In this paper, the calculation of the dynamic response to harmonic transverse excitation of cantilever Euler-Bernoulli beam carrying a point mass with the mode superposition method is presented. This method uses mode shapes and modal coordinate functions which are calculated by separation of variables and Laplace transformations. The accuracy of defined expressions is confirmed on examples with and without Rayleigh damping. All calculations are in good agreement with the results of commercial software based on finite element methods.

Eigenfrequencies of a nanobeam with a point mass interacting with a heavy fluid are calculated using Bernoulli-Euler kinematics and consistent nonlocal elasticity model. The proposed approach is applicable to a variety of nanotechnology materials and structures, especially mass nanosensors. Eigenfrequencies are compared with those of local theory and conclusions are drawn. Influence of nonlocal effects, heavy fluid interaction and added point mass on dynamic responses is analyzed and the results are documented. Size phenomena are noted and discussed.

The International Maritime Organization (IMO) mandates a reduction in carbon dioxide emissions from 2008 levels by at least 40% by 2030, prompting the widespread adoption of slow steaming and engine de-rating strategies. This study investigates the fatigue life of marine propulsion shafts under slow steaming conditions, focusing on the interplay between torsional and bending vibrations. A finite element (FE) model of a low-speed two-stroke propulsion system is developed, incorporating torsional and lateral excitation sources from both the engine and propeller. Vibrational stresses are computed for multiple operating conditions, and fatigue life is assessed using both the conventional Det Norske Veritas (DNV) methodology and a proposed biaxial stress approach. Results indicate that while torsional vibrations remain the primary fatigue driver, bending-induced stresses contribute marginally to the overall fatigue life. The proposed methodology refines high-cycle fatigue (HCF) assessment by incorporating a corrected S-N curve and equivalent von Mises stress criteria. Comparisons with classification society standards demonstrate that existing guidelines remain valid for most cases, though further studies on extreme alignment deviations and dynamic bending effects are recommended. This study enhances understanding of fatigue mechanisms in marine shafting and proposes a refined methodology for improved fatigue life prediction.

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.