An Equality-Based Weighted Residual Formulation for the Vibration of Systems with Two-Dimensional Friction.
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Bibliographic record
Abstract
An equality-based weighted residual formulation is proposed for the periodic responses of vibrating systems subject to two-dimensional dry friction on a plane. Coulomb's law is expressed as two coupled nonsmooth equality conditions which augment the equations of motion, resulting in a mixed displacement-friction force formulation whose periodic solutions are sought using a standard Ritz-Galerkin procedure in time. The shape functions considered are the classical Fourier functions, and a quasi-analytical expression for the Jacobian of the friction terms is derived in a piecewise linear fashion and computed in a weighted residual sense. The method is based on an exact equality representation of Coulomb's law for interfaces with mass, thus avoiding common hypotheses such as regularization, penalization, or massless interfaces. It is entirely carried out in the frequency domain, contrary to existing frequency-time methods which require the calculation of contact forces in the time domain at each iteration of the nonlinear solver. The method is compact and found to be robust and accurate. It is void of convergence or other numerical issues up to very large numbers of harmonics of the response in all cases considered. Periodic responses featuring complex two-dimensional interface motions and multiple stick-slip transitions are calculated accurately at various resonant and sub-resonant excitation frequencies, at a reasonable computational cost. Since the only approximation in the procedure is the finite number of terms in the Ritz-Galerkin expansion, the intricate behavior of the two-dimensional friction force dictated by Coulomb's law can be captured with a high degree of accuracy.
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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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| 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 |
Machine scores (provisional)
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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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