonics. Attempts to introduce two-dimensional models were related to the possibility of one transducer working at several frequencies, including both thickness vibrations and those resulting from the transducer...