A compact, equality-based weighted residual formulation for periodic solutions of systems undergoing frictional occurrences.
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Bibliographic record
Abstract
A very compact weighted residual formulation is proposed for the construction of periodic solutions of oscillators subject to frictional occurrences. Coulomb's friction is commonly expressed as a differential inclusion which can be cast into the complementarity formalism. When targeting periodic solutions, existing algorithms rely on a procedure alternating between the frequency domain, where the dynamics is solved, and the time domain, where friction is dealt with. In contrast, the key idea of the present work is to express all governing equations including friction as equalities, which are then satisfied in a weak integral sense through a weighted residual formulation. The resulting algebraic nonlinear equations are solved numerically using an adapted trust region nonlinear solver and basic integral quadrature schemes. To increase efficiency, the Jacobian of the friction forces is calculated analytically in a piecewise linear fashion. The shape functions considered in this work are the classical Fourier functions. It is shown that periodic solutions with clear multiple sticking and sliding phases can be found with a high degree of accuracy. The equality-based formulation is shown to be effective and efficient, convergence being achieved in all cases considered with low computational cost, including for large numbers of harmonics. Importantly, this new friction formulation does not suffer from the typical limitations or hypotheses of existing frequency-time domain methods for non-smooth systems, such as regularization, penalization, or massless frictional interfaces.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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