arXiv subjects

Grzegorz Kudra

Publications and source records attributed to Grzegorz Kudra.

8 recordsLinked to original sources

arXiv2026

Large post-critical dynamics of an inextensible spinning fluid-conveying pipe with pinned-roller supports: high-order Galerkin and a modified Hencky bar-chain framework

This paper investigates the stability and large post-critical dynamics of an inextensible spinning fluid-conveying pipe with pinned-roller supports. Replacing the pinned-pinned support of the extensible counterpart with a sliding support removes the axial-stretching restoring mechanism and fundamentally changes the governing equations of motion. Derived here for this configuration, these equations contain a different set of nonlinear terms -- arising from the inextensibility constraint and the bending curvatures rather than the single axial-stretching term -- that drives a post-critical regime with large deflections. The regime is analysed with two complementary methods. The first is a Galerkin discretisation in which the bending curvatures are Taylor-expanded to ninth order, shown to be the lowest order resolving the post-critical amplitude; the standard cubic truncation overestimates the deflection significantly by missing the geometric stiffening from inextensibility. The second is a modified Hencky bar-chain model with a global angular description: a closed, $n$-independent matrix framework with exact trigonometric kinematics, directly implementable in any standard programming environment with matrix routines and adaptable to both extensible and inextensible configurations through a single boundary-condition reduction. The linearised dynamics give an ellipse-like stability boundary in the flow-velocity--rotational-speed plane with semi-axes $U=π$ and $Ω=π^{2}$; three damping regimes are identified, including a high-rotation instability driven by rotating damping. Close agreement between the two methods across linear-stability, bifurcation, and time-history comparisons confirms the ninth-order Galerkin truncation and establishes the modified Hencky bar-chain as a reliable general-purpose discrete framework for spinning fluid-conveying pipes.

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arXiv2026

Analytical and numerical study of a parametrically excited 2DOF oscillator with nonlinear restoring magnetic force and rotating rectangular rod

This study investigates a detailed analytical and numerical investigation of a nonlinear two-degree-of-freedom (2DOF) mechanical oscillator subjected to parametric excitation, magnetic stiffness nonlinearities, and dry friction. The considered system consists of two coupled oscillators, both of which are connected to a rotating rectangular beam that induces a time-periodic stiffness variation. The Complex Averaging (CxA) method is employed to derive approximate analytical solutions, which are thoroughly validated through time-domain simulations and bifurcation analyses. The dynamic analysis reveals a rich spectrum of nonlinear behaviors, including periodic, quasi-periodic, and chaotic responses. Detailed bifurcation diagrams, Lyapunov exponent analysis, and Poincaré maps demonstrate the influence of nonlinear stiffness degree, mass symmetry, and frictional effects on system stability and response amplitude. The obtained results give a significant understanding of the dynamic behavior of coupled nonlinear systems and establish a conceptual framework for the development of complex vibration abatement strategies, energy harvesting devices, and advanced mechanical systems.

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arXiv2025

Bifurcations and synchronization of coupled translational-rotational stick-slip oscillators

This paper presents mathematical modeling and numerical analysis of bifurcation and synchronization phenomena in a system of coupled oscillators driven by a finite-power energy source and generating two-dimensional stick-slip translational-rotational vibrations. The mechanical system consists of rigid disks placed on moving belts and asymmetrically connected to a support via springs, with each disk simultaneously performing translational and rotational motion. The belts are driven by a common DC motor. The disk contacts the belt over a finite contact area, resulting in mutually coupled frictional force and torque through their dependence on the linear and angular slip velocity. The paper utilizes special approximations of the resultant frictional forces and torques based on generalizations of Padé's developements, in which special smoothing elements are introduced to avoid singularities when the relative motion between the disk and belt disappears. Furthermore, the model also provides a smooth approximation of the friction model, in which static friction is greater than kinetic friction. Numerical analysis was conducted based on direct numerical simulations, Poincare maps, and bifurcation diagrams. It was shown that the system can exhibit a two-dimensional stick-slip phenomenon, and the system exhibits predominantly periodic dynamics, sometimes with a very long period. In this work, a system of two oscillators driven by a common, limited-power motor was studied, which can synchronize and oscillate in phase or in counterphase.

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arXiv2025

Dynamic behavior of shear-thickening fluids under harmonic excitation: an experimental investigation

Shear-thickening fluids (STFs) become more viscous under shear stress, which makes them useful for many engineering and scientific applications. However, their behavior under normal forces, especially when these forces are applied harmonically, is less understood. Here, we examine the dynamic response of STFs under harmonic excitation. Our experimental setup features an unbalanced rotor attached to a vibrating plate submerged in an STF-filled container. We monitored the rotor's speed, the system's displacement, and the STF force. Data from experiments without STF were used to identify system parameters, and measurements of the STF force revealed the dynamic nature of the STF force. By comparing responses with and without STF, we identified three regions of behavior: in the pre-resonance region, the STF force is negligible; at resonance, it acts as a significant damping force; and in the post-resonance region, it behaves like an on-off force. Overall, the STF effectively reduces resonance amplitudes. These results can inform the design of complex structures incorporating STFs.

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arXiv2024

Overcoming stretching and shortening assumptions in Euler-Bernoulli theory using nonlinear Hencky beam models: applicable to partly-shortened and partly-stretched beams

This paper addresses the challenges of the Euler-Bernoulli beam theory regarding shortening and stretching assumptions. Certain boundary conditions, such as a cantilever with a horizontal spring attached to its end, result in beams that partly shorten or stretch, depending on the spring stiffness. The traditional Euler-Bernoulli beam model may not accurately capture the geometrical nonlinearity in these cases. To address this, nonlinear Hencky's beam models are proposed to describe such conditions. The validity of these models is assessed against the nonlinear Euler-Bernoulli model using the Galerkin method, with examples including cantilever and clamped-clamped configurations representing shortened and stretched beams. An analysis of a cantilever with a horizontal spring, where stiffness varies, using the nonlinear Hencky's model, indicates that increasing horizontal stiffness stiffens the system. This analysis reveals a transition from softening to linear behavior to hardening near the second resonance frequency, suggesting a bifurcation point. Despite the computational demands of nonlinear Hencky's models, this study highlights their effectiveness in overcoming the inherent assumptions of stretching and shortening in Euler-Bernoulli beam theory. These models enable a comprehensive nonlinear analysis of partly shortened or stretched beams.

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arXiv2024

Nonlinear dynamics of spinning fluid-conveying pipes with structural damping: stability analysis and post-instability behavior

Nonlinear dynamics of fluid conveying pipe, rotating with constant velocity about its longitudinal axis is analyzed. Considering boundary conditions and internal damping, the nonlinear equation of motion is derived, and it is discretized via the Galerkin method. Afterward, the stability of the system is investigated by characterizing the eigenvalues under the action of two control parameters: rotational speed and flow velocity. Then using direct numerical simulation, instability and stability regions are distinguished in a map as the control parameters vary. It is shown that due to the presence of internal damping in the system, both rotational speed and flow velocity determine the critical speeds. Post-instability behavior is characterized by non-zero equilibrium points, representing deflection in the rotating frame, which correspond to forward whirling motion in the inertial frame. Finally, Hencky Bar-chain Model was employed to verify the results.

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arXiv2022

Study the Bifurcations of a 2DoF Mechanical Impacting System

Impacting mechanical systems with suitable parameter settings exhibit a large amplitude chaotic oscillation close to the grazing with the impacting surface. The cause behind this uncertainty is the square root singularity and the occurrence of dangerous border collision bifurcation. In the case of one degree of freedom mechanical systems, it has already been shown that this phenomenon occurs under certain conditions. This paper proposes the same uncertainty of a two-degree freedom mechanical impacting system under specific requirements. This paper shows that the phenomena earlier reported in the case of one degree of freedom mechanical systems (like narrow band chaos, finger-shaped attractor, etc.) also occur in the two degrees of freedom mechanical impacting system. We have numerically predicted the narrowband chaos ensues under specific parameter settings. We have also shown that the narrowband chaos can be avoided under some parameter settings. At last, we demonstrate the numerical predictions experimentally by constructing an equivalent electronic circuit of the mechanical rig.

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arXiv2019

Properties of impact events in the model of forced impacting oscillator: experimental and numerical investigations

The paper deals with the studies of forced impacting oscillator when are taken into account the dry and viscous resistance, as well as the generalized Hertz contact law during an impact. The numerical treatments of mathematical model are accompanied with the validations on the base of experimental rig. To study the solutions of the mathematical model, we construct the sequences of impacts, when the system is evolved in periodic and chaotic modes. The statistical properties of chaotic impact events are considered in more details. In particular, we analyze the successive iterations of impact map, autocorrelation function and coefficient of variation for the impact train, the histograms for the inter-impact intervals and values of obstacle penetrations. It is revealed that the impact sequence is stationary but non-Poissonian and contains temporal scales which do not relate to the external stimulus. This sequence can be described by a bimodal distribution. These findings are confirmed by the analysis of experimental data.

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