Influence of the longitudinal displacement on nonlinear principal parametric resonance of the woodworking bandsaw

The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice

Acoustic response analysis of a rectangular panel

Stijenke prostorija raznih namjena često se izvode kao paneli s periodičkim ukrepama. U slučaju kad na panel djeluje ravninsko akustičko polje i kad je visina perioda panela najmanje tri puta veća od njegove širine, beskonačni panel periodički oslonjen u jednom smjeru može se modelirati kao beskonačna periodički oslonjena greda. Akustički odziv izračunat je metodom virtualnog rada uz definiranje pomaka redom prostornih harmonijskih valova. Rezultati za konačni panel dobiveni su ograničavajući površinu udarnog vala i radijacije zvuka na konačne dimenzije panela koristeći metodu prostornog isječka. U sustavu s dovoljnim prigušenjem maksimalan broj potrebnih prostornih harmonijskih valova za traženu točnost pojavljuje se na najvišoj frekvenciji izračuna međutim ukoliko sustav nije dovoljno prigušen maksimalan broj potrebnih prostornih harmonijskih valova pojavit će se na nižoj frekvenciji. O prigušenju u sustavu ovisit će i granični broj prostornih harmonijskih valova do kojeg se njihove amplitude kod slobodnih vibracija panela brže izračunavaju pomoću metode virtualnog rada nego s razvojem u red. U dijagramu prikaza prostornih harmonijskih valova amplitude i fazne brzine valova definirane su za frekventni pojas što se postiglo definiranjem amplitude vala bojom. U sklopu disertacije izvedena je funkcija oblika slobodnih vibracija grede periodički oslonjene na jednostavne oslonce čija je točnost verificirana primjenom metode virtualnog rada. Rezultati eksperimentalnog modela dobro se slažu sa rezultatima numeričke simulacije.Room walls in which people reside are often built as panels with periodical stiffeners because the aim is to have walls with sufficient rigidity and minimum mass. That kind of panels have specific acoustic response which can be modelled with simplified methods because of periodic repetition of boundary conditions which causes minimising of calculation time with optimal accuracy. In the case when the panel is excited with plane acoustic waves and when the height of the panel period is minimally three times larger than its width, infinite panel periodically supported in one direction can be modelled as infinite periodically supported beam which is the case in this dissertation. Acoustical response is calculated with the method od virtual work with displacement defined with series of space-harmonic waves. Concerning that the model is infinite and that a goal is to calculate finite panel the acoustic and structural coupling effects are limited to the area of finite panel using spatial windowing method. In this dissertation a theoretical description and numerical results of influence of damping in periodical stiffeners on the panel acoustic response is given. The influence of the number of space-harmonic waves on the calculation accuracy of the panel acoustic response and panel free vibrations is analysed. When there is sufficient damping in the system the maximum number of space-harmonic waves for gaining an appropriate accuracy appear on the maximum frequency as it is published in the literature but if the damping is low, the maximum number of space-harmonic waves appear on lower frequency. Damping in the sistem also influences the number of space-harmonic waves bellow which there amplitudes of free vibrations are faster calculated with the method of virtual work than with the series expansion. Space-harmonic waves in the literature are presented in diagrams with amplitudes and phase velocities for one frequency. In this dissertation a diagram with amplitude and phase velocities for an frequency spectrum is used where wave amplitudes are defined with color. With this approach the changes in the frequency domain can be easily seen. Also, free vibrations of infinite periodically supported beam on simple supports are calculated where a new shape function is derived and its accuracy is confirmed with the method of virtual work. In accordance with numerical simulation within the current computer possibilities experimental boundary conditions are chosen and the experiments where done. The results of eksperimental model are in a good agreement with the results of numerical simulation

SOUND FIELD ANALYSIS AROUND VIBRATING PLATE

U radu su prikazani analitički, numerički i eksperimentalni pristup u rješavanju problema stvaranja zvučnog polja oko vibrirajuće ploče. Analitički su određene vlastite vrijednosti vibracija ukliještene pravokutne ploče. Numeričkom metodom konačnih elemenata izračunate su vlastite vrijednosti ploče te odziv ploče pri zadanom opterećenju jednakom eksperimentu te vlastite vrijednosti te odziv zvučnog polja oko ploče. Strukturni valovi približno su određeni pomoću pomaka dok su valovi u okolnom zraku definirani pomoću tlaka. U eksperimentalnom dijelu izrađen je odgovarajući model ukliještene pravokutne ploče na koju djeluje sinusna pobuda po z-osi u točki te nosači beskontaktnih senzora.In this paper analytical, numerical and experimental approaches in dealing with problem of created sound field around vibrating plate are presented. Eigenvalues of wedged plate are analytically calculated. Finite element method for calculating plate eigenvalues and plate response to point sinusoidal force and eigenvalues of sound field and response of sound field around vibrating plate is applied. Structure waves are defined with displacements and fluid waves with acoustical pressure. Experimental model of wedged plate with sinusoidal exciter and corresponding sensor support is built. Sound-protecting shield for cancellation of surrounding noise influence on plate sound was built

Noise Pollution – Introduction to the State of the Research and the Implementation in the Horizon 2020 Project Pixel

Noise pollution is a significant factor in the modern world. There are many researches dealing with the noise modelling, differing in their purpose, studied influence and the scope of the research, either in geographical terms or in the terms of studied noise sources. The objective of this paper is to give insight in the current state on the field of negative influence that noise has on the environment and/ or human health. Special attention was given to the influence that ports’ noise pollution has on the surroundings, both on the population and on the environment. Within the influence on the environment, underwater noise, which is often excluded when the main priority is the influence on general population, was also included. As it is one of the main tools for analysing the influence of noise, several examples of assessing the influence of the noise using noise maps are presented. Several objectives of the EU Horizon 2020 project (PIXEL), dealing with the reduction of ports’ pollution, were covered. The segment of the project dealing with the noise is linked with the previous achievements, which forms the basis to give some guidelines for the further research of the noise influence on the environment

USPOREDNA PROCJENA NEKIH METODA RJEŠAVANJA SLOBODNIH VIBRACIJA ELASTIČNO OSLONJENIH GREDA

This paper presents a comparison between two analytic methods for the determination of natural frequencies and mode shapes of Euler-Bernoulli beams. The subject under scrutiny is the problem of a beam supported by an arbitrary number of translational springs of varying stiffness, which is solved first by the method of Laplace transform and then by the Green’s function method. The two methods are compared on the level of algebraic equivalence of the resulting formulas and these are compared to the results obtained by FEM analysis for the case when the beam is supported with a single spring. It can be shown that both methods result in equivalent algebraic expressions, whose results are to be regarded as accurate for any given set of boundary conditions. This is also verified by means of FEM analysis, whose solutions converged to almost identical values. Hence, the two methods are found to be equally accurate for the calculation of natural frequencies and natural modes. This paper also brings a new formulation for the mode shape equation, obtained by the Laplace transform method, which to the best of the authors’ knowledge has not been reported in existing literature. Additional comments about the advantages and disadvantages of the two employed analytical methods are given from the standpoint of mathematical structure and practicality of implementation, which is also supplemented by a comment on the comparative advantages and disadvantages of FEM analysis.U radu je prikazana usporedba dviju analitičkih metoda za određivanje vlastitih frekvencija i oblika vibriranja Euler-Bernoullijevih greda. Predmet istraživanja je problem grede oslonjene na proizvoljnom broju translacijskih opruga različitih krutosti, što je riješeno prvo metodom Laplaceovih transformacija, a zatim metodom Greenovih funkcija. Te dvije metode uspoređene su na razini algebarske ekvivalentnosti dobivenih izraza, čiji su rezultati zatim uspoređeni s rezultatima analize zasnovane na MKE-u za slučaj kada je greda oslonjena na jednoj opruzi. Pokazano je da obje metode rezultiraju algebarski ekvivalentnim izrazima, čije se rezultate za zadane rubne uvjete može smatrati točnima. To je potvrđeno i rezultatima analize pomoću MKE-a, čija su rješenja konvergirala na gotovo identične vrijednosti. Za dvije je metode, stoga, utvrđeno da su jednako točne u izračunu vlastitih frekvencija i oblika vibriranja. U radu je iznesena i nova formulacija jednadžbe oblika vibriranja koja prema saznanjima autora nije objavljena u postojećoj literaturi. U radu su dodatno uspoređene prednosti i nedostaci dviju analitičkih metoda sa stanovišta matematičke strukture i praktičnosti implementacije, što je nadopunjeno i komentarom o komparativnim prednostima i nedostacima analize pomoću MKE-a

Acoustic response analysis of a rectangular panel

Stijenke prostorija raznih namjena često se izvode kao paneli s periodičkim ukrepama. U slučaju kad na panel djeluje ravninsko akustičko polje i kad je visina perioda panela najmanje tri puta veća od njegove širine, beskonačni panel periodički oslonjen u jednom smjeru može se modelirati kao beskonačna periodički oslonjena greda. Akustički odziv izračunat je metodom virtualnog rada uz definiranje pomaka redom prostornih harmonijskih valova. Rezultati za konačni panel dobiveni su ograničavajući površinu udarnog vala i radijacije zvuka na konačne dimenzije panela koristeći metodu prostornog isječka. U sustavu s dovoljnim prigušenjem maksimalan broj potrebnih prostornih harmonijskih valova za traženu točnost pojavljuje se na najvišoj frekvenciji izračuna međutim ukoliko sustav nije dovoljno prigušen maksimalan broj potrebnih prostornih harmonijskih valova pojavit će se na nižoj frekvenciji. O prigušenju u sustavu ovisit će i granični broj prostornih harmonijskih valova do kojeg se njihove amplitude kod slobodnih vibracija panela brže izračunavaju pomoću metode virtualnog rada nego s razvojem u red. U dijagramu prikaza prostornih harmonijskih valova amplitude i fazne brzine valova definirane su za frekventni pojas što se postiglo definiranjem amplitude vala bojom. U sklopu disertacije izvedena je funkcija oblika slobodnih vibracija grede periodički oslonjene na jednostavne oslonce čija je točnost verificirana primjenom metode virtualnog rada. Rezultati eksperimentalnog modela dobro se slažu sa rezultatima numeričke simulacije.Room walls in which people reside are often built as panels with periodical stiffeners because the aim is to have walls with sufficient rigidity and minimum mass. That kind of panels have specific acoustic response which can be modelled with simplified methods because of periodic repetition of boundary conditions which causes minimising of calculation time with optimal accuracy. In the case when the panel is excited with plane acoustic waves and when the height of the panel period is minimally three times larger than its width, infinite panel periodically supported in one direction can be modelled as infinite periodically supported beam which is the case in this dissertation. Acoustical response is calculated with the method od virtual work with displacement defined with series of space-harmonic waves. Concerning that the model is infinite and that a goal is to calculate finite panel the acoustic and structural coupling effects are limited to the area of finite panel using spatial windowing method. In this dissertation a theoretical description and numerical results of influence of damping in periodical stiffeners on the panel acoustic response is given. The influence of the number of space-harmonic waves on the calculation accuracy of the panel acoustic response and panel free vibrations is analysed. When there is sufficient damping in the system the maximum number of space-harmonic waves for gaining an appropriate accuracy appear on the maximum frequency as it is published in the literature but if the damping is low, the maximum number of space-harmonic waves appear on lower frequency. Damping in the sistem also influences the number of space-harmonic waves bellow which there amplitudes of free vibrations are faster calculated with the method of virtual work than with the series expansion. Space-harmonic waves in the literature are presented in diagrams with amplitudes and phase velocities for one frequency. In this dissertation a diagram with amplitude and phase velocities for an frequency spectrum is used where wave amplitudes are defined with color. With this approach the changes in the frequency domain can be easily seen. Also, free vibrations of infinite periodically supported beam on simple supports are calculated where a new shape function is derived and its accuracy is confirmed with the method of virtual work. In accordance with numerical simulation within the current computer possibilities experimental boundary conditions are chosen and the experiments where done. The results of eksperimental model are in a good agreement with the results of numerical simulation

DETERMINATION OF CRITICAL ROTATIONAL SPEED OF CIRCULAR SAWS FROM NATURAL FREQUENCIES OF ANNULAR PLATE WITH ANALOGOUS DIMENSIONS

It is suitable to reduce thickness of circular saw when trying to enhance usability of wood raw material, but reducing thickness also causes reduction of permissible rotational speed which reduces sawing speed. If one increase circular saw rotational speed over permissible one the quality of machined surfaces will reduce because of enhanced vibrations. Permissible rotational speed can be calculated from critical rotational speed which can be defined from natural frequencies of the saw. In this article critical rotational speeds of standard clamped saws (with flat disk surface and without slots) are calculated by using finite element method and classical theory of thin plates on annular plates. Mode shapes and natural frequencies of annular plates are determined by using Bessel functions and by using polynomial functions. Obtained results suggest that standard clamped circular saws without slots and with relatively small teeth can be determined from classical theory of thin plates for annular plates with accuracy depending on clamping ratio

Influence of the longitudinal displacement on nonlinear principal parametric resonance of the woodworking bandsaw

The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice

Applying New Algorithms for Numerical Integration on the Sphere in the Far Field of Sound Pressure

For some sound sources, the function of the square of sound pressure amplitudes on the sphere in the far field is an integrable function or can be integrated with geometrical simplifications, so an exact or approximated analytical expression for the sound power can be calculated. However, often the sound pressure on the sphere in the far field can only be defined in discrete points, for which a numerical integration is required for the calculation of the sound power. In this paper, two new algorithms, Anchored Radially Projected Integration on Spherical Triangles (ARPIST) and Spherical Quadrature Radial Basis Function (SQRBF), for surface numerical integration are used to calculate the sound power from the sound pressures on the sphere surface in the far field, and their solutions are compared with the analytical and the finite element method solution. If function values are available at any location on a sphere, ARPIST has a greater accuracy and stability than SQRBF while being faster and easier to implement. If function values are available only at user-prescribed locations, SQRBF can directly calculate weights while ARPIST needs data interpolation to obtain function values at predefined node locations, which reduces the accuracy and increases the calculation time

Fault detection based on instantaneous angular speed measurement and variational mode decomposition

Rotating machinery encounter throughout their lifetime various problems. Among them, a rotor-stator rubbing problem is one of the most common. This paper proposes a procedure, which applies the instantaneous angular speed (IAS) measurement as a starting step for rotor-stator partial rub detection. There are various approaches regarding counting techniques and processing of signal. In this paper, an application of analog signals from toothed wheel encoder or zebra tape encoder is considered at low to moderate sampling rates. As the rubbing process is nonlinear, this paper is proposing a variational mode decomposition (VMD) as the second step of the detection procedure. The VMD is relatively new method with promising results especially interesting for machinery fault detection. Detection tool is tested on laboratory test rig at two different rotor operating conditions i.e. without rotor-stator rubbing and with light partial rotor-stator rub. Measurements were performed with non-contact eddy current displacement sensors pointed to toothed wheel encoder. Results are presented in the shape of rotor orbits, IAS signals, FFT spectra of IAS signals and VMD spectrograms. Developed fault detection procedure based on IAS measurement and VMD decomposition was successfully tested on laboratory test rig for no rubbing and light rotor to stator partial rub condition