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Original scientific paper 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. Keywords: hybrid boundary conditions; method of multiple scales; numerical examples; principal parametric resonance; travelling Timoshenko beam Utjecaj uzdužnog pomaka na nelinearne glavne parametarske rezonancije tračne pileIzvorni znanstveni članak U radu su istražene poprečne vibracije aksijalno pomične Timoshenkove grede, kao podesnog modela za matematičko modeliranje tračnih pila. Posebno je obrađen utjecaj aksijalnog pomaka koji je, za razliku od većine modela koji se susreću u literaturi, uveden preko integro-parcijalne diferencijalne jednadžbe. Također su izvedeni izrazi za forme vibriranja za slučaj hibridnih oslonaca s različitim krutostima torzijskih opruga u točkama oslonaca. Ispitivan je utjecaj srednje brzine gibanja grede kao i aksijalnog naprezanja na vlastite frekvencije i oblike vibriranja. Razvijen je i nelinearni model na temelju kojega su izračunati amplitudni odzivi nelinearnih ravnomjernih vibracija za slučaj parametarske rezonancije. Izrađeno programsko rješenje testirano je na određenom broju ranije poznatih primjera. Iznijeta teorijska razmatranja, uz pomoć programskog rješenja, poslužila su i za analizu jednog od dostupnih primjera iz industrijske prakse.

Dynamic response of structures can be determined by mode superposition method which uses mode shape functions and modal coordinate functions. If the structure can be modelled with a uniform Euler-Bernoulli beam and a combination of clamped, free, pinned or sliding boundary conditions, analytical expressions for mode shape functions and modal coordinate functions are used. If analytical expressions for higher-order mode shapes or analytical expressions for the modal coordinate functions at high value of time are numerically evaluated, errors occur since the programming language cannot recognise the resulting value as a number, due to ill-conditioned analytical expressions. An analytical expression is ill-conditioned if small errors in the data may produce large errors in the solution. In this paper, we present a procedure for the exact numerical evaluation of analytical expressions for modal coordinates at high value of time while existing modified analytical expressions are used for exact calculation of higher-order mode shapes of the Euler-Bernoulli beam. The proposed procedure can be applied in the design of structure elements whose response is calculated with mode superposition method, e.g. for designing of metal frames for sawing machines in woodworking industry.

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