Search

This paper is concerned with the problem of free vibration of an Euler-Bernoulli beam supported with an arbitrary number of translational springs of different constant stiffnesses. The natural frequencies and mode shapes are calculated by means of the Green's function method. Seven different boundary conditions are considered at both ends of the beam: fixed, pinned, sliding, free, translational spring, rotational spring and combined translational-rotational spring. Each boundary condition on one end is combined with all other boundary conditions on the other, yielding a total of 49 possible cases. It has been shown in this paper that a Green's function for any set of boundary conditions can be represented as a linear combination of function coefficients calculated for homogeneous boundary conditions, and also a new form of the general solution is formulated using this principle. Green's functions are calculated and tabulated for each of the 49 possible cases, which in essence closes the question of analytical solutions for this particular kind of problem using Green's functions. The tabulated solutions were used as the starting point in the derivation of the general formulae and to provide a new insight into the underlying mathematical structure of the solutions formulated with Green's functions. This can be of practical interest to the end user who wishes to implement them in computer code. The validity of the obtained solutions is demonstrated with examples, whose solutions are compared to the results in the published literature and to the results obtained with FEM.

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.

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.

This paper outlines the optimum selection procedure for a two-speed planetary gear train with four external shafts based on two design and operational criteria: radial dimensions and efficiency. The paper outlines how to quickly determine the structure and important basic parameters of two-speed planetary gear trains that meet predefined transmission requirements. The procedure is followed by a numerical example in which the optimal two-speed planetary gear train is selected, defined by the teeth number of the central gears, modules and transmission ratios. The relevance of the presented results is reflected in the fact that these transmissions offer significant advantages in systems which require speed change under load.

This paper investigates the behaviour of a receding contact when a cylindrical indenter presses an unbonded layer resting on a substrate. The problem is analysed by using FEM within the scope of the linear theory of elasticity and under the assumption of plane strain. This paper presents new and original results in the analysis of influence of load intensity and indenter geometry on the contact parameters. In addition, in the investigation into material properties a reference analysis was carried out for the case of material similarity between all three bodies, and material properties were subsequently varied for each body. This class of problems shows nonlinear behaviour, with both contact pressure distributions and contact half-widths found to depend nonlinearly on the applied load. The experimental analysis was carried out by employing the digital image correlation method and the ARAMIS 4M system was used. The obtained measurement results show good agreement with the numerical results.

This paper presents experimental results in which the deformation behaviour of an elastic layer was ascertained for the case of a receding contact problem. The employed optical measurement is based on the digital image correlation method and the optical system ARAMIS 4M was used in the experiment. Based on the measured differences of displacements of the points along the layer, the occurrence of receding contact is clearly observed.

U ovom su radu, uz uvod u problematiku preciznog pozicioniranja, izložene osnovne teorijske postavke Hsieh-Panova modela statičkog trenja. Na primjerima je prikazana sposobnost simuliranja svih relevantnih pojava opažanih u eksperimentima te ovisnost ponašanja modela o promjenama vrijednosti njegovih parametara, pri čemu se kroz kratak osvrt na problematiku kvantitativne procjene parametara modela uvodi razmatranje važnosti njihova pojedinačnog poznavanja kod pozicioniranja s impuslnim upravljanjem., In this article, along with an introduction into precision positioning, basic concepts of the Hsieh-Pan static friction model are presented. Through the use of examples, the ability of the model to simulate all relevant phenomena observed in experiments is shown and variations in model behaviour due to parameter changes are investigated, where the issue of quantitative estimation of the model parameters is briefly commented upon, through which a consideration of the importance of their individual estimation is introduced with regard to impulsive control in positioning.

An effort to optimise both mechanical and electronic/control components of ultra-precision devices is presented. The considered mechanics is compliant, which overcomes the non-linearities of conventional devices. Design guidelines for hinge optimisation are given and a preliminary consideration of the scaling effects is performed. The developed control system is based on a rapid controller prototyping platform consisting of a Compact-PCI system running under the Linux RTAI real-time extension.