The dynamics of a hanging chain pendulum, long treated as a textbook problem in classical mechanics, are revisited from a fresh and rigorous analytical perspective. By systematically deriving and comparing the continuum and discrete formulations, subtle but significant differences in the vibrational spectrum, particularly in the high-frequency regime are uncovered. Using asymptotic expansions, boundary layer theory, and matched scaling arguments, a comprehensive description of the eigenmodes and their scaling behavior is developed. In the discrete model, we reveal a striking two-regime structure: low-frequency modes governed by Bessel-type equations, and high-frequency modes localized near the free end, described by Airy-type asymptotics. The transition between these regimes emerges naturally from a balance of competing terms in the governing equations, yielding a characteristic crossover scaling. This analysis clarifies the limitations of discrete and continuum approximations and exposes the deeper mathematical structure underlying the system. Ultimately, the followed approach provides a dual perspective and case study, demonstrating how rigorous asymptotics bridge discrete and continuum models and yield fresh insight into seemingly well-understood mechanics of the chain pendulum.

This study presents a mathematically rigorous derivation and spectral analysis of the string shooter system, a closed-loop continuum driven at high axial speeds. While often regarded merely as a scientific toy, the system exhibits rich nonlinear dynamics located at the intersection of drag-dominated string behavior and stiffness-dominated rod mechanics. To capture this duality, we establish a unified continuum framework modeling the loop as an axially moving, geometrically nonlinear Kirchhoff rod. The derivation proceeds from first principles using an Arbitrary Lagrangian–Eulerian (ALE) kinematic description, strictly separating kinematic constraints from balance laws to resolve the interplay between gyroscopic transport terms, aerodynamic drag, and flexural rigidity. A distinct novelty of this work lies in the rigorous treatment of the system’s topology: we formulate a dimensionless strong-form finite difference scheme that enforces C^1 -geometric continuity for the loop closure while simultaneously permitting physical discontinuities in shear and tension forces at the drive singularity. This approach allows for a direct solution of the generalized eigenvalue problem, revealing the spectrum of the underlying steady-state solution. We validate the nonlinear boundary value problem for the steady-state and its modal analysis against simple (analytical) benchmarks and present examples that characterize the transition from stable stationary equilibria to flutter instabilities (self-excited vibrations) driven by non-conservative drag forces.

A framework for the dimensional reduction of transient thermo-structural problems is presented in which the cross-sectional heat diffusion is encoded as an autonomous evolution system for the thermal stress resultants. A through-the-thickness temperature ansatz satisfying the essential boundary conditions is postulated and the heat equation is projected via a weighted-residual Galerkin method, yielding coupled first-order evolution equations for the thermally induced membrane force and bending moment. The resulting 1D model requires no resolved cross-sectional temperature computation at runtime, yet preserves the diffusive time scales of the parent 2D problem; the structural response follows from Euler–Bernoulli beam theory driven by the evolved thermal resultants. The projection procedure is methodologically general; its accuracy is demonstrated here for a homogeneous rectangular beam under mixed Dirichlet–Neumann thermal conditions, for which an exact 2D Fourier series solution is derived independently. A three-way numerical validation against the exact solution and a high-fidelity 2D finite element reference confirms that the reduced 1D model reproduces deformationfields to within a few percent, although the reconstructed cross-sectional stress underestimates the peak thermal eigenstress by approximately 43

A physics-informed neural network (PINN) is developed to identify a closed-form moment–curvature constitutive law for elastoplastic Euler–Bernoulli beams. The fiber-level von Mises response integrated through the thickness is recast as a structural plasticity problem with a single yield surface in moment space. The primary novelty is the data-driven identification of the structural kinematic hardening closure, an emergent property of the dimensional reduction, absent from the local material law, embedded inside the exact algorithmic return mapping, enforcing algorithmic admissibility by construction rather than by penalty terms. A cyclic maturity indicator and architectural constraints ensure Masing-rule consistency between virgin and stabilized branches and positivity of the hardening modulus. Trained weights are extracted to explicit linear algebra, enabling microsecond-scale constitutive evaluation in standard finite element codes. Two boundary value problems validate the 1D model against 2D plane-stress references and closed-form analytical solutions, confirming quantitative agreement in force–displacement response and deflection profiles across cyclic loading–unloading. Computation times are orders of magnitude shorter than the plane-stress reference, demonstrating the practical efficiency of the model reduction.

The magnetic track brake is a mechanical contact (with friction) based braking system that is typically actuated electromagnetically and used as an emergency brake in railway transport. Within this paper, the practically relevant task of predicting the effective local and global forces of the contacting bodies and the respective deformations during the quasi-static braking process is addressed. Therefore, a simplified, yet efficient and accurate numerical contact model is developed to treat the frictional sliding contact problem. In order to verify and validate the model a couple of numerical experiments are carried out. The proposed model and algorithm are first tested against an analytic benchmark problem of a parabolic indenter indenting an elastic half-space. The developed model is then compared against a reference Abaqus finite element simulation in application-oriented braking simulations that treat the contact problem between a single braking element (pole shoe) and the rail. The results demonstrate and highlight the applicability and efficiency of the proposed model but also show the current limitations and shortcomings that hint at possible future augmentations.

We propose a nonlinear shell finite element model to simulate sheet metal roll forming, a continuous forming process to produce endless metal profiles. A mixed Eulerian–Lagrangian kinematic description is employed to overcome the drawbacks of the common Lagrangian parametrization. The finite element mesh is detached from the particle motion in axial direction and, thus, facilitates a two-step solution procedure to capture the continuous forming process: First, an equilibrium is sought with the account for contact and plastic flow. Secondly, the material transport is taken into account, which amounts to the integration of an advection problem for the plastic variables. The continuum plasticity model with through-the-thickness integration for the stress resultants guarantees a precise resolution of the forming process in each cross section of the Kirchhoff–Love shell. A series of simulations is carried out to ascertain the convergence of the numerical scheme, to highlight the impact of characteristic parameters and to establish a correspondence to a reference computation with the commercial software Abaqus in a simplified static setting. A physical experiment is devised on an actual roll forming mill to assess the quality of the current computational model.

The proposed Kirchhoff-Love shell stress resultant plasticity model extends a previously reported model for plates by complementing the constitutive law of elastoplasticity with membrane effects. This enhanced model is designed for bending dominant settings with small to moderate membrane forces. It is thus implemented in a purpose-built nonlinear mixed Eulerian–Lagrangian finite element scheme for the simulation of sheet metal roll forming. Numerical experiments by imposing artificial strain histories on a through-the-thickness element are conducted to test the model against previously reported stress resultant plasticity models and to validate it against the traditional continuum plasticity approach that features an integration of relations of elastoplasticity in a set of grid points distributed over the thickness. Results of actual roll forming simulations demonstrate the practicality in comparison to the computationally more expensive continuum plasticity approach.

Modeling of roll forming process of sheet metal requires efficient treatment of plastic deformations of thin shells and plates. We suggest a new general approach toward constructing the governing equations of bending of an elastic‐plastic Kirchhoff plate on the structural mechanics level. The generic function of the isotropic hardening law is formulated in terms of variables, which are meaningful for a material surface, namely resultant bending moments and dissipative work. This function is then identified by comparison of the exact solutions for a uni‐axial or isotropic bending experiment within the structural model and with the continuum model of a through‐the‐thickness element. We validate the approach against both the fully three‐dimensional simulations as well as the results of the traditional elastic‐plastic plate analysis, the latter one featuring the continuum laws of plasticity treated in the integration points along the thickness direction. We considered benchmark problems from the literature as well as a prototype for the roll forming problem. Besides good agreement regarding the shape of plastic zones and force‐deflection characteristics, experiments also demonstrate higher computational efficiency of the new stress resultant model.

Roll forming is a continuous process in which a moving metal sheet passes through numerous pairs of opposing forming rolls. The shafts of the roll forming mill are equipped with these rolls and must be set up and aligned to achieve the required final profile of the sheet. The practically relevant task of predicting the profile geometry of this incremental rolling process with varying characteristics of the metal sheet entering the mill requires an accurate description of the stiffness behavior of the shaft with rolls, which is the most compliant part of the roll forming mill. In this paper, the measured force-deflection characteristic of the shaft without rolls is compared with predictions of various theoretical models, followed by the adoption of the shear deformable beam model of the shaft with nonlinear elastic supports in the bearings. The coefficients of the cubic stiffness characteristics of the rotational springs as well as the effective length between the supports are identified based on the experimental data for the deflections, measured along the shaft for various loading levels. The theoretical predictions are obtained via the nonlinear finite element model of the shaft. The model thus provided shows high accuracy compared with the measurements. The paper’s results serve as a foundation for models to predict the stiffness of shafts with rolls.